Computer Aided Design / Computer Aided Manufacturing (CAD/CAM)
Computer Hided and Integrated Manufacturing Systems fl S-Volume Set Cornelius
T Leondes
Vol.4 Computer Aided Design / Computer Aided Manufacturing (CAD/CAM)
Compurer Hided m Integrated Monuficruring Siisfems fl S-Volume Ser
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Vol.4 Computer Aided Design / Computer Aided Manufacturing (CAD/CAM)
C o m p u t e r A i d e d and Integrated Manufacturing Systems H S-Volume Set
Cornelius T Leondes Umrnly of California, Los Angeks, USA
fj|)p World Scientific NEW JERSEY • LONDON • SINGAPORE • SHANGHAI • HONGKONG • TAIPEI * BANGALORE
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
COMPUTER AIDED AND INTEGRATED MANUFACTURING SYSTEMS A 5-Volume Set Volume 4: Computer Aided Design/Computer Aided Manufacturing (CAD/CAM) Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-238-339-5 (Set) ISBN 981-238-980-6 (Vol. 4)
Typeset by Stallion Press
Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore
Preface
Computer Technology This 5 volume MRW (Major Reference Work) is entitled "Computer Aided and Integrated Manufacturing Systems". A brief summary description of each of the 5 volumes will be noted in their respective PREFACES. An MRW is normally on a broad subject of major importance on the international scene. Because of the breadth of a major subject area, an MRW will normally consist of an integrated set of distinctly titled and well-integrated volumes each of which occupies a major role in the broad subject of the MRW. MRWs are normally required when a given major subject cannot be adequately treated in a single volume or, for that matter, by a single author or coauthors. Normally, the individual chapter authors for the respective volumes of an MRW will be among the leading contributors on the international scene in the subject area of their chapter. The great breadth and significance of the subject of this MRW evidently calls for treatment by means of an MRW. As will be noted later in this preface, the technology and techniques utilized in the methods of computer aided and integrated manufacturing systems have produced and will, no doubt, continue to produce significant annual improvement in productivity — the goods and services produced from each hour of work. In addition, as will be noted later in this preface, the positive economic implications of constant annual improvements in productivity have very positive implications for national economies as, in fact, might be expected. Before getting into these matters, it is perhaps interesting to briefly touch on Moore's Law for integrated circuits because, while Moore's Law is in an entirely different area, some significant and somewhat interesting parallels can be seen. In 1965, Gordon Moore, cofounder of INTEL made the observation that the number of transistors per square inch on integrated circuits could be expected to double every year for the foreseeable future. In subsequent years, the pace slowed down a bit, but density has doubled approximately every 18 months, and this is the current definition of Moore's Law. Currently, experts, including Moore himself, expect Moore's Law to hold for at least another decade and a half. This is impressive with many significant implications in technology and economies on the international scene. With these observations in mind, we now turn our attention to the greatly significant and broad subject area of this MRW.
VI
Preface
"The Magic Elixir of Productivity" is the title of a significant editorial which appeared in the Wall Street Journal. While the focus in this editorial was on productivity trends in the United States and the significant positive implications for the economy in the United States, the issues addressed apply, in general, to developed economies on the international scene. Economists split productivity growth into two components: Capital Deepening which refers to expenditures in capital equipment, particularly IT (Information Technology) equipment: and what is called Multifactor Productivity Growth, in which existing resources of capital and labor are utilized more effectively. It is observed by economists that Multifactor Productivity Growth is a better gauge of true productivity. In fact, computer aided and integrated manufacturing systems are, in essence, Multifactor Productivity Growth in the hugely important manufacturing sector of global economies. Finally, in the United States, although there are various estimates by economists on what the annual growth in productivity might be, Chairman of the Federal Reserve Board, Alan Greenspan — the one economist whose opinions actually count, remains an optimist that actual annual productivity gains can be expected to be close to 3% for the next 5 to 10 years. Further, the Treasure Secretary in the President's Cabinet is of the view that the potential for productivity gains in the US economy is higher than we realize. He observes that the penetration of good ideas suggests that we are still at the 20 to 30% level of what is possible. The economic implications of significant annual growth in productivity are huge. A half-percentage point rise in annual productivity adds $1.2 trillion to the federal budget revenues over a period of ten years. This means, of course, that an annual growth rate of 2.5 to 3% in productivity over 10 years would generate anywhere from $6 to $7 trillion in federal budget revenues over that time period and, of course, that is hugely significant. Further, the faster productivity rises, the faster wages climb. That is obviously good for workers, but it also means more taxes flowing into social security. This, of course, strengthens the social security program. Further, the annual productivity growth rate is a significant factor in controlling the growth rate of inflation. This continuing annual growth in productivity can be compared with Moore's Law, both with huge implications for the economy. The respective volumes of this MRW "Computer Aided and Integrated Manufacturing Systems" are entitled: Volume 1: Computer Techniques Volume 2: Intelligent Systems Technology Volume 3: Optimization Methods Volume 4: Computer Aided Design/Computer Aided Manufacturing (CAD/CAM) Volume 5: Manufacturing Process A description of the contents of each of the volumes is included in the PREFACE for that respective volume.
Preface
vn
There is really very little doubt that all future manufacturing systems and processes will utilize the methods of CAD/CAM (Computer Aided Design/Computer Aided Manufacturing), and this is the subject of Volume 4. Key to the processes of CAD/CAM is the generation of three dimensional shapes, a subject treated at the beginning of this volume, 2D assembly drawings are what are generally utilized for conversion to 3D part drawings in the CAD process in order to generate three dimensional shapes for the CAM process, and this is treated in depth and rather comprehensively in this volume. The evolution of a design process and product is often referred to as an adaptive growth representation in the CAD process and this receives necessary treatment in this volume. Fixture designs for the manufacturing process utilize modular elements, and the CAD methods for this essential process are treated rather comprehensively in this volume. Finite element techniques are becoming a way of life for CADS and CAE (Computer Aided Engineering) and rather powerful optimization techniques for processes involved here are also treated in depth in this volume. Rapid prototyping techniques are now a way of life in manufacturing systems, and CAD techniques for this are presented in this volume. These and numerous other techniques are treated rather comprehensively in this volume. As noted earlier, this MRW (Major Reference Work) on "Computer Aided and Integrated Manufacturing Systems" consists of 5 distinctly titled and well-integrated volumes. It is appropriate to mention that each of the volumes can be utilized individually. The significance and the potential pervasiveness of the very broad subject of this MRW certainly suggests the clear requirement of an MRW for a comprehensive treatment. All the contributors to this MRW are to be highly commended for their splendid contributions that will provide a significant and unique reference source for students, research workers, practitioners, computer scientists and others, as well as institutional libraries on the international scene for years to come.
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Contents
Preface Chapter 1 G e n e r a t i o n of T h r e e - D i m e n s i o n a l S h a p e s in C A D / C A M S y s t e m s using A r t - t o - P a r t Technique C. K. Chua and K. Y. Chow Chapter 2 C o m p u t e r Techniques a n d Applications of C o n v e r t i n g 2D Assembly Drawings into 3D P a r t Drawings in C o m p u t e r Aided Design Masaji Tanaka, Kenzo Iwama, Atsushi Hosoda and Tohru Watanabe Chapter 3 C o m p u t e r Techniques a n d Applications of A d a p t i v e - G r o w t h - T y p e R e p r e s e n t a t i o n in C o m p u t e r Aided Design ( C A D ) /. Nagasaka, K. Veda and T. Taura Chapter 4 C o m p u t e r - A i d e d M o d u l a r F i x t u r e Design Yiming (Kevin) Rong Chapter 5 O p t i m i z a t i o n in Finite E l e m e n t a n d Differential Q u a d r a t u r e E l e m e n t Analysis Techniques in C o m p u t e r Aided Design a n d Engineering C.-N. Chen
v
1
35
73
101
171
Chapter 6 C o m p u t e r Techniques a n d Applications in R a p i d P r o t o t y p i n g Gill Barequet
281
Index
297
CHAPTER 1 G E N E R A T I O N OF T H R E E - D I M E N S I O N A L S H A P E S IN C A D / C A M SYSTEMS U S I N G ART-TO-PART T E C H N I Q U E
CHUA C. K. and CHOW K. Y. School of Mechanical & Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 In some industries, products have elements of complex engraving or low relief on them. Traditionally, such work is carried out by skilled engravers working from 2D artwork manually. This process is costly, open to unwanted misinterpretations and lengthens the design cycle. This research presents the Art-to-Part technique which relies on computers and automation from the scanning of 2D artwork, to 3D surface and relief generation, and finally to the fabrication of the model by rapid prototyping. The technique links design to manufacturing stages together and reduces the whole production time. Furthermore, the quality is increased and reproducibility and reliability are ensured, as demonstrated in the 3 case studies. Keywords: 3D relief; Art-to-Part; CAD/CAM; rapid prototyping.
1.
Introduction
There are presently numerous commercially-available software for product design for a particular range of industries which include ceramics, glassware, bottle making, b o t h plastic and glass, jewelry, packaging and food processing for molded products and products produced from forming rolls, coins and badges, and embossing r o l l e r s . 1 - 3 All of these industries share a common problem: most of their products have elements of complex engraving or low relief on t h e m . 4 Traditionally, such work is carried out by skilled engravers either in-house or more often by a t h i r d - p a r t y sub-contractor, working from 2D artwork. This process is costly, open to unwanted misinterpretation of the design by t h e engraver and most importantly, lengthens the time of t h e design cycle. Advances in manufacturing technology allow many industries t o upgrade and change their usual production practices from labor-intensive to a u t o m a t e d and computerized methods. W i t h these changes, the production cycle time and cost 1
2
Chua C. K. and Chow K. Y.
could be reduced tremendously with an improvement in the quality of the product. In recent years, computer-aided design and computer-aided manufacturing (CAD/CAM) have become very popular, especially in the manufacturing industries. It links the designing and manufacturing stages together and thus reduces the whole production time. It is a significant step toward the design of the factory of the future. 5
2. Art-to-Part Process The use of CAD/CAM and Stereolithography Apparatus (SLA) reduces the time required for design modifications and improvement of prototypes. The steps involved in the art-to-part process include the following: 1. 2. 3. 4. 5. 6.
Scanning of artwork Generation of surfaces Generation of 3D relief Wrapping of relief on surfaces Converting triangular mesh files to STL file Building of model by the SLA.
The flow of this series of stages is illustrated using coin design as a case study. Figure 1 shows the steps involved in the art-to-part process.
2.1. Scanning
of
artwork
The function of scanning software is to create a 2D image from 2D artwork automatically or semi-automatically. It would normally be applied in cases where it would be too complicated and time consuming to model the part from a drawing using existing CAD techniques. The 2D artwork is first read into ArtCAM, the CAD/CAM system used for the project, using a Sharp JX A4 scanner. Figure 2 shows the 2D artwork of a series of Chinese characters and a roaring dragon. This combination of hardware and software allows the direct production of a standard image from the artwork, which can be read directly into ArtCAM. The 2D artwork in such instances represent the designs to be used on the face of the coin. In the ArtCAM environment, the scanned image is first reduced from a colour image to a monochrome image with the fully automatic "Gray Scale" function. Alternatively, the number of colours in the image can be reduced using the "Reduce Colour" function. A colour palette is provided for colour selection and the various areas of the images are coloured, either using different sizes or types of brushes or the automatic flood fill function. Figure 3 illustrates the touched-up image.
Generation
of 3D Shapes in CAD/CAM
Systems
Scanning of artwork
_^_ Generation of surfaces (eg.coin shape)
XL Relief generation using ArtCAM
?e
^L Wrapping of relief onto surfaces
\l/ Viewing of final model
OK yes
^L Model building by SLA
OK? yes
_^_ Final model
Fig. 1.
2.2. Generation
of
S t e p s involved in t h e A r t - t o - P a r t p r o c e s s .
surfaces
The shape of a coin is generated to the required size in the CAD system for model building. Figure 4 shows the shape of a coin model generated. A triangular mesh file is produced automatically from the 3D model. This is used as a base onto which the relief data is wrapped and later combined with the relief model to form the finished part.
Chna C. K. and Chow K. Y
4
Fig. 2.
Fig. 3.
2*3* Generation
of 3D
21) artwork.
Touched-up image.
relief
The next stage in creating the 3D relief is to assign each colour in the image a shape profile. There are various fields which control the shape profile of the selected coloured region, namely, the overall general shape for the region, the curvatures of
Generation
of 3D Shapes in CAD/CAM
Fig. 4.
Systems
5
Shape of a coin model.
Selected colour
• Plane
• Round
• Convex
• Concave
Max. height:
|
Base height:
j 0.0
Angle :
| 45.0
Scale:
I i.o
J • Square
| |
0.0
Heij|ht Array Calculate I Apply
Fig. 5.
Zero Ileset
Close j
Control pane! for the shape profile.
the profile (convex or concave), the maximum height, base height, angle and scale. Figure 5 shows the control panel for the shape profile. There are three possibilities for the overall general shape; a plane shape profile will appear completely flat, whereas a round shape profile will have a rounded cross section and lastly, the square shape profile will have straight angled sides. Figure 6 illustrates the various shapes of the 3D reliefs. For each of these shapes, there is an option to define the profile as either convex or concave.
Chua C. K. and Chow K. Y.
6
The square and round profiles can be given a maximum height. If the specified shape reaches this height? it will 'plateau' out at this height giving in effect a l a t region with rounded or angled corners, depending on whether a round or square shape was selected for the overall prolle respectively (see Fig. 6). The overall prolle height, which covers the respective region, can be controlled by specifying the required angle of the profile which represents the tangent angle of the curve at the edge of the region. Figure 7 further illustrates the concept of the overall profile height. An alternative to control the overall profile height is to use the 'scale5 function to flatten out or elevate the height of the shape profile
$qM$*£
fmm
p^tm
SF0*S&
i
^
fiat &tQim ***i&
mm m%r#
mm
L
m^ht
X
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mm* rstight
« ? « « pet * ^ M * " * * *
r m
&r*dflft&3s**£t*?8ftfci|ghl
Fig. 6.
^ t l Z t
nod mzk^m^m my
%$Various shapes of the 3D relief.
&?$Qk* * ro-
seate * i .0
aogl& m 20" t06
Fig. 7.
An illustration of the overall profile height.
Generation
of 3D Shapes in CAD/CAM
• « fT£$
Systems
7
&3&& Insight 1 mm r i round profila *w$h angle 30 s*sd
U
t^&& h^gSht 1 mm asd max tw§M ^ mm
r - r^yr^d proiiifc w#H £^$8 W and
Fig. 8.
An illustration of the definition of shape profiles on different regions.
Fig. 9.
3D relief of an artwork.
(see Fig. 7). The relief detail can be examined in a dynamic Graphic Window within the ArtCAM environment itself. Figure 8 shows an illustration of the definition of shape profiles on different regions. Figure 9 illustrates the 3D relief of an artwork. 2*4. Wrapping
of relief on
surfaces
The 3D relief is next wrapped onto the triangular mesh file generated from the coin surfaces using the command Wrap (see Fig. 10). This is a true surface wrap and not a simple projection. The wrapped relief is also converted into triangular mesh files (see Fig. 11). The triangular mesh files can be used to produce a 3D model suitable for colour shading and machining. The two sets of triangular mesh files, of the relief and the coin shape, are automatically combined (see Fig. 12). The resultant model file can be colour-shaded and used by the SLA to build the prototype (see Fig. 13). 2.5* Converting
of triangular
mesh file into STL
file
The STL format is originated by 3D System Inc. as the input format to the SLA, and has since been accepted as the de facto standard of input for Rapid Prototyping
8
Chua C. K. and Chow K. Y.
Fig. 10. 3D relief wrapped onto coin surface.
Fig. 11. Wrapped relief converted Into triangular mesh lies.
(EP) systems. 6 " 8 Upon conversion to STL, the object's surfaces are triangulated, which means that the STL format essentially consists of a description of inter-joining triangles that enclose the object's volume. The triangular mesh files are also triangulated surfaces, however, of a slightly different format (see Fig. 14). Therefore, an interface programme written in Turbo-C language was developed for the purpose of conversion. The converted triangular file adheres to the standard STL format as in Fig. 15. It has the capability of handling triangular files of huge memory size. 2.6. Building
of model by
SLA
Californian company 3D System Inc. pioneered the Rapid Prototyping (EP) technology when they released their commercial E P system in December 1938 — the
Generation
Fig. 12.
of 3D Shapes in CAD/CAM
Systems
9
2 sets of triangular mesh iiles •- relief and coin shapes are automatically combined.
Fig. 13.
Colour-shaded resultant model file.
SLA-250 model of their Stereolithography Apparatus (SLA). 6 ' 7 Stereolithography technology was Irst developed by Chuck Hall, SD's founding president, in 1982. Stereolithography works by using a low-power Helium-Cadmium laser or an Argon laser to scan the surface of a vat of liquid photopolymer which solidiies when struck by a laser beam. The SLA process chamber consists of a vat containing liquid photopolymer resin, a platform on which the object is to be built and whose height is controlled by an elevator mechanism, a re-coating blade wiper and a Helium-Cadmium or Argon laser subsystem. At the start of the object building process, the platform is positioned at a depth of one layer's thickness below the resin level The laser will trace over areas of the resin surface defined by vectors as the cross-section of the first layer. The area where the resin is struck by the laser beam solidifies to form the first layer of the object. Subsequently, the platform is lowered by a distance equal to the layer thickness, pauses for about 15 seconds to allow the resin level to settle and the re-coating blade wipes over the resin surface to prepare the construction of the next layer as the process repeats itself. When the object has been completely built, the platform is raised above the vat of the resin to drain off the excess liquid resin that has adhered to the object. Figure 16 illustrates the building of prototype using the SLA.
10
Chua C. K. and Chow K. Y.
DUCT 5.2 TRIANGLE BLOCK P 18 AUG 1993 21.43.28 * 1 @1 1 GREEN Paint Duct @1 1 0 4 2 0 0.00000 10.00000 0.00000 20.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 0.00000 1.00000 0.00000 1.00000 0 0 0 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 10.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 20.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 10.00000 20.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 3 1 4 Fig. 14.
The original triangular file format.
solid print facet normal -0.00000e+00 2.00000e+02 -0.00000e+00 outer loop vertex 0.00000e+00 0.00000e+00 2.00000e+01 vertex 0.00000e+00 0.00000e+00 0.00000e+00 vertex 1.00000e+01 0.00000e+00 2.00000e+01 end loop endfacet facet normal 0.00000e+00 2.00000e+02 0.00000e+00 outer loop vertex 1.00000e+01 0.00000e+00 2.00000e+01 vertex 0.00000e+00 0.00000e+00 0.00000e+00 vertex 1.00000e+01 0.00000e+00 0.00000e+00 end loop endfacet Fig. 15.
The converted triangular file to follow the STL format.
Generation
of 3D Shapes in CAD/CAM
Systems
11
LASER
ELEVATOR PLATFORM VAT
RESIN ' / / / / / / / / /
Fig. 16.
s-7-7
Building of prototype using the SLA.
The object that comes out from the SLA's process chamber is approximately 96% solidified and there are minute gaps between the laser's cross-hatching vectors in between the top and bottom layers which hold uncured, liquid resin. A postcuring apparatus, which comes in the form of an oven containing ultraviolet lamps and a rotating turntable, is used to post-cure the object to make it totally solid. Support structures are required at the base of the object so that it does not adhere directly to the platform. They are also needed to support overhanging features of the object to prevent them from collapsing during the building process. These support structures are removed from the object when it has been completely built. The SLA makes use of a variety of photopolymers with different properties suited for different requirements. The properties of the cured photopolymers should allow SLA prototypes to be used for making soft tools like rubber moulds for mass production. Research has shown that feasible rubber moulds can be made from SLA-produced jewellery rings. 9 The SLA is capable of a 0.125 mm minimum layer thickness and an accuracy of within 0.5%. 3. Advantages of Art-to-Part Process The introduction of the scanner, the CAD/CAM system and the SLA provides a list of specific advantages to the art-to-part process: (1) save time, (2) easy to amend and (3) easy to master and apply. 3.1. Save
time
The existing technique of hand-carving takes about two weeks to complete a plaster mould. However, relief can be created in the CAD/CAM system in two hours' time and the prototype will be ready for examination in the next morning after going through the SLA. Most companies that manufacture a product invest considerable time and money in developing a prototype or model. Typically, it is common that the prototyping process could take weeks or months. The time to market has become a competitive issue in the need to prototype quickly.10,11 Besides SLA, other methods are also available for RP. 1 2 - 1 4
Chua C. K. and Chow K. Y.
12
3.2. Easy to
amend
Very often, there is a need to amend the design of the prototype. Serious amendments will result in discarding of the plaster mould and doubling of the time needed to produce a model. The CAD/CAM system allows amendment to be done quickly and easily, and rebuilding of the model is also a simple task.
3.3. Easy to master
and
apply
The whole package is relatively user friendly and the procedures for generating relief are short and simple. The fear of making mistakes in the design becomes an unjustified worry. There is also a high potential in further extending the application into other areas such as the jewellery and ceramics tableware industries.
4. Development of STL File Interface ArtCAM is a 3D CNC engraving software produced by Delcam International and is used to convert a two-dimensional picture into a three-dimensional relief format. The two-dimensional picture can be a scanned picture in bitmap format, any picture file in graphics format like BMP, TIFF, GIF, JPEG or PCX format. The software converts this picture into a three-dimensional format (file extension *.rlf) by colouring the picture and assigning different colours to each part of the picture. The colours are then given an altitude or height so that when a relief is calculated and displayed each of the colours is transformed into a relief and the whole image is viewable as a three-dimensional format called the relief. The output relief format (*.rlf) is specific to the ArtCAM software which is used for CNC engraving. However the relief format is not suitable for RP Systems. In order to create the 3D part using RP technology, it is required to transform the relief format into a STL file. Rapid prototyping (RP) is a key technology of the 1990s. More than two dozen RP techniques have emerged since the first RP technique, stereolithography, was commercialised in 1988.15 The most commonly used input to a RP system is the de facto stereolithography file (STL). All vendors of RP systems accept this format and practically all major suppliers of CAD/CAM systems today provide an interface between their CAD model and the STL file.
4.1. Format of relief and STL
files
The formats of the relief and STL file, which are the input and the output files, are respectively discussed in detail. The structure of their internal detail is explained with the help of figures.
Generation
of 3D Shapes in CAD/CAM
Systems
13
4.1.1. Format of relief file (input file) A relief file consists of 3D image in x, y and z coordinates. The x and y coordinates represent the in-plane data and the z coordinate represents height measurements in pixels. The 3D relief image is bounded by a rectangular frame. The height of the pixel gives the z coordinates, which is the most important part of the relief. The coordinates represent the internal structure of the relief. The file extension is .rlf. The contents of the relief file can be binary or ASCII. A relief is represented internally as a 2 dimensional array of 16 bit signed integers along with a scaling factor used to transform the integer value into a floating point height. This representation halves the memory requirements of a relief when compared to storing the values as floats.
4.1.2. Format of STL file (output file) The STL file format is the most commonly used file format for input in rapid prototyping technologies and is also the de facto industry standard. The STL file defines the surface of an object as a set of interfacing triangles or facet. Each facet as shown in Fig. 17 is defined with three vertices and a normal, which identifies which side faces out and which side faces in. In the STL file, solid models are represented as an unordered collection of facets and each facet has an outward directed facet normal associated with it. 16 The generation of these facets depends on the information contained in the STL file. It should be noted that in the format of the STL file, the coordinates of the vertices are ordered according to the right hand screw rule. That is in an
V2 V 4
—7
Edge y j / V
1
\
^^srV Facet 1
Edge
\ /
V3
Vertex
Direction of Facet Normal
Fig. 17.
r
Description of facet.
14
Chua C. K. and Chow K. Y.
anti-clockwise direction such that the normal of the facet is being directed away from the model as shown in the Fig. 18. Another important information that can be derived from the STL file is that for every facet edge, there must be another facet and only that facet sharing the same edge. Since the vertices of a facet are ordered, the direction on one facet's edge is exactly opposite to another facet sharing the same edge, this necessary condition is also known as Mobius rule as shown in Fig. 19. A facet can reference the three edges which bound it. 17 Each edge can reference the two vertices which define it. Vertex points can contain the connectivity information to all edges or faces which share it. The STL format only contains facets with minimum information necessary to define the image or solid object.
Fig. 18.
Fig. 19.
Right hand screw rule.
Mobius rule = Edge shared by two facets.
Generation
of 3D Shapes in CAD/CAM
Systems
15
For each vertex that is present in the STL file, it is absolutely necessary to calculate the normal in order to determine which way the facet is facing, whether inwards or outwards. To calculate the normal, the essential information required is the vertices which bound the facet. The normal is calculated by the cross product of the vertices. In order to understand the format of a STL file, it is important to know the basic internal structure of a STL file. There are two different formats of STL files. One is the ASCII format which is human readable, and the other is the binary which is totally unreadable. During the developmental stages of this project, the ASCII format was used for debugging purposes. The binary format is used only in the final release version. The reason for using the binary file format is because of its compactness. To illustrate with the example of the bear, the size of the STL file in ASCII is 12 MB whereas the size STL file in binary is only 900 kilobytes. The size of the file differentiates these two formats.
4.2. STL
conversion
To convert the relief file into STL output, a number of problems must be overcome. They are mainly related to the size of the image file being converted. The major concerns that will affect the image size are the resolution of the relief image (input file), the size of the output file with respect to testing, and the reduction of triangles which are directly dependent on the resolution of the image. Each problem will be explained in detail as follows: 4.2.1. Limitation of the STL format In order for the file size not to be too large, the ASCII version of the STL file was used only for verification purposes and the binary version was used for testing and in the final release version. The size of the converted STL file (binary) should not exceed a size greater than 50 MB. This is a limitation of the STL format amongst several other disadvantages of the STL format. This is one of the major problems affecting the testing of the file because if the file could not be tested, it would not be possible to detect the errors. This puts a major constraint on the image size of the relief file. The example used for testing purposes was that of the face of a bear. The output of the STL ASCII file size was 12 MB whereas the size of the binary file was 900 kilobytes. 4.2.2. Resolution of the relief image The resolution of the relief image also affected the output. If the resolution of the relief image was higher, then the STL output would grow in direct proportion to the resolution of the relief image. The primary reason being that, for a high-resolution image the number of pixels used to describe the image would be far greater than required. Hence in order to keep the size of the file under control, it is necessary to
16
Chin C. K. and Chow K. Y.
keep the resolution of the bitmap image used as input to the ArtCAM software to an acceptable level of visibility. The acceptable level of visibility means that the bitmap image can be viewed without raising the resolution measured in dots per inch (DPI). The DPI has to be as minimum as possible and at the same time the image detail should be clear. Once this has been done the relief obtained from the bitmap image would also be of reasonable resolution. This would help in reducing the size of the output STL file. An example of a relief with low resolution and high resolution is shown in Fig. 20.
Fig. 20.
Low and high resolution relief.
Generation
of 3D Shapes in CAD/CAM
Systems
17
4.2.3. Triangle reduction The next factor affecting the size of the output file is the number of triangles used in describing the STL output. The technique used in reducing the number of triangles was by searching for triangles having the same pixel height. Then, these large triangles with the same pixel height will be grouped in an orderly manner and reduced to only two triangles. An algorithm in the next section explains the implementation of this step. Thus by reducing the STL output file to fewer triangles, it would result in fewer points in the STL output file, thereby reducing its overall size. 4.3. Conversion
algorithm
The algorithm has the following steps: 1. Converting each set of points on the relief into triangles. Checking for points with same pixel height. Formation of triangles with same pixel heights. 2. Checking for gaps in the relief image. 18 3. Formation of box so that the STL output is closed with a variable height. The relief is first treated as a whole with rectangular mapping of points as shown in Fig. 21 like a XY graph. The points on the XY graph represent the pixels (Z plane). Each pixel may or may not have a height depending on its location on the image. The direction of its traversal path starts from the origin zero. The following steps are carried out using the notation (x,y): Start with the first point (0, 0) and its counterpart (0, 1) which is on top left. These two points are considered as a set of points for comparison. 1. Select the next point along the x-axis (1, 0) and its counterpart (1, 1) which is on top right. These two points are considered the next set of points. 2. Check if the heights of the first and second sets of points along with their counterparts are equal. There are three possible cases that arise from the comparison. a. If the heights of the points are equal, then there are two possibilities to be considered. The first possibility is the heights of the pixel for these points is zero. Triangles are not formed for this possibility because this part of the image does not contain the relief points. The second possibility is that the heights are equal but the heights of the pixel are greater than zero. Once the heights are equal and greater than zero, a comparison with the next point in sequence is made. Here the next point is the third point. A comparison is made between the first and the third set of points. A continuous comparison is made until the end of the x-axis is reached or when the heights are not equal. b. If the heights are not equal then triangles are formed as shown in the Fig. 22. c. Once a set of triangles is created between two sets of points A {(0,0), (0,1)} and B {(1,0), (1,1)} as shown in Fig. 22, then the second set of points in this
18
Chua C. K. and Chow K. Y. (X=Maximum, YsMaximum)
(1,0)
Origin (x,y) (0,0)
(0,1)
(End of x axis, y=0) Formation of triangles when the heights of points are equal until the end of the x~axis.
Fig. 21. Case A traversal path showing formation of triangles when heights of two consecutive points are equal.
case B {(1,0), (1,1)} is used as the reference origin from which the heights are compared with the next set of points. 3. This is done for the entire image, covering the entire x and y-axis. The next step involves detection of gaps for those areas of images where the surface of the image protrudes outwards with its upper surface not connected to the base of the relief on all sides. One side of the surface is open whereas the other side is connected to the relief as shown in Fig. 23. The above algorithm does not handle this kind of problem and so the entire relief image is scanned for such gaps and if they exist, these gaps are covered by formation of triangles using the triangle reduction technique. The inal step is the formation of a box. Giving a fixed height measurement to the inal image does this. The image is enclosed by a rectangular boundary based on the relief image in pixels. An exact shaped boundary is provided at the bottom with enclosures on all the four sides. The final image is a box with fixed width and the STL image located on the top surface of the box. Figures 24 and 25 show the
Generation of 3D Shapes in GAD/CAM
Systems
19
(X=Maximum, Y=Maximum)
(End of x axis, y=l)
I (1,0)
(2,0)
Origin (x,y) (0,0)
Starting point after triangles have been formed with previous points
(End of x axis, y=0)
Formation of triangles when the heights of the points are not equal and so triangles is formed with these points. The process starts again with the last point being starting point as shown above Fig. 22. Case B traversal path showing formation of triangles when heights of two consecutive points is not equal.
Gap or uncovered portion of image Fig. 23.
Gaps in relief image.
20
Chua C. K. and Chow K. Y.
Fig. 24.
Image of the sculptor shown in STL format without rectangular box.
Sculptor model before and after the formation of the box respectively. This is done so that the whole relief image is closed and no openings are shown.
4.4.
Verification
The verification of the converted STL file is carried out by a visual check and a physical check through the SLA model. 4.4.1. Visual check In this method, the converted STL file can be viewed using any CAD/CAM software, which is capable of viewing STL files. A close examination would reveal any gaps or unclosed triangles. 4.4.2. Physical check In this method, the actual binary STL file is fed into the SLA. The converted file is fed into a computer where the model is sliced into cross sections or layers. Here the liquid resin is used as the raw material to form a solid version of the image, which has been converted from relief into STL format. Before doing the physical check, it is advisable to go through an extensive check using the visual method.
Generation
Fig. 25.
of 3D Shapes in CAD/CAM
Systems
21
Shaded image of the sculptor converted to STL file enclosed by a rectangular box.
5, Case Studies Three case studies are selected to illustrate the significant advantages of using the proposed art-to-part technique over the conventional tools and processes. These case studies are done to cover various types of artwork designs including animals, a human face, flowers as well as Chinese characters, and at the same time to show the feasibility of replacing the current plaster mould prototype with the resin model prototype. Alongside the advantages obtained in adopting the use of the proposed prototyping technique, the case studies also revealed shortcomings which provide scope for future work.
5.1. Chinese
Legend mnd
Tpaditi&n
The Chinese Legend and Tradition is a coin design consisting of a series of Chinese characters and a roaring dragon. In the conventional method, a trained craftsman would require 30-40 hours to make a plaster mould prototype of diameter 200 mm (the size of the plaster mould should be made sufficiently large for easy carving to be carried out). In the new prototyping technique, the diameter and thickness of the resin model can be reduced substantially to save production cost using the zooming function in the software. The total processing time from scanning in of the artwork up to building of the model using the SLA takes about 13 hours to complete, which
22
Chum C. K. and Chow K. Y.
Fig. 26. Resin prototype of Chinese characters and roaring dragon.
is equivalent to 63% in time saving! The resin prototype which is painted in red is shown in Fig. 26. The reliefs of the Chinese characters and the border surrounding the dragon are designed to form a plane surface with the base height of 1.0 mm. Whereas the dragon is assigned with a convex relief at a starting angle of 35 degrees and base height of 0.5 mm. After examining the resin model built from the SLA, it is found that all the relief details turned out as defined except for the body of the dragon which is slightly higher than the desirable convex profile. This can be overcome by changing the colour of that portion and reassigning a relief profile with a smaller angle, say 25 degrees. Steps are observed on the dragon profile, particularly on the slope, due to the constraint of the minimum layer size obtainable on the SLA (0.125 mm). In this case, the craftsman has to spend a little time on removing the stepped appearance of the surfaces for a smoother, better-looking finish. However, this compares favourably because the new prototyping technique is faster and less tedious than carving the prototype from a plaster block. 5.2. Hummm fmce This case study illustrates the formation of a human face using the relief generating method and building of the resin model using the SLA. The scanned-in image is originally a full coloured picture consisting of more than one hundred different colours. Eeducing this huge amount of colours to eight colours representing different parts of the face seems tedious. However, the "Reduce Colour" function helps to simplify the job. The 40 minutes spent on editing the colours is relatively long as compared to that of the black and white dragon image. Nevertheless, the total processing time of 14 hours is far shorter than the conventional prototyping method
Generation of 3D Shapes in CAD/CAM Systems
23
which requires §6 hours to produce a human facial feature plaster mould prototype. The edited image is shown in Fig. 27. The complex shape of the human facial expression, however, presents two main problems when it comes to building the prototypes using the SLA. Firstly, as mentioned earlier, steps are clearly seen on the relief at almost every portion of the face. There are two factors contributing to this shortcoming, the Irst Is the large area occupied by the artwork on the coin surface. Flaws can be easily detected in a large object, likewise, the steps on the face (see Fig. 28) Is enhanced in this case where the
S
^^^^^^^^^^^^^^^^^™^ Fig. 27. Edited image of human face.
Fig. 28. Steps on the face.
Chum C. K. and Chow K. Y.
24
Pig. 29. Resin prototype of the human face.
convex profiles ccwer a large area (relief area of 120 x 120 mm 2 on a 180 mm diameter coin surface) as compared to that of the dragon's body (relief area of 60 x 50 mm 2 on a 100mm diameter coin surface). The second contribution to the distinct step appearance is the gradual slope (ranging from 5 to 25 degrees) of the relief defined. The more gently the slope is defined, the further is the distance between subsequent layer edges, in order for the relief to fit into the curvature. Therefore, it is strongly advisable to define a higher relief angle or to scale down the diameter of the coin surface so that the relief profile covers a smaller area. The second main problem is the reality of the facial expressions, especially the eyes, that can be brought out by the resin prototype. Figure 29 shows the resin prototype. This is largely dependent on the skill and experience of the designer in manipulating the relief as close to the ideal situation as possible.
5,3.
Omhid
Flowers are one of the most popular features used in designing coins. The orchid, being the national flower of Singapore, is thus selected as a case study in prototype making. Similar to that of the human face image, the original photograph (see Fig. 30) of this orchid lower consists of more than a hundred colours, which poses quite a tedious job in reducing the number of colours on the scanned-in image and raising the possibility of the final image (see Fig. 31) diverging unacceptably from the original design, after editing work has been carried out.
Generation of 3D Shapes in CAD/CAM Systems
25
p • • a s s H B £ B s i BB a s a s a s LLU * BBBBBBSBBBaBBBBBUBBJJJJI J v.
U ] j U J J •>
v ^ w v ? •.
.5
%
: ! ] j K I | 1 T J
w^ % w. • • ~ w ~ J , » ^ ¥ vwvwi » ™ ^
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J
v. %v~
i%%%%
*
Fig. 80. Original photograph of the orchid.
^ ^ ^ ^ ^ ^ ^ ^ ^ A
Fig. 31. Edit image of the scanned-in orchid.
The number of hours spent on building the orchid model using the SLA is close to that for the human face model. The duration of model building depends largely on the number of layers required and the size of the model.
26
Chua C. K. and Chow K. Y.
Fig. 32. Resin prototype of the orchid.
The level of complexity in the orchid design is less than that of the human facial design. The reliefs produced are close to the pattern done by the craftsman. However, steps axe seen on the slope of the convex profile, resulting in a substantial amount of polishing work needed to be carried out to improve the surface finish. The convex curvature of the flower pattern appears to be slightly higher than the ideal size, however, this can be easily amended by redefining the angle of the relief in the Art CAM environment, which is actually the main advantage of using this technique in prototyping. A new model can be rebuilt within 10 hours. Texts of all font types can be added to the design by scanning the required texts together with the main design. The reliefs' of the texts are usually defined as a plane surface. The CAD/CAM system has the full capability of generating such reliefs. The resulting model built by the SLA (see Fig. 32) has the advantage of avoiding undesirable steps due to its plane surface nature.
6. Application of A r t - t o » P a r t Technique In M i n t I n d u s t r y The mint industry has traditionally been regarded as very labour-intensive and craft-based. It relies primarily on the skills of trained craftsmen. At present, automation in this industry has been restricted to the use of machines at certain individual stages of the manufacturing process. Several of these stages are not linked up and thus slow down the whole process. The flexibility of CAD/CAM enables the modelling and manufacturing of working dies needed for stamping coins. It is able to link up the design and die-making stages and to provide a common database.
Generation
6.1. Current
of 3D Shapes in CAD/CAM
Systems
27
practices
This section briefly outlines the various processes involved in the design and manufacturing cycle of coin items (circulating and noncirculating). Based on the discussion held with a coin manufacturer as well as some literature, 19 ' 20 it is found that advanced technologies have been implemented in upgrading the production speed and quality of the coin manufacturing processes, especially in the designing stage whereby artworks are prepared using computer graphics software. However, the prototyping of a coin model is still as traditional as one hundred years ago, which is done by hand-carving a plaster mould. This section focuses on the suitability of applying CAD/CAM techniques to replace the conventional practices in prototyping coin models. Therefore, the discussion on the coin manufacturing processes will concentrate on the initial stages up to the making of the working die. The design and manufacturing cycle can be broken down into the following stages: design, plaster mould engraving, making of rubber mould, making of epoxy mould, making of master die and making of working die. 6.1.1. Design At the first stage of the coin production cycle, the designer prepares a 2D artwork based on the Aldus Freehand 3.1 graphical software package which accepts scanned images as input. Majority of the work is spent on touching up the scanned-in picture and editing the text which may not be found in the original image. The output is directed to a laser printer, the resolution of the print-out can be improved by camera shooting it, developed into a photograph, rescanned into Aldus Freehand environment and edit to yield a better image. It is essential to do a few iterations to produce a piece of high resolution artwork so as to reduce the probability of the craftsman's misinterpretation of the artwork while making the plaster mould. The designer's concern is concentrated on the aesthetic issues of the design rather than the mundane issues of precise geometry and the dimensions of the design. 6.1.2. Plaster mould engraving The creation of coin prototypes from a circular plaster block using simple tools such as small chisels involves a high level of skill and experience. The designer's artwork of a coin piece is interpreted by the craftsman who builds a prototype from this interpretation. The craftsman is responsible for dimensioning the various parts of the design based on the proportions provided by the designer, who does not dimension his artwork. These interpretations are greatly influenced by his skills and experience, as a direct consequence, misinterpretations often result. Generally, a number of iterations is carried out within the first two stages (i.e. design and prototyping) before a design is approved for manufacturing. The designer assesses the prototype and suggests modifications to be made to the prototype if it does not turn out in the expected form. The quintessential aspect of prototype building is
Chua C. K. and Chow K. Y.
Fig. 33. Engraving of plaster mould by a craftsman.
the skill of the craftsman, which ultimately determines the quality of the prototype, and consequently, that of the finished product. It takes about one to two weeks for the craftsman to complete one piece of plaster mould prototype. Any crack or mistake on the mould will result in a great deal of amendment work and a more serious case will require the whole work-piece to be discarded. Figure 33 illustrates the engraving of plaster mould prototype by a craftsman.
6.1.3. Making of rubber mould This is a negative side of the plaster mould produced by pouring liquid rubber over the surface of the plaster mould. The rubber mould is removed by peeling off from the plaster mould when solidified. The time taken for a rubber mould to solidify varies from 5 to 8 hours, depending on the amount of hardener mixed in the solution.
6.1.4. Making of epoxy mould The epoxy mould is similar to the plaster mould in terms of its shape and size, except that the epoxy mould is made up of a harder material The epoxy mould is produced by solidifying liquid epoxy over the rubber mould for about one day. It will be used for making the master die.
Generation
of 3D Shapes in CAD/CAM
Systems
29
6.1.5. Making of master die Steel is employed as the working material and processed by an instrument called the pantograph. The pantograph, an instrument for the mechanical copying of a drawing or diagram on the same, an enlarged or a reduced scale, is used for the engraving of coin dies. It consists of an arm that is used to trace an enlarged design at one end and at the other end, a revolving engraving tool simultaneously cuts an exact reduction of the original, to produce a hub from which working dies are made. 19 The whole process takes about two days to complete, after which it undergoes a polishing and touching up treatment.
6.1.6. Making of working die The final working die, a negative face, is produced by the hobbing process using the master die. The hobbing process takes one day to complete, excluding the annealing process which requires another 8 hours of the time. Polishing of the working die is necessary between hobbing processes. The same procedures are carried when processing the reverse surfaces of the working die of a coin. Flow A of Fig. 34 illustrates the steps involved in making coin prototype by the traditional method.
6.2. Use of CAD modeling
and CNC
machining
Once the client has given the requirements (usually in the form of artwork or photographs such as in Fig. 35), the designer would scan in the design and store as an image file in the workstation called Silicon Graphics Iris. The surface modeller, DUCT5.2, is used to map this 2D pattern onto 3D surfaces. The method used to create surfaces is to form four boundaries for each surface needed using the scanned image in the background as a guide for outlining. By doing so, the whole design would be made of many surfaces but is still 2D. Flow B of Fig. 34 gives a brief outline of the process. Flow C of Fig. 34 outlines the art-to-part process as described in Sec. 2 for coin manufacturing. Using the same software, the surfaces could be manipulated in all directions so as to edit on the shape, size (as in Fig. 36) and most important of all, give them a third dimension. Each surface is made up of many meshes. The number of meshes is proportional to the flexibility in manipulating the surfaces. The intersections of the meshes are known as points. DUCT5.2 defines meshes using laterals and longitudes which are perpendicular to each other at the points. DUCT5.2 uses NURBS to define surfaces. Thus, the control of each point is local, that is, moving the position of a point would not affect the rest of the points, even those that are near to that point. After some manipulation of the 3D surfaces, the design on the top surface of the coin is then ready for addition of text. After the top surface has been completed, the shape of the coin is then modelled.
Chua C. K. and Chow K. Y.
30
B
Use scanned in image as background to outline the picture so as to create surface
Surface Generation using DUCT
Relief Genertion using ArtCAM Create 3D coin with surfaces (real time) using DUCT
WrapArtwork with reliefs created onto 3D Coin
Surface painting to visualise the actual model of the coin before producing the die
Surface apinting to visualise the actual model of the coin before producing the prototype Create prototype using Stereolithography
Create machining path
Making of Epoxy Mould Pentograph Machining of Master Die
CNC machining
Fine Touching up and Polishing of Master Die
(?)
Create Machining Path
(?)
(?)
CNC Machining
Hobbing Process to produce Working Die
(?)
Yes T ->f Final Polishing
(?)
= ^ _ Working Die Fig. 34.
->- Ready to Manufacture Coin
Current and proposed coining processes.
After the coin has been completely modelled, surface painting is done to visualise the actual model of the coin before producing the die. Lighting on the model could be done to enhance the appearance of the model. Up till this stage, the soft prototype has been produced on the workstation. Before investing substantial sum of money in making the mould, the manager could actually see the prototype of the coin and comment on it. Alterations could be made and money would not be spent unnecessarily in making moulds.
Generation
of 3D Shapes in CAD/CAM
Systems
Fig. 35.
Artwork of a Dendrobium Singa Snow orchid.
Fig. 36.
Manipulation of a petal surface of the orchid.
31
After the higher authority has approved the overall design, the designer would then create the machining path and send the NC codes to the CNC machine for machining. At this stage, two alternatives arise, either to machine the actual working die (Fig. 37) or the master die (Fig. 38), or both. By using the above methods of producing the dies, it reduces many existing processes and replaces the use of a copying machine. The main disadvantage of the copying method using pantograph is the time spent in producing the master die, which Is made without using computerised equipment. With NC and CNC, the master die is not required because a working die could be machined at anytime if the NC codes are available. 21
Chna C. K, and Chow K. Y.
32
Fig. 37.
Fig. 38.
Machined working die using plastic.
Machined master die using plastic.
7. Conclusion The CAD/CAM software allows the formation of complicated and time consuming reliefs on models such as jewellery, ceramics tableware, pewter ware, coin dies, etc. to be semi-automatically or automatically created. The software employs the technique of colour segmentation in generation of three-dimensional relief. The software provides realistic viewing function to see the colour shaded Inal model and permits amendments to be made easily. The art-to-part technique has facilitated the creation of complex surfaces. Experiments on building models using the SLA have been carried out to study the application of the relief generating software system. Three models, the Chinese
Generation of 3D Shapes in CAD/CAM Systems
33
Legend and Tradition, the human face and the orchid were built and examined. It was found that substantial amount of polishing work is needed to improve the surface finish of the resin models. Steps were observed on the slope of the relief profile which was caused mainly by the minimum layer size obtainable on the SLA, which is not small enough to produce a smooth surface finish. The major advantage of this prototyping technique is the ability to create more prototypes for less time and cost.
References 1. C. K. Chua, W. Hoheisel, G. Keller and E. Werling, Computing and Control Engineering Journal 4 (1993) 100-112. 2. H. B. Lee, S. H. K. Micheal, R. K. L. Gay, K. F. Leong and C. K. Chua, International Journal of Computer Applications in Technology 5 (1992) 72-80. 3. "2D to 3D Software (ArtCAM) 'Queen's award' case study", Carl Jury International Sales, Delcam International, UK (1992). 4. "ArtCAM 'Rose' case study", Carl Jury International Sales, Delcam International, UK (1992). 5. Y. Koren, Computer Control of Manufacturing Systems (McGraw Hill, Singapore, 1983). 6. J. C. Fidoora, 49th Annual Technical Conference - ANTEC '91. (May 1991) 5-9. 7. I. Mueller, Die Casting Engineer 36, 3 (1992) 28-33. 8. F. E. DeAngelis, Laser in Microelectronic Manufacturing (1991) 61-70. 9. H. B. Lee, Computer-aided and Manufacturing for Jewellery Industry, a thesis for Degree of Master of Engineering in Nanyang Technological University, Singapore, 1993. 10. P. L. Ulerich, Proceeding of the 1992 ASME International Computer in Engineering and Exposition, Computer in Engineering, ASME (1992) 275-281. 11. C. Bradley, G. W. Vickers, and J. Tlusty, CIRP Annals 4 1 , 1 (1992) 437-440. 12. D. Kochan, Computers in Industry, 20, 2 (1992) 133-140. 13. D. L. Bourell, H. L. Marcus, J. W. Barlow and J. J. Beaman, International Journal of Powder Metallurgy (Princeton, New Jersey), 28, 4 (1992) 369-381. 14. L. L. Kimble, Winter Annaul Meeting of the American Society of Mechanical Engineers (1991) 73-80. 15. C. K. Chua, K. F. Leong and C. S. Lim, Rapid Prototyping: Principles and Applications (World Scientific, 2003). 16. C. T. Lee, Implementation of Repair Algorithms for Faulty Stereolithography Files, Final Year Project School of Mechanical and Production Engineering, Nanyang Technological University, Singapore, 1996. 17. Stephen J. Rock and Michael J. Wozny, Solid Freeform Fabrication Symposium Proceedings, the University of Texas at Austin, Texas, 12-14 (1991) 28-36. 18. K. F. Leong, C. K. Chua, and Y. M. Ng, International Journal of Advanced Manufacturing Technology 12 (1996) 407-414. 19. G. Hoberman, The art of coins and their photography, Harry N. Abrams, New York, 1982. 20. W. J. Zimmetman, The coin collector's fact book (AMO, New York, 1974). 21. Y. Koren, Computer Control of Manufacturing Systems (McGraw Hill, Singapore, 1983).
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CHAPTER 2 C O M P U T E R T E C H N I Q U E S A N D A P P L I C A T I O N S OF C O N V E R T I N G 2D ASSEMBLY D R A W I N G S I N T O 3 D PART DRAWINGS IN C O M P U T E R AIDED DESIGN MASAJI TANAKA Okayama University of Science, Dept. of Mechanical Systems Engineering, Ridai-cho 1-1, Okayama 700-0005, Japan Tel. & Fax: +81-86-256-9594 E-mail:
[email protected] KENZO IWAMA Engicom Corp., Jimpo-cho 2-32, Kanda, Chiyoda, Tokyo 101-0051, Japan ATSUSHI HOSODA and TOHRU WATANABE Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Siga 525-0058, Japan Presently, though solid modelers are introduced to CAD systems, 2D CAD systems are often used for designing products that do not need complex curved surfaces. However, solid models will be needed for many cases such as CAM, CAPP, catalogs and manuals in the future. As the result, it is necessary to convert 2D drawings in to solid models. Many research efforts have been conducted to automatically reconstruct solid models from orthographic views of one solid. However, previously proposed algorithms require a large amount of computational time when the solid model becomes complex because many combinatorial searches of geometric elements such as faces and blocks are needed. In this chapter, a method is proposed that can minimize the combinatorial search of blocks called solid elements. The method can very fast generate solid models from orthographic views and also generate solid models of parts from 2D assembly drawings simply. Detailed explanation of the idea in the method and many examples are indicated in this chapter. Keywords: Orthographic view; assembly drawing; part drawing; solid element; solid model. 1. Introduction CAD (Computer Aided Design)/CAM (Computer Aided Manufacturing) systems have advanced automated design and manufacturing. In particular, solid modeling 35
36
Masaji Tanaka et al.
enables the manipulation of 3D (three-dimensional) models of objects such as mechanical products. As a result, beginners who have little experiences about design and manufacturing can make solid models of various objects. 3D models become popular in CAM systems to generate NC (Numerical Control) programs and in other applications to make assembly planning and robot programs. However, 2D (two-dimensional) CAD systems are still preferred by expert designers for designing correct and precise products. Therefore various input devices provide engineers with choices in 2D drawings and improve flexibility for designers. Perhaps many designers will use both 2D CAD systems and solid modelers. They have desired automatic conversion programs of 2D drawings into solid models, and the automatic conversion is still an important research issue. Many research efforts have been conducted to automatically reconstruct solid models from orthographic views of one solid. Especially Idesawa,1 and Wesley and Markowsky2 led the studies at the first stage of the research. Idesawa constructed (3D) wireframe models and (3D) surface models from orthographic views. Wireframe models consist of edges of solid models. Surface models consist of faces each of which is a closed loop of the edges. Wesley and Markowsky constructed blocks each of which is a closed region of the faces. Those edges, faces and blocks are virtual geometric elements because all of them do not form the objects drawn as orthographic views. There can be ghost elements that do not actually exist in the solid models of the objects. Therefore, the solid models of the objects are obtained by removing the ghost elements. They take combinatorial search to remove the ghost elements from the reconstructed elements. However, the number of combinations of geometric elements is exponential of the number of the elements. For example, when the number of faces is n, the number of all combinations becomes 2 n . All of the combinations must be examined because plural solutions can exist in orthographic views. As a result, their methods were not practical. In addition, they limited the shapes of the objects to polyhedrons. Later Sakurai et al.3 constructed blocks that include cylindrical, conical, spherical and toroidal faces. In this paper, real elements of the objects are called true elements and ghost elements are called false elements, and the blocks are called solid elements. Solid models were represented as B-Reps (Boundary Representation) in the three methods. 1 " 3 On the other hand, several methods have reconstructed solid models as CSG (Constructive Solid Geometry) by recognition of elementary objects such as cube, cylinder and fillet from orthographic views. 4 " 6 However, the orthographic views that can be applied to the methods are very restricted, and it is difficult to apply the methods to plural solutions. At the next stage of the research, many methods were proposed to effectively search for the solutions. For example, Sasaki et al.7 applied pseudo-Boolean algebra to distinguish the truth of edges and faces. Nishihara et al.8 applied heuristics to distinguish the truth of faces. Gujar and Nagendra 9 ' 10 surveyed the study and developed a more systematic method than Wesley and Markowsky's to generate solid elements. Though those methods could effectively search for the solutions,
Converting
2D Assembly Drawings into 3D Part Drawings in CAD
37
they required a large amount of computational time when the shapes of solid models become complex. The authors proposed a method to realize a practical system that can very fast generate solid models from orthographic views. 11 ' 12 The method uses solid elements to reconstruct solid models from orthographic views. Generally there are two kinds of geometric elements that can reconstruct solid models from orthographic views. One is face and the other is solid element. Solid elements are superior to faces in the combination because the number of solid elements is much fewer than the number of faces in solid models. In addition, it is more difficult to convert surface models to solid models than to convert solid elements to solid models because each of the solid elements already forms a solid model. The authors' method established a set of equations that represent relationships of the truth among solid elements by comparing the solid elements with orthographic views. Since the method solves the equations, it does not employ a mechanism of searching suitable combinations of faces or solid elements. The method runs faster than those which employ search mechanisms. The computational time of the method can be linear to the complexity of a solid. This paper describes a method of automatic conversion of 2D assembly drawings into 3D part drawings. 13 The method is developed as an application of the authors' method described above. Though there are many methods to reconstruct solid models from orthographic views, nobody has attempted to reconstruct solid models of more than one part from orthographic views to the knowledge of the authors. In general, product design proceeds from conceptual to detailed design. Designers are observed first drawing 2D assembly drawings of products, and then drawing each 2D part drawing despite the fact that there are methods of designing products by composing designed parts in CAD systems by superimposing layers of 2D part drawings. The 2D part drawings are transformed into solid models for CAM systems and others. It is routine and takes much time to decompose 2D assembly drawings into 2D part drawings and transform them into solid models. Therefore, these transformations are very often processed by operators other than designers. The automatic decomposition system is important for designers for the following two reasons: • It takes much time to draw each 2D part drawing from a 2D assembly drawing in proportion to the number of parts. • If operators other than designers process the decomposition, the operators may fail to correctly recognize each part of the 2D assembly drawing. In this method, wireframe models, surface models and solid elements are constructed from 2D assembly drawings that are drawn as orthographic views. Solid elements become the components of solid models of parts of assemblies. The method generates solid models of each part by classifying solid elements into elements of some parts and false elements that do not actually exist in any parts. If there is more than one solution, this method can generate all of the solutions. The domain
38
Masaji Tanaka et al.
y 1^ 0
Top
z
Z A
A
0
-" A
Front Fig. 1.1.
0
Side
Coordinate system of orthographic views.
of 2D assembly drawings is limited to orthographic views consisting of front, top and side views, and cross-sectional views. Also, the types of faces are limited to planar, cylindrical, conical and spherical faces. This paper firstly explains a method that reconstructs solid models from orthographic views of one part by using cubic elements. The cubic element is a particular solid element and forms a cube. This method conveys the basic idea in this study, but it is only applied to orthographic views that consist of only straight lines that are parallel to the axes of the coordinate system. Secondary, the method is extended to include oblique lines and curved lines in orthographic views, and to convert cubic elements to solid elements. Finally, this paper explains the method that decompose 2D assembly drawings into solid models of parts. Figure 1.1 illustrates the coordinate system of orthographic views of this study (front: x-z, top: x-y, side: y-z). 2. Conversion of Orthographic Views into Solid Models by Cubic Element Equations 2.1. Construction
of cubic elements
from lattice
faces
When the orthographic view of a part is drawn by only straight lines that are parallel to the axes of the coordinate system, it is possible to convert the view to that consisting of three lattice faces by extending the lines. Here, solid lines and dotted lines are not distinguished. For example, Example 2.1 illustrated as in Fig. 2.1 can be converted to the lattice faces as in Fig. 2.2. Lattice faces obtained from an orthographic view can be regarded as the projections of a set of cubes. In this case, each square of lattice faces corresponds to a face of a cube. When an orthographic view can be converted to three lattice faces, the solid models of the part are obtained by searching suitable combinations of the cubes. Because all of
Converting 2D Assembly
Drawings into 3D Part Drawings in CAD
Fig. 2.1.
Fig. 2.2.
39
Example 2.1.
The lattice faces in Example 2.1.
the faces of the solid models are parallel to x-z, x-y or y-z planes, the set of the cubes form a three dimensional matrix. However, all of the cubes are not elements of the solid models. Let cubes that are elements of the solid models be true cubic elements and let cubes that are not elements of the solid models be false cubic elements. Then our goal is to obtain the solid models by distinguishing the truth
40
Masaji Tanaka et al.
Fig. 2.3.
The wireframe model of the cubic elements.
of all of the cubic elements. Figure 2.3 illustrates the wireframe model of the cubic elements constructing from Fig. 2.2. 2.2. Relationships
of the truth among cubic
elements
The relationships between cubic elements and squares of lattice faces can be recognized since all the cubic elements form 3D matrices. For example, when the cubic elements in Example 2.1 are numbered as in Fig. 2.4, the numbers of cubic elements corresponding to the squares of the lattice faces in Fig. 2.2 are illustrated as in Fig. 2.5. The relationships of the truth among the cubic elements are obtained by comparing original orthographic views and the lattice faces corresponding to the cubic elements. The relationships are classified into five conditions explained in the following. Conditions of Cubic Elements • Solid Line Condition If a line exists between two squares of a lattice face and this line exists as a solid line in an original view, two cubic elements corresponding to these squares are not both true elements. The reason for this is that the solid line in the original view cannot be there or must be a dotted line if both of the two cubic elements are true. When the two cubic elements are Sx and Sy, the relationship is expressed as a Solid Line Condition, Sx x Sy. For example, 5i x S2 is obtained from the front view in Example 2.1. • Temporary Line Condition If a line exists between two squares of a lattice face and this line does not exist in the original view, two cubic elements corresponding to these squares are both true elements or both false elements. In short their truth values are the same. The reason for this is that a new solid line appears in the original view if their
Converting
2D Assembly Drawings into 3D Part Drawings in CAD
Fig. 2.4.
Fig. 2.5.
41
The cubic element in Example 2.1.
S?
Ss
S9
S4
Ss
S6
Si
S2
S3
Si
S2
S3
S3
S6
S9
Sio
Su
Sl2
Sl2
Sis
Sis
The relationship between lattice faces and cubic elements in Example 2.1.
truth values are not the same. When the two cubic elements are Sx and Sy, the relationship is expressed as a Temporary Line Condition, Sx — Sy. For example, Su — S12 is obtained from the front view in Example 2.1 and also Su — S15, Si7 — Sis a r e obtained because these elements inherit the relation of S u — S12 in the direction of the front view.
42
Masaji Tanaka et al.
• Dotted Line Condition If a line exists between two squares of a lattice face and this line exists as a dotted line in the original view, the truth values of two cubic elements corresponding to these squares are the same. The dotted line must be a solid line if their truth values are not the same. For a dotted line to exist, cubic elements must exist that make edges corresponding to the dotted line. When the two cubic elements are Sx and Sy, the relationship is expressed as a Dotted Line Condition, Sx — Sy. For example, S\o — Su is obtained from the front view in Example 2.1 and the dotted line must be formed by the combinations of the truth in set {S13, S14} or set {516,517}.
• Existence Condition For each line segment that is an edge of squares of lattice faces, there must be one true cubic element in all of the cubic elements that have edges corresponding to one line segment. For example, though the line segment exists that is horizontal and corresponds to the top edge of S 7 in the top lattice face in Example 2.1, the corresponding line segment does not exist in the original top view. Therefore, S7 and Si6 are false elements. Also, since the line segment exists that is vertical and corresponds to the left edge of Si in the front lattice face in Example 2.1 and the corresponding line segment exists in the original front view, all elements of {Si,S4,Sr} are not false. • Mass Condition Cubic elements that form a solid model cannot be separated into two or more solids, nor can the cubic elements be connected only through vertices or edges. 2.3. Search of solutions
by cubic element
equations
The relationships of cubic elements corresponding to lattice faces are organized as a system of equations that is called a cubic element equation. For example, three systems of equations are organized from Fig. 2.5 as in Fig. 2.6(a) and they can be combined as in Fig. 2.6(b). The cubic element equation in Example 2.1 is organized as in Fig. 2.6(c) by adding S10 — S13, S u — S14, Si7 — Sis that do not appear on the lattice faces in Fig. 2.5 and simplifying the equations. Cubic element equations simplify the relationships of all cubic elements. In Fig. 2.6(c), two sets of cubic elements A = {S 1 ; S 3 , . . . , Sis} and B = {S2, S 5 , S 6 } are made, and the truth values of the cubic elements of each set are equal. As a result, solid models are obtained in cubic element equations by searching not suitable combinations of all cubic elements but suitable combinations of several sets of cubic elements. Therefore, cubic element equations can minimize the amount of the combinatorial search of cubic elements. In Example 2.1, the solution is obtained as the following. When Existence Conditions are applied to the cubic elements, Si and S10 become true elements. Therefore set A becomes true, and S7 and set B becomes false. All of Dotted Line Conditions are satisfied in set A. Therefore, the solution is given as set A. Figure 2.7 illustrates the solution.
Converting
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2D Assembly Drawings into 3D Part Drawings in CAD
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ons in each view.
44
Masaji Tanaka et al.
Fig. 2.7.
The solution of Example 2.1.
Since the method using cubic elements need not construct wireframe models and surface models from orthographic views, solid models are easily obtained. However, the method is applied to only orthographic views that can convert to lattice faces. So the authors apply the method to solid elements that are constructed from various orthographic views such as including curved lines as in the following chapter. 3. Conversion of Orthographic Views into Solid Models by Solid Element Equations 3.1. Construction
of solid
elements
This subsection gives the definitions of the following seven geometric elements, and our method generates these geometric elements to construct solid elements. (1) 2D vertex and 2D edge 2D vertices are defined in each of the orthographic views, and are classified into the following three types: • Crossing points of straight lines and arc lines. • Tangency points between straight lines and arc lines, and tangential points between arc lines. • Maximum or minimum points of arc lines in the vertical or horizontal directions in orthographic views, and the center points of circles. The 2D edges are defined in the orthographic views as the following two types: • Straight lines and arc lines existing between two 2D vertices. • Vertical or horizontal straight lines that correspond to 2D vertices selected from tangency points and maximum or minimum points described in the other views. Those edges are called silhouette edges. Figure 3.1 illustrates Example 3.1 and all of the 2D vertices and 2D edges are illustrated as in Fig. 3.2.
Converting 2D Assembly
Drawings into 3D Part Drawings in CAD
45
r
Fig. 3.1.
Example 3.1.
<•
1»
•
>t—it Fig. 3.2.
II
• — 1 »
i
1
•
The 2D vertices and 2D edges in Example 3.1.
(2) Vertex and edge A vertex is defined from the three 2D vertices in each view. Let a front 2D vertex be (fx, f z ) , a top 2D vertex be (tx,ty) and a side 2D vertex be (sy, sz). If fx = tx, ty = sy and fz = sz, (3D) vertex (fx,sy, fz) can exist as a virtual vertex of the solid. An edge is defined from two vertices. Let two vertices be (xi, j/i, z{) and (x2,2/2) £2)If 2D edge (x\, z{) — (X2, ^2) exists, or Xi = x%, z\ = 22 and 2D vertex (x\, z{) exists in the front view, (xi,yi,zi) — (x2,V2,Z2) has the projection on the front view.
46
Masaji Tanaka et al.
Fig. 3.3.
The wireframe model in Example 3.1.
In the same way, if an edge (XI,J/I,.ZI) — (^2,2/21 z2) has projections on each view, the edge can exist as a virtual edge of the solid. All of edges form wireframe models of solids. Figure 3.3 illustrates the wireframe model obtained from Fig. 3.2. (3) 2D face and face Except for the silhouette edges, a 2D face is a closed loop of 2D edges. Figure 3.4 illustrates all of the 2D faces obtained from Fig. 3.2. A face is a closed loop of edges. All of faces form surface models of solids. Figure 3.5 illustrates all of the faces obtained from Fig. 3.3. (4) Solid element A solid element is a closed region of faces. Figure 3.6 illustrates each of the solid elements (S-y, 52,^3) obtained from Fig. 3.5. A solid element that actually exists in the solid models is called a true solid element, and a solid element that does not exist in the solid models is called a false solid element. Solutions are obtained by distinguishing the truth of all solid elements. 3.2. Relationships
of the truth among solid
elements
When cubic elements correspond to solid elements, squares of lattice faces correspond to 2D faces. A 2D face corresponds to a face of a solid element in the direction of a view, but a face of a solid element does not always correspond to a 2D face in the direction of a view. For example, Fig. 3.7 illustrates the relationships between 2D faces and solid elements in Example 3.1. It is found that Si corresponds to three 2D faces in the front view. Solid elements corresponding to 2D faces are found by
Converting
2D Assembly Drawings into 3D Part Drawings in CAD
Fig. 3.4.
Fig. 3.5.
47
The 2D faces of Example 3.1.
The faces in Example 3.1.
searching the relationships between 2D edges and edges. The relationships of the truth between solid elements are obtained by comparing 2D faces with the faces of solid elements that correspond to them. The relationships are expressed as in the following four conditions.
Masaji Tanaka et al.
48
Fig. 3.6.
T h e solid elements in Example 3.1.
S2 Si
Si
S3 Fig. 3.7.
S2
Si
The relationship between 2D faces and solid elements in Example 3.1.
Conditions of Solid Elements • Solid Edge Condition If a 2D edge existing between two 2D faces is a solid line, the corresponding two faces of two solid elements must not be tangent to each other at the edge that corresponds to the solid 2D edge. If the two faces are the same, the solid element
is a false element. If the two faces are tangent to each other, the two solid elements are not both true. When the two solid elements are Sx and Sy, the relationship is expressed as a Solid Edge Condition, S X X Oy .
• Dotted Edge Condition If a 2D edge existing between two 2D faces is a dotted line, the corresponding two faces of two solid elements must be tangent to each other at the edge that
Converting
2D Assembly Drawings into 3D Part Drawings in CAD
49
corresponds to the dotted 2D edge. Since the two faces are tangent to each other, the two solid elements are both true or both false. If the two faces are not tangent to each other, the two solid elements are not both true. For a dotted 2D edge to exist, solid elements must exist that make edges corresponding to the dotted 2D edge. When the two solid elements are Sx and Sy, the relationship is expressed as a Dotted Edge Condition, Sx — Sy. • Existence Condition For each 2D edge, there must be one true solid element in all of the solid elements that have edges corresponding to one 2D edge. • Mass Condition Solid elements that form a solid model cannot be separated into two or more solids, nor can the solid elements be connected only through vertices or edges. When those conditions are applied to Fig. 3.7, Si x S 2 is obtained from the top view and S\ x S 2 , S\ x 53 are obtained from the side view. As the result, S 2 x Si x 53 is obtained that is the solid element equation in Example 3.1. Solid element equations correspond to cubic element equations. The solution is obtained as the following. When Existence Conditions are applied to the solid elements, Si becomes true. Therefore S 2 and 53 are false. All of Dotted Edge Conditions are satisfied in Si. Therefore, the solution is given as Si.
3.3. Two
examples
Figure 3.8(a) illustrates the orthographic view of Example 3.2 that has plural solutions. The wireframe model and solid elements(Si, S 2 ,. • •, S n ) are illustrated as in Fig. 3.8(b), (c). Figure 3.8(d) illustrates the relationships between 2D faces and solid elements. The equations of the solid elements in each view are illustrated as in Fig. 3.8(e). The solid element equation is made as in Fig. 3.8(f) by combining the equations. The solutions are obtained as the following. When Existence Conditions are applied to the solid elements, S@ becomes true and Si, S 2 , S5, S7, S$, Su become false. Then when Existence Conditions are applied again, S3 and S9 become true. Therefore S4 and S5 become false. S10 does not appear in the solid element equation and all of the conditions cannot distinguish the truth of S i 0 . It is called undecided solid elements that cannot be distinguished by all of the conditions such as S10 in this example. Undecided solid elements generate plural solutions. The two solutions are obtained as in Fig. 3.8(g). S10 is false in solution (I) and it is true in solution (II). Figure 3.9(a) illustrates the orthographic view of Example 3.3 that form very complex shapes. The wireframe model and solid elements (Si, S 2 , . . . , S 27 ) are illustrated as in Fig. 3.9(b), (c). Figure 3.9(d) illustrates the relationships between 2D faces and solid elements. The equations of the solid elements in each view are illustrated as in Fig. 3.9(e). The solid element equation is made as in Fig. 3.9(f) by combining the equations. The solutions are obtained as the following. When Existence Conditions are applied to the solid elements, set A = {Si, S 4 , . . . , S27}
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Masaji Tanaka et al.
(b) Wireframe model.
(a) Orthographic view.
(c) Solid elements. Fig. 3.8.
Example 3.2.
becomes true and S2, SQ, S12 become false. Though S3 can be true in the solid element equation, it connects into set A at only edges. Therefore, S3 becomes false by Moss Condition. All of Dotted Edge Conditions are not satisfied in set A. Set B = {S\3, S14, S\e, Sn, S22, S23, S25, S26} is the set of solid elements that do not appear on the solid element equation. All of Dotted Edge Conditions are satisfied
Converting 2D Assembly Drawings into 3D Part Drawings in CAD
S7
51
Ss S6
Si
S2
Si y
S 2
\
S2
Ss Ss
S3 Ss
Ss
S4
Su
(d) Relationships between 2D faces and solid elements. S?
X X
Ss X
S6
X Si
X X
S? XSsX XXX
S2
Su
S6 Si
S2
X
X X
S4
X
X Se
X SS
XXX Si XS2X XXX
SS
X
S3
X
S2
X
S3
X
Ss
Su
(e) Equations in each view. Fig. 3.8.
Ss
X
X
S4
SS
(f) Solid element equation. Continued
by suitable combinations of the elements of set B. Since the suitable combinations are not obtained by all of the conditions, 2 8 = 256 combinations of the truth are all attempted. The elements of set B are similar to S\Q in Example 3.2 because all of their combinations must be attempted to obtain the solutions.
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Masaji Tanaka et al.
(I)
(II)
(g) Two solutions. Fig. 3.8.
Continued
r
i
(a) Orthographic view. Fig. 3.9.
(b) Wireframe model. Example 3.3.
Therefore, they are undecided solid elements in the method. As the result, it is found that twelve combinations of them satisfy all of Dotted Edge Conditions. Therefore, twelve solutions are obtained. Figure 3.9(g) illustrates two examples of the solutions.
Converting
53
2D Assembly Drawings into 3D Part Drawings in CAD
s?
Ss
S9
S4
Ss
Se
Si
S2
S3
Si
S2
S3
Si
S2
S3
Sio
S11
S12
S12
Sis
Sis
Sl9
S20
S21
S21
S24
S27 t
(d) Relationships between 2D faces (c) Solid elements.
and solid elements.
S? - Ss — S9 I I X S4-SsX Ss I X X S1XS2 X S3 S1XS2X S3 I X X Sio - S11 X S12 I I X S19 — S20 X S21
S3 X X X S2 Se S12 X X X S1-S4-SS-S7-S8
S3XS6X S9 I X X ' S12-S1SXS18 S18-S1S-S11-S10-S9 X S21 - S24 X S27 Sl9 — S20 — S21 - S24 — S27
(e) Equations in each view. Fig. 3.9.
(f) Solid element equation. Continued
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Masaji Tanaka et al.
(g) Two examples of solutions. Fig. 3.9.
Continued
4. Conversion of 2D Assembly Drawings into 3 D Part Drawings 4.1. The difference between 2D assembly orthographic views of one part
drawings
and
2D assembly drawings include all the information about the shapes of all the parts in the assembly. The main differences between orthographic views of one part and 2D assembly drawings are the following two points. • Border lines between parts are drawn in 2D assembly drawings. • The part numbers are indicated to each part in 2D assembly drawings. Even if two faces of different parts are tangent to each other at an edge, this edge is drawn as a solid or dotted line in 2D assembly drawings. Therefore, it is possible to generate all of the solid models of the parts from solid elements that are constructed from 2D assembly drawings. The two points of 2D assembly drawings are applied to the conditions in the relationships between 2D faces and solid elements described above. When the number of parts is M and the number of solid elements is N in an 2D assembly drawing, the number of all combinations of the solid elements is (M + 1)^. Therefore, it terribly increases against the complexity of shapes of assemblies and the number of parts compared with orthographic views of one part. In addition, consideration of actual possibility of assembling is needed to obtain the solutions. As Subsec. 4.4 explains the effectiveness of cross-sectional views for reducing the number of solutions, our method makes use of cross-sectional views. The input to the method consists of orthographic views of an assembly that include cross-sectional views and part numbers. Each part number is given to some surfaces in orthographic views. Figure 4.1 illustrates Example 4.1 that is changed from Example 3.1 in a way
Converting
2D Assembly
Drawings into 3D Part Drawings in CAD
55
WD —
Fig. 4.1.
Example 4.1.
that Example 4.1 consists of two parts. Note that solid elements of Example 4.1 and Example 3.1 are the same, but Example 4.1 has parts Pi and P 2 while Example 3.1 has only part P\. In our method, parts in an assembly are called Pi, P 2 , and so on. 4.2. Relationships of the ti~uth among solid elements assembly drawings
in
The following conditions are applied to solid elements that are constructed from 2D assembly drawings. Conditions of Solid Elements in Assembly • Solid Edge Condition In Assembly If a 2D edge existing between two 2D faces is a solid line, the corresponding two faces of two solid elements must not be tangent to each other at the edge that corresponds to the solid 2D edge except where the two solid elements are not both elements of a part. If the two faces are the same, the solid element is a false element. If the two faces are tangent to each other, the two solid elements are neither both true nor both elements of a part. When the two solid elements are Sx and Sy, the relationship is expressed as a Solid Edge Condition In Assembly, • Dotted Edge Condition In Assembly If a 2D edge existing between two 2D faces is a dotted line, the corresponding two faces of two solid elements must be tangent to each other at the edge that corresponds to the dotted 2D edge. Since the two faces are tangent to each other,
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Masaji Tanaka et al.
the two solid elements are both elements of a part or both false. If the two faces are not tangent to each other, the two solid elements are not both true. For a dotted 2D edge to exist, solid elements must exist that make edges corresponding to the dotted 2D edge. When the two solid elements are Sx and Sy, the relationship is expressed as a Dotted Edge Condition In Assembly, Sx — Sy. • Existence Condition In Assembly For each 2D edge, there must be one true solid element in all of the solid elements that have edges corresponding to one 2D edge. In addition, for each 2D face marked by some part numbers, there must be one solid element in all of the solid elements that have faces corresponding to one 2D face marked by the part number. • Mass Condition In Assembly Solid elements that form a part cannot be separated into two or more solids, nor can the solid elements be connected only through vertices or edges. Figure 4.2 illustrates the relationships between 2D faces and solid elements in Example 4.1. The solution is obtained by applying those conditions to Fig. 4.2. Firstly Si x S2 is obtained from the top view, and Si x S2, S\ x 5 3 are obtained from the side view. As a result, 52 x Si x S3 is the solid element equation. It is found that Si is an element of Pi, and S2 is an element of P2 due to the existence of two 2D faces marked Pi and P2. 5 3 is false due to Si x S3 and Mass Condition In Assembly. As a result, Pi is Si, and P2 is S2 as in Fig. 4.3. 4.3. Search
algorithm
Figure 4.4 illustrates our search algorithm to solve a system of solid element equations. The input is a set of conditions that consist of every Solid Edge Condition In
S2 Si
Si
S3 Fig. 4.2.
S2
Si
The relationships between 2D faces and solid elements in Example 4.1.
Converting
2D Assembly
Fig. 4.3.
Drawings into 3D Part Drawings in CAD
57
The solution of Example 4.1.
Assembly from all the solid 2D edges, Dotted Edge Condition In Assembly from all the dotted 2D edges, and Existence Condition In Assembly from all the 2D edges and part numbers on some 2D faces (the proposed program gets the part numbers). An outline of the algorithm is as follows: (1) Finding true or false elements Find solid elements that are trivially false by Solid Edge Conditions In Assembly and Dotted Edge Conditions In Assembly, and find true solid elements by Existence Conditions In Assembly. (2) True elements exist If true solid elements do not exist, develop combinations of all solid elements. (3) Combining elements If the number of parts is M and the number of undecided solid elements is K, make (M + l)K combinations. (4) Applying all conditions Apply Solid Edge Conditions In Assembly and Dotted Edge Conditions In Assembly to find true or false solid elements. After the application of these two kinds of conditions, select a false element and check to see if there is a true element where the true element has an edge of the false element. Finally, apply Mass Condition In Assembly to check and see if a part includes more than one separated solid. (5) Unsatisfied conditions exist If unsatisfied conditions exist, the drawing must have mistakes.
Masaji Tanaka et al.
58
(
O
START )
V
Finding true or false elements
® Combining ^ elements
® Output of error
Output of solutions Fig. 4.4.
The algorithm used to solve a system of solid element equations.
(6) Output of error Output a message "Drawing error". (7) Undecided elements exist If solid elements that are undecided in terms of belonging to parts or truthfulness exist, combine them. (8) Output of solutions Solutions are formed by connecting the solid elements in each part.
Converting 2D Assembly Drawings into 3D Part Drawings in CAD
4.4. Effect of cross-sectional
views and assembly
59
knowledge
The number of solutions rapidly increases as part geometries become complex and the number of parts becomes large. This subsection explains that knowledge about assembly sequences and cross-sectional views reduce the number of solutions. The knowledge of assembly often reduces the solutions by removing solutions that cannot be assembled. There are cases where the cross-sectional views show border lines between parts while the borders do not appear in orthographic views. In such cases the cross-sectional views reduce ambiguities in 2D drawings. Figure 4.5(a) illustrates a 2D assembly drawing of Example 4.2 that consists of two parts. Figure 4.5(b), (c), (d) illustrate the wireframe model, solid elements, and the relationships between 2D faces and solid elements. The solid element equation becomes S\ x S2 x 5s. It is found that 52 is an element of Pi and S$ is an element of Pi by Existence Conditions In Assembly. Therefore Si becomes false. 53 and S4 are undecided elements but neither of them are elements of Pi. As the result, three solutions are obtained as in Fig. 4.5(e) by the combination of S3 and S4. They are both false in solution (I). S3 is an element of P\ and S4 is false in solution (II). They are elements of P\ in solution (III). However, it is unstable to assemble the two parts in solution (I), and it is impossible to assemble them in solution (III). Therefore solution (III) is removed from final solutions of Example 4.2, and solution (I) may be marked as an unstable
O ;
r
\
/
i
I 1
/
/
/
/
1
\
\
\
*-f2 \
\
/ 1
/
/
1
\ \
\
\
(a) Orthographic view. Fig. 4.5.
Example 4.2.
(b) Wireframe model.
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Masaji Tanaka et al.
e*
\S2J &
Ss \ Ss r t
/
i
Ss
Ss \
\
\
\
(d) Relationships between 2D faces and solid elements.
(c) Solid elements. Fig. 4.5.
Continued
solution of Example 4.2. Cross-sectional views are often drawn by designers when orthographic views include ambiguities in reconstructing solids from the views. An example is shown in Fig. 4.5(f) that illustrates the full sectional view in Example 4.2. Figure 4.5(g) illustrates the relationships between 2D faces and solid elements in the full sectional view. Those relations are obtained by cutting solid elements at the cross-section. Figure 4.5(h) illustrates the solid element equation in the full sectional view. By solving the equation, one gets that Pi consists of 52 and 53, and Pi has only S5. As the result, only solution (II) in Fig.4.5(e) is obtained as the solution of Example 4.2. 4.5. Effect
of design
knowledge
Design knowledge is also able to reduce the number of solutions. There are many mechanical parts that form fixed shapes. Examples are bolts, nuts, gears, etc. If such
Converting
2D Assembly
Drawings into 3D Part Drawings in CAD
61
©
r
V
r
"*
©
( I) (i) (in) (e) Three solutions. Si
X
S3 X X S4 X Ss « - @ (g) Relationships between 2D faces and (h) Solid element equation (f) Full sectional view.
solid elements in full sectional view. Fig. 4.5.
in full sectional view.
Continued
parts are drawn in 2D assembly drawings, recognition of the parts assist to decide the solution. Consideration of machinability of parts restrict reconstructed solids to machinable ones. Figure 4.6(a) illustrates the 2D assembly drawing in Example 4.3 that consists of two parts. Figures 4.6(b), (c), (d) illustrate the wireframe model, solid elements, and the relationships between 2D faces and solid elements. It is found that Si is an element of P\ and 5s is an element of P^ by Existence Conditions In Assembly. S3 can be false or an element of P2. Therefore, S2 and S4 become the elements of P2 because of Mass Condition In Assembly. As the result, two solutions are obtained as in Fig. 4.6(e). If it is recognized that 52, S3 and 54 form the rail of Pi, only solution (I) is selected. Here a functionality of a part is used. Furthermore
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Masaji Tanaka et al.
I)
(a) Orthographic view.
(b) Wireframe model. Fig. 4.6.
(c) Solid elements. Example 4.3.
if the difficulty of machining P2 consisting of S2, S4, and S$ is recognized, only solution (I) is selected. P2 includes two acute edges. However, design knowledge has not been well formalized yet, and further research is required for its application to reconstructing solids from orthographic views. Knowledge about machinability and feature recognition has some success, and may be included into the reconstruction techniques.
Converting
S2
2D Assembly Drawings into 3D Part Drawings in CAD
Si Si Si 'Si\ Si Si Si Si Si
63
S4
Ss
Si S4
Ss
(d) Relationships between 2D faces and solid elements.
T~^ ( I)
(I) (e) Two solutions. Fig. 4.6.
4.6. Two complex
Continued
examples
Lastly, two complex examples of assemblies are indicated in this paper. Example 4.4 in Fig. 4.7 is an example of complex shapes in assemblies. Example 4.5 in Fig. 4.8 is an example of assemblies that consists of many parts. Figure 4.7(a) illustrates the assembly drawing in Example 4.4 that consists of two parts and a cross-sectional view (section AA'). Figures 4.7(b), (c) illustrate the wireframe model and solid elements. Figure 4.7(d) illustrates the set of cut solid elements in
64
Masaji Tanaka et al.
IV N
I
\
\ I -X X ,
\
i
/
Z—NI A I
Section AA'
-© (a) Orthographic view and sectional view. Fig. 4.7. Example 4.4.
section AA'. Figure 4.7(e) illustrates the relationships between 2D faces and solid elements. The equations of the solid elements in each view are illustrated as in Fig.4.7(f). It is clearly found that S u is false in the equations of section AA'. The solid element equation are made as in Fig.4.7(g) by combining the equations. As the result, Pi consists of {S 2 , S 3 , . . . , 5 1 6 }, and P 2 consists of {Si, S 6 , . . . , Si 9 }. Figure 4.7(h) illustrates the solution. In this case, the number of all combinations of the solid elements is 3 1 9 = 1.16 x 10 9 . But the method combines them to only the solution. Figure 4.8(a) illustrates the assembly drawing in Example 4.5 that consists of seven parts. The image of Example 4.5 is a chair. Figures 4.8(b), (c) illustrate the wireframe model and solid elements. The solid element equation becomes 5 8 x 5 2 S3 — S5 x Sg that is obtained from the top view. 5 2 , £5, Ss, SQ, 5 n , Sn, S13, Su and 5IQ become true by Existence Conditions In Assembly. Then 5 2 , S3, S5 become elements of P6. Each of S u , S 1 2 , S13, Su is an element of P 2 , P 5 , P 3 , P 4 . S 9 is an element of P7. Since S 7 is not an element of P3 because of Dotted Edge Conditions In Assembly, S8 and S 4 must be elements of P7. As the result, Pi is S19, P2 is S u or { S n . S i s } , F3 is S13 or {Si 3 ,Si 7 }, P 4 is Su or {Si 4 ,Si 8 }, and P5 is S12 or {Si 2 ,S 1 6 }. S 2 , S 3 , S 5 are elements of P 6 . Si and S 6 are false or elements of P 6 . S 4 , S 8 , S 9 are elements of P7. S 7 and S10 are false or elements of P 7 . Therefore,
65
Converting 2D Assembly Drawings into 3D Part Drawings in CAD
S3
S6
S9
Sl2
Sis
Sl8
(c) Solid elements.
(b) Wireframe model. r ^
^ """""--]
Si
Ss
S2
Ss
S7
Su
Ss
Su Sl2
Sl3
Sl7
Sl4 Sl9
Sl7 Sl8
S6
(d) Cut solid elements at section AA'. Fig. 4.7.
Continued
Masaji Tanaka et al.
66
Si/
YS3 *
" °* N
Su \\
:' Si \
Y S4\
Ss
Si
Se
Sn
\
Ss/ / /
s2
S*
k
S7
SJI
Sl3 Sl7
s4 ; ss
Se
Sw
S12
Sl6
Su Sl7
Sn
MD
Sl8
Se
Ss ! Ss
Sl2
Su
Ss
Su Sl2
Sl4 SJ7
Sl9
Sis
Section AA'
a. f-f'-'-
S9 Sl5
, Si7
(e) Relationships between 2D faces and solid elements. Si X S2-
S3
I S4-SsX
Ss S2— S5— SS— Sl4
XSuXSu Si — S7 — S13—S17 I
S4I
SsX Se X I
SJOXSJIXSI2<—(2)
I
X
S16XS17-S18
I
SeX Ss- S3 I X I
S6-S12-S18-S19
S12XS11XS9
I X I Sis-SnXSis (f) Equations in each view. Fig. 4.7.
Continued
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2D Assembly Drawings into 3D Part Drawings in CAD
67
9 S2-S3
-
S4— S5-S8-S9—
S10- Si4— Sis - Sie
X Si - Se-
S7-S12-S13—
S17— S18-S19
(g) Solid element equation.
(h) Solution. Fig. 4.7.
Continued
2 8 = 256 solutions are obtained. In this case, the number of all combinations of the solid elements is 8 19 = 1.44 x 10 17 . But the method can reduce the combination to 256 solutions. By the way, the assemblies seldom exist in which the parts are connected only through vertices or edges. Therefore, when a condition is applied
Masaji Tanaka et al.
68
>
L~>:-:
'.-'
(a) Orthographic view. Fig. 4.8.
(b) Wireframe model. Example 4.5.
that parts are connected only through faces, Si, SQ, SJ, SIO, S15, 5i6, S17, >Si8 become true. Finally, only one solution can be obtained as in Fig. 4.8(d).
5. Conclusion This paper describes a method of converting 2D assembly drawings into 3D part drawings. The conversion is an application of the method that converts orthographic views of one part into solid models by solid element equations. The solid element equations represent the relationships among the truth of solid elements, and the equations can be solved as algebraic equations. To explain the idea behind the equations in detail, the paper first introduces cubic element equations and then describes techniques of converting 2D assembly drawings into 3D part drawings. There are many studies to convert orthographic views of one part into solid models. 1_10,14 ~ 16 Since their methods employ some form of combinatorial search
Converting
Si
2D Assembly Drawings into 3D Part Drawings in CAD
S2
0 O S9&
&
S5
S6
f Si
0
69
S4
@Sio
£^\
/Z7\ Su B
Sn
SM
S12
s„0 s«0 s „ 0
»«0>
(d) Solution.
(c) Solid elements. Fig. 4.8.
Continued
for solids from faces or solid elements, the computational time of the methods is exponential of the complexity of solids. If one applies those methods to the conversion of 2D assembly drawings into 3D part drawings, the explosion of computational complexity for the search is expected in the case of 2D assembly drawings that form complex shapes and consist of many parts. Our method applies combinatorial search to undecided elements for finding plural solutions of solid element equations. If the number of solutions is small, the solutions are obtained with a small number of combinations of the undecided elements even if the number of solid elements and the number of parts are very large. Suppose that the number of solutions is in proportion to the complexity of solids. Then the computational complexity of our search is linear to the complexity of the solids. It
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can be concluded t h a t our method obtains the solutions with a computation time of the third power of t h e complexity of solids, namely t h e number of solid elements. Designers draw 2D assembly drawings as unambiguously as possible, and they use cross-sectional views in many cases. Our method makes use of t h e crosssectional views by cutting solid elements t h r o u g h t h e crossings, and reduces undecided solid elements. T h e knowledge of assembly and design are also able t o reduce t h e m . T h e assembly knowledge is already formalized to generate assembly sequences. T h e authors have also proposed methods to automatically generate assembly sequences, 1 7 ' 1 8 and this knowledge can be integrated into the method. Since the design knowledge is not formalized yet, t h e use of t h a t is a subject for future research. On the whole, t h e issue of automatic conversion of 2D drawings into 3D drawings is not only a geometric issue but also a combinatorial issue because orthographic views inherently include ambiguities. For t h e future research, it seems important t o devise an element suitable for representing relationships among geometric elements in drawings and for being composed to form solids. Perhaps there are issues similar to the issue in this paper in t h e field of C A D / C A M .
References 1. M. Idesawa, A System to Generate a Solid Figure from a Three View, Bull. JSME 16 (1973) 216-225. 2. M. A. Wesley and G. Markowsky, Fleshing Out Projections, IBM J. Res. & Develop. 25, 6 (1981) 934-954. 3. H. Sakurai and D. C. Gossard, Solid Model Input Through Orthographic Views, Computer Graphics 17, 3 (1983) 243-252. 4. B. Aldefeld, On Automatic Recognition of 3D Structures from 2D Representations, Computer-Aided Design 15, 2 (1983) 59-64. 5. Z. Chen and D. B. Perng, Automatic Reconstruction of 3D Solid Objects from 2D Orthographic Views, Pattern Recognition 2 1 , 5 (1988) 439-449. 6. K. Kitajima and M. Tasaka, A Method to Reconstruct a CSG Solid Model from a Set of Orthographic Views, IEICE Trans., D-II J75-D-2, 9 (1992) 1526-1538. 7. Y. Sasaki, K. Itoh, and S. Suzuki, Solid Generation from Orthographic views by Nonlinear Pseudo-Boolean Algebraic Solution, Trans. IPS Japan 30, 6 (1989) 699-708. 8. C. Kim, M. Inoue, and S. Nishihara, Heuristic Understanding of Three Orthographic Views, Journal of Information Processing 15, 4 (1992) 510-518. 9. I. V. Nagendra and U. G. Gujar, 3D Objects from 2D Orthographic View - A Survey, Comput. & Graphics 12, 1 (1988) 111-114. 10. U. G. Gujar and I. V. Nagendra, Construction of 3D Solid Objects from Orthographic Views, Comput. & Graphics 13, 4 (1989) 505-521. 11. M. Tanaka, Construction of Solid Models from Orthographic Views by Solid Element Equations, Trans. IPS Japan 34, 9 (1993) 1956-1966. 12. M. Tanaka, K. Iwama, and T. Watanabe, A Method of Constructing Solid Models from Orthographic Views by Using Solid Elements, Proc. of the 1st ISICIMS'94 (1994) 219-224. 13. M. Tanaka, K. Iwama, A. Hosoda, and T. Watanabe, Decomposition of A 2D Assembly Drawing into 3D Part Drawings, Computer-Aided Design 30, 1 (1998) 37-46.
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14. Q. Yan, C. L. Philip, and Z. Tang, Efficient Algorithm for the Reconstruction of 3D Objects from Orthographic Projections, Computer-Aided Design 26, 9 (1994) 699717. 15. H. Masuda and M. Numao, A Cell-based Approach for Generating Solid Objects from Orthographic Projections, Computer-Aided Design 29, 3 (1997) 177-187. 16. B. S. Shin and Y. G. Shin, Fast 3D Solid Model Reconstruction from Orthographic Views, Computer-Aided Design 30, 1 (1998) 63-76. 17. M. Tanaka and K. Iwama, A Method of Generating Assembly Plans from Assembly Drawings by Disassembly Equations, IPS Japan Trans. 35, 9 (1994) 1912-1921. 18. M. Tanaka, T. Kaneeda, K. Iwama, and T. Watanabe, A Method of Generating Optimal Subassemblies from Assembly Drawings by Disassembly Equations, Trans. Jpn. Soc. Mech. Eng., C 6 5 , 635 (1999) 2965-2972.
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CHAPTER 3 COMPUTER TECHNIQUES AND APPLICATIONS OF ADAPTIVE-GROWTH-TYPE REPRESENTATION IN COMPUTER AIDED DESIGN(CAD) I. NAGASAKA Faculty of Letters, Kobe University, Kobe, Japan E-mail:
[email protected] K. UEDA Research into Artifacts, Center for Engineering, The University of Tokyo, Tokyo, Japan E-mail: uedaQrace-u.tokyo.co.jp T. TAURA Department of Mechanical Engineering, Kobe University, Kobe, Japan E-mail:
[email protected] In our study, the design of geometrical shapes of function carriers and their layout in a given space is called configuration design. The constraint satisfaction problem in configuration design may be very difficult to solve due to the lack of tight constraints and the countless combinations of the layout, a diversity of solutions which satisfy the constraints should be allowed. Therefore, in order to allow such diversity, we directed our attention to developmental processes in biology and proposed an adaptive-growth-type 3D representation based on evolutionary algorithms. Here, the adaptive-growth type means the shape expressed in the process, which develops through interaction with an outside environment, like shape generation of a living organism in the natural world. The usefulness of the representation was verified by applying it to the component layout problem in the early stage of satellite design. Keywords: Geometric representation; spational design; genetic classifier system.
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1. Introduction All artifacts that surround us are designed to fulfil some tasks and possess the structure of a system with numerous function carriers — functional elements that have physical substances such as modules of a satellite and rooms of a house. By fulfiling the function of each carrier, the artifact functions as a whole. In the design of such artifacts, the design process of forming them plays an essential role. In this research, such a design process is called configuration design, and it is considered to be a process of constraint specification and satisfaction, as stated in Thornton, 1 and Michalewicz et al.2 For example, product specifications define the first constraints on a new design and the rest of the design process can be seen as the search for a solution that satisfies these constraints. The very characteristic of configuration design is that it is not suitable to determine the shapes and the layout of the function carriers separately, because the shape and the layout are closely interrelated. For example, in the design of a machine, configuration design involves determining the shapes of its components and arranging them in the given space: there are strict constraints on the space. On the other hand, since such processes were often conducted by making a sketch or a physical prototype, they were time-consuming and expensive as soon as the composition of the artifact became complex. Thus, to support configuration design, there have been many studies on optimizing 3D shapes of the function carriers e.g. Smith et al.3 and on generating their layout within the given space e.g. Szykman and Cagan 4 . However, in many of those studies, two aspects of configuration design, that is, designing 3D shapes and their layout, are handled separately. Due to a countless number of combinations of the layout, the very high degree of freedom of 3D shapes and sometimes it is lack of constraints, it can be very difficult to solve constraint problems with many function carriers. In addition to this, in those studies, each problem was treated as a constraint optimization problem (COP) which aimed at finding only a single optimal solution. However, in the case of configuration design, for the above reason, it can be very difficult to generate and determine the best solution which satisfies all constraints. Therefore, most configuration designs should not be viewed as COPs. Therefore, in this research, to support configuration design, it is assumed that a diversity of solutions which satisfies constraints, should be allowed and it is important to determine 3D shapes and their layout in the same framework so that to leave the designers the option of selecting from among solutions by presenting variant under tentative constraints. In the process of constraint specification and satisfaction, constraints are refined gradually. In order to allow such diversity and gradual refinement, we directed our attention to developmental processes in biology and proposed adaptive-growth-type 3D representation based on evolutionary algorithms (EAs). Recently, there has been many application of EAs to engineering problems, 5 and there also several studies on embryologies within evolutionary computation. 6 However, these studies mainly dealt with 2D shapes.
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The expression "adaptive-growth-type" means that the shape expressed in the process, and developed through interaction with an outside environment like shape generation of a living organism in the natural world. For example, the growth of the bones is autonomous provided the right hormones are present, but the growth of the muscles and tendons are dependent on bone growth. If for example, the growth of a bone is delayed, so will be that of the associated muscles and tendons too. In this way, the lengths of muscles, tendons, and bones grow. At the same time, they are adapted to each other. 7 With shape representation, by generating shapes through interaction with constraints that represent the environment, it becomes possible to determine 3D shapes and their layout in the same framework. A diversity of design solutions is obtained by changing the environment or by generating the alternatives of the shapes with using genetic operations of EAs. Here, diversity in shape representation means that a variation of shape can be made while still preserving the characteristics of the shape. From positional signals provided by simple diffusion of chemicals,7 in the development process of the living organism, the cells could "know" how and when to change shape or move. This implies that the cells have the ability to sense their environment, and according to their genetic information, adapt autonomously by changing their shape or function. If the representation mentioned above is achieved at such a level that the shape of each function carrier is autonomously adapted to the environment, as in the case of the living organism, it would be possible to solve the shape and layout problems simultaneously. However, in our study, as a first step toward achieving such results, various solutions have been derived by moderating the constraints of configuration design and by a search process that adapts and develops various design solutions.
2. Configuration Design 2.1. Steps of configuration
design
Configuration design, in our study, is defined as developing preliminary shape designs and layouts for main function carriers at the early stage of embodiment design — the design phase in which elaboration of configuration, shapes, sizes, materials and manufacturing processes are made. In general, the procedure in embodiment design, 8,9 consists of the following steps: (i) Identify embodiment-determining requirements: e.g. in the case of house design, ask clients to clarify their requirements for their house, (ii) Produce scale drawings of constraints on the space: e.g. make drawings that shows the space where the house is allowed to build, (iii) Identify embodiment-determining main function carriers: e.g. in the case of house design, the function carriers would be entrance, living room or bedrooms that have main influence on the layout of house.
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(iv) Develop preliminary layouts and shape designs for the embodimentdetermining main function carriers: e.g. make many sketches or physical prototypes and examine the candidates of preliminary layouts of rooms. (v) Select suitable preliminary layouts. 2.2. Constraints
in configuration
design
In general terms, each constraint represents a relationship of properties (e.g. of volumes of elements) that should be valid in each consistent combination of the elements. However, when limiting the problem to configuration design, it is clear that these elements are function carriers. Values of outline shape, volume, size and position are adopted as configuration design constraints which should be consistent among function carriers. Since general configuration design — i.e. design not limited to a specific domain or design object — is the topic of our research, these general characteristics are adopted as constraints. The constraints are classified in accordance with the two issues of configuration design, that is, "three-dimensional shape of function carrier" and "layout of function carrier", as follows. A. Constraints concerning three-dimensional shape of function carrier: • Outline shape • Volume • Size B. Constraints concerning layout of function carrier: • Position • Interference In general, the function carriers have physical substances and are designed not to interfere: hence, interference is added as a characteristic to be considered as a constraint in the configuration design. The constraints concerning these two issues of configuration design are called, respectively, "geometrical constraints" and "layout constraints" hereafter. 3. Adaptive-Growth-Type Shape Representation As explained above, the constraint satisfaction problem in configuration design often becomes extremely difficult. Therefore, it is necessary to allow local optimal solutions and to generate various solutions to leave the designers room for selection.
Thus, we directed our attention to developmental processes in biology that realize the diversity of living objects. In nature, the shape and other characteristics of a living organism (phenotype) is determined by the genetic information (genotype) and by the environment. The genotype contains information that is descriptive through the execution of a set of developmental rules and through information, encoded in the developmental process
Adaptive-Growth-Type
Representation
in CAD
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Nature genetic information (genotype) —•
It
"
living organism (phenotype)
environment
CAD representation
Fig. 1.
—- •
form
Shape generation in nature and CAD.
itself: the shape is generated through the interaction with the environment. This brings about diversity in the shapes of the living organisms (Fig. 1) along with the other characteristics. Shape — /(genetic information,
environment).
In this research, the adaptive-growth-type shape representation is adopted for configuration design. In analogy to biological development, in our representation, a shape is determined by the interaction between the rules of generation and the environment. This representation aims not so much at the precise representation of shape, as usual in current CAD systems, but at the generation of various design solutions, that satisfy the constraints in configuration design. The authors proposed a Shape Feature Generating Process Model (SFGP Model),10 based on a similar analogy to biological development. However, in SFGP model, to produce an independent shape representation, the internal environment of the shape was given as environment. In the case of configuration design, since each shape here is to adapt to the constraints given by two or more other function carriers, the mechanism of adaptation to the external environment should be reconsidered, too. Therefore, in our study, the SFGP model is extended as an adaptive-growthtype 3D representation to make it adaptable to an external environment. 3.1. Outline of SFGP
model
In the SFGP model, shape generation starts with a primary shape (sphere) and rules are selected and applied according to the position and local conditions, that is, the internal environment of the shape. After a number of generations, a final shape is
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primary
final Fig. 2.
Cell division model.
generated. The cell division model, since it is the basis for our SFGP model, and the new shape generation process are explained briefly below.
3.1.1. Cell division model In our model, the set of rules consists of rules of the division for a dot (which we call a cell in analogy to biological development) on a sphere. The shape feature generation process is the series for cell divisions. As shown in Fig. 2, in the beginning, there are few cells on the sphere. According to the rules, they divide into two or more cells and spread over the sphere. Consequently, after a number of generations, the cell division results in a distribution of cell density at the surface of the sphere. As Fig. 3 illustrates, the shape is derived by processing the density of cells: O is the center of the sphere and d is the density of cells near point A. Density d is converted to a distance from O to a point on the actual surface in the direction of OA. By proportioning the density of cells to the distance to the surface of the shape, the cell division model can display fairly complicated shapes. The positions of points on the sphere where the density is measured are arbitrary, but clearly if more points are set, the resolution of representation of the shape increases.
3.1.2. Shape generation process However, applying all rules to every single cell on the sphere surface is not only very inefficient, requiring an enormous amount of computational time, it also makes it very difficult to determine which rules are responsible for the generation of the various design features of the shape. Therefore, as shown in Fig. 4, the sphere is divided into a certain number of parts and, depending on the location (A, B, C, . . . ) of the cell on the sphere, a group of rules is selected (B). After the density around the cell is measured, a rule which matches the criterion of the density, that is, the condition in the rule, is applied (Rule B-2) to prompt cell division with parameters specified in the action side of
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79
Point cm Surface
d = density of cells near point A Fig. 3.
Correspondence between shape and distribution of density in cell division model.
the rules. As shown in Fig. 5, actual sets of rules (classifiers) are binary codes with conditions referring to parts of the sphere (A, B, C, . . . ) . After a number of cell divisions, the distribution of the density of cells is measured and converted into a free form shape. Consequently, the shape is generated by applying the set of cell division rules through the classifier system. Figure 6 shows the correspondence between the genotype (set of cell division rules) and the phenotype (shape). 3.2. Extension
of SFGP
model
As previously mentioned, the SFGP model has been extended to make it adaptable to an external environment. With respect to the two issues of configuration design, the shape can be adapted to the environment of the given space in the following ways: (i) reshapes according to the environment, (ii) repositioning the shape, according to the environment. In our study, the SFGP model is first extended with taking (i) into consideration. To begin with, it is necessary to extend the model so that it adapts to the constraint of configuration design since this is one of the elements of the environment of the shape. The same process that determines the generation rule in the SFGP model can be directly applied to the algorithm that satisfies the constraints. In the SFGP model, the evaluation function is the degree of similarity to the shapes given
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set of rules
< Rule >
Fig. 4.
= cell density : cell division (condition) (action)
Rules applying to cell division.
by the designer. Here, in the extended model, the evaluation function is the degree of satisfaction of the constraints. Secondly, the SFGP model has been extended to adapt to the constraints among function carriers. Constraints between function carriers belong to the two categories as described in Sec. 2. Among the layout constraints, the most related constraint to (i) is the mutual interference among function carriers. For instance, an interference can be cancelled if it is possible to grow one of the shapes as shown by the interfering parts in the right side of Fig. 7. Therefore, the rule that is responsible for making interfered part is replaced: thanks for the characteristic of the SFGP model, it is possible to specify which rules are responsible for the generation of the various features of the shape. The following three techniques are sequentially applied, and the rule is selected if the interference is cancelled while the geometrical constraint is still satisfied. (i) Partial replacement of the rule is responsible for the interfering part with an other rules. For instance, as shown in Fig. 8, the rule which is responsible for making interfered part is specified in the set of rules of B, and exchanged with a same part of rule in the set of rules of other form with the same evaluation value for its geometric constraint.
Adaptive-Growth-T'ype
000010 000111 101101 001000
101010 100000 111010 010010
011000 000000 101000 000101
Dioioio 010110 110000 010000 000111 011111 011011 000011 101101 011110 110100 001011 000100 000111 011010 010001
110011 101100 101000 010000
100010 000001 100001 001101
D
001000 110010 111000 001001 001111 100100 010110 101001 001100 110110 101010 010011
^
Representation
001110 100001 001100 001100
111110 111100 100100 010111
in CAD
81
100110 010010 001000 011010
110010 100101 001101 010110 010000 001000 linn 001101 101110 100011 001001 101110 101101 111101 000011 uIOOIOO
101100 011000 100000 100100
nC 011011 001011
000100 001000 101000 011111
moooK
100001 110110 000011 110001 001001 000101 110111 100010 000010
110111 100010 000010 density (condition ) : Fig. 5.
(action )
Set of rules in binary codes.
(ii) Replace the whole rule with another rule. For instance, as shown in Fig. 9, the whole set of rules of B is exchanged with the other set of rules that has the same evaluation value, (iii) Generate a new rule. For instance, as shown in Fig. 10, a new set of rules is generated from other rules with the same evaluation value as the interfered shape. The adaptation of shape is achieved by such strategies of modifying the constraint, that represent the environment. However, no mechanism of repositioning of the shapes have been implemented yet. As an alternative, we have adapted a method that repositions the shape as an external algorithm.
4. Methodology for Supporting Configuration Design 4.1. Implemented
system
A basic strategy is to generate shape gradually by using the adaptive-growth-type shape representation, and to search for the configuration design solution by adapting shapes to the situation of layout in a given space. There is a similarity between this
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Genotype
EJbenQtype
rules of cell division
Shape mmiBf^^^-
> -
)
) Cell Division
Fig. 6.
Genotype and phenotype.
Fig. 7.
Shape adaptation.
strategy and the work on skeleton-based techniques in which rays from a skeleton are used to determine shapes. 11 As shown in Fig. 11, the support system consists of three parts: a geometric constraint solver (GCS), a layout constraint solYer (LCS) and a configuration design unit (CDU). Each solver searches for configuration design solutions by interaction with the designer. The low of the constraint satisfaction process is in the GCS and the LCS, rough solutions are obtained, and the shape and layout are finally adapted by the CDU to each other within the same frame. This means that neither the 3-dimensional shape of the function carrier nor the layout are directly generated in parallel, but the outline of the solution of the layout problem is obtained by LCS using genetic algorithms (GAs). The outline shape is obtained by the GCS
Adaptive-Growth-Type
Representation
in CAD
•> •"-/ , • set of rules of other form
Fig. 8.
set of rules of B
Partial replacement of a rule.
set of rules of other form
Fig. 9.
set
of rules of B
t>
^-»*
Replacement of a rule as a whole.
•HUM set of rules of other form
Fig. 10.
set of rules of B
Generation of a new rule.
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geometrical constraints
Spatial layout constraints
(2)
(1) Geometric Constraints Solver
Spatial Layout Constraints Solver
Spatial Design Unit
Fig. 11.
Flowchart of the system.
using adaptive-growth-type shape representation based on CS. Moreover, possible contradictions between the outlines of the solutions of the two constraint solvers are adjusted by exploiting the ability of generating a diversity of shapes in the same frame for the CDU. Such an approach is impossible with conventional shape representation. 4.2. Configuration
design solution
generation
algorithm
In this section, the algorithms, the specific role and solution generation methods of the three solvers are explained, mainly in terms of the CDU which plays the center role in the support system. 4.2.1. Geometric constraints solver The GCS searches for the outline of the solution for the geometrical constraints and obtains a rough shape before the CDU adjusts and generates the design solution. As mentioned previously, the process of finding the generation rule in the SFGP model is directly applied to satisfying the constraints.
Adaptive-Growth-Type
Representation
85
in CAD
4.2.2. Layout constraints solver The LCS searches for the outline layout which satisfies the layout constraints before obtaining the configuration design solution made by the CDU. Therefore, this solver only deals with the centerpoint of the shape of the function carrier, and from this centerpoint, the shape obtained by the GCS is generated. This solver is based on a similar technique in layout design of VLSI chips (e.g. Ref. 12) and utilizes GAs.
4.2.3. Configuration design unit After the outline shape of the function elements and the positions of the outline are obtained by the two solvers as discussed above, the configuration design solution is generated using this configuration design unit. The algorithm for the CDU is used not to achieve constraint satisfaction, but to integrate the outline of the solutions generated by the two solvers and to determine 3D shapes and their configuration layout in the same framework. Here, searching in the "same framework" means that a design solution is generated by searching for a point of compromise and correcting the contradictions — if there are any — between the outline of the solutions that were generated by the GCS and the LCS in one framework within the CDU. Each step is explained with reference to the numbers in Fig. 12 as follows. In Fig. 12-(1), the centerpoints of the function carrier generated by the LCS are obtained (Fig. 13-1).
0
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Flowchart of the configuration design unit.
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Next, a shape is generated by centering on coordinates obtained in (1), as shown in Fig. 13-2. The shape is gradually generated without any farther operations because there is no interference between function carriers in this step [Fig. 12-(2),(3)]. As shown in Fig. 13-3, as a shape is generated, the adjoining function carriers interfere and an overlapping region arises. In this case? as shown in Fig. 12-(4),(5),(6), the adjustment of the generation rules and the repositioning of the centerpoints of interfered shapes take place in the same framework. As shown in Fig. 14, the positions of centerpoints of interfered shapes A and B are adjusted by moving them in the directions of NA and NB (Fig. 14, left). The distance of movement is defined as,
\NA\ = \NB\ =
tfvi
(i)
where V* is the volume of the common region. The layout constraint is concerned with the positional relationship between all function carriers involved. Therefore, after moving the positions of the centerpoints of just two function carriers in the directions of NA and iVjg, in most cases, the overall positional relationship among the function carriers, including those two that were actually moved, may not satisfy some criteria of the layout constraint any
Fig. 13.
Growing shapes of function carriers. overall boundary
Fig. 14.
Movement of function carriers shape.
Adaptive-Growth-Type
Representation
87
in CAD
more. Thus, it is necessary to readjust the centerpoints of each function carrier again after moving some of them. Therefore, the center positions are readjusted so that they satisfy the constraints again (Fig. 14, right). At the same time, adjustment of the rules of the shape takes place, as described in the foregoing section (Fig. 12). After these processes, configuration design solutions are obtained in the CDU through the correction of the contradictions between the shape and its layout and they are presented the designer. The designer observes and examines the solutions, and if the designer accepts them, the configuration design process is finished. If the designer accept none of them, he/she modifies the constraints of the GCS or the LCS and the process repeats. In this way, the configuration design proceeds with interaction between the support system and the designer, and eventually, a configuration design solution can be obtained.
5. Prototype System for Supporting Configuration Design 5.1. Configuration
of the
system
The prototype of the supporting system is based on our method for configuration design as explained in the preceding sections. Figure 15 shows the overall configuration of the system. This system has been implemented using C + + on a Sun Workstation. It consists of five parts: the constraint specification unit, the configuration design unit, the constraint solver, the user interface, and the geometry engine.
5.2. Application
to configuration
design of a satellite
design
5.2.1. Configuration design in satellite design In most cases of configuration design, the task and the specification of the design object are already determined before this step of the design process. In satellite design, it is called the "mission" which is the purpose of the satellite (e.g. weather observation, communication, etc.). After the mission is given, the configuration design is made as follows. (1) Assume externals of the satellite according to the installed mission equipment. (2) Distribute functions necessary for accomplishing the mission, and determine the specifications of each component, and assume the size of the equipment. (3) Identify embodiment-determining main function carriers in each component. (4) Develop preliminary layout. (5) Select suitable preliminary layout as a result of the examination. After finishing the configuration design, the designer moves to a detailed embodiment design.
Nagasaka et at
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5.2.2. Example In configuration design of a satellite, since reliability is the most important issue, components are selected from among those in a catalog. However, despite the required reliability of the satellite, the fuel tank may have a comparatively high degree of freedom in the shape and is usually newly designed for each satellite, because it has little iniuence on reliability. Therefore, this example focuses on the application of our configuration design method to determining the outline shape of the fuel tank, as it is obtained by considering its volume, and the outline space of the installation and the layout of each component. External constraints of the satellite, as determined from the payload of the mission and the launch rocket, are shown in Fig. 16. Geometric constraints Geometric constraints in this example are the installation space necessary for the 11 main components. For the fuel tank, it is the volume of the fuel. Examples of these constraints for the fuel tank, battery charge control unit (BCCU) and battery are as follows. Fuel tank (volume) 10,000 cc BCCU 40.0 x 30.0 x 30.0 cm Battery 50.0 x 30.0 x 40.0 cm
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in CAD
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Solar battery panel
,
Fig. 16.
1.000 m
.
Externals of a satellite.
Layout constraint The layout constraints concern the layout of the main components as follows: • Communication system <—* Power supply system: These elements should be close to make the cord short. • Power supply system <—> Fuel tank: These elements should be close so that heat generated by the power supply system will prevent freezing of the fuel. • Put similar systems close to each other to make the cord short. • Equipments such as Power supply system and the Posture control system which generate heat can be arranged neither on the earth side of the satellite nor on the side of the mission equipment, since the sides are not suitable for radiation of heat. Select preliminary layout After obtaining the outline shapes of components (using GCS), layouts of components are generated by the LCS. Figure 17 shows an example of the output of the LCS. The straight lines connecting components show that there is a constraint relation of the relative positions of the components. Subsequently, generation of configuration design solutions was attempted by correcting the discrepancies between the outline solutions as generated by the GCS and the LCS. Figure 18 shows the process of adjustment. Adjustment of the generation rules and the repositioning of the centerpoints of interfering shapes are also performed. Thus, shape is generated by adjusting the two components of configuration design in the same shared framework, and the configuration design solution is
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Fig. 17. Example output of the layout constraints solver.
Fig. 18. Adjustment of the shapes and the layout of components.
obtained when the generation of shapes is completed. Figures 19 and 20 show horizontal and vertical sections of the solution and outline shape of the fuel tank, respectively. Figure 21 shows the variations of configuration design solutions. Table 1 shows the evaluation value of each type of constraints — layout and geometric constraints, the maximum dimensions and volume of the fuel tank in each conlguration design solution. Solution 1 has the best evaluation value for each constraint and has the volume that is also very close to the given constraint. The maximum dimensions show that even each solution has similar volume, each of the maximum dimensions are quite different, especially on i^axis. This means that the solutions are, in fact, diverse. As shown in the above figures, it is confirmed that the configuration design solutions generated by these processes show a diversity of preliminary layouts. Moreover,
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Fig. 19. Horizontal and vertical sections (solution 1),
Fig. 20. Outline shape of fuel tank (solution 1).
it can be confirmed that the fuel tank has the required volume and does not interfere with other components. After examining and Yerifying these solutions, the designer may take further constraints into consideration, and proceed with the detailed embodiment design. Examples of the results were presented to the designer of the satellite and he had the comments as follow: • The shape and layout of the fuel tank of the best solution (solution 1) would be acceptable, but layout of other components such as battery and BCCU should be refined further. • There are a number of constraints which are more important than geometrical constraints or layout constraints such as weight balance and strength of materials.
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Fig. 21. Variations of generated solutions (solution 2, 3, 4, 5). Table 1.
Generated configuration design solution
Sol. (no.)
Eval. (layout)
Eval. (geometric)
X (cm)
Y (cm)
Z (cm)
Volume (cc)
1 2 3 4 5
2.3100 1.8914 1.8235 1.6239 2.0161
1.7855 1.5253 1.6228 1.7345 1.4879
39 34 34 35 37
51 44 42 45 47
5.5 6.0 3.7 6.0 5.6
9882 10820 8876 9257 10660
• It is nice for designers to be able to see other candidate solutions. It is similar to seeing the solutions designed by some other designers. 6. Adaptive-Growth-Type Assembly Structure Representation If the representation mentioned above is achieved at such a level that the shape and layout of each function carrier is autonomously adapted to the environment like a living organism in the natural world, it would be possible to solve the shape and layout problems simultaneously. In previous section, as a first step toward achieving such objectives, the mechanism of adapting the layout to the environment has not
been implemented within the shape representation itself, but it has been conducted as an external algorithm [Fig. 22 (left)]. Thus, in this section, we propose a methodology to solve the shape and layout problems simultaneously by extending the adaptive-growth-type 3D geometric representation with the implementation of a mechanism that adapts the layout and the relation between function carriers [Fig. 22 (right)] — i.e. assembly structure. The
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function carriers
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artifact
arranged by ,, outer algorithms artifact
\ fT\ (TJ \ t j O —• \2J
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1
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(H) layout and assembly structure are implemented in representaion
Extension of adaptive-growth-type representation.
extended representation is called adaptive-growth-type assembly structure representation (AGTAS representation). With this approach, it becomes possible to apply this representation to design process from the early design stage to the manufacturing of products. 6.1. Assembly
structure
In general, artifacts consist of numerous function carriers and, in many cases, they take the form of hierarchy. For instance, in the case of a satellite, the satellite consists of many module such as a power module, a communication module, a data processing module, etc. The data processing module also consists of a control unit, interface unit, data recorder, etc. In this research, this hierarchical relationships of between function carriers are called assembly structure. The structure of an assembly consists of information about the function carriers composing the assembly and their spatial relationships. However, in this study, among many aspect of assembly structure, only the spatial relationships between them are considered. 6.1.1. Model of assembly structure In this research, the design process of the assembly structure is assumed to have three steps as follow: (1) Combine assemblies. (2) Make assembly hierarchy. (3) Check interference of assemblies. This assembly structure is determined during the configuration design, i.e. the 3D shapes, their layout and assembly structure are determined in the same framework, with interaction between a product design support system using the
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representation and a designer. In order to complete the geometrical representation with assembly information, the tree structure is adopted to represent the relationships between function carriers, and genetic programming (GP) 1 3 is applied to manipulate and search for better assembly tree structures. GP is a branch of genetic algorithms (GAs) 14 and is the extension of the genetic model of learning into the space of programs. These programs are expressed in genetic programming as parse trees, rather than as lines of code. Thus, for example, the simple program "a + b * c" would be represented as shown in Fig. 23 (left). As shown in Fig. 23 (right), the assembly structure is represented in the similar manner, but the operators in the nodes of GP representation are changed into vectors that indicate where the shapes of function carriers are generated. Figure 24 shows gradual generation of three function carriers — / i , J2 and jz — in two dimensions for the sake of simplicity by following the tree representation mentioned above. The vectors indicate centerpoints of each shape since this adaptive-growth-type representation only needs the centerpoints of function carriers to specify their location. In this way, the mechanism of adapting the layout
GP representation of individual
Representation of assembly structure
a,b,c,d,e: vectors Fig. 23.
Application of GP representation to assembly structure.
Fig. 24.
Generation of assembly structure and shapes.
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ED
Representation
i
in CAD
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to the relation between function carriers is implemented in adaptive-growth-type representation and is called AGTAS representation. This tree representation and manipulation of assembly structure is also an analogue to biological development: in the development of a nomatod called Caenorhabditis elegans, some of the mutations that have been found support their developmental process is similar to the approach proposed here. For example, as shown in Fig. 25, a typical lineage may result in cell type A giving two cell types B and C, and only C continues to divide and give types D and E. Mutants can result in the substitution of type Z for B; or make the lineage symmetrical — B now giving D and E; or even generating a stem cell line with B being replaced by A which repeats the pattern of cell divisions and cell differentiation.
7. Methodology of Configuration Design Using Adaptive-Growth-Type Assembly Structure Representation 7.1. Implemented
system
The basic strategy of the support system is to generate the shape of function carriers gradually in a position specified by the assembly structure and to adapt them to the constraints given by the designer. As shown in Fig. 26, the support system mainly consists of two parts: AGTAS Generation Unit and Shape Generation Unit. ASTAS Generation Unit generates numerous variation of the assembly structures using GP and Shape Generation Unit generates each function carriers according to the assembly structure. After the shapes generation, the system calculate a fitness of each assembly structure with the evaluation function of GP based on the degree of satisfaction of the constraints, i.e. geometrical constraints and layout constraints mentioned previously. After these processes, configuration design solutions are obtained. The designer observes and examines the solutions. If the designer accepts them, the configuration design process is finished. If the designer accept none of them, he/she modifies the constraints and the process repeats. In this way, the configuration design proceeds with interaction between the support system and the designer. Eventually, a configuration design solution can be obtained.
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AGTAS Generation Unit (GP)
Fig. 26.
7.2. Shape adaptation
Flowchart of the system.
process
After the assembly structures are obtained by the ASTAS generation unit utilizing GP, the adaptation of the assembly structure and the shapes of function carriers is achieved in the shape generation unit. Each step is explained with reference to the numbers in Fig. 27 as follows. In Fig. 27-(l), the assembly structure of the function carrier generated by the AGTAS generation unit are obtained (Fig. 28-1). Next, a shape is generated by centering on coordinates defined in assembly structure, as shown in Fig. 28-2. The shape is gradually generated without any further operations because there is no interference between function carriers in this step [Fig. 27-(2),(3)]. As shown in Fig. 28-3, as a shape is generated, the adjoining function carriers interfere and an overlapping region arises. In this case, as explained in previous section, the rule of the adaptive-growth-type representation that responsible for making interfered part is replaced [Fig. 27-(4)]. After these processes, configuration design solutions are obtained and the fitness value of the assembly structures that generates these
configuration are measured based on the degree of satisfaction of the constraints. According to this fitness value, the AGTAS generation unit generates offsprings of the better solution by the mechanism of GP. These process repeats until the solutions satisfy the constraints to a certain degree specified by the designer beforehand and they are presented to the designer. The designer observes and examines the solutions. If the designer accepts them, the configuration design process is finished.
Adaptive-Growth-Type
Representation
in GAD
m
AGTAS Generation Unit
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Fig. 27.
Flowchart of the shape generation unit.
overall boundary
Assembly structure Fig. 28.
Growing shapes of function carriers.
8. Application t o Configuration Design of a Satellite Design 8.1. Simulation
of configuration
design
As previously mentioned, the ASTAS generation unit generates numerous assembly structures and send them to the shape generation unit. Figure 29 shows an example of the output of the ASTAS generation unit. The straight lines connecting components show the tree representation of the assembly structure. Subsequently, generation of coniguration design solutions was attempted by generating shapes according to the assembly structure and evaluating them by referring
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Fig, 29.
Fig. 30.
Example output of the assembly structure.
Generated primary configuration design solution.
to the geometric constraints and the layout constraints. Figure 30 shows the primary solution. Since the ASTAS generation unit initializes assembly structure and the adaptive-growth-type representation randomly, there are large interference parts between function carriers observed. After certain iteration of the ASTAS generation unit and the shape generation unit, the solution eventually satisfy the constraints specified by the designer. Figure 31 shows the variations of configuration design solutions. Figure 32 shows horizontal sections of the solution. Figure 33 shows the variations of configuration design solutions. As shown in the above figures, it is confirmed that the configuration design solutions generated by these processes show a diversity of preliminary
Adaptive-Growth-Type
Fig. 31.
Representation
in CAD
m
Example of configuration design solution.
Fig. 32.
Horizontal section of the solution.
layouts. Moreover, it can be confirmed that the fuel tank has the required volume and does not interfere with other components. After examining and verifying these solutions, the designer may take further constraints into consideration, and proceed with the detailed embodiment design. 9. Conclusions a n d F u t u r e Directions We proposes a new approach to determine 3D shapes and their layout in the same framework, by adaptive-growth-type 3D representations and AGTAS representations that is the extension of adaptive-growth-type 3D geometric representation with the implementation of a mechanism that adapts the assembly structure. The usefulness of the representation was verified by applying it to the component layout problem in the early stage of satellite design: 3D shapes and layout could be handled simultaneously and a diversity of configuration design solutions was generated.
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Fig. 33.
Variations of generated solutions
Acknowledgments This s t u d y has been conducted in "Methodology of Emergent Synthesis" Project (JSPS-RFTF96P00702) supported by t h e J a p a n Society for t h e Promotion of Science. References 1. A. C. Thornton and A. L. Johnson, CADET: A software support tool for constraint processes in embodiment design, Research in Engineering Design 8 (1996) 1-13. 2. Z. Michalewicz, D. Dasgupta, R. G. Le Riche and M. Schoenauer, Evolutionary algorithms for constrained engineering problems, Computers and Industrial Engineering Journal 30, 2 (1996) 851-870. 3. R. Smith, S. Warrington and F. Mill, Shape representation for optimization. Proceedings of the 1st IEE/IEEE International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications GALESIAf95 (1995) 112-117. 4. S. Szykman and J. Cagan, Automated generation of optimally directed three dimensional component layouts, Proceedings of the 19th ASME Design Automation Conference 65, 1 (1993) 527-537. 5. D. Dasgupta and Z. Michalewicz, Evolutionary Algorithms in Engineering Applications (Springer Verlag, 1997). 6. H. Garis, Artiicial embryology: The genetic programming of cellular differentiation, Artificial Life III Workshop, Santa Fe, New Mexico, USA, June 1992.
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7. L. Wolpert, The Triumph of the Embryo (Oxford University Press, 1991). 8. G. Pahl and W. Beitz, Engineering Design: A Systematic Approach (Springer, Berlin, 1988). 9. Verein Deutscher Ingenieure, VDI Guidelines — Systematic approach to the design of technical systems and products, VDI 2221, 1987. 10. T. Taura, I. Nagasaka and A. Yamagishi, An application of evolutionary programming to shape design, Computer-Aided Design 30, 1 (1998) 29-35. 11. D. M. Stal and G. M. Turkiyyah, Skeleton-based techniques for creative synthesis of structural shapes, Artificial Intelligence in Design 1996, eds. J. S. Gero and F. Sudweeks (Kluwer Academic, 1996) 761-780. 12. V. Schnecke and O. Vornberger, A genetic algorithm for VLSI physical design automation, Proceedings of 2nd International Conference on Adaptive Computing in Engineering Design and Control, ACEDC'96 (1996) 53-58. 13. J. R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (MIT Press, Cambridge, MA, 1992). 14. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, MA, 1989).
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CHAPTER 4
COMPUTER-AIDED MODULAR FIXTURE DESIGN
YIMING (KEVIN) RONG Worcester Polytechnic Institute, Manufacturing Engineering, 100 Institute Road, Worcester, MA 01609-2280 http://www.wpi. edu/ E-mail: [email protected] Manufacturing planning makes a significant contribution to production cycle. This chapter studies a feature analysis based fixture and setup planning system to enhance the flexibility of production systems and reduce the manufacturing planning time. Manufacturing features are defined where the operational information is included in feature descriptions. Manufacturing is modeled as a process of transferring a workpiece blank into final product by subtracting manufacturing features from the workpiece model. A backward reasoning method is developed for setup planning. Based on geometric and operational information, manufacturing features are identified and clustered into setups with a certain sequence. Fixturing features are defined and locating surfaces are automatically selected based on accuracy relationships and geometric accessibility. Fixture configuration design is used to verify the setup planning. Examples are given to show the effectiveness of the method. Keywords: Fixture design; CAD/CAM; setup planning. 1. I n t r o d u c t i o n Advanced manufacturing is characterized by t h e ability to allow a rapid response t o continuous changes of customer requirements. T h e very core is flexible manufacturing systems (FMS) which leads t o a reduced manufacturing lead time, increased quality, and flexibility for t h e changes in design. 1 Manufacturing planning is a key issue in the integration of product design and manufacturing, which makes a significant contribution t o t h e production cycle. 2 As CNC techniques and machining centers are developed and widely utilized in industry, multiple operations in a single set-up are quite common and desired t o save production time and cost. Machine motion can be controlled in many axes with high accuracy for processing workpieces with complex geometry in b o t h rough and finish processing. 3 F i x t u r e design becomes a major restriction for setup planning with multiple operations and 103
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influences the operation planning. 4 Conventional computer-aided process planning (CAPP) systems do not include the fixture design function as desired.5 Flexible fixturing has become an important aspect in FMS and computerintegrated manufacturing systems (CIMS). 6 There are several different categories of flexible fixtures such as phase-change materials, modular, adjustable, and programmable fixtures where modular fixtures are widely used in industry. 7,8 Modular fixtures were originally developed for small batch production to reduce the fixturing cost, where the dedicated fixtures were not economically feasible. The flexibility of the modular fixture is derived from the large number of fixture configurations from different combinations of the fixture element which may be bolted to a baseplate. 9 Modular fixture elements can be dis-assembled after processing a batch of parts and re-used for new parts. Modular fixture configuration design is a complex and highly experience-dependent task. This impedes further applications of modular fixtures. Lack of skillful fixture designers is a common problem in industry. The development of Computer-Aided Fixture Design (CAFD) systems is necessary to make manufacturing systems truly flexible. Figure 1 shows an outline of fixture design activities in manufacturing systems, including three steps: setup planning, fixture planning, and fixture configuration design. The objective of the setup planning is to determine the number of setups needed, the orientation of workpiece in each setup, and the machining surfaces in each setup. The setup planning could be a subset of process planning. Fixture planning is to determine the locating, supporting and clamping surfaces and points
Product Design
Fig. 1.
Fixture design in manufacturing systems.
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Modular Fixture
Design
105
on the workpiece. The task of fixture configuration design is to select fixture elements and place them into a final configuration to locate and clamp the workpiece. As more and more CNC machines and machining centers are employed, many operations can be carried out within a single setup, which needs to be ensured by a well designed fixture configuration. Some previous research on setup planning and fixture planning can be found in the computer-aided process planning (CAPP) area, including: a generateand-evaluate strategy studied to determine the orientation of workpieces for milling operation; 10 a method for automated determination of fixture location and clamping derived from a mathematical model; 11 an algorithm for selection of locating/clamping positions which provided a maximum mechanical leverage;12 kinematic analysis based fixture planning; 13 ' 14 and rule-based systems developed to design modular fixtures for box-type workpieces. 15,16 An automated selection of setups was presented with consideration of fixture designs, where tolerance factors of orientation errors were used with several rules as the basis of determining locating surfaces and setups. 17 Fixturing features were studied, which need to be extracted from a product design for the fixture design purpose where surface features of locatable surfaces and inter-relationships between fixturing surfaces were analyzed. 18 The fixturability of a workpiece as part of manufacturability was studied where fixturing grade and dependency grade were defined for fiat and form fixturing features which were orientation-dependent. 19 In the area of automated fixture configuration design (AFCD), relatively less literature can be found. Given locating and clamping points on workpiece surfaces, fixture elements can be selected to hold the workpiece based on CAD graphic functions.20 A 2-D modular fixture synthesis algorithm was developed for polygonal parts. 21 Whybrew and Ngoi presented a method to automatically design the configuration of T-slot based modular fixturing elements. 22 The key feature of the system was the development of a matrix spatial representation technique which permitted the program to search and identify both objects and object intersections. It was also able to determine the position of objects during the design process. However, the limitation of the method was that only the blocks whose edges were parallel or perpendicular to each other could be represented. Therefore, the design system could only layout the fixture elements in such a way that all the edges of fixture elements were parallel or perpendicular to each other. Trappey et al. presented a methodology for determining the location and orientation of dowel-pin based modular fixture in a 2-D projection basis. 23 It only presented a detailed research on selecting the fixed point between baseplate and bottom modular fixture elements and did not describe the rule to select the suitable modular fixture elements and the way to combine them together. The fixture design methodology in the case of 3-2-1 fixture layout method was applied in the study. In this study, manufacturing features are first described with operational or non-geometric information. Fixturing features are discussed with a consideration of
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accuracy relationship and surface accessibility. A backward reasoning methodology is applied to the setup planning. In the development of AFCD technique, fixture structure is decomposed into fixture units, fixture elements, and fixturing functional surfaces. A fixture unit generation algorithm is developed based on a modular fixture element assembly relationship graph (MFEARG) which can be automatically identified from fixture element CAD models. A fixture unit/element placement scheme is also developed to ensure the fixture element assembly relationships. Implementation of the setup and fixture planning system as well as the modular fixture configuration design system is presented in this chapter.
2. Manufacturing Features Manufacturing planning starts with manufacturing information extraction from computer-aided design (CAD) models of products. 24 Feature recognition and feature based design are two basic approaches of accessing the information. 25 The former involves a form of 3-D matching between feature definitions and a geometric representation of a solid model. This method may be used in dealing with CAD data in a standard format (e.g. IGES, PDES, STEP, etc.). Therefore, it can be applied to different companies. However, the features which can be recognized are limited and many complex geometric features cannot be identified. It is hard to handle non-geometric information such as tolerance and operational information. The latter is a relatively straightforward approach which allows the designers to use directly a set of pre-defined primitive features to perform designs. In manufacturing planning applications, a manufacturing feature-base needs to be built up in advance. One disadvantage of this approach is that the feature definitions may vary in different types of industry and different companies, which becomes a major limitation of applications.
2.1. Production
model and backward setup
planning
Manufacturing features can be defined as high level geometric entities representing volumes of material removed from a workpiece (or forming the workpiece geometry). Once a geometric model of a workpiece is built up, manufacturing processes are actually the processes of removing manufacturing features from the workpiece blank model with a certain sequence and accuracy so that a product model is approached. That is: W = W 0 - E F f c , fc = l , 2 , . . . , N ;
(1)
where W is the product model; Wo is the workpiece blank model; Ffc is a manufacturing feature removed from the workpiece; and N is the number of manufacturing features.
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107
Geometrically, when operations under j t h setup are considered, Eq. 1 is decomposed into: W^Wj-i-EFy,
i = l,2,...,ni; j = l,2,...,r;
(2)
where Wj is workpiece model after operations under j t h setup; Fij is a manufacturing feature removed from the workpiece under j t h setup; n_,- is the number of features processed under j t h setup; and r is the number of setups. When a backward setup planning approach is considered, Eq. 2 can be written as: W,-_i = W,-+ EFtf,
i = l,2,...,n,-; j = 1, 2 , . . . ,r.
(3)
In this approach, the setup planning starts from the finished product model (i.e. the product design model). Once the planning for setup j is finished, the product model for setup j — 1 is generated by applying the add-material technique. When the setup planning is finished for all setups, the workpiece blank model is reached. 2.2. Manufacturing
feature
classification
and
feature-base
Manufacturing features are defined to transmit geometric and non-geometric information for setup planning. The geometric information includes: feature shape and dimensional parameters, and feature position and orientation, while the nongeometric information includes accuracy and operation information. In order to recognize manufacturing features for setup planning, the features need to be clearly defined and classified into certain types according to their geometry and operations used to generate these features. Figure 2 shows a sketch for such a manufacturing feature classification. Figure 3 shows several examples of manufacturing features. This feature classification can be used to build up a manufacturing feature-base and to cluster the features into setups with a sequence. 2.2.1. Protrusion features One technical problem in feature analysis is the treatment of protrusion features. Protrusion features include boss, fillet, and rib features which are necessary in constructing workpiece models. Usually protrusion features are decomposed into depression features. 26 The decomposition is not unique. Actually in many designs, operational shoulders are considered for a machining convenience based on the design for manufacturing principle. In this case, a parametric expression of depression features can be applied to the corresponding protrusion features, as shown in Fig. 3 (RIB and BOSS). 2.2.2. Non-geometric
information
In setup planning, non-geometric information becomes necessary in operation selection and sequencing. In order to make such decisions, manufacturing features were defined with the accuracy requirement information and operational information. 27
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*flatplanes • end planes
ilane features i itep features Depression features
1 » straight steps fc? angular s(eps shoulders N.hnnM^c
straight slots ./jjangular slots through slots,"^T-T-slots > ^••V-slots blind slots,
slot features
through keyways ^* blind keyways « + >
hole features
Manufacturing features
special holes edge cut boss features Protrusion features
fillet features rib features
Fig. 2. Manufacturing feature classification.
\
<
Y
I.P.I T-SLQT
RIB Fig. 3. Samples of manufacturing features. T h e accuracy information includes feature geometric accuracy such as dimensional and form tolerance, and inter-feature accuracy relationships such as orientation tolerance (e.g. perpendicularity and parallelism) and position tolerance (e.g. t r u e position). Operational information refers t o work material, possible manufacturing
Computer-Aided
Modular Fixture
Design
109
methods and machine tools, manufacturing tool types and size, feed motion directions of the tool, etc. For example, tool information is the most important information in operation selection and NC programming (including interference checking). Motion directions of the tool include approaching, feeding, and backing directions. Once a machine tool (e.g. a horizontal or vertical machining center) is selected for a specific setup, the tool axis direction relative to the axis of machine tool spindle is also an important information for setup planning. The information is included in manufacturing feature description as attributes. It should be noted that for a specific feature, the tool used and the axis directions may not be uniquely determined where multi-attributes are assigned for different selections in setup planning. 2.2.3. Manufacturing feature information
description
In summary, manufacturing features can be represented by a CAD model with geometric and non-geometric attributes, i.e. F = (X0,X1,X2,X3,X4,X5)
(4)
where XQ indicates the feature type and index; X\ is a set of feature parameters representing the geometric shape and dimensions; X2 is a matrix indicating the feature position and orientation (origin and cosine directions of the local coordinate system) relative to the workpiece coordinate system; X3 is a set of data representing feature tolerances (both dimensional and form tolerances) respective to Xi; X4 is a set of data representing inter-feature tolerances (dimensional, orientation, and position tolerances with datum references); and X5 represents operational information such as work material, possible machining methods and machine tools of feature processing, machining tool types and sizes, feed approaching direction, and tool axis relative the machine spindle. Table 1 shows an example of the information organization of manufacturing features, which is used in setup planning to determine feature groups in different setups and the setup sequence. 2.3. Feature
accuracy
Accuracy requirements of manufacturing features are the most important consideration of determining the sequences of setups and feature processing. The ranges of dimensional tolerance, form tolerance, and surface finish are related to the geometrical sizes of the features. In order to automatically access the accuracy information, they should be represented in a compatible format so that the accuracy requirements of different manufacturing features can be compared and assigned with manufacturing sequence priorities. A tolerance grade (IT: ISO tolerance) was defined by ISO. When dimensional and form tolerances are considered, according to the natural relationship between the tolerance grade and tolerance range with the nominal dimension, the tolerance grade of a manufacturing feature can be determined in the
110
Yiming (Kevin) Table 1.
Rong
Manufacturing feature information.
Workpiece material: Casting-iron; Workpiece hardness: HB 250 F
Xo Type
& No.
02
index
plane 110101
x2
Xi
Parameters
x3
Position & orientation
Dimensional & form tolerance
1 = 10.0
"-5" 'xo" yo — -2 .0 . .zo. orientation:
b = 4.0
0
1
0
x4 feature tolerance
reference
Operational information
//:0.001
plane 110102
machining methods:
R a = 63
_1_: 0.0002
plane 110103
D : D.0004
distance: 6.0 ± 0.0002
3.0 ± 0.0006
plane 110102 hole 151103 hole 151101
milling tools: face mill T401002 approaching direction: [1111110] T tool axis: direction: -Z
/ / : 0.0003
hole
machining
151201
methods:
-5 4.4 ± 0.0007
z = 0.1
"10 0 - 2 " 0 0 - 1 0 .0 0 0 1 _ '2.0" yo — 2.4 .3.1. .zo.
x5
Inter-feature tolerance
Xo'
d 0 = 2.4
R a = 63
distance: 2.0 ± 0.0008 22
Endhole
ho = .66
151300
d i = 3.0
h i = 0.3
c rientation:
+0.0005 2.4 - 0 . 0 0 0 2 3.5 ± 0.0009
"-0 - 1 0 2 0 0 1 2.4 - 1 0 0 3.1 . 0 0 0 1
3.0 ± 0.0006
plane 110103 hole 151201 plane 110101 hole 151201
boring tools: borer T301001 T302002 approaching direction: [000010] T tool axis direction: + y
1.5 ± 0.0005
following way. A tolerance unit is denned as: (5)
where D is the nominal value of the feature dimension; and a and b are constants. According to ISO description, the first term represents the uncertainty caused by the manufacturing errors and the second term is the uncertainty due to the measuring errors. In product design, the standard tolerance range with a tolerance grade should be given by: Tol = kji
(6)
where kj is a coefficient and j is the tolerance grade. Table 2 shows different kj for tolerance range calculation. In current engineering designs, a tolerance range is usually given to each dimension while the selection of manufacturing processes and machine tools are based on the tolerance grade.
Computer-Aided Table 2.
Modular Fixture Design
111
ISO Tolerance grade and tolerance values.
IT
5
6
7
8
9
10
11
12
13
14
15
16
Tol.
7i
10 i
16 i
25 i
40 i
64 i
100 i
160 i
250 i
400 i
640 i
1000 i
In most CAPP systems, the tolerance grade is determined by looking up a prestored data table. In this research, the tolerance grade is estimated based on the given tolerance range, which can uniformly be used with other accuracy or nonaccuracy factors for setup planning decisions. From Table 2, it can be seen that kj+\ is approximately increased from kj in a constant ratio of 1.585. Therefore, the dimensional and form tolerance grades can be obtained by taking tolerance grade 6 as a reference: r d (or Tf) = Int
'log(100Tol)-log(i) log(1.585)
6
(7)
In order to determine the values of constant a and b in Eq. 6, feature dimensions can be divided into several ranges. The constants can be pre-estimated in these ranges with a desired precision. By following a similar idea, the surface finish of a feature can be taken into account by converting it to an equivalent tolerance grade: Tr = Int[2.88fl° 2 ]
(8)
where Ra is the roughness height measure of surface finish. Besides feature dimensional/form tolerances and surface finish, the positional and orientation tolerances need to be considered in fixturing surface selection of fixture planning, which relates to the accuracy relationships between features. If there is a feature which has a tight dimensional tolerance relationship with a machining feature, that implies the feature may be potentially used as an operational datum, i.e. a locating surface in the setup. The orientation tolerance grade can be calculated as in Ref. 28: Tp = Val/L
(9)
where Val is the orientation tolerance value and L is the maximum feature dimension. In order to evaluate the accuracy of a feature and utilize it efficiently in fixture planning, a generalized feature accuracy grade is applied in this investigation, which is defined as: Tg = (WlTd + w2Tp) * {w3Tf + wATr)
(10)
where T^, Tp and Tf are the dimensional tolerance grade, positional tolerance grade and form tolerance grade, respectively; TT is the tolerance grade equivalent to the surface finish of the feature. wi,w2,w3, and w± are the weight factors. The multiple operation "*" represents a dominant relationship where a zero value can contribute
112
Yiming (Kevin)
Rong
to the final result, while the operation "+" represents a relatively weak relationship with preferences. 2.4. Fixturing
features
In a particular operation setup, the features used for fixturing the workpiece can be defined as fixturing features, or fixturing surfaces since most fixturing features are plane and cylindrical (internal and external) surfaces. According to the fixturing functions, the fixturing surfaces can be classified into three categories: locating, clamping, and supporting surfaces. Unlike the design and manufacturing features, fixturing surfaces are orientation-dependent. They do not play the same role throughout the manufacturing processes. A set of surfaces may serve as fixturing surfaces in a setup, but may not be used for the fixturing purpose or have the different fixturing functions in another setup. The concept of fixturing features allows the fixturing requirements of the workpiece to be associated with the workpiece geometry. On the other hand, the feature information in the feature-based workpiece model can also be used directly for fixture design purpose. Similar to manufacturing features, the information necessary for describing a fixturing feature contains geometric and non-geometric aspects. The former includes feature type, shape and dimension parameters, position and orientation in the workpiece coordinate system. The latter includes the surface finish, accuracy level and accuracy relationship with the machining features, and surface accessibility. It should be noted that fixturing surface accessibility is an important property of fixturing features. 29 The information describing fixturing features is retrieved in fixture planning where fixturing surfaces and points are determined. 3. Fixture Planning In setup planning, the selection of locating datum is the most important task to ensure the manufacturing quality. In the case of multiple operations in a single setup with machining centers, fixture configuration design becomes a major constraint in the setup planning because possible collisions or interference between the workpiece, fixture components, and tool path have to be avoided. 3.1. Basic
requirements
of fixture
planning
In practice, fixture planning is governed by a number of factors, including: (1) workpiece design, which mainly involves information of geometry and tolerance; (2) setup planning, which identifies the machining features, the machine tool and cutting tools to be used in each setup; (3) initial and resulting forms of the workpiece in each setup; and (4) available fixture components. To ensure that the fixture can be used to hold the workpiece in a proper position so that the manufacturing process can be carried out according to design specifications, the following conditions should be satisfied for a feasible
Computer-Aided
Modular Fixture
Design
113
fixture plan: 1. The degrees of freedom (DOF) of workpiece are totally constrained when the workpiece is located. 2. Machining accuracy specifications can be ensured in current setup. 3. Fixture design is stable enough to resist any effects of external force and torque. 4. Fixturing surfaces and points are easily accessed by available fixture components. 5. There is no interference between the workpiece and the fixture, and between the cutter tool and the fixture. In this investigation, we focus on the first four requirements. The fixture planning is carried out based on the following considerations: • Type of the surfaces on the workpiece that can be selected as fixturing surfaces Although the workpiece geometry could complex in real production, in most fixture designs, planes and cylindrical surfaces (including holes) are usually used as the locating and clamping surfaces due to the ease of access and measurement of the features when the workpiece is fixed. In this investigation, those two types of surfaces are utilized in fixture planning. • Orientations of fixturing surfaces respect to cutting tool axis Many CNC machines, especially machining centers, can be used to perform various operations within one setup. But in most cases, the cutting tool axis of the machine tool is unique. Once the workpiece is fixed in a specific setup, the orientation and position of the workpiece in the coordinate system are determined, which is associated with the machine tool. Considering the fixturing stability, the locating surfaces are preferably those ones with normal directions opposite or perpendicular to the cutting tool axis. For clamping features, the normal directions should be concordant or perpendicular to cutting tool axis too because it is desired in fixture design that the clamping forces are against locators. • Accuracy of candidate fixturing surfaces For the surfaces to be machined, there exist datum surfaces serving as the positional and orientation references based on which other dimensions and tolerances are measured. The surface accuracy level is certainly an important factor in locating surface selection. A generalized accuracy expression is necessary for different types of tolerances and surface finish associated with the surface. In fixture planning, those surfaces with higher generalized accuracy grades should be selected as locating surfaces with priority such that the inherited machining error can be minimized and the required tolerances of the machining features can be easily attained. • Surface combination status of candidate fixturing surfaces In fixture planning, more than one surface of the workpiece generally require to be selected as the locating and clamping surfaces for restricting the DOF of the
114
Yiming (Kevin)
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workpiece in a setup. Therefore, besides the conditions of individual surfaces, the combination status of the available locating surfaces is also important for accurate locating of the workpiece. For example, two planes with a perpendicular relationship should be considered with priority as locating surfaces in side locating due to the fact of accurate, reliable, and convenient locating of the workpiece in horizontal directions where two points are in one surface and one point is in the other surface. • Fixturing stability Fixturing stability is a very important consideration in fixture planning, especially when the fixturing positions of locators and clamps are determined. Since the locators and clamps are in contact with the workpiece, the distribution of fixturing points plays a critical role in ensuring the fixturing stability. For example, in order to locate the workpiece steadily, the resting area composed of three bottom locators should be as large as possible and the projection of the workpiece gravity center should be inside of the area. In the aspect of clamping, the clamps should be placed against corresponding locators to ensure the fixturing stability. • Accessibility of fixturing surfaces For a feasible fixture design, the fixturing surfaces must be accessible to fixture components. The usable (effective) area of the fixturing surface should be large enough to match with the functional surfaces of the locators and clamps. The effective area of the surface should exclude the part obstructed by other surfaces since the workpiece geometry could be complex. Besides considering a fixturing surface, the accessibility of potential fixturing points on the surface is also important for the determination of the final fixturing point distribution. 3.2. Strategy
of
fixture
planning
The overview of the automated fixture planning system is shown in Fig. 4. The procedure of fixture planning can be mainly divided into five stages, i.e. input, analysis, planning, verification, and output. Input data include workpiece CAD model which contains the geometric and tolerance information of the features on the workpiece, and setup planning information including the features to be machined and the machine tool type in the specific setup. The data can be either extracted from a CAD database or entered interactively by user with a CAD system. Analysis involves in an extraction of the candidate fixturing features with accuracy information and an evaluation of accessibility of the fixturing features. In this study, planar and cylinder surfaces are mainly considered for fixturing purpose. The task of planning is to automatically determine the primary locating direction and select the optimal locating/clamping surfaces and points in current setup. Algorithms are developed for the planning of bottom (top) and side locating/clamping.
Computer-Aided
Modular Fixture
Design
115
Input specifications • Workpiece model • Setup plan — Feature list in the setup — Machine tool information
• workpiece modeling > setup planning
Feature data processing for fixturing • Extraction of fixturing features • Generation of generalized accuracy
T Accessibility analysis for fixture features • Accessibility analysis for the surfaces • Accessibility analysis for the points
Determination of primary locating direction • Tool axis and feature orientation analysis • Determination of primary locating direction
T
Vertical fixturing planning • Bottom locating planning • Top clamping planning
Horizontal fixturing planning • Side locating planning • Side clamping planning
Locating accuracy analysis • Generation of fixture unit • Locating accuracy verification
> Fixture components > Assembly relationship
Production of output files • Illustration of fixturing surfaces/points • Generation of fixture plans Fig. 4.
The procedure of fixture planning.
Accurately locating is the major contributor to ensuring the machining accuracy of the workpiece. Once the locating/clamping scheme is determined, the fixture units corresponding to the fixturing points can be generated in fixture configuration design. A comprehensive program has been developed to analyze the final configuration of the fixture in terms of cumulative tolerances of the fixture components
116
Yiming (Kevin)
Rong
and the effects on the workpiece accuracy. These two steps are completed in the stage of verification. Output of the fixture planning is the fixturing surfaces/points in a format of the fixture plan which can be used in fixture configuration design. Although the fixture plan is generated based on some optimization rules, alternative fixture plans are also provided for further optimization or user confirmation. Once the fixture plan is obtained, the fixture configuration can be generated for current setup of the workpiece. Following sections will address the key parts of the fixture planning, including primary locating direction determination, locating planning and clamping planning. 3.3. Determination
of primary
locating
direction
In fixture design, there are usually three locating reference surfaces to determine the position and orientation of the workpiece. There exists one locating surface, namely primary locating surface, perpendicular to other locating surfaces. This is especially true when modular fixture systems are utilized. The primary locating surface is the major locating datum for determining the major spatial position and orientation of the workpiece in the current setup and constrain at least three DOF of the workpiece. In a general case, the primary locating surface could be a single plane surface or several planes in the same direction with the same or different heights. The normal direction of the primary locating surface, called primary locating direction, needs to be determined first in fixture planning, which should be parallel or perpendicular to the cutting tool axis of the machining operations. Assume that the tool axis is known as Vt = (Vx, Vy, Vz). The surfaces with the normal directions parallel or perpendicular to the tool axis are extracted from the workpiece model. They are grouped as follows: Sfn = {MV^Tgi, At) \Vt±Vt
or VJ = -Vu Nf > i > 0}, N, > n > 1
(11)
where Sfn expresses a group of surfaces with a normal direction potentially in the primary locating direction; fi(Vi,Tgi, Ai) represents a feature with normal vector Vi, generalized accuracy grade Tgi, and usable (effective) area A,; Nf is the number of the features in the group; and Ns is the number of the feature groups. Suppose the primary locating direction is selected as Vl(Vlx, Vly, Vlz). Obviously, VI € {Vi}. The following index is used to identify VI with a priority order: In-Vl = max{WA * SAJ
max SA + WT\ * STn/ max ST, Ns > n > 1}
(12)
where WA and WT\ are the weight factors of importance on surface area and accuracy respectively. SAn = X^=i Aj, STn = J2jli Tgj, maxSA is the maximum area in the group, max ST is the maximum value of the generalized feature accuracy grade in the group. Once the InJVl is obtained, the normal vector corresponding to the InJVl is selected as the primary locating direction.
Computer-Aided
3.4. Planning
Modular Fixture
for bottom locating and top
Design
117
clamping
The task of fixture planning in this stage is to determine the surfaces suitable for primary locating purpose and the locating point distribution on the surface as well as the clamping surfaces and points corresponding to the primary locating, as shown in Fig. 5. As stated above, the primary locating surface could be a single plane or multiplanes in the direction. Besides the normal direction of the planes, other factors such as the generalized accuracy grade and the accessibility of the planes are taken into account when the planes are selected for bottom locating purpose. The set of the planes which meet the bottom locating requirements can be expressed as: LV = {fi(Vu Tgi, Ci) | Vi = -VI, Nf>i>0}
(13)
where fj(Vi, Tgi, Ct) represents a feature with normal vector Vt, generalized accuracy Tgi, and contours C» which is characterized by the lines and arcs, Nf is the number of the features in the set. When more than one plane are involved, they are projected along the primary locating direction to form a virtual plane surface which is represented by its boundary entities such as line segments and arcs. The potential locating points are apparently in the region enclosed by the boundary. As the surface is sampled into discrete
Fixturing feature extraction Virtual locating surface formation
^r Surface discretization - locating region establishing - candidate locating points ^r Locating surface/point selection - fixturing stability - surface accuracy grade - surface/point accessibility - uniform locating height 1
'
Clamping surface/point selection - against the locating surface - against locator or locating region - surface accessibility
i
r
Output planning results Fig. 5.
A procedure of fixture planning in vertical direction.
Yiming (Kevin)
118
Rong
points, the outer-bounding rectangular region is generated in the virtual plane. Considering that the locating points cannot be very close to the outer edges of the workpiece, the size of the rectangular region is decreased by moving the boundary toward its center with T. It is obvious that the projection of the final locating points will be in this new region. However, some points may be outside the surface boundary. A standard algorithm is employed for detecting if a point is in the specific region. In the primary locating direction, three points (or equivalent) need to be selected to constrain three DOF, one translation and two rotations. The three points can be used to construct a triangle where the center of workpiece gravity should locate inside the triangle in order to guarantee the locating stability. The optimal locating points are selected from based on following factors: a. The area of the triangle is as large as possible, which is calculated as: TA = ^S{S-11){S-12){S-13)
(14)
where S = 0.5 * (11 + 12 + 13), and 11, 12, 13 are the edge lengths of the triangle. b. The distance from the gravity center of the workpiece to three edges of the triangle is as large as possible, which is calculated as: 3
TL = '%2 Di
(15)
i=l
where Di is the distance form zth edge of the triangle to the gravity center of the workpiece. c. The generalized accuracy of the planes in which locating points locate is as high as possible (the tolerance value is as small as possible), which is calculated as: 3
TT = ^ T
s i
(16)
i=l
where Tgi is the generalized accuracy grade of the plane in which the locating point Pi locates. d. The accessibility of the three locating points is as large as possible, which is calculated as: TC = min{Accj, Ace,-, Acc^}
(17)
where Ace,, Ace,-, Acc^ are the accessibility values of the three locating points. e. The locating height equalization is as much uniform as possible, which is evaluated as:
{
1, 2,
if Zi ± Zj ^ zk if Zi = Zj or Zi = Zk or Zj = Zk
3,
if zt = Zj = zk
(18)
Computer-Aided
Modular Fixture
Design
119
When the values of above factors are obtained, following index is used to identify the optimal locating points, which has the maximum value: In_Pl = W s * (TA/ maxTA + T L / maxTL) + W T 2 * T T / m a x T T + W C i * T Q 4- W H * TH/3
(19)
where Ws, Wx2, Wei and W H are the weight factors for the importance of fixturing stability, accuracy, accessibility, and uniform height respectively; max TA, max TL, and max TT are the normalization factors for all candidate vertical locating planes. Once the final locating points are determined, the planes corresponding to the three locating points are obtained. It should be noted that by using this procedure one or more than one planes may be selected as the primary locating planes. Selection of clamping type is related mainly to the direction of machining force and the surfaces available to place clamping devices. The top clamping surfaces are determined based on the following criteria: • Surface is opposite to the bottom locating surfaces. • Surface cannot be the machining surface in current setup. • There is an overlap area if the surface is projected into the bottom locating triangle region. • Surface is easy to be accessed by the clamp (e.g. has a high value of accessibility). Once the clamping surface is determined, the optimal clamping point is selected such that the clamping force is in the direction against one of the bottom locates or inside the bottom locating triangle. After the steps stated above, all fixture plans available for bottom locating and top clamping are generated and recorded sequentially with priority determined by In_Pl. Each fixture plan file contains fixturing information such as fixturing functions, locating/ clamping surface IDs, and the coordinates of the locating/clamping points. 3.5. Side locating/clamping
planning
Fixture planning in horizontal direction includes side locating and clamping planning. Side locating is to select or determine the non-primary locating surfaces and points. The most common method of the side locating is the standard 3-2-1 locating principle. In this case, side locating planning is to select two planes which are perpendicular as the secondary and tertiary locating surfaces where the secondary locating plane contains two locating points and the tertiary locating plane contains one locating point. This locating scheme is easy to design fixture configuration and to control locating accuracy because of the independent constraints in different DOF. Therefore it is widely applied in fixture design. However, there are many cases where it is hard to find such mutually perpendicular locating planes in fixture design. For a more general situation, cylindrical surfaces and the non-perpendicular planes may also serve as the locating surfaces and sometimes, the three side locating
120
Yiming (Kevin)
Rong
points may distribute on three different surfaces. In this study, general solutions are generated, which includes the standard 3-2-1 situation as a priority solution. The first step of side locating planing is to select locating surfaces where planar and cylindrical surfaces are considered as the candidate surfaces for side locating. To select proper side locating surfaces, the normal direction, generalized accuracy grade, accessibility value, and the shape of the candidate surfaces are taken into account. The set of the features which meet the side locating requirements can be expressed as: LH = {fi(Vi,Tgi, Acci,Ci) | Vi _L VI for planes, Vi // VI for cylinders Nf > i > 0} (20) where /i(Vj,Tj, ACQ, Q ) represents a feature with normal vector V,, generalized accuracy grade T 9 j, accessibility ACQ, and contour CJ; Nf is the number of the features in the set. In order to constrain three DOF (two translations and one rotation) left from the primary locating, more than one surfaces is needed for the side locating. As previously stated, besides the condition of individual surfaces, the combination status of the candidate locating surfaces is also an important factor affecting the locating of the workpiece. For the two kinds of locating features, there are many combinations which can be used in side locating. Following is a partial list of the combinations in a preferable order: (1) two planes perpendicular to each other, (2) two planes which are not perpendicular, (3) three planes, (4) one plane and one cylindrical surfaces, (5) two cylindrical surfaces, and 6) one plane and two cylindrical surfaces, as shown in Fig. 6. Based on these types of combinations, feature groups can be constructed and expressed as: LHC,„ = {fi|t = 1,2 or 1,2,3,^ e L H } ,
m = l,2,...Nm;
(21)
where fj is a selected feature in the group and N m is the number of feature groups. Each feature group contains two or three features and evaluated for a suitability of side locating in order to select a set of surfaces which satisfies the requirements for side locating. The criteria used for evaluating the feature group includes: a. Feature Combination Status A weight HF is assigned for the different types of combination of locating surfaces. HF is the highest if the feature group is comprised of two perpendicular planes. If the feature group is comprised of three cylindrical surfaces, HF is the lowest. b. Generalized accuracy grade of the feature group Generalized feature accuracy grade is considered for all the surfaces in the group, HT — ]TTj, where Tj is the generalized accuracy grade of the surface i in the feature group, and i = 1,2, and 3. c. Accessibility value of the feature group Accessibility of each surface in the group is considered, HC = min{Acc,|i = 1,2 or 3}, where ACQ, is the accessibility values of the feature in the candidate horizontal locating surface group.
Computer-Aided
^
z
't? /
Modular Fixture
£ r&6
1^
3
/
/
/ / /
1. 2. 3. 4. 5. 6.
121
Design
WJ
Two perpendicular planes Two non-perpendicular and non-parallel planes Three non-perpendicular and non-parallel planes One plane and one cylindrical surfaces Two cylindrical surfaces One plane and two cylindrical surfaces Fig. 6.
The feature combination types.
The following index is used to identify the optimal locating surface group when the values of above factors are attained. In_Hl = HF + W T 3 * HT i > 1
(22)
where W T 3 and Wc2 are the weight factors of the importance on surface accuracy and accessibility respectively, and max HT is a normalization factor of feature accuracy. When the candidate locating surfaces are classified into groups, the locating height needs to be considered. It is desired that all the side locators as well as clamps are placed in an identical height or the difference of the side fixturing point heights are minimum. If there is an overlap in height between the features in the group, the locating height can be decided for the group at the lower part of the overlap height. When the locating height is determined, available locating region in the locating surfaces becomes 2D lines and arcs or circles. Those 2D locating "region" can be obtained by using a virtual plane(s) perpendicular to primary locating direction to intersect the surfaces of the feature group at the locating height(s). When a surface is intersected by the virtual plane, more than one intersecting segment may be generated if there are pockets or extrudes on the feature, as shown in Fig. 7. These segments can be obtained by calculating the intersections of the virtual plane and the contour segments of the selected surface. The position of locating points on the intersecting segments are determined based on the different surface combination status and point accessibility. It should be mentioned that two conditions must be
Yiraing (Kevin)
fe* ^ ^
s /
- ^J. ^ ^^ \^
— 'Ql v ^fe
i
<&>-.^
Rong
r
\ >
<5^
<-"' N<x^ \ ^ \ ^
-V
.^3«"
locating surface
Pocket Slot
Extrude (b) Fig. 7.
Workpiece model and intersection plane for side locating.
satisfied for a feasible solution of side locating planning. 30 The first one is that the normal directions of locating surfaces cannot be all parallel, which is ensured in the feature grouping process. The second one is that the normal directions from the three locating points cannot meet at one point, which gives an uncertain location of the workpiece and needs to be checked during the locating point determination. For a fixturing stability consideration, the side clamps should be applied to the side clamping surfaces which are opposite to the locating surfaces. The side clamping surfaces can be determined in terms of the position of the side locating surfaces. A complete solution have been developed for determining the side clamping surfaces and feasible regions of clamping points. 31 Figure 8 shows the procedure of side locating/clamping procedure. When the side locating/clamping plans are generated with a priority sequence. These data together with the vertical fixture plans are
Computer-Aided
Modular Fixture Design
123
Fixturing feature extraction Fixturing surface grouping
Locating surface selection - surface combination status - generalized accuracy grade - surface accessibility
Locating point selection - locating height - locating region segment extraction - locating point determination
Side clamping planning - clamping surface selection - clamping point determination
Output planning results Fig. 8.
A procedure of fixture planning in horizontal direction.
used as the input for the fixturing accuracy analysis to verify if those fixture plans can meet the requirements for machining accuracy. Once the verification is passed, the fixture configuration can be generated using those fixture plans. 3.6. Development
of fixture planning
systems
A fixture planning system, Fix-Planning, is developed based on the methodology presented above. This system can be integrated with a commercial CAD system and an automated fixture configuration system. The CAD system is used as the platform to provide the system with input information necessary for fixture planning. Figure 9 shows a system main menu which contains eight functional modules. SysSetup is used to initialize the system before the system performs the planning tasks. An example of system initialization is shown in Fig. 10 where customized planning conditions are setup, such as the clamping type, minimum size of locators and minimum placement height of locators in horizontal locating, and the priority sequence of the major factors which affect the vertical locating. File is used to deal with reading the workpiece specification from CAD database, and store the fixture plans for fixture configuration design. LocatingDir is for determining the workpiece primary locating direction. Accessibility is for evaluating the accessibility of fixturing features and points. The algorithms of side and bottom(top) locating/clamping are embedded in modules HorLocating, HorClamping, VerLocating and VerClamping. When fixture planning is completed, the planning results are displayed with priority by system. Users can either chose the first result which has the highest priority in the system, or browse the next ones and then make the final choice.
Yiming (Kevin) Rong
124
i>
lc-"ak*.Of,^«e«bfc> ,E^w!>i£ V^Xoc^^S H^O^ybj V$ pampas :&»$*H# H*^
/jHl
* T | ^ I •*[ **]*-*] J l V j
"Fixture Design li f ?v?| ^4»l
s§&
Version IS C0&rt&®\im-ii&
•
Fig. 9. Overview of the Fix-Planning system.
••••Iliii
Fig. 10. An example of initialization of the system.
An example workpiece is shown in Fig. 6a where the step surface F46 is to be machined. Table 3 shows the results of the accessibility evaluation of candidate fixtaring surfaces and Fig. 11 shows a point accessibility distribution of a candidate bottom locating surface. The results of fixture planning in horizontal and vertical directions are shown in Fig. 12. The results may not be unique. Alternative planning is also provided when it is necessary. The fixture planning result can be
Computer-Aided Table 3.
Modular Fixture
Design
125
Results of accessibility analysis.
Face-id
Normal direction
Area
Function
Valid
OA
Fl F8 FIO Fll F12 F13 F14 F15 F16 F17 F18 F23 F28 F35 F36 F38 F40 F44 F59
(0, - 1 , 0) (1, 0, 0) (0, 1, 0) ( - 1 , 0, 0) (1, 0, 0) (0, 0, - 1 ) (0, 1, 0) (1, 0, 0) (0, - 1 , 0) ( - 1 , 0, 0) (0, 0, - 1 ) (0, 0, - 1 ) (0, 1, 0) (0, 1, 0) (0.707, 0.707, 0) (0, 0, 1) (0, - 1 , 0) (0, 1, 0) (0, - 1 , 0)
6095.04 1900 2322.58 6464.19 5126.26 3462.37 563.77 614.14 563.77 614.14 875.73 12342.47 3109.46 1008.58 1996.16 9942.9 3415.2 2322.58 1578.54
SL/SC SL/SC SL/SC SL/SC SL/SC BL SL/SC SL/SC SL/SC SL/SC BL BL SL/SC SL/SC SL/SC TC SL/SC SL/SC SL/SC
Yes No No Yes Yes Yes No No No No No Yes Yes Yes Yes Yes Yes Yes Yes
1.312395 N/A N/A 6.983632 4.819779 0.8750)0 N/A N/A N/A N/A N/A 16.962994 4.090943 0.967515 1.743387 19.599906 1.219500 0.579375 2.114299
BL-bottom-locating; SL-side-locating; SC-side-clamping; TG-top-clamping.
Z\° i
0
1
(a)The distribution of sample points on surface F23 (b) PA values of all sample points on F23 Fig. 11.
PA values of sample points on a bottom-locating candidate surface F23.
used in automated modular fixture design system. 32 Figure 13 shows the fixture configuration design using the fixture plan from Fix-Planning.
4» S e t u p P l a n n i n g Setup planning is to determine the number and sequence of setups as well as the number of operations performed in each setup. A recursive backward setup
126
Yiming (Kevin) Rong
Fig. 12a. An example of horizontal locating/clamping Locating surfaces Locating points Gravity of center of workpiece Clamping surface Clamping point.
planning system has been developed in this research where setups are generated and sequenced based on the information of feature accuracy, machining methods including heat treatment requirement, tool axis direction, etc. Although current work is limited to the planning for machining processes, the method can be generally expanded to other processes. Manufacturing accuracy is the major consideration in setup planning where features with lower accuracy requirement are usually machined in very i r s t setups while features with higher accuracy are machined in the very last setups. Although sometime rough and finish machining may be performed in a single setup when a machining center is utilized, in the case of high precision, they may need to be separated where the large machining force involved in rough machining may cause serious vibration and damage the product quality generated in finish machining.
4.1. Heat treatment
mnd feature
voiume
Besides feature accuracy, several other factors need to be considered. When a heat treatment is required in manufacturing processes, the setup groups are separated by the heat treatment. In our research, three setup groups are defined as: (1) rough machining setups, (2) semi-finish machining setups, and (3) finish machining setups. These setup groups are separated by either the heat treatment requirement or fine
Gompmter-Aided Modular Fixture Design
BIHII^HHHHBHiH^^HBBHH
Fig. 12b. An example of vertical locating.
Fig. 12c. An example of vertical clamping corresponding to the vertical locating.
127
128
Yiming (Kevin) Rong
(a) 2-D top view
(b) 3-D View after removing hidden lines
Fig. 13. The final result of fixture configuration design.
manufacturing accuracy requirement. One example is the case that a grinding operation is necessary for a feature processing, which is usually performed in setup group three. Examples of planning rules for carbon-steel material are designed as shown in Table 4. In practice, features with small volumes removed (such as screw-holes) are usually machined in the very last setup since the machining force involved is little and has no effect on the accuracy of other features, although the feature accuracy may not be high. Therefore the feature volume is considered as a factor in our setup planning. In order to count a relative feature volume in an equivalent scale with feature accuracy calculations, the relative feature volume (V) is first defined as: V = Vc/V0
(23)
Computer-Aided Table 4.
Modular Fixture Design
129
Example of rules for feature assignment t o setup groups.
IF
HB < 350 and heat-treatment = 0 T H E N group = 1 ELSE IF HB < 350 and heat-treatment=l (1-steel, normal) IF IT < 9 or Ra < 250 or V < 0.00253 THEN group = 2 ELSE group = 1 ELSE IF HB > 350 or heat-treatment = 2 ( 2 - carbon steel, quenching) IF IT < 6 or Ra < 32 or V < 0.00063 THEN group = 3 ELSE IF (IT < 9 and IT > 6 ) or (Ra < 250 and Ra > 32) or (V < 0.00253 and V > 0.00063) THEN group = 2 ELSE group = 1
where V c is a feature volume calculated based on dimensional parameters of the feature; Vo is the volume of the workpiece. In our setup planning system, the following assumptions are made to consider the feature volume effect. When V < 1 x 10~ 3 in 3 , V c is small and the machining force would not affect the machining accuracy. When V < 4 x 10~ 3 in 3 , the feature may be machined in semi-finish machining setups because the machining force may affect the machining accuracy in a certain extent. If a feature accuracy is high (e.g. IT < 7), it needs to be machined in two setups in different groups. Finally when V is greater than the critical value, the best feature accuracy in rough machining setups is IT 9 and up. By following a similar procedure presented before, the feature volume factor can be calculated by: ,„
T
VI = I n t
TlogfVx 10- 4 )1
[ log(1.585) J
„ +1
-
,
N
<M>
Therefore, when feature volume factor is considered in setup grouping, a priority feature selection in the backward setup planning becomes: F a c = w t T g + w v VI. 4.2. Backward
setup
(25)
planning
Once the tool approaching and feeding directions are given to manufacturing features and fixturing features are identified, feature groups can be formed and sequenced for setup planning based on the considerations of feature accuracy and other factors. In order to overcome the problems of multiple and/or unreasonable approaching directions under certain feature combinations, the tool axis is considered with machine tool information to make the feature grouping decision. Only a group of features are assigned to a single setup, which can be processed in the same workpiece orientation and a feasible fixture configuration design can be realized (e.g. interference free).
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Yiming (Kevin)
Rong
Figure 14 shows a recursive backward setup planning algorithm where the planning starts from the last setup with a product model and ends up with the first setup and workpiece blank model. In one setup planning cycle, following procedure is implemented: (1) A critical manufacturing feature is identified usually with a highest feature accuracy; (2) According to the manufacturing method and machine tool required for the feature processing, the workpiece orientation is determined; (3) Based on the machining tool axis which is determined by feature processing requirement and machine tool information, other features are grouped into the current setup usually with a same tool direction; (4) Fixturing features are identified and locating surfaces/points are selected for the setup based on the feature accuracy relationship and geometric accessibility; (5) Fixture configuration design is conducted to verify the setup planning. If a fixture configuration design with quality and interference-free cannot be generated, modification information is feedback to feature grouping and locating datum selection; (6) Operation details are generated for each operation within the setup where the depth of cut is determined and CNC programming is worked out; (7) When the setup planning is carried out for the specific manufacturing features, the material volumes removed in these operations are calculated in terms of machining parameters. These volumes are "added" to
•
Product model in setup #1 Critical manufacturing feature identification and analysis Machine tool information workpiece orientation Manufacturing feature groupingl Fixturing feature identification locating/clamping design Fixture configuration design
.+
Operation planning CNC programming
i Feature recovering (add material)
Finish
Product model generation for setup #i-l
Fig. 14.
Backward reasoning algorithm for setup planning.
Computer-Aided
Modular Fixture
Design
131
the workpiece model to form a "product" model for the setup prior to the current setup; (8) If the setup planning is not finished (i.e. the blank model is not reached), another cycle of the setup planning starts. It should be noted that during the setup planning feedback exists in each step for the setup modification. A manufacturing feature database, a machining tool database, and a manufacturing process database are necessary for setup planning decision making. 4.3. Implementation
examples
Figure 15 shows a virtual workpiece used to illustrate the setup planning method presented in this chapter, which is similar to the one used for fixture planning (Fig. 7). Because of the high feature accuracy and heat treatment requirement, three setup groups are necessary, as shown in Table 5. Within the first and second setup groups, the recursive backward planning algorithm is applied to generate the setup plans where a horizontal machining center is assumed available. In each setup groups, locating surfaces for the manufacturing features with major tolerance requirements are machined in the very first setups. Machining tools and the number of fixtures required are also determined. In the third setup group, grinding operations are concerned where the planning rules are much different with machining operations. The planning for grinding operations is not included in this chapter. Figure 16 shows another example of workpieces with a relatively complex geometry. Table 6 shows the setup planning results. The framework of setup planning includes manufacturing feature description and feature-base development, fixturing feature definition and locating surface selection, recursive backward setup planning algorithm, and fixture design and verification. These functions are integrated into a single package in the environment of AutoCAD and C + + platform. The inter-feature accuracy relationship is taken into account so that the manufacturing accuracy can be ensured. Geometric constraints are analyzed in fixture planning and fixture design. Other factors such as heat treatment and feature volumes are also considered in the setup planning. Therefore, a feasible setup plan can be generated for complex workpieces. The recursive backward planning algorithm leads to an automated setup planning. When this system is implemented, the lead-time of manufacturing planning would be significantly reduced.
5. Analysis of Modular Fixture Structures In order to develop an automated modular fixture configuration design system, the fundamental structure of dowel-pin based modular fixture and fixturing characteristics of commonly used modular fixture elements are first investigated. Figure 17 sketches a dowel-pin type modular fixturing system which includes a library of a large number of standard fixture elements. 33 With combinations of the fixture elements, an experienced fixture designer can build fixtures for
Yiming (Kevin)
132
others
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9150 through
MATERIAL- CARBON STEEL HEAT-TREATMENT:
QUENCHING
HARDNESS' HRc 38-45
Fig. 15.
An example workpiece for setup planning.
Computer-Aided Table 5.
Modular Fixture
Design
133
Setup Planning for the First Example Part.
Material: Carbon steel GROUP: 1 SETUP1: Tool axis direction: X + ; locating surface: Ra: 500 Feature 11: plane IT: 15 SETUP2: Tool axis direction: Z - ; locating surface: Feature 1 : plane IT: 12 Ra: 125 SETUP3: Tool axis direction: Y - ; locating surface: Feature 8 : plane IT: 12 Ra: 500 SETUP4: Tool axis direction: X - ; locating surface: Feature 9 : plane IT: 12 Ra: 125 Ra: 250 Feature 10: slot IT: 12 SETUP5: Tool axis direction: Y + ; locating surface: Feature 4 : slot IT: 10 Ra: 125 Ra: 1000 Feature 3 : hole IT: 12 SETUP6: Tool axis direction: Z+; locating surface: Feature 6 : plane IT: 12 Ra: 500 Feature 5 : step IT: 12 Ra: 500
I, (6), 9, 2 mill I I , (9), 6, 2 mill 1, (6), 2. 11 mill 1, 11, 8 mill mill 1, 11, 8 mill drill 8, 1, 11 mill mill
VISE F I X T U R E D: 2 L: 4 VISE FIXTURE D: 2 L: 4 VISE FIXTURE D: 2 L: 4 FIXTURE 2 D: 2 L: 4 D: 1/2 L: l _ l / 4 FIXTURE 2 D: 27/32 L: 1.5 D: 1-1/2 L: 4 - 7 / 8 FIXTURE 1 D: 2 L: 4 D: 2 L: 4
1, (6), 8, 9 mill 1, (6), 8, 9 mill 1, 11, (9), 8 mill 1, 8, 11 mill mill bore 8, 1, 11 mill drill
VISE FIXTURE D: 2 L: 4 VISE FIXTURE D: 2 L: 4 VISE FIXTURE D: 2 L: 4 FIXTURE 2 D: 2 L: 4 D: 63/64 L: 1.5
Heat treatment: Normal GROUP: 2 SETUP 1: Tool axis direction: X + ; locating surface: Feature 11: plane IT: 12 Ra: 125 SETUP2: Tool axis direction: Y—; locating surface: Feature 8 : plane IT: 9 Ra: 125 SETUP3: Tool axis direction: Z—; locating surface: Feature 1 : plane IT: 9 Ra: 250 SETUP4: Tool axis direction: Y + ; locating surface Feature 2 : plane IT: 9 Ra: 63 Feature 4 : slot IT: 7 Ra: 63 Feature 3 : hole IT: 9 Ra: 250 SETUP5: Tool axis direction: Z+; locating surface: Feature 5 : step IT: 9 Ra: 125 Feature 7 : hole IT: 8 Ra: 125
FIXTURE 1 D: 2 L: 4 D: 11/16 L: 3 _ l / 8 bore
Heat treatment: Quenching , Hrc 38-45 Feature Feature Feature Feature
GROUP: 3 1: plane 2 : plane 3 : hole 4 : step
GRINDER IT: 6 IT: 6 IT: 6 IT: 6
Ra: Ra: Ra: Ra:
32 16 32 32
grinder grinder grinder grinder
1 1 2 1
a variety of workpieces. In order to automatically generate a fixture configuration design, the issues for the following problems are presented in the following sections: (1) The selection of suitable fixture elements and combinations of these elements into desired functional units; (2) the methodology to mount (position) the fixture units (or elements) in appropriate positions and orientations on a baseplate without interference with the
134
Yiming (Kevin)
Fig. 16.
Rong
Second example workpiece for setup planning.
space already occupied by the workpiece, machining envelope or other fixture units mounted in advance. It should be noted that kinematic constraints, locating accuracy, fixturing stability, and fixturing deformation are also important in fixture planning and fixture configuration design. Once a fixture configuration design is finished, these design performances, which are not presented in this chapter need to be verified.34"37 Verification results are the feedback information to the fixture configuration design module for alternative designs, if necessary. Fixturing features of a workpiece have been analyzed, including geometric, operational, and fixturing surface information. 38 Once a fixture structure is decomposed into functional units, fixture elements and functional surfaces, the fixture design process becomes a search for a match between the fixturing features and fixture structure. 18 In application of modular fixtures, a fixture element assembly relationship database is built up based on the analysis of the fixture structure.
5.1. Decomposition
of modular
fixture
structure
The advantage of modular fixtures is its adaptability for various workpieces by changing the configuration combinations of fixture elements. Modular fixture structures can be decomposed into functional units, elements, and functional surfaces. By applying Set Theory, a fixture body can be defined as a set or an assembly of fixture elements. Let F denote a fixture and e; (i = 1,2,..., n e ) a fixture element
Computer-Aided Table 6.
Modular Fixture
135
Design
Setup planning for the second example part
Material: Casting iron GROUP 1 SETUP1: Tool axis direction: Y - ; locating surface: 1, 25, 2 Feature 25: plane IT: 12 Ra: 1000 mill SETUP2: Tool axis direction: X - ; locating surface: 1, 25, 2 Feature 17: plane IT: 12 Ra: 1000 mill mill Feature 21: plane IT: 12 Ra: 1000 SETUP3: Tool axis direction: Z - ; locating surface: 25, 17, 12 Feature 1 : plane IT: 12 Ra: 250 mill SETUP4: Tool axis direction: X + , locating surface: 1, 25, 21 mill IT: 12 Ra: 500 Feature 2 : plane mill IT: 12 Ra: 500 Feature 6 : plane R_ bore IT: 9 Ra: 250 Feature 7 : hole SETUP5: Tool axis direction: Z + ; locating surface: 1, 25, 2 mill Feature 12: plane IT: 15 Ra: 500 mill Feature 3 : plane IT: 12 Ra: 500 mill Feature 18: plane IT: 12 Ra: 500 drill Feature 4 : step hole IT: 12 Ra: 250 drill Feature 5 : step hole IT: 12 Ra: 250 drill Feature 19: step hole IT: 12 Ra: 250 drill Feature 20: step hole IT: 12 Ra: 250 SETUP6: Tool axis direction: Y + ; locating surface: 1, 25, 2 F _ bore Feature 24: en_hole IT: 9 Ra: 500 drill Feature 23: step hole IT: 10 Ra: 250
FIXTURE D: 2 FIXTURE D: 2 D: 2 FIXTURE D: 2 FIXTURE D: 2 D: 2
1 L: 4 1 L: 4 L: 4 3 L: 4 1 L: 4 L: 4
FIXTURE 2 L: 4 D: 2 L: 4 D: 2 L: 4 D: 2 L: 2 _ l / 4 C_bore D: 1/2 D: 1/2 L: 2 - 1 / 4 C - b o r e D: 1/2 L: 2 - 1 / 4 C - b o r e D: 1/2 L: 2 _ l / 4 C_bore FIXTURE 1 D: 7/8
L: l _ 3 / 8 C_bore
Heat Treatment: Normalization GROUP: 2 SETUP 1: Tool axis direction: Z - ; locating surface: 25 , 12, 2 Feature 1 : plane IT: 8 mill Ra: 63 SETUP2: Tool axis direction: Z+; locating surface: 1, 25, 2 Feature 12: plane IT: 12 Ra: 125 mill IT: 12 Feature 13: hole Ra: 250 drill IT: 12 Feature 14: hole Ra: 250 drill Feature 15: hole IT: 12 Ra: 250 drill Feature 16: hole IT: 12 Ra: 250 drill SETUP3: Tool axis direction: X + ; locating surface: 1, 25, 2 IT: 12 Feature 6 : plane mill Ra: 125 Feature 7 : hole IT: 6 R_ bore Ra: 63 Feature 8 : hole IT: 12 Ra: 250 drill Feature 9 : hole IT: 12 Ra: 250 drill Feature 10: hole IT: 12 drill Ra: 250 Feature 11: hole IT: 12 Ra: 250 drill SETUP4: Tool axis direction: Y + ; locating surface: 1, 25, 2 Feature 24: en_hole IT: 6 Ra: 125 F_bore Feature 23: step-hole IT: 7 Ra: 63 drill ,
FIXTURE D: 2 FIXTURE D: 3/4 D: 5/16 D: 5/16 D: 5/16 D: 5/16 FIXTURE D: 2 D: 1-1/2 D: 5/16 D: 5/16 D: 5/16 D: 5/16 FIXTURE
3 L: 4 2 + ANGLE PLATE L: 5/16 t a p D: 3/8 L: 1 - 1 3 / 1 6 tap D: 3/8 L: 1 - 1 3 / 1 6 tap D: 3/8 L: 1 - 1 3 / 1 6 tap D: 3/8 L: 1 - 1 3 / 1 6 1 L:4 L:4_7/8 tap D 3/8 L:l_ 13/16 tap D 3/8 L:l_ 13/16 tap D 3/8 L:l_ 13/16 t a p D 3/8 L:l_ 13/16 1
-
-
C-bore
D:l L : l _ 3 / 8
in F, where n e is the number of fixture elements in F, i.e. F={ei|iene}. This is a representation of a fixture at the level of fixture elements.
(26)
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Yiming (Kevin)
Fig. 17.
Rong
A sketch of Bluco Technik modular fixturing system.
A fixture consists of several sub-assemblies. Each sub-assembly performs one or more fixturing functions (usually one). These kinds of sub-assemblies in a fixture are considered as fixture functional units. In a fixture unit, all elements are connected one with another directly where only one element is connected directly with the baseplate and one or more elements in the subset are contacted directly with workpiece serving as locator, clamp or support. Let Uj denote a fixture unit in a fixture. From above description we have: Ui = {eij\j
enei}
(27)
where n e j is the number of elements in unit U^. Therefore a representation of a fixture at the level of fixture units can be written in the following way: F = {Ul|i€n„},
(28)
F = {{eij\jenei}\ienu}
(29)
where nu is the number of units in fixture F. Dividing a fixture structure into functional units and giving detailed analyses on the functional units plays a key role in automated modular fixture designs. A fixture element consists of several surfaces which can either serve as a locating, clamping or supporting surface in contact directly with workpiece (which is named a fixturing-functional surface) or serve as supporting or supported surfaces in contact with other fixture elements (which are called assembly-functional surfaces). Therefore an element can be represented by: e» = { s i f c | k e n s i }
(30)
where Sik denotes the functional surface k on fixture element i and nsi is the number of functional surfaces the element i contains.
Computer-Aided
Modular Fixture Design
ci
Fixture Structure
y
\,
k\
I
\
6 6 \
i
4 6 Functional Surfaces
Fig. 18.
137
Functional Units
Fixt«re Elements
Fixture structure tree.
By combining Eqs. 29 and 30, a fixture can be represented at the level of fixture surfaces in the following form: F
= {{{Sijk\k
e nsij}\j
€ nei}\i
e nu}.
(31)
In this way, a fixture structure is decomposed into three levels, i.e. unit, element, and functional surface levels. A conceptual sketch of the fixture structure decomposition is shown in Fig. 18. Based on the investigation of various application examples of dowel-pin modular fixtures and also for the purpose of automated fixture configuration design, a fixture structure can be classified into seven types of units (sub-structures): Vertical Locating Unit (VLU), Horizontal Locating Unit (HLU), Vertical-Horizontal Combination Locating Unit (VHCLU), Vertical Claming Unit (VCU), Horizontal Clamping Unit (HCU), Vertical Supporting Unit (VSU) and Horizontal Supporting Unit (HSU). Fixture units are composed of modular fixture elements. The functional surfaces of a fixture element perform the task of locating, supporting and clamping. All the above units are mounted on a baseplate. Figure 19 shows the fixture structure decomposition for dowel-pin modular fixture systems. 5.2. Fixture
units
and
elements
In general, a fixture unit consists of several fixture elements where usually only one element is in contact with the workpiece by its fixturing-functional surface to serve as a locator, supporter, or clamp. All fixture elements in a fixture unit are connected together through their assembly-functional surfaces. This fixturingfunctional surface in a fixture unit is defined as an acting surface of the fixture unit. Each unit must have at least one acting surface which performs the fixturing function. Usually the acting surface is a plane or a cylindrical surface. The acting plane of a fixture unit can be described by a point on the plane and the external normal vector of the plane. The center of the fixturing plane is chosen as the point to describe the plane. The acting cylindrical surface of a fixture unit can be described by a point on the axis of the cylinder and the vector of axis. The center point of the acting surface is defined as an acting point of the unit and the distance between the surface of baseplate and the acting point is defined as an acting height of the fixture unit. The acting direction of a fixture unit can also be defined by the direction of the external normal vector of the acting surface.
Yiming (Kevin)
138
. Vertical Locating. Unit (VLU)
. Horizontal Locating Unit (HLU)
Fixture __ Structure
Vertical and Horizontal ' Cmbination Locating — Unit(VHCLU) , Vertical Clamping Unit (VCU)
Horizontal Clamping — Unit (HCU)
, Vertical Supporting Unit (VSU)
. Horizontal Supporting— Unit (HSU)
Unit Level
Fig. 19.
Rong
Surface and Edgd3ar — Adjustable LocatinjBar
' Top Surface Side Surface
• Adjustment Stop V-Pad
' Surface and Edge Bar Dual Surface and Edge Block
— Top Surface • Side Surface
Clamping Support Clamping Bar Speed Clamp with Adjustable Block Serrated Edge Clamp
' Adjustable Bar V-Pad
Adjustable Stop
Element Level
Surface Level
D e c o m p o s i t i o n of m o d u l a r f i x t u r e s t r u c t u r e s .
For fixture units, the most important parameter in fixture design is the acting height. Figure 20 shows the acting heights of different fixture units in a fixture design. In general cases, several fixture elements need to be assembled together to achieve the acting height. The acting heights of fixture units are the parameters to know before the suitable fixture elements can be selected. The fixture element selection to form a fixture unit is based on a fixture element assembly relationship analysis as shown in the next section. Fixture configuration design is a process of selecting fixture elements from a fixture element library and allocating them together in space according to a certain sequence. In AFCD, a fixture element database needs to be built up, in which the geometry information such as geometric profile, the edges and surfaces of a fixture element is represented in its own (local) coordinate system. To represent the position and orientation of a fixture element in the fixture system, global and local coordinate systems need to be defined. If the global coordinate system which is associated with the fixture baseplate is defined by 0(X, Y, Z), the local coordinate system of fixture
Computer-Aided
Modular Fixture
139
Design
Workpiece
Acting Height of VCU
^ ,
/
Acting Height of HLU
Acting Height of VLU Fig. 20.
Acting heights of fixture units.
• X
Fig. 21.
Coordinate systems in automated fixture configuration design.
element i can be denned by three orthogonal unit vectors (u^v^w,) with a local origin pi(x,y,z) as seen in Fig. 21. Once a fixture configuration is built up, the position and orientation of each fixture element needs to be determined. Parameters (Pi,ax,ay,az,bx,by,bz) are used to represent the position and orientation of the fixture element i in the global coordinate system, where Pi is the origin of the element local coordinate system and symbol ax,ay,az,bx,by,bz are the directional cosines of unit vector u^ and v, respectively. The unit vector w;(c x ,c y ,c z ) is not independent and can be determined by: Wi =
UiXVi.
(32)
140
Yiming (Kevin)
Rong
During AFCD, the bottom element of a fixture unit is first placed on the fixture baseplate, i.e. the position and orientation of the bottom element is first determined relative to the global coordinate system, although this relationship may be adjusted later on. Then other fixture elements in the fixture unit are, in turn, allocated until the acting height is reached. This bottom-up approach has been applied to the fixture unit mounting algorithm in the AFCD system. 5.3. Assembly
characteristics
of modular fixture
elements
The methodology of selecting fixture elements and assembling them together to form a fixture functional unit is the key issue in automated fixture configuration design. If a detailed examination is made on the level of fixture functional units from many practical application cases, it is found that there are some commonly used fundamental structures in various fixture bodies. These fundamental structures have the properties of adaptability, rigidity, simplicity, ease of loading, etc. Studying the assembly relationship between fixture elements and extracting basic combinations of the elements is a way to achieve automated fixture configuration design. In fact, the assembly relationships between modular fixture elements are not arbitrary but constrained. A fixture element can be only assembled with a fraction of other modular fixture elements and usually it can only be used in one or several units. Following are examples showing the fixturing characteristics of some commonly used modular fixture elements and their possible assembly relationship with other modular fixture elements. Figure 22(a) shows a console, which is usually used as a riser to raise other fixture elements up to the necessary acting height. Two adjacent sides have an alternating pattern of clearance and tapped holes for accurately mounting the console to baseplate or other support elements. The other two sides have bushed and tapped holes for mounting locating or clamping elements. A console can be mounted on the top of another console of its kind, which is named as a self-supportable fixture element. Because a console is relatively larger than other locating and clamping elements, many elements can be mounted on the top of a console. But, a console usually can be only mounted on a baseplate or another console. A console may be used in building up different kinds of fixture units and it is one of the most adaptable fixture elements. Surface/edge bar and dual surface/edge block shown in Fig. 22(b) are used as rises or locators either individually or in a combination with other fixture elements. The slot edge can serve as a vertical-horizontal combination locator. The surface/edge bar can be assembled on the top of dual surface/edge block and both of them can be stacked on the top of a console to achieve an appropriate height. Figure 22(c) shows several surface locator towers, including locating tower, multi-surface tower, ground spacer, and tipped screw. All of these towers are only used as locators in VLU or VHCLU. They may be mounted at the top level of fixture units and contact directly with surfaces of the workpiece. These towers cannot
Computer-Aided
Modular Fixture
141
Design
(b) Surface/edge bar and block
(a) Console
(c) Surface locating towers
(d) Adjustable locating bars
(e) Adjustable stop
(f) V-blocks
(h) Edge clamps (g) Clamping stop Fig. 22.
Typical modular fixture elements.
support any other fixture elements. A ground spacer is a self-supportable fixture element, which may provide a precise establishment of the acting height. Adjustable surface bar and adjustable locating bar, as shown as in Fig. 22(d), can be used in VLU as locators. Since these adjustable bars are fixed by a screw
142
Yiming (Kevin)
Rong
along a T-slot, the actual locating positions can be adjusted to any desired positions and orientation in the range which can be reached. Therefore, they are very useful in the case of a strict locating point position required. Other commonly used fixture elements in the modular fixture system include adjustment stop (Fig. 22(e)), V-bar, V-block and adjustable V-tower (Fig. 22(f)), clamping support (Fig. 22(g)), and edge clamps (Fig. 22(h)). Assembly characteristics of these fixture elements are similar to those analyzed previously. In order to automatically select and generate fixturing units in fixture configuration designs, the assembly relationships between fixture elements need to be analyzed and represented in a computer compatible format, which is the foundation of forming fixturing units with elements. A Modular Fixture Element Assembly Relationship Graph(MFEARG) has been developed to represent the assembly relationships in building fixture units. Figure 23 is a partial MFEARG composed of real fixture elements, showing assembly relationships of the fixture elements for possibly building a VLU. It should be noted that for the purpose of explicitness, only a few typical fixture elements are shown in Fig. 23. A total MFEARG for assembling a VLU may contain more fixture elements and more assembly relationships. MFEARG can be further represented by an abstract graph. A mathematical model and computer implementation of MFEARG will be introduced in the next section.
Fig. 23.
Modular fixture element assembly relationship graph for a VLU.
Computer-Aided
5.4. Modular fixture element (MFEARG)
Modular Fixture
assembly
Design
relationship
143
graph
An MFEARG can be denned, without loss of generality, as a directed graph (digraph) G, as shown as in Fig. 24, i.e. G = (V,E) and
V = {v | v G fixture elements}; E — {e | P(VJ, Vj) A (v*, Vj € V)}; (33)
where V is a set of vertices representing fixture elements used in building a specific fixture unit; and E is a set of directed pairs of members of V and is an edge representing the assembly relationship between fixture elements (i and j). The edge e(vj -^> Vj) presents that fixture element v^, the start-vertex of edge e, can be mounted on the fixture element v^, the end-vertex of edge e. The number of edges going from other vertices to an end-vertex denotes an indegree of the vertex and the number of edges coming from a start-vertex to other vertices denotes an outdegree of the vertex. An edge e(vj —» Vj) is called a self-loop if fixture element Vj can be assembled with a fixture element of its own kind. Consoles and adopter blocks discussed before are such kinds of fixture elements. A directed-path is a sequence of edges v,i —^-> VJ2 —^ v,3 —^» . . . such that the end-vertex of ej_i is the start-vertex of e^, which represents the possible assembly relationship for building a fixture unit. If the indegree of a vertex in MFEARG (vi, V2, or V3, in Fig. 24) is zero, that means that no fixture element can be mounted on the fixture element. Locating tower, multi-surface tower, etc. are such kinds of fixture elements. Similarly the outdegree of vg is zero in Fig. 24, which means there is no other fixture elements can be mounted to it except the baseplate. Therefore, a complete directed-path represents a possible formation of a fixture unit. In the AFCD system, a modular fixture element assembly relationship database (MFEARDB) is established to represent the MFEARG information where the
Fig. 24.
A sketch of MFEARG models.
144
Yirning (Kevin)
Rong
relative positions and orientations between any two fixture elements are specified according to their possible assembly relationships (e.g. Fig. 23). Once the MFEARDB is built up, it can be used in fixture configuration design.
6. Establishment of M F E A R D B The MFEARG is stored in an MFEARDB. Based on the MFEARG model, algorithms were implemented to choose all suitable fixturing unit candidates and mount fixture units on a fixture baseplate. Since different fixture systems have different modular fixture elements, the corresponding MFEARGs will be different. In order to generally implement the AFCD system, the MFEARDB should be automatically constructed for various fixture systems. Figure 25 outlines the approach to automatically construct MFEARDB. For a modular fixture system, all modular fixture elements are first represented by CAD models with specified assembly features. Then, modular fixture element assembly relationship reasoning engine is applied to find all the possible assembly relationship between any element pairs. The reasoning results are used to construct the MFEARDB, which is based on MFEARG model. The MFEARDB needs to update only when any fixture element is added to or canceled from the fixture system.
6.1. Modular fixture element
modeling
Geometric information of fixture elements is used when interference of two elements is checked in specific spatial positions and orientations. Since the geometry of fixture elements is relatively simple and pre-known, a primitive instancing scheme 39 is used to model the fixture element geometry. Some geometry simplifications are made when modeling fixture elements to avoid time-consuming in intersection checking for complex geometry. Geometric information of a fixture element includes the shape type of the element and dimensional parameters. Figure 26 shows some examples of fixture element shape geometry: block, cylinder, bracket. Block-type elements are defined by three parameters (/, w, h), cylinder-type elements are represented by two parameters (r,h), and bracket-type elements are described by five parameters (li,l2,w,hi,h2).
Modular Fixture Element Database
Fig. 25.
Modular Fixture Element Assembly Relationship Reasoning Engine
Modular Fixture Element Assembly Relationship Database
System for constructing modular fixture element assembly relationship database.
Computer-Aided
Modular Fixture
Design
145
Z
AZ
X Block ( 1, w, h )
Y Cylinder ( r, h)
Bracket ( 11 , 12 , w, h ,, h2 ) Fig. 26.
Three categories of modular fixture element.
To reason assembly relationship between fixture element, assembly features together with the geometric information need to be defined and used to represent modular fixture elements. Following functional surfaces are defined as assembly features of fixture elements: (1) Supporting faces; (2) Supported faces; (3) Locating holes; (4) Counterbore holes; (5) Screw holes; (6) Fixing slots; (7) Pins; and (8) Screw bolts. Figure 27 shows the fixture assembly features. A supporting face is the surface that can be used to support other fixture elements or workpiece. A supported face is the surface that is supported by other fixture elements in a fixture design. A locating hole is the hole machined to a certain accuracy level and can be used as a locating datum with locating pins. Counterbore holes and fixing slots are used to fasten two elements with screw bolts. In a modular fixture system, assembly features of elements such as locating hole, counterbore hole, screw hole, pin and screw are designed with standard dimensions. Other parameters of an assembly feature are the position and orientation of the feature in the element's local coordinate system. The homogeneous transformation is used in this research to describe the position and orientation of features.
146
Yiming (Kevin)
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Let F denote the feature position and orientation of an element, which can be represented by: F = (V,P) 1
(34)
where V = (vx vy vz 0) is the homogeneous representation of the orientation vector V of feature F and vx,vy,vz are the directional cosines of V. P = (x y z 1) is the homogeneous coordinate of origin of feature F. If F is a face type feature, its origin P is a point on the face, and the orientation vector V is normal to the face and points out from it (Fig. 27a). If F is a hole type feature, its origin P is the center of the hole end circle, and V points outwards along the axis of the hole (Fig. 27b). If F is a pin type feature, its origin P is the center
Fig. 27a.
Face type assembly feature.
Fig. 27b.
Hole type assembly features.
V" p
V' I
I
f—
Fig. 27c.
Pin type assembly features.
Computer-Aided
Modular Fixture
Design
147
t I Fig. 27d.
Fixing slot assembly feature.
on the tip of the shaft and V points outwards along the axis of the shaft (Fig. 27c). In the case of fixing slots, the origin P and vector V are defined as shown in Fig. 27d. In modular fixture systems, locating holes, counterbore holes, screw holes and fixing slots are designed to perpendicular to the supporting or supported face of an element. The locating holes, counterbore holes and fixing slots of a supported element are used to locate and fix the supported element to a supporting element. They are defined as associate assembly features with the supported face. Because of the standard design, their relative positions and orientations are known in the local coordinate system of the fixture element and can be extracted from the vector of the supported face. Similarly, locating holes and screw holes of a supporting element are used to locate and fix a supported element on the supporting element. They are also defined as associate assembly features with the supporting face. Their positions and orientations can be extracted from the vector of the supporting face in database. It should be noted that a fixture element may serve as a supporting element to a supported element in a fixture and may serve as a supported element to another supporting element. Since the number of assembly features on a face may vary, a linked lists structure is used in MFEARDB to represent the fixture elements (Fig. 28). In the MFEARDB, fixture element information is organized into four levels, i.e. an element list, element records, functional surfaces, and associate assembly features. In an element record, a fixture element identification code and shape type is first defined. The geometric dimensions are retrieved from element parameters. Associate assembly features are represented in terms of their assembly features on a functional surface, which provides a convenient way to find all associate assembly feature information for a specific surface. This will benefit in reasoning assembly relationship, which is mainly carried out according to supporting-supported face pairs. In the data structure, if there are no more assembly features associated with a functional face, the pointer just points to a symbol NIL which represents the end of list. Therefore this approach has the advantage of saving memory space. Figure 29 shows an example of the data structure for an edge-bar element where two functional surfaces (supporting and supported faces) and three types of associate assembly features (two locating holes, two screw holes, and one counterbore
148
Yiming (Kevin) Element List
Element Record ID Name Shape Type # of Parameters Parameter 1 Parameter N # of Supported Face Supported Face 1 Pointer Supported Face M Pointer # of Supporting Face Supporting Face 1 Pointer Supporting Face P Pointer
Fig. 28.
Rong
Supported Face 1 Record Index/ID Vx
Associate Locating Hole 1 — • Next ID X
Associate Locating Hole Pointer Associate Counterbore Pointer Associate fixing Slot Pointer Supporting Face 1 Record ID Vx Vy Vz Associate Locating Hole Pointer Associate Screw Hole Pointer
y z Associate Counterbore Next ID
Associate Locating Hole 1 — • Next —• ID X
y z
Associate Fixing Slot Next ID
Associate Screw hole 1—* Next ID X
y z
A linked list data structure representing fixture.
hole) can be identified with position and orientation information. The dimensions of the assembly features are standardized with a specific series of modular fixture systems. 6.2. Mathematical
reasoning
of assembly
relationships
When a data structure is designed to represent fixture element and mating relationships are defined between fixture elements, the assembly relationships between fixture elements can be obtained through a reasoning or inference procedure. Actually, the fixture configuration design is similar to an assembly process. Some previous work in assembly area provides valuable information for analyzing assembly relationships between modular fixture elements. 40 ~ 42 6.2.1. Mating relationship between assembly features Mating relationships have been used to define assembly relationships between part components. Researchers defined their own mating assembly relationship according to the application area. In this research, five types of relationships are defined between assembly features for the purpose of reasoning the assembly relationship between modular fixture elements (Fig. 30).
Computer-Aided
Modular Fixture
Supported Face 1 Record 1 0 0 1 LHptr CBptr FSptr -» Nil
Element Record 310020 Surface and Edge Bar Block 3 90 30 20 1 SPDF 1 Ptr 1 SPGF 1 Ptr
Supporting Face 1 Record 1 ' 0 0 -1 LHptr SHptr
^ ^ ^^5S&% Ky0^^ \J>^ Fig. 29.
149
Design
Associate Locating Hole — i Next 1 15 15 20
Next 2 75 15 20
•Nil
Associate Locating Hole Next 1 15 15 0
Next 2 75 15 n
Nil
Associate Screw Hole N«rt 1 30 15 0
Next 2 60 15 0
•Nil
Associate Counterbore Next —»Nil 1 45 15 20
D a t a structure representing.
(1) Against. Face 1 is against face 2 when they are coplanar and with opposite normals. This is the assembly relationship between a sporting face of an element and a supported face of another. Let Fi = (Vi, P i ) T and F2 = (V2, P2) T denote the positions and orientations of face 1 and face 2 respectively. Against condition can be represented by following equations: V 2 * M = Vi, and Vi * (P 2 - Pi) = 0
(35)
where M is mirror transformation matrix. (2) Align. A hole aligns another hole when their vectors lie along the same line but in opposition. This is the assembly relationship between two holes. Similarly let Fi = (Vi, P i ) T and F 2 = (V 2 , P2) T denote the positions and orientations of hole 1 and hole 2 respectively. Align condition can be represented by: V 2 * M = Vi, and K * (P 2 - P i ) = Vi or Pi = P 2
(36)
where K is a constant. (3) Fit. A pin fits a hole when their vectors lie along the same line but in opposition. This is an assembly relationship between a pin and a hole. In the same way, let Fi = (Vi, P i ) T and F 2 = (V 2 , P 2 ) T denote the positions and orientations
150
Yiming
(Kevin)
Rong
V2A
ft
^ (a) Aginst
^
^
(b) Align
£s^ (c) Fit
fi
(d) Screw fit
vector V2 points to reader.
(e) Across Fig. 30.
F i v e basic r e l a t i o n s h i p b e t w e e n
fixture,
of the pin and the hole respectively. Fit condition can be represented by: V 2 * M = Vi, and K * (P 2 - P i ) = Vi or Pi = P 2
(37)
(4) Screw Fit. A screw blot fits a screw hole when their vectors lie along the same line but in opposition. Let Fi = (Vi, P i ) T and F 2 = (V 2 , P 2 ) T denote the positions and orientations of the screw blot and the hole respectively. Screw fit condition can be represented by: V 2 * M = Vi, and K * (P 2 - P x ) = V x or Pi = P 2
(38)
(5) Across. A fixing slot crosses a screw hole when the vector of the fixing slot and the vector of the screw hole are coplanar and perpendicular. Let F x = (Vi, P i ) T and F 2 = (V2, P2) T denote the positions and orientations of the fixing slot and the screw hole respectively. Across condition can be represented by: Vi * V 2 = 0, and Vi * (Pi - P 2 ) = 0
(39)
These five types of mating relationship may cover the assembly relationships between assembly features of fixture elements in most fixture designs.
Computer-Aided
Modular Fixture
Design
151
6.2.2. Assembly criteria between fixture elements In order to establish the MFEARDB, possible assembly relationships between fixture elements need to be evaluated. By examining typical fixture assembly structures, the following criteria in four cases for assembling two fixture elements are employed in modular fixture configuration design (Fig. 31). Let Ei donate a supporting fixture element and E2 a supported element. Case 1. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) A supporting face of Ei is against a supported face of E2. The face on El covers most part of the face on E2; (2) At least two locating holes of Ei align with locating holes of E2 respectively; (3) One or more counterbore holes of E2 align with the screw holes of Ei; and (4) Body of El does not intersect body of E2. Second half of condition 1 is a fuzzy condition. It ensures a firm connection between elements. Condition 2 ensures high locating accuracy between two elements since locating pins can be inserted into accuracy locating holes. Condition 3 ensures that two elements can be fixed together by using screws. Condition 4 is obviously an important criterion for interference free. Once these conditions are satisfied, an assembly relationship between fixture element Ei and E2 is identified and can be added to the MFEARDB. Case 2. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) The same as condition 1 in case 1; (2) The same as condition 3 in case 1; and (3) The same as condition 4 in case 1. The case is the same as last one except the requirement of locating hole alignment. In this case, locating accuracy can be only ensured in the direction of vector of supporting or supported face. Case 3. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) The same as condition in case 1; (2) A fixing slot of E2 is across a screw hole of Ei; and (3) The same as condition 4 in case 1. This case is similar to case 2. Again, in this case, the locating accuracy can be only ensured in the direction of vector of supporting or supported face. Case 4. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) A screw of E2 fits screw hole of Ei when E 2 is a crew bolt; and (2) The same as condition 4 in case 1. This kind of assembly case is usually used in adjustable locating fixture unit. The relative position between two elements is fixed by a nut.
6.2.3. Inference assembly relationship between element pairs Suppose two fixture elements Ei and E2 are an assembly pair. Assembly features and geometry of the two fixture elements are retrieved from MFEDDB. Let Fi = (Vi, P i ) T denote a supporting face of Ei and F2 = (V2, P2) T a supported face of E 2 . Assume P 1 1 ( P i 2 are any two locating holes on the supporting face and P 2 i, P22 are any two locating holes on the supported face. Note that V 1 ; P i , P n , P12 and
152
Yiming (Kevin)
counterbore holes
locating holes I I
Rong
Ej
counterbore hole
is*
~r
E,
L
y
case 2: Locating tower on edge block
o case 1: Edge block on console
screw holes E2.
© o
w-^ z*-t- fixing slot
o
o
o
E2
sm
screw bolt
w case 4: Adjustable locating sto
E2
E.
case 3: Surface bar on console Fig. 31.
Four cases of assembling two fixture elements.
V2, P2, P2I) P22 a r e represented in the fixture element local coordinate systems. If we can find a position and orientation that satisfies the following conditions: (1) Fi against F 2 and (2) P n , P12 align P 2 i, P22 respectively, the assembly position and orientation of E2 on Ei can be obtained from solving assembly mating equations.
Computer-Aided
Modular Fixture
153
Design
Our purpose is to find the position and orientation of E 2 on Ei in Ei 's local coordinate system. The local coordinate systems of Ei and E 2 are first made coincidence. Then, after a series of transformations, E2 is to be translated and rotated to a position and orientation that the relationship between E2 and Ei satisfies above conditions. Based on the mating conditions, we have: V 2 * T * M = Vi, P 2 i * T = P n , and P 2 2 * T = P i 2
(40)
where T is a transformation matrix calculated from: T = ROT x (a) * ROTy{(3) * ROT z ( 7 ) * TRAN(x,
y, z)
including rotation transformation matrices about the x, y, and z axes and a translation transformation matrix. T is further represented as: cos (3 sin 7 sin (3 0\ cos (3 cos 7 — sin a sin (3 cos 7 — cos a sin 7 sin a sin (3 sin 7 + cos a cos 7 sin a cos (3 0 T = — cos a sin (3 cos 7 + sin a sin 7 cos a sin j3 sin 7 — sin a cos 7 cos a cos (3 0 y z \) V /
The solution of above equations implies a potential assembly relationship between Ei and E 2 . Solution (x, y, z) is the position coordinate of E 2 on Ei in Ej local coordinate system, and solution (a, (3,7) is the orientation coordinate of E 2 on Ei in E^ local coordinate system. Furthermore, we should check whether the conditions 3 and 4 in case 1 are satisfied for Ei and E 2 in above position and orientation (x,y,z,a,P,j). If the checking is pass, (x, y, z, a, (3,7) will store into the MFEARDB as an assembly relationship between Ei and E 2 . Similar approach can be used to test if other assembly criteria are satisfied. 6.3. Assembly
relationship
reasoning
system
and
examples
Figure 32 shows the architecture of automatically reasoning assembly relationship engine. Once the MFEDDB is available, the reasoning engine will examine all element pairs to find their assembly relationship. The results are store in an MFEARDB, which is based on MFEARG model discussed in Ref. 39. This information is used to automatically design modular fixture configuration. To illustrate the implementation of the method, an example is given where a console and a surface/edge block are chosen as Ei and E 2 (Fig. 33). Vi, the vector of the supporting face Fj of Ei, is (0, 0, 1,0) and V2, the vector of the supported face F 2 of E 2 is (0, 0, - 1 , 0). P n = (60,75,120,1) and P 1 2 = (30,45,120,1) are the two locating holes on F i . P 2 i = (45,15,0,1) and P 2 2 = (15,45,0,1) are the two locating holes on F 2 . According to Eq. 40, one solution can be identified: x = 75, y = 30, z = 120, a = 0, /? = 0, 7 = 90. The solution shows that there is a potential assembly relationship between Ei and E 2 , which satisfied the conditions: (1) Fj against F 2 , and (2) P n , P i 2 align
154
Yiming (Kevin)
Nfodular Fixture Element Database (MFEDB)
Select El as supporting Eerrent Select E2 as supported Element *
*
Rong
Select a Face Fl in El L* Select aFaceF2 in E2 *
Search Assembly Position Satisfin Gisel yes
Search Assembly Position Satisfin Case2
Store Assembly Reationship yes
Modular Fixture Element Assembly Relationship Database (MFEARDB)
Fig. 32.
Architecture of assembly relationship reasoning.
Computer-Aided
(a)
Modular Fixture
155
Design
Console
(b) Surface/edge Block Fig. 33.
An example of reasoning assembly relationship between console and surface/edge block.
Locating tower Fig. 34.
Surface/edge block
Console
Example fixture elements.
P2ij P22 respectively. It is obvious, in further checking, that conditions 3 and 4 are also satisfied. Therefore there is an assembly relationship between Ei and E 2 with a high locating accuracy. The result can be stored in the MFEARDB. When more than two fixture elements are considered, the assembly relationships can be established in pair. Figure 34 shows three fixture components which are a console, a surface/edge block, and a surface locating tower. Table 7 shows the reasoning result between the console and surface/edge block when the former serves as
Yiming (Kevin)
156 Table 7.
Assembly relationship reasoning results of the console and surface/edge block. Relative position
Potential assembly relationship 1 2 3 4 5 6 7 8
Relative orientation
X
y
z
a
b
g
Locating direction
75 15 45 105 90 30 30 90
-l 59 -1 59 -1 59 -1 59
120 120 120 120 120 120 120 120
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
90 270 0 180 90 270 0 180
All All All All All All All All
Relative iorientation
Relative position
Potential assembly relationship
Interference checking pass?
Percentage of covered area
Is this a assembly relationship?
Yes Yes Yes Yes Yes Yes Yes Yes
98% 98% 98% 98% 98% 98% 98% 98%
Yes Yes Yes Yes Yes Yes Yes Yes
Assembly relationship reasoning results of console Sz locating 'tower.
Table 8.
1 2 3 4 5 6 7 8 9 10
Rong
X
22 30 45 45 60 60 75 75 90 98
y
z
a
P
7
Locating direction
29 44 14 44 14 29 14 44 44 29
120 120 120 120 120 120 120 120 120 120
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
* * * * * * * * * *
z z z z z z z z z z
Interference checking passed?
Percentage of covered area
Is this a assembly relationship?
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
100% 100% 100% 100% 100% 100% 100% 100% 100% 100%
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
unconstrained
a supporting element and the latter a supported element. The assembly criteria can be satisfied with several possible relative assembly positions, which are identified and stored into the MFEARDB. It should be noted that for different assembly position and orientation, the effective fixturing function (e.g. locating direction) may be different. When the console serves as a supported element and the surface/edge block is a supporting element, the assembly criteria are not satisfied, especially the area of supporting face is not large enough. Therefore there is no possible assembly relationship identified. Tables 8 and 9 show the assembly relationships between the console (supporting element) and the locating tower (supported element), and between the surface/edge block (supporting element) and the locating tower (supported element). There is no assembly relationship which can be identified if the function role of the elements are changed (e.g. supporting element to supported element). Figure 35 sketch the MFEARG based on the reasoning results among the fixture element pairs where the allow show the assembly direction from a supported
Computer-Aided Table 9.
Modular Fixture
157
Design
Assembly relationship reasoning results of surface block and locating tower. Relative position
Relative orientation
Potential assembly relationship
x
y
z
a
f3
1 2 3
30 30 45
15 30 30
20 20 20
0 0 0
0 0 0
7
Locating direction
Interference checking passed?
Percentage of covered area
Is this a assembly relationship?
* * *
z z z
Yes Yes Yes
100% 100% 100%
Yes
Yes Yes
unconstrained
Fig. 35.
Second example of constructing MFEARG.
element to a supporting element and the number shows the possible assembly positions between the two fixture elements. Once all fixture elements are checked in pair for the assembly relationships, the MFEARDB is actually established. 7. Automated Fixture Configuration Design Figure 36 shows a prototype AFCD system. The input specifications of the system are workpiece and operational information including geometry representation, workpiece orientation, positions of locating and claming points, and machining envelope. The information is extracted from a CAD model of workpieces with process planning information. The outputs of the system are a fixture assembly drawing and a list of modular fixture elements with their positions and orientations. The AFCD system includes three core modules: fixture unit generation and selection module, fixture unit mount module, and interference checking module. The AFCD is carried out in the following manner: (1) based on the coordinates of locating and clamping points an algorithm is applied to calculate all acting heights of fixture units by considering the least clearance between the workpiece and baseplate, which is usually required for a minimum height of machine tool operations; (2) the
Yiming (Kevin)
158 Part Geometry Representation
Workpiece Orientation
Rang
Fixture Planning
Machining Envelope
Locator/Clamp Selection Module
< Fixture Unit Generation
Fixture Unit Mount Module
Modular Fixture Element Assembly Relationship Database
Interference Check Module
Postprocessor
List of Elements and Their Positions and Orientation
Fig. 36.
Fixture Assembly Drawing Display
Modular Fixture Element Drawing Database
Automated fixture configuration design (AFCD) system.
fixture unit generation and selection module is used to generate suitable fixture units according to the acting heights; (3) The mounting algorithm is used to calculate a position that is suitable for a fixture unit mounted on the baseplate; (4) finally the interference checking module is called to check whether the fixture unit at this position interferes with the machining envelope, the workpiece and other fixture units that have been mounted. If interference checking is not passed, the fixture unit is adjusted to the next candidate position. The mounting and interference checking procedure continues until the interference checking is passed. In some cases, no candidate mounting position is acceptable. Another fixture unit candidate is chosen to ensure that the final output be a collision-free fixture design. To display the design result visually on the computer screen or to get a hard-copy from a plotter,
Computer-Aided
Modular Fixture
Design
159
a post processor is used to calculate the positions and orientation of all fixture elements used in AFCD. 7.1. Fixture unit generation
module
In the algorithm for generating and selecting a fixture unit, all possible assembly relationships in building fixture units are presented in correspondence with MFEARGs. When the acting height of a fixture unit is input, a fixture unit forming algorithm is applied to search all possible combinations by a tree-search approach and find out all fixture unit candidates which satisfy the acting height. In the algorithm, locators and clamps (the fixture elements directly in contact with the workpiece) are first selected. The fixture elements are then selected from the one next to the locator (or clamp) to the bottom element which is directly mounted to the baseplate. Therefore, this is a so-called top-down fixture unit formation algorithm. Assuming a locator or clamp is selected as v», we get a sub-digraph G' of G: G' = (V',E')
(41)
where V ' C V and E'E. In G', Vj is the only fixture element with a zero indegree. All the directed path originally starts from Vj. Sub-digraph G' represents all possible fixture element assembly relationships as v, is chosen as the locator or clamp. The process to generate a fixture unit becomes a search process in G' with an objective of finding the directed paths Vj —» Vji —> Vj2 —>...—> v j m which satisfy the following acting height constraint: m
H = h(vi) + 2 h (vifc)
(42)
fc=i
where h(v) is the acting height of fixture element v and H is the acting height desired for the fixture unit. Fixture unit candidates are listed in three sequences according to: (1) the number of fixture elements used in the fixture unit; (2) the total weight of the unit; and (3) the volume of the unit. When a specially high accuracy or stiffness is required, the fixture unit with the least number of elements is chosen with priority. In case a light fixture body is desired, the lightest fixture unit is first selected. If the spatial restriction becomes a big problem in the process of fixture unit mounting, the fixture unit with the smallest volume is the one to be selected. Other optimization methods may also be applied with different criteria. 7.2. Fixture
unit mount
module
At the fixture unit generation stage, only fixture elements are selected and the topdown assembly relationships between fixture elements are determined. The exact positions and orientations between fixture elements need to be further determined
Yiming (Kevin)
160
Rong
at the fixture unit mounting stage. The mounting procedure can be conducted in two steps: (1) mounting the bottom element of fixture unit onto the baseplate; (2) determining the positions and orientations of other fixture elements, which is presented in next section. To mount the bottom element onto the baseplate, the following factors are taken into consideration: the position and orientation of workpiece, the suggested locating or clamping points, the machining envelope, the position possessed by other mounted fixture units, and the positions of bushed and tapped holes on the baseplate. The mounting requirements include a satisfaction of the acting point position of the unit to the desired fixturing point and the assembly relationship between the bottom element of the unit and the base plate. The algorithm for mounting bottom elements is similar to that presented in Refs. 23 and 43. When fixture units are mounted on a baseplate, the baseplate size is selected from the fixture component database based on the workpiece size and an estimation of the space required for fixture configuration design, although it may be changed later. Figure 37 shows a typical baseplate with locating holes and tapped holes. As it is discussed before, the global coordinate system is associated with the baseplate. Two parameters are used to indicate the positions of center of locating or tapped holes on the surface of baseplate, which are integers u and v in the ranges of (-N, N) and ( - M , M). For the modular fixture system, the screws and holes are alternatively and evenly distributed in two dimensions (X and Y). The center positions of tapped
1
( T 4
•
/
\
© O © O © C> © o © 6 © (M-DO © O © O C » O © O © O ••• © o © o © c ) © O © O © i O © O © O C» O © O © O f i i n Hi O A o © O © O © Cj1 %J> KJ %J) KJ *y • 1 O O O O O 0 O O O O O ... © 0 © 0 © 0 © 0 © 0 © -(M-DO © 0 © 0 © 0 © 0 © 0 M
-M© ..u -N
Fig. 37.
X
O © O © C) O O C 3> (D © -(N-l) ...
-2-1
0
1
2 ...
(N-l)
N
Representation of baseplate in dowel-pin based fixture system.
Computer-Aided
Modular Fixture
Design
161
holes on the baseplate can be represented parametrically as: xs = 2Tu + T((u + 1) mod 2), V. = Tv.
(43)
The center positions of locating holes on the baseplate can be represented as: a;h = 2Tu + T(umod2), Vh = Tv
(44)
where u = - N , . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . , N; and v = - M , . . . , - 3 , - 2 , - 1 , 0 , 1 , 2, 3 , . . . , M. T is a spacing increment between the tapped and locating holes in the row or column directions. In the modular fixture system there are three series of modular fixtures with a uniformed spacing increment, e.g. T = 30, 40 or 50 mm. In mounting a fixture unit onto the baseplate, a fixturing point (x*, y*, z*) and direction is the target to be approached by the acting point and acting direction of the unit. The acting height of the unit is designed to approach the target in z direction, which is presented by Eq. 42. Therefore the fixturing point is projected onto XOY plane with the target (x*, y*). The two parameters are determined for the center position of the tapped hole on the baseplate which is nearest to point (x*, y*)\ v* =div(y*/T + 0.5), u* = div((x* - T((div(i/*/T + 0.5) + l)mod 2))/2T + 0.5).
(45)
The coordinates of the nearest tapped hole can be calculated with Eq. 18 where u* and v* are the variables. The determination of the center position of the locating hole follows a similar procedure and sometimes is not necessary when standard modular fixture elements are utilized because these holes are evenly distributed. The mounting range of a fixture unit largely depends on the fixturing direction. Once the fixturing direction is specified, an acceptable mount range can be determined by considering the information of the fixture unit, mainly the bottom element of the unit. The geometric and assembly relationship information of the bottom element is recalled to match with the holes on the baseplate as represented above. 7.3. Position
of fixture unit acting
point
After the position and orientation of the bottom element of a fixture unit is decided, positions and orientations of other fixture elements in the fixture unit can be determined. Then the position of fixturing unit acting point can be determined, which is desired to be at the closest to the required locating/clamping position. There are usually a number of assembly positions between two fixture elements. For different fixture elements selected to build a fixture unit, there may be many combinations between those fixture elements. Figure 38 shows a sketch of all possible position assembly combinations between the fixture elements, where a series of matrices A
162
Yiming (Kevin)
No. t (bottom) element Fig. 38.
No. t-1 element
Rong
No. 1 (top) element
Position assembly combinations of fixture.
present position relationships between two fixture elements. Ai^ (i\ = 1, 2 , . . . , ni) are the position relationship matrices between the top element and the element supporting it. Atit (it = 1,2,..., n t ) are the position relationship matrices between baseplate and the bottom element which are obtained from bottom element mounting. The number ni and nt are the numbers of candidate mounting locations for the top and bottom elements. Therefore, we can see that the first subscript of the transformation matrix A is a sequential index of a fixture element from the top element and the second subscript indicates the possible assembly relationship of the current element with the element under it. In automated fixture design process, the assembly relationship between every two connecting elements in a fixture is retrieved from the fixture element assembly relationship database. The assembly relationship includes relative position and orientation between two fixture elements. Let us assume element i and i + 1 are two directly connecting elements in a fixture unit. Element i is supported element and element i + 1 is supporting element. If A; denotes the transformation matrix between element i and i + 1 local coordinate systems, it can be described by a 4 x 4 homogeneous matrix in the following form:
where U is a 3 x 3 matrix representing a rotation of the two coordinate systems; and d is 1 x 3 vector representing the translation of the assembly pair coordinate systems. By recalling the fixture element representation described previously, the relative position of element i in element i + 1 coordinate system (pi+i,ui+i,vi+i,Wi+i) is represented by the coordinates of its origin p^x^y^z^) in (pi+i,Ui+i,Vi+\,Wi+i) system and the orientation of element i in (pi+i,Ui+i,Vi+i,Wi+i) is represented by the directional cosines of the unit vectors of Uj, Vi and Wi in (pi+\,Ui+i,Vi+i,Wi)
Computer-Aided
Modular Fixture
Design
163
system as shown in Fig. 8. Aj can be expressed as: &
&
Ki T
K T Ki "r c' .
xi
Ai=
c' .
yi
^Xl
l<
y'i
<
00" ?0 0 1
(47)
By using the transformation matrix A, the transformation between local coordinate systems can be computed through composing A matrices. If the transformation is from the ith coordinate system to jth (assuming j > i), the final transformation matrix becomes as T,j which is given by: Tij=AiAi+i...Ai_i.
(48)
When i = 1 and j = n where n is the number of elements in a fixture unit, we get transformation matrix between top element and baseplate. Equation 48 gives the transformation relationship between the acting point of a fixture unit and the global coordinate system of the baseplate in a specific combination of fixture element assemblies. When all possible combinations are considered, a best fixture unit candidate can be selected to approach the desired acting point with accuracy. Assume that (x*,y*, z*) are coordinates of suggested point of locating or clamping in the baseplate (global) coordinate system and (xa,ya,za) are coordinates of the contacting point (or acting point) of locator or clamp with the workpiece in its own (local) coordinate system. A set of acting point coordinates of the locator or clamp in the baseplate coordinate system, (x,y,z), can be calculated as following: (x,y,z,l)
= (xa,ya,za,l)AlilA2i2
•••Atit,
h = 1,2,..., m ; i2 = 1,2,..., n 2 ; . . . ; it = 1,2,... ,nt.
(49)
For different assembly combinations, the coordinates of the acting point of locator or clamp may be changed. The combination that makes the acting point of locator or clamp closest to the suggested locating or clamping point are the ones we want to choose, i.e. (x* - x)2 + {y* - y)2 + (z* - z)2 - • min.
(50)
Once the best combination is found, the position and orientation of fixture elements in the baseplate coordinate system can be calculated based on the bottom-up calculation procedure. 7.4. Determination
of spatial positions
of fixture
elements
In fixture unit generation algorithm, fixture unit mounting algorithm and interference checking algorithm, we need to transfer the position and orientation of fixture elements from local coordinate systems into the global coordinate system. Let (xi+i,yi+i,zi+i) denote the coordinates of the origin pt+\ of fixture element i + 1 and &Xi+i,a,yi+i,aZi+i and bxi+i,byi+i,bZi+i be the direction cosines of the
Yiming (Kevin) Rong
164
coordinate axes in the global coordinate system 0(X, Y, Z). Then the coordinate of Pi(xi,yi,Zi) in 0(X, Y, Z) is calculated by applying transformation matrix:
(xi,yi,Zi)
= (x-,2/-,2-)
O-xi+l
Qyi+1
a
bxi+l
byi+1
bzi+1
c
Cyi+l
Czi+1
Vi+1
Zi+l
xi+l Xi+l
zi+l
0 0 0 1
(51)
To determine the orientation of fixture element i in 0(X, Y, Z), the direction cosines of the first two axes of the local coordinate system are calculated as: T
=K
aZi.
a
'yi
dyi+l
bxi+l
byi+1
bzi+1
Cyi+l
c
.Cxi+l a
Vxi Oyi
tbzi.
=K
Ki
O-zi+1
O-xi+1
zi+l_
xi+l
&yi+l
a
b'zi] bxi+\
byi+1
bzi+1
Cyi+l
Czi+1.
Cxi+1
(52a)
zi+l
(52b)
The direction cosines of the third coordinate axis is not independent and can be calculated as presented in Eq. 32. Suppose there are t fixture elements in a fixture unit and we want to determine the position and orientation of each element in the fixture unit. First, the position and orientation of bottom element of fixture unit in 0(X, Y, Z) is determined by fixture unit mounting algorithm. The bottom element is considered as the tth element in the fixture unit. By using Eqs. 51 and 52, and the information about the assembly relationship between bottom element and the element t — 1 (the element which rests on bottom element directly), the position and orientation of element t — 1 in 0(X, Y, Z) can be determined. Repeating this procedure (a bottom-up procedure), we can determine the positions and orientations of all fixture elements in the fixture unit. 7.5. Interference
checking
module
Interference checking is a necessary step in AFCD which is quite different from a real fixture design process where fixture elements can not be placed on the position other objects already possess. Without interference checking function, a CAFD system may not generate practically useful fixture configurations. Since a large amount of calculations are needed for interference checking in AFCD, a fast algorithm is important to the design process. In our system, the interference checking is performed in three 2-D projection views. When a fixture unit is placed on a position by the mounting algorithm, the geometry of each fixture element in the unit is projected onto the three orthogonal coordinate planes. Standard 2-D interference checking algorithms are used to check whether the projections of the fixture element penetrate the machining envelope, workpiece, and each fixture element in other fixture units. 44 Once it is found that there is no penetration in a 2-D plane, an
Computer-Aided
Fig. 39a.
Fig. 39b.
Modular Fixture
Design
165
Example of modular fixture design without workpiece.
Example of modular fixture design with workpiece.
interference-free condition is identified for the fixture unit. The interference checking is performed from one 2-D plane to another until the interference-free condition is identified for all fixture units. The checking result is sent to the fixture unit mount or fixture unit generation module for a proper response.
166
Yiming (Kevin) Table 10.
Rong
Input d a t a file format.
Input data file format
Explanation
LM 1 NV 0.00 0.00 - 1 . 0 0 NLP 6 C 260.00 60.00 0.00 10.00
3-2-1 locating method; Normal vector of primary locating surface; Circle on primary locating surface;
L 220.00 180.00 0.00 220.00 20.00 0.00 Boundary line segment of primary locating surface; A 275.00 20.00 0.00 300.00 45.00 0.00 27.32 292.68 0.00
Boundary arc segment of primary locating surface;
LP 280.00 96.00 0.00 LP 40.00 70.00 0.00 LP 40.00 120.00 0.00 E NV 0.00 - 1 . 0 0 0.00
Coordinates of locating point on primary locating surface; Coordinates of locating point on primary locating surface; Coordinates of locating point on primary locating surface;
NV 1.00 0.00 0.00
Normal vector of tertiary locating surface;
NCS 1 NV 0.00 0.00 1.00 CP 280.00 100.00 40.00 CP 50.00 100.00 40.00
The number of clamping surface; Normal vector of clamping surface; Coordinates of clamping point; Coordinates of clamping point;
WPP XYL 275.00 180.00 45.00 180.00
Workpiece profile segment on X-Y projection plane;
ZXL 20.00 40.00 100.00 40.00
Workpiece profile segment on Z-X projection plane;
YZL 180.00 0.00 180.00 180.00
Workpiece profile segment on Y-Z projection plane;
MEP XYL 24.00 196.00 124.00 196.00
Machining envelope segment on X-Y projection plane;
ZXL 250.00 188.00 80.00 188.00
Machining envelope segment on Z-X projection plane;
YZL 190.00 182.00 190.00 154.00
Machining envelope segment on Y-Z projection plane;
7.6. Fixture
configuration
Normal vector of secondary locating surface;
design
example
Figure 39 shows a fixture configuration design example by using the AFCD system. The input information is extracted from a CAD model of the workpiece with process planning information. Table 10 sketches the input file format. Eight fixture units are generated in the fixture design. The AFCD system provides two kinds of outputs. One is the fixture assembly document which lists the elements used and their positions and orientations. The other is the fixture assembly drawing. Table 11 shows an example of fixture assembly document. The corresponding fixture assembly drawings are shown without the workpiece in Fig. 39(a) and with the workpiece in Fig. 39(b).
Computer-Aided Table 11.
Modular Fixture Design
An example of fixture assembly document.
Baseplate number Fixturing unit # 1
450 Acting direction
Acting point 135.00
Element # 1 033 052 052
-15.00
60.00
0.00
135.00 150.00 150.00
-15.00 0.00 0.00
1.00
Orientation 40.00 20.00 0.00
Unit # 2
-105.00
-45.00
60.00
-105.00 -120.00 -120.00
-45.00 -30.00 -30.00
40.00 20.00 0.00
Unit # 3
-105.00
0.00
60.00
033 052 052
-105.00 -90.00 -90.00
0.00 0.00 0.00
40.00 20.00 0.00
Unit # 4
-45.00
-95.00
90.00
119 058 145 145
-45.00 -45.00 -30.00 -60.00
-135.00 -120.00 -150.00 -150.00
90.00 50.00 25.00 0.00
Unit # 5
75.00
-95.00
90.00
119 058 145 145
75.00 75.00 90.00 60.00
-135.00 -120.00 -150.00 -150.00
90.00 50.00 25.00 0.00
55.00
-15.00
80.00
119 058 052 052
195.00 180.00 180.00 180.00
-15.00 -15.00 -30.00 -30.00
80.00 40.00 20.00 0.00
Unit # 7
135.00
15.00
100.00
117 118 112 188 119 170 112 166 151c
180.00 180.00 180.00 180.00 135.00 218.48 198.97 217.95 180.00
30.00 30.00 30.00 30.00 15.00 42.83 40.14 39.28 30.00
145.00 140.00 77.00 90.00 101.00 118.00 99.00 78.00 0.00
Unit # 8
-95.00
-15.00
100.00
-150.00 -150.00 -150.00 -150.00
0.00 0.00 0.00 0.00
145.00 140.00 77.00 90.00
117 118 112 188
0.00
Position
033 052 052
Unit # 6
167
0.00 0.00 0.00
0.00 0.00 0.00
--1.00 1.00 1.00
0.00 0.00 0.00
--1.00 --1.00 •-1.00
0.00 0.00 0.00
0.00 0.00 0.00 --1.00
0.00 0.00 0.00 0.00
-1.00 1.00 1.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 --1.00
0.00 0.00 0.00 0.00
0.32 0.32 0.32 0.95 0.32 0.32 0.32 0.32 0.32
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00 1.00
0.00 -1.00 1.00 0.00
0.26 0.26 0.26 --0.97
0.00 0.00 0.00 0.00
0.00 0.00 0.00
0.00 -1.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 1.00
0.00 -1.00 1.00 0.00
0.00 1.00 0.00 0.00
0.00 0.00 -1.00 -1.00
--0.32 --0.32 --0.32 -0.95 --0.32 --0.32 --0.32 --0.32 --0.32
0.95 0.95 0.95 -0.32 0.95 0.95 0.95 0.95 0.95
-1.00 0.00 0.00 0.00
0.00
0.00
0.00 -0.97 -0.97 -0.97 -0.26
0.00 0.00 0.00
1.00 1.00 1.00 1.00
0.00
1.00 --1.00 0.00 0.00
0.00 0.95 0.95 0.95 -0.32 0.95 0.95 0.95 0.95 0.95
0.00 0.00 0.00 0.00
0.00 0.00 0.00
1.00
1.00
-1.00 0.00 0.00 -1.00 -1.00
0.00 0.00 0.00
1.00
0.00 1.00 -1.00 1.00 0.00
1.00 --1.00 --1.00
0.00
0.00 1.00 -1.00 1.00 0.00
-1.00 1.00 1.00
0.00
0.00 0.00 0.00 0.00
0.00 0.00 0.00
0.00 --0.26 --0.26 --0.26 0.97
-1.00 0.00 0.00 0.00
-1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
-1.00 -0.97 -0.97 -0.97 -0.26
0.00 0.00 0.00 0.00
Yiming (Kevin) Rong
168
Table 11. Continued Baseplate number Fixturing unit # 1
450 Acting point 135.00
Element # 1 119 170 112 166 151c
8.
-15.00
Acting direction 60.00
0.00
0.00
Position -95.00 -179.90 -169.30 -188.59 -150.00
-15.00 8.15 5.26 10.53 0.00
1.00
Orientation 101.00 118.00 99.00 78.00 0.00
-0.97 -0.97 -0.97 -0.97 -0.97
0.26 0.26 0.26 0.26 0.26
0.00 0.00 0.00 0.00 0.00
--0.26 --0.26 --0.26 --0.26 --0.26
-0.97 -0.97 -0.97 -0.97 -0.97
0.00 0.00 0.00 0.00 0.00
Summary
Computer-aided modular fixture design is a means t o implement flexible fixturing methodology in F M S and CIMS. This chapter introduces t h e newly developed a u t o m a t e d fixture design technique, including s e t u p planning, fixture planning, and modular fixture configuration design. In setup planning, based on manufacturing feature analysis and a backward reasoning algorithm, t h e number and sequence of setups are determined, including t h e machining features and workpiece orientation in each setup. T h e a u t o m a t e d fixture planning involves t h e determination of locating d a t u m s and clamping positions in each setup where t h e locating accuracy, surface accessibility, and fixturing stability are considered. Finally, an a u t o m a t e d fixture configuration design (AFCD) system is developed for dowel-pine based moduoar fixtures, which is based on analyses of modular fixture structure and fixture element assembly relationships. Algorithms are developed t o automatically search and select fixture elements t o form fixture units, mount t h e units onto a baseplate, and determine spatial positions of each fixture element in the fixture configuration design. T h e a u t o m a t e d fixture design system can be integrated with a C A P P and NC programming system, which may significantly enhance the flexibility of production systems and reduce t h e manufacturing planning time. Implementation examples are given t o show t h e functions of setup planning, fixture planning, and fixture configuration design.
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26. M. A. Chamberlain, A. Joneja and T. C. Chang, Protrusion-features handling in design & manufacturing planning, Computer-aided Design 25, 1 (1993) 19-28. 27. C. L. P. Chen and S. R. LeClair, Integration of design and manufacturing: Solving setup generation and feature sequencing using an unsupervised-learning approach, Computer-aided Design 26, 1 (1994) 59-74. 28. Y. Rong, X. Liu and A. Wen, Feature reasoning based manufacturing planning, Manufacturing Systems 25, 3 (1996) 271-276; presented at the 27th CIRP Seminar on Manufacturing Systems, Ann Arbor, MI, May (1995) 320-328. 29. J. Li, W. Ma and Y. Rong, Fixturing surface accessibility analysis for automated fixture design, International Journal of Production Research 37, 13 (1999) 29973016. 30. Y. Wu, Y. Rong, W. Ma and S. LeClair, Automated modular fixture design: Geometric analysis, Robotics and Computer-integrated Manufacturing 14 (1998) 1-15. 31. Y. Wu, Y. Rong, W. Ma and S. LeClair, Automated modular fixture design: accuracy analysis and clamping design, Robotics and Computer-integrated Manufacturing 14 (1998) 17-26; partially presented at ASME IMECE, Dallas, TX, Nov. 1997. 32. Y. Rong and Y. Bai, Automated generation of modular fixture configuration design, ASME Transaction: Journal of Manufacturing Science and Engineering 119 May (1997) 208-219; partially presented at ASME Design Automation Conference, Boston MA, Sep. (1995) 681-688. 33. E. G. Hoffman, Modular Fixturing (Manufacturing Technology Press, Lake Geneva, Wisconsin, 1987). 34. Y. Rong and Y. Bai, Machining accuracy analysis for computer-aided fixture design, ASME Transaction: Journal of Engineering for Industry 118 Aug. (1996) 289-300. 35. Y. Rong, W. Li and Y. Bai, Locating error analysis for computer-aided fixture design and verification, ASME Computers in Engineering, Boston, MA, Sep. (1995) 825-832. 36. Y. Rong, S. Wu and T. Chu, Automatic verification of clamping stability in computeraided fixture design, ASME Computers in Engineering, Minneapolis, MN, Sep. (1994) 421-426. 37. Y. Zhu, S. Zhang and Y. Rong, Experimental study on fixturing stiffness of T-slot based modular fixtures, NAMRC XXI, Stillwater, OK, May (1993) 231-235. 38. Y. Rong and Y. Zhu, Computer-aided fixture design and management, 9th International CAD/CAM, Robotics and Factories of the Future Conference, Newark, NJ, Aug. (1993). 39. M. Mantyla, An Introduction to Solid Modeling (Computer Science Press, Rockville, Maryland, 1988). 40. A. P. Ambler and R. J. Popplestone, Inferring the positions of bodies from specified spatial relationships, Artificial Intelligence 6 (1975) (1975) 157-174. 41. K. Lee and D. C. Gossard, A hierarchical data structure for representing assemblies: Part 1, Computer-aided Design 17, 1 (1985) 15-19. 42. K. Lee and G. Andrews, Inference of the position of components in an assembly: Part 2, Computer-aided Design 17, 1 (1985) 20-24. 43. Y. Bai and Y. Rong, Automated modular fixture design, Research Report, Southern Illinois University at Carbondale, Dec. 1993. 44. J. D. Foley, A. Dam, S. K. Feiner and J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, MA, 1989). 45. A. Joneja and T. C. Chang, A generalized framework for automatic planning of fixture configuration, Advances in Manufacturing Systems Engineering ASME WAM, San Francisco, CA, Dec. (1989) 17-28.
CHAPTER 5 OPTIMIZATION IN FINITE ELEMENT AND DIFFERENTIAL QUADRATURE ELEMENT ANALYSIS TECHNIQUES IN COMPUTER AIDED DESIGN AND ENGINEERING CHANG-NEW CHEN Department of Naval Architecture and Marine Engineering National Cheng Kung University, Tainan Taiwan The analysis of continuum mechanics problems is frequently necessary in computer aided design and engineering. The finite element method and differential quadrature element method are effective numerical methods for analyzing continuum mechanics problems having arbitrary domain configuration. This chapter introduces some finite element improvement procedures and differential quadrature element method related numerical techniques. Keywords: Multiplicative and additive corrections; GSR-based accelerated equilibrium iteration; differential quadrature; generic differential quadrature; extended differential quadrature; differential quadrature element method; differential quadrature finite difference method.
1. I n t r o d u c t i o n Analysis is usually necessary in carrying out t h e engineering design and treating generic scientific problems. Following t h e increasing requirement for optimum design, accurate discrete analysis of scientific and engineering problems becomes even important. Consequently, in order to compete for t h e leading position in the area of computational science and engineering internationally, it is necessary to spare no efforts in developing accurate, reliable and efficient discrete analysis techniques. Following the advance in computer technology, t h e numerical technique has made significant progress in the past fifty years. Numerical approach has from then on become a major branch in the field of engineering or scientific research. Among t h e major techniques for numerically analyzing continuous problems, finite difference method (FDM) is most early developed. This method uses divided difference expressions established from a local Taylor series t o replace differential or partial differential operators appearing in a mathematical t e r m in discretizing an engineering or scientific problem. Though t h e discretization is straight, it is difficult to deal with problems showing nonrectangular or complex curvilinear geometries by using this method. 171
172
C.-N.
Chen
The finite element method (FEM) can consistently discretize problems showing generic geometries since it uses interpolation and mapping techniques. This method employs the variational calculus or weighted-residual along with the divergence theorem to carry out a weak formulation which results in an integral statement valid for a discretization. The discretization is performed on the domain of an element, which can have different shape configurations, to result in a computable algebraic form. This method has been successfully applied to the solution of various problems in many engineering or scientific areas. Recently some differential quadrature (DQ) based discrete numerical methods, which are original works and have perspectives, were proposed by the author. These methods are generic differential quadrature (GDQ), extended differential quadrature (EDQ), differential quadrature finite element method (DQFEM), differential quadrature finite difference method (DQFDM), differential quadrature element method (DQEM) and generalized differential quadrature element method (GDQEM). These methods have excellent numerical efficiency and reliability. Moreover, they have the advantage of consistently discretizing continuum problems having arbitrarily complex geometries. They are all suitable for vector and parallel processing. In addition to the efficient refinement analysis techniques of FEM, FDM and DQ related numerical methods, the efficient solution of linear and nonlinear algebraic systems is also important for modern engineering design and scientific analysis. This chapter deals with these topics.
2. Improvement of Finite Element Solutions In industrial applications of the finite element method, it is necessary to produce accurate results at a reasonable cost. However, even an experienced finite element analyst may find that it is difficult to design an analysis model which can, from the outset, efficiently produce accurate results. For this reason a number of algorithms have been introduced to adaptively alter finite element models and modelling procedures to obtain improved solutions based on a preliminary discretization. The most promising procedures seem to be the p-method, the /i-method, the repositioning method, the Loubignac iterative method and the finite element-difference method. In the p-method, 1,2 the polynomial degree of the underlying finite element approximation is increased, on the elements of the preliminary grid, to produce convergence. The hierarchical method is a particularly useful p-method in which the discrete equations for the higher order elements can be obtained by adding the higher order terms related discrete equations into the discrete equations for the lower order approximation. 3-5 In the /i-method, adaptive techniques for mesh refinement are introduced to obtain improved finite element solutions. 6,7 The convergence can be assured by successively carrying out the refinement analysis which adopts a certainly defined refinement indicator and a convergence criterion. The
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refinement indicator can be the absolute or relative local error of the variable function or a certain physical quantity denned by the derivatives or partial derivatives of the variable function, or an absolute or relative error norm defined by the error of the variable function or a physical quantity defined by the derivatives or partial derivatives of the variable function. There are three methods for solving the overall algebraic system. The first one is to use the direct method to all refinement stages. The second one is to use the iterative method to all refinement stages. The last one is to use the direct method to the initial refinement stage following the use of the iterative method to the other refinement stages. Prom the computation cost point of view, the two iterative refinement techniques are effective for solving generic problems. They are especially effective for solving non-linear problems. In the repositioning techniques, the nodes of the preliminary grid are moved in order to minimize the errors inherent in the finite element approximation. 8 ' 9 In the two-level finite element computation algorithm, rational basis functions are used to improve the lower level finite element solution. 10 In the Loubignac method, a stress redistribution procedure is introduced to iteratively reduce the error caused by the standard finite element discretization. 11 A continuous stress field can be obtained by using this method. However, the displacement might converge to a value which is too flexible compared to the exact solution. In the finite element-difference method, partial approximations are introduced to iteratively improve the results obtained by standard finite element methods. 12,13 In particular, in the postprocessing phase, finite difference representations are used for the strain gradient expressions. Because of the use of the finite difference approximations, this method may not have the generality of standard finite element methods. A multiplicative and additive correction procedure has been introduced to automatically improve preliminary finite element solutions. 14 The procedure utilizes multiplicative and additive corrections based on the preliminary finite element solution. The algorithms are implemented using a hierarchy of meshes and an analysis procedure based on a multi-level discretization. These algorithms are designed to improve stresses and displacements without the loss of the generality inherent in the standard finite element methods. This section deals with this correction procedure. 2.1. Multiplicative
and additive
corrections
In the algorithms to be developed in this work, procedures are introduced to iteratively improve the finite element solution defined on an initial mesh. The concept is based on the premise that there are two types of errors in the initial finite element solution. These are either shorter period-high frequency errors or longer period-low frequency errors. If the response variable function approximated by the initial finite element discretization cannot properly express the locally high gradient response distribution, shorter period errors exist. Consequently, the computed gradients will deviate from the true results significantly.
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Longer period errors are another matter. In the presence of longer period errors, the entire solution may be off by a substantial amount. Closer examination may indicate that the response variable function is in error, from mesh point to mesh point, by slowly varying values. This may occur, for example, if we model a thin cantilever beam with bilinear elements. Then, because of the use of lower order elements, longer period errors exist in the solution. There is a procedure which is used in engineering practice and which is useful in discussing the issue of shorter period-longer period errors. This procedure involves the solution of a problem on an initial mesh followed by the solution of a second problem on a subset of the original mesh, where detailed and accurate results are desired. In the second problem a very detailed model of a relatively small region in the problem domain is constructed. The boundary conditions in the second problem are defined by the solution of the initial problem. This procedure can only reduce shorter period errors. Longer period errors will make the assumed boundary conditions of the second problem erroneous. There are, however, some interesting characteristics involved in this technique. This procedure makes use of the initial grid solution to proceed to a more accurate solution, and it makes use of the idea of working with correction on blocks or substructures of the initial mesh. Certain existing techniques for iterative improvement of finite element solutions can reduce both longer period and shorter period errors. An ideal way to implement the idea of reducing longer period and shorter period errors is through the use of multi-level solution procedures and multiplicative and additive correction techniques. In the multi-level solution method multi-level meshes are first defined, perhaps by a mesh generation procedure. The original finite element mesh at the lowest level is the fine grid. At higher levels there can exist coarser grids for defining block corrections. The correction solution at higher levels can be obtained by multiplicative and additive correction procedures based on a particular lower level solution. Two methods seem to have the most obvious application. In each method the starting point is the original finite element solution on the lowest level mesh. Assume that the lowest level is level p having the original finite element solution denoted U^(XJ). In the first method the correction procedure at a given level involves the use of additive and multiplicative corrections which are always based on the original U?(XJ) finite element solution. In this case the degree of the polynomials in the correction solution can continuously increase with level by continuously increasing the order of block interpolation functions. In the second method the correction procedure involves a sequence of higher level corrections in which, at a given level, multiplicative and additive corrections are based on the approximate solution at the next lower level. In this method the degree of the polynomials in the approximation increases with level so that a series of approximations V%(XJ),V%+ (XJ),U?+ (XJ),... is generated as the algorithm proceeds through levels. These methods attempt to divide the finite element solution errors into longer period and shorter period categories. If one block contains only one original finite element it results in a p refinement
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procedure. The correction solution can also be a refined piecewisely continuous function. This correction procedure provides a link with classical series solutions which adopt the series expansions such as trigonometric, exponential, polynomial, Fourier and certain other series expansions. The convergence can be assured by successively carrying out the iterative correction analysis which adopts a certainly defined correction indicator and a convergence criterion. The correction indicator can be the absolute or relative local errors of the variable function or a certain physical quantity defined by the derivatives or partial derivatives of the variable function, or an absolute or relative error norm defined by the error of the variable function or a certain physical quantity defined by the derivatives or partial derivatives of the variable function. The adaptive approach can also be used to the correction analysis. In the adaptive correction process, longer period errors can be first significantly reduced by carrying out a rather large block based correction. Successive corrections can be carried out by gradually reducing the block size and concentrating further block corrections to certain subregions in which certainly defined correction indicators detect that shorter period errors are still significant. In gradually reducing the shorter period errors, the longer period errors can be further reduced. The philosophy inherent in the multiplicative and additive correction procedure based on reducing the solution errors was also discussed before.14 The adaptive h refinement procedure can also reduce the errors existing in the initial finite element solution, sequentially.
2.2. Interpolation functions correction procedures
for multiplicative
and
additive
Initially consider the correction procedure on the one-dimensional block. The spatial co-ordinate on the block is denoted x. Let up(x) be the original level p finite element solution of this block. It is necessary to define a procedure which, using the original finite element solution and block based corrections, will provide an improved finite element approximation for the problem. Let un(x) be the level n correction solution. It is assumed that this correction solution can be expressed as follows: un(x)=g?(x)u'(x)+g2(x).
(1)
Here g™(x) is a multiplicative correction distribution function. It is designed to scale the original level p solution up(x) using a function with a longer period spatial variation. The term g% (x) is the additive correction term. The key point in defining the correction solution is the choice of nodal variables and associated interpolations for the correction distribution functions gi(x) and g£(x). Generally, a mixed representation could be used in the interpolation. Suppose that the interpolation functions ^"R(x) and ^ 2 s ( x ) which can have different orders are used to discretize g"(x) and
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functions can be in standard or hierarchic forms. g% {x) can also be a refined piecewisely continuous function added to the block. Then the multiplicative and additive correction functions can be defined as follows: 0?(*) = * ? A ( * ) < ? 8 ,
«#(*) = *2s(a:)CS
(2)
where CJj and Cg are nodal correction constants at level n. Introducing (2) in (1), the level n correction solution would take the following form: u B (x) = 9?R{x)CW(x)
+* M .
(3)
If the order of the block interpolation functions is higher than one and they are in hierarchical form, the interpolation functions can be a linear expansion specified by the standard level p + 1 linear interpolation functions adding a series of polynomials always so designed as to have zero values at the ends of the block. These bubble functions can be defined in terms of integrals of Legendre polynomials or in certain other forms. If both block interpolation functions \&™fi and ^JJS are bubble functions the correction solution is also a bubble function. In real application, one of the two correction terms can be neglected. Meanwhile, one block can contain only one element. Thus this method has the option to carry out the standard p refinement analysis. It just uses the one-element block and neglects the multiplicative correction term. In addition, if the hierarchic p refinement adopting the interpolation functions of which the derivatives of them have the orthogonal property, the coupling between successive solutions disappears. The hierarchical interpolation functions defined in terms of integrals of Legendre polynomials have this property. There are two block nodes. Suppose that equal order interpolation functions are used for the multiplicative and additive corrections. Then the block interpolation functions are ^" f l (x) = *&2R(X) = ^R(X) anc ^ * n e correction solution is defined as follows: un(x) = *5(x)Cgu"(x) + *£(*)<%•
(4)
As an example, consider the level p + 1 linear block shown in Fig. 1 in which TV equals p + 1 is coordinate within the block is £™. Then the correction solution can be expressed as uN(ZN)
= [*f ( ^ ) C f + * 2 "(£")C 2 "] «*(£") + * ? ( £ " ) C f + *?(£N)C?
(5)
where 1 - £"
1 + £n
Adopting the standard interpolation functions, the multiplicative and additive correction constants C^ and C^ can be related to the nodal correction displacement Uft by applying the constraint conditions at the nodes. These constraint conditions
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A ->- C n , u P ( C n ) , Q n (C n ) n n r 11 2'U2'C2
un,Oj,c?
U nn
Fig. 1.
A typical one-dimensional block.
take the following form: (6) where UR = u p (££). Then substituting (4), the multiplicative and additive &R = &R(6)~ into ^R^(R) correction solution can be obtained.
un(C) = *S(D
«p(0
*?*)
eg + *£(£")#£
or (7) where $ ^ are bubble type interpolation functions defined as follows:
*£(£") = * R ( D [up(C) - u?R)] • The multiplicative and additive correction solution expressed by the form of (7) can be generalized (8)
«n(0 = *iR(r)cs + *55(D^5-
It should be mentioned that the orders of ^ R and * ^ s can be different. Moreover, the block interpolation functions #£g have the standard form while # ? R used to construct the bubble type interpolation functions $ ? R can be standard or hierarchical. In real application, certain terms of the bubble type interpolation functions can be neglected. In addition, one of the two correction terms in (8) can be neglected. Meanwhile, one block can contain only one element. For the one-element block model, if only the bubble type interpolation functions are adopted, it results in a hierarchic p refinement model. The second correction term added to a block can also be a piecewisely continuous function. 2.3. Multi-dimensional
multiplicative
and additive
corrections
The multiplicative and additive correction procedure defined in (7) for a onedimensional problem can be extended to deal with multi-dimensional problems. Let up(xj) be the original level p finite element displacement solution on the elements
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of the multi-dimensional block. Also let U"(XJ) be the level n correction solution. This level n correction solution can be defined as follows:
W )
= *i*(tf )CTK«? 0 (#) + *2s(#)C?5-
(9)
Like the one-dimensional model, (9) represents various correction versions. It should also be mentioned that the block can have irregular shape. Using equal order block interpolation functions, the constants C"R can be obtained by evaluating M™(£") at the block nodal points: C?R = U?R - UflR)C?R.
(10)
Substituting (10) into (9), a final multi-dimensional multiplicative and additive correction solution can be obtained.
<(£?) = *&)*(#)£?« + *£(#)£&
(ii)
where
*?«(#) = n($) [«?(#) - u*{R)]. The interpolation functions $"#(£") are bubble functions. Equation (11) can also be generalized to obtain (80) like multi-dimensional multiplicative and additive correction solution. The previously introduced multiplicative and additive correction techniques define the correction solution at a general level n in terms of the level p finite element solution. Essentially as n is varied a sequence of correction solutions is developed on a series of adding block corrections to certain subregions in which the correction indicator detects that the solutions have not converged yet. Another type of correction procedure can also be defined. In this procedure the fundamental solution on which the multiplicative correction operates is allowed to change. In this case the degree of correction solution can be made to increase as the level number n is increased. Consider a method which utilizes level p,p + 1, p + 2 , . . . , etc. as successively higher levels with each level having the same linear block interpolation functions. In this algorithm the level n correction solution is obtained from the level n — 1 correction solution through the following relationship: * ? & ) = ^RiW^RU^1^)
+ n(Zj)C?R,
n = p+l,...,N
(12)
where N is the highest level allowed. In terms of the block nodal variables this correction solution would take the following form: « ? & ) = %)R(ii)CiR
+ *R&)C?R
(13)
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where ©£R(£J) can be written as
p+i ©S2fe) = *«&) *5(^)Cffi«?Ki) + <M^)C?2
JV-1
.T,JV/
©&&)=*£&)
n *s(Wos «?&•)
t=p+l JV-1
N-l
n *5fe)q?)s
k=p+l
**M(Zi)c;M
,m=fc+l
In real application, the second term of the correction solution can be neglected. The correction solution of this type of correction procedure can also be defined by referring to (11). (14) where
U=P+I
2.4. Multi-level
finite
element
J
procedure
The nonlinear elasticity problem is selected for analysis. The total Lagrangian formulation is used. Let T^J denote the second Piola-Kirchhoff stress in the elastic body with domain CI. It is assumed that in this analysis stress is linearly related to strain through a matrix representing the elasticities of the solid. Then, if ers represent the strain, the stress-strain relationship is Tkj
=
^kjrs^-rs-
In this analysis, the strain measure known as the Green-Saint Venant strain is used. This strain measure is defined as follows:
Let fi denote the body force, T; denote the specified boundary traction on the natural boundary dCla, and rij denote the outward unit normal vector to the boundary
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surface d£l. The problem is described by the following equilibrium equation: [Tkjikj + Ui,j)]k = -fi
in Q.
(15)
The natural boundary condition is Ti = nkTkj{5ij + Uij)
on dQ„.
(16)
A general incremental loading procedure adopting the equilibrium iteration technique is utilized to accurately update the loading history. In the current nonlinear iteration, following each level p incremental solution the higher level correction solutions are included in the equilibrium iteration procedure to improve the level p solution. Let u? be the displacement increment on level p for a specific load stage K, and u° be the converged displacement for the previous load stage. Then the updated displacement uf can be expressed as
«? = < + *?•
(17)
The incremental solution up for an element e with element domain Q,p,e and element boundary dQP'e can be discretized by using the level p interpolation functions ^ ^ ,
«? = nura.
(is)
Using the inner products (u, v) = I
uvdfl;
(u, v) =
vudS
and referring to (15)-(18), the Galerkin residual equation can be expressed by (fi + [r^iSij + < , ) ] , * , *S) + ((fi - n^Sij
+ «JV), *S> = 0
(19)
where T, are the surface tractions and fi the body forces, at the present load stage. It should be noted that the integration is performed over the undeformed volume of the element. Integrating by parts in (19) leads to the following discrete equilibrium equation: (rPkj(Sij + < , ) , K,k) = (/i- *Z) + (fi,^)-
(20)
Now we derive a set of incremental equations which can be used to update the deformation history by solving a series of linear algebraic equations. Let the following incremental relations be denned: TPkj=T°kj+fPkj,
f^Tf
+ fi,
fi = f° + fi
(21)
where f?., Ti and fi are increments of the second Piola-Kirchhoff stress, surface traction and body force, respectively. Introducing (15), (18) and (21) into (20), the
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following incremental stiffness equation can be obtained:
K':hA=Pia where the tangent stiffness matrix k^h/3 defined by
(22)
and the incremental load vector peia are
and PL = (fi,n)+
&,*>)•
(24)
It should be noted that the incremental constitutive law has been assumed to be T
kj ~
^kjrs^rs
where the strain increment e£s is
The level p original finite element solution is judged to be converged if a certain correction indicator which measures the solution error is small enough. Assume for the moment that the correction indicator specifies that the level p solution has not converged. Then consider the construction of an iterative correction procedure. Based on the distribution of the correction indicator, the mesh of block can be designed which is used to construct the iterative correction. For a specified block, let u"' m and u™'n be the updated displacements of the last iterative solution step and the correction displacements of the current iterative solution step, respectively. Then the updated displacements of iteration step m can be defined as follows: n.m
V
~n,m
=V
,
~n,m
+V •
Though there are various correction versions, the variational simulation adopting the correction solution (11) is illustrated. This correction field takes the following form:
where t / ^ m and C"^71 are nodal incremental parameters defining the correction solution of iteration step m. Let f^'"1 denote the updated stresses of the last iterative solution step. By using the same Galerkin procedures, the stiffness equations of
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level n correction solution in iteration loop m can be defined as follows: ri
-iRhSuhS
rL
u
iRhS hS
+
"+"
^iRhS^hS ^iRhS^hS
~ ~
n
iR
>
/„,.-)
u
iR
where
UC
K?£S
= Wr^fasj**,
cu
K"£s
= Wr*h5*>
^Kms
= Wr*hsA*)>
n,k)
+( * W * y
+ u7;P(Srh
unh?)*WS,s^R,k),
+
*««,*) + (EkJrs&j + ZZnVrh +
+ u?f)(5rh
+
*&«,*)' n
u h^(h)s,s,^i)R,k) (26)
and
-A.T = (/„ *»)+&, n) - wr&i+
(27)
The block based correction will reduce longer period errors. However, the local element to element shorter period errors may be still significant. That is, the solution may be closer to the exact solution in the overall sense, but some readjustment in the solution may be needed locally. Let u™ be the updated displacements and u™ the incremental element based correction displacements, of iteration step m. u™ can be defined as follows:
The following expression for the element based correction solution will be used:
u? = *£ff£
(28)
where ^ P are the interpolation functions for the level p solution and t/?£ are the incremental displacements. Let it™ and fj£j be the updated displacements and stresses, respectively, of the last iteration solution. Then by using the Galerkin method, the following incremental stiffness equation can be obtained:
krahpur0 = f?a
(29)
where kZhf, = (f%*P0,M> Kk) + (^irsiSij
+ u^){8rh
+ < r ) # £ , 5 , *pa,k)
(30)
and r?a = (/<,*£) + <£,*£> - (f^iSij
+5&)> *»,*)•
(31)
It should be noted that the above formulations for correction solutions are based on the general Newton-Raphson procedure. For simplicity, we use the modified Newton-Raphson procedure to carry out the numerical tests.
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examples
In order to verify the multiplicative and additive correction procedure, a series of numerical tests was carried out. The first problem solved concerns the iterative improvement of a tapered bar subjected to a uniformly distributed load, which is shown in Fig. 2. Four analyses were carried out. Two of them use the two-node linear element while the other two solve the problem using the correction procedure. Only the bubble type block based correction displacement is added. It is a kind of hierarchic p type refinement analysis. The first iterative improvement solution uses one two-node element and one two-node block to model the tapered bar. The levels are restricted to two. The other iterative improvement solution uses two two-node elements and two two-node blocks to model the bar. Each block has one element. The results obtained are summarized and listed in Table 1. They are compared with the exact solution. It shows that the bubble type correction displacement can improve the original finite element solution. In the numerical tests, all two bubble type block based interpolation functions are considered. However, one of them could be neglected. The geometrically nonlinear analysis of a cantilever beam subjected to a lateral tip load P was also carried out. Let L, h, I and E denote the length of beam, depth, moment of inertia of cross section with respect to the neutral axis and Young's modulus, respectively. The beam has the nondimensional constants L/h = 10 and PL2/EL = 0.1. Two rectangular eight-node quadratic serendipity elements are used to model the beam. One four-node block and one eight-node block containing the p(x)=l
Ai = l
E=l
/ A3 =2
Fig. 2.
Table 1.
The tapered bar modelled by various meshes.
Displacements of the tapered bar problem.
Element 1 2-node element 2 2-node elements 1 2-node element 2 2-node elements Exact solution
Block
1 2-node block 2 2-node blocks
Point 2
Point 3
0.1666667 0.3000000 0.3076923 0.3117136 0.3109302
0.3333333 0.3714285 0.3846154 0.3851341 0.3862942
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70<»65 60 •M
C
55
e
50
a u o a
i
•H •P
c 0)
E
40
30
rH
25
01 •iH T3 M-l
20
& o
e S-l
15 -
o c
10
>-3
5 0
Fig. 3.
2 8-node elements and 1 8-node block
35
n)
2 8-node elements and 1 4-node block
45
1 °* £ I o T It 2 tg +« ^ *i * 1« i. I 7 9 11 13 15 17 19 21 23 Number of Iterations
Convergence for the iterative correction of a beam with geometrically non-linear deflection.
two elements are used to carry out two two-level analyses, separately. The changes of L,2 norm of incremental displacement vector and the lateral tip displacement, following the increase of iteration number, are plotted and shown in Figs. 3 and 4, separately. The figures show that the eight-node block based correction analysis not only converges faster than the four-node correction analysis but also converges to a better result. Figure 4 also shows that the tip displacement reaches a value very close to the converged value after the first iteration. However, Fig. 3 shows that the value of Z/2 norm is still rather large after the same iteration step. The inherent sense is that the shorter period errors are still significant though the longer period errors become very small. It is caused by the existence of nonlinearity and the block correction. It is not a general character. The last example is the two-level analysis of a plate structure with vacant regions, shown in Fig. 5. The eight-node serendipity blocks are used to improve the original finite element solution obtained by using the four-node elements. The variation of the computed displacement at point C with iteration is plotted in Fig. 6. It is shown that only few iterations are necessary for the results to converge to a degree of enough accuracy. In this figure the correction solution is compared to two results
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1.0 — o
So IS
0.
m >
e
0.5
Fig. 4.
o
0.
I
Converged s o l u t i o n u s i n g 2 8-node e l e m e n t s 2 8-node e l e m e n t s and 1 4-node block solution 2 8-node elements and 1 8-node block s o l u t i o n I I I I L I L _L l _ l 3 4 5 6 7 8 9 10 11 12 Number of I t e r a t i o n s
Displacements at the tip of a cantilever beam with geometrically non-linear deflection.
1000
E=3*107 psi v=0.25
|
|: Element Block
^
Fig. 5.
Vacant region
A thick beam model with vacant regions.
computed with a fine grid of eight-node elements adopting two different integration orders. In Fig. 7, the computed normal stress distribution along cross section BB in the structural model is presented. The results show that the additive and multiplicative correction is effective. 3. Solution of Static and Dynamic Finite Element Problems Efficient and reliable solution of a static or dynamic nonlinear finite element system with equilibrium iteration is an important topic in the area of scientific and engineering computation since advanced design is a challenging trend in current and future technology. The global secant relaxation-based accelerated constant stiffness iteration technique can be incorporated into the static or dynamic nonlinear finite element solution procedure to construct an efficient and reliable algorithm. The choice of an appropriate direct time integration scheme can also reduce the computational
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,106 .104 .102 40 8-node elements with 2x2 integration order
.100 .098
4 0 8-node elements with 3x3 intagration order
U .096 >
4 0 4-node elements and 10 8-node blocks
.094.092.090
Fig. 6.
J 1
I
I
L J_
2 3 4 5 Number o f
J_
_[_
6 7 8 9 Iterations
_L 10
Improvement of the displacement for the thick beam model with vacant regions. 3
40 8-node elements with 2x2 integration order o : 40 4-node elements • : 40 4-node elements and 10 8-node blocks
in
a
ira
o
CQ
cn C o
3V
4 5 B •Position (xioin)
-1
-3 Fig. 7.
Normal stress distribution for the thick beam problem.
cost and increase the accuracy of the computed results. It is generally admitted in the computational mechanics community that implicit direct schemes are more economical in inertial problems, while explicit direct schemes are more economical in shock loading and wave propagation problems. However, for high speed dynamic
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187
problems certain implicit direct schemes can be more efficient if highly non-linear response behaviors do exist. 15 An efficient and reliable incremental/iterative solution procedure, based on using a global secant relaxation (GSR) technique to adjust the constant stiffness prediction and improve the convergence, for non-linear finite element problems which have geometric and/or material nonlinearities has been proposed by the author. 1 6 - 1 7 In this section, the results of static and dynamic elastic-plastic material failure finite element problems considering non-linear deformation solved by direct time integration and the global secant relaxation-based accelerated constant stiffness iteration technique are presented. It has been proved that the algorithm is efficient and reliable. It is also thought that the accuracy is high due to the adoption of equilibrium iteration. In order to further reduce the computational cost, the differential quadrature finite element method (DQFEM) 1 8 - 2 0 proposed by the author can be used to carry out the finite element discretization. This approach can reduce the arithmetic operations in calculating the static or dynamic incremental equilibrium equations.
3.1. Integration
of dynamic
equilibrium
equation
By assembling the element dynamic equilibrium equation for the time stage t+1 over all elements, the incremental FEM dynamic equilibrium equation can be obtained MrsUts+1 + CrsUts+1 + KlrsAUl+1
= F r t+1 - Qlr
(32)
where MTS is the mass matrix and Crs the damping matrix; K*s and Q* are the tangent stiffness matrix and internal force vector, respectively, updated at the end of time stage t F^+l is the external force vector, f/*+1 the acceleration vector and Ul+l the velocity vector at the end of time stage t + 1; AUl+1 the incremental displacement vector of time stage t + 1. Considering the direct integration method, the following three equations represent an implicit scheme 20 : (1 - aM)MrsUl+1
+ aMMrsUl
+ (1 - ac)CrsUts+l
+ acU\ + (1 -
= Fl+l - Q\
aK)KtrsAUts+1 (33)
+1
Ut
+1
= Ul + (1 - 7)AtC7* + jAtUl
Ul+l = Ul + At£7s' + (A - 0) At2Ul + /?At 2 £/* +1
(34) (35)
where CUM, OLQ, OIK, P and 7 are five constants and At the time increment from time step t to time step t + 1. The above two equations give
t/f+i _ _A AE/*+1 - AtUl ~(\-p) pAt2 jjt+i
=
At2C/s*
2 _J_ pAt AUr - (l - f ) Atl>j - Q - t) At ^'
(36) (37)
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The substitution of (36) and (37) into (33) leads to the following incremental equivalent stiffness equation for determining the incremental displacement vector Kl AUt+1 =
AFt+1
Kls = (1 + aK)Kts + l~~^Mrs + (1 -
(38)
ac^Cr
(39)
is the equivalent stiffness matrix, and 1 - aM P + At (1 - ac)(j (7 - (3 - aci)Ut +
AFrt+1=Frf+1-Q*+Mrs + Cr
a
j i
(l-aM)(l-20)tTt - 2/3)
AtUl
(40)
the equivalent incremental load vector. If etc = <*K it represents the generalised-a trapezoidal scheme.21 If Q M = 0 and ax ^ 0, it results in the Hilber-HughesTaylor scheme. 22 If a j j ^ 0 and a^ = 0, it leads to the Wood-Bossak-Zienkiewicz scheme. 23 If QM = 0 and ax = 0, it reduces to the Newmark @ scheme.24 For the Newmark (3 scheme, if (3 = \ and 7 = | , it results in the constant acceleration scheme while (3 = | and 7 = | leads to the linear acceleration scheme. An alternative efficient time integration scheme is the generalized-^ mid-point scheme. 25 3.2. Equilibrium
iteration
A generic non-linear equation system can be solved by the non-linear iteration. The generalized non-linear iteration techniques such as the non-linear Jacobi, GaussSeidel, SOR, Peaceman-Rachford iterations, etc. are typical iteration methods. The generalized-linearized methods such as the Newton, secant and Steffensen iterations are simplified non-linear iterative methods. The non-linear iteration can be carried out by combining a generalized-linearized method with a certain linear iterative method. By adopting a linear iterative procedure as the primary iteration and a generalized-linearized method as the secondary iteration, in the non-linear iteration, it results in a linear-nonlinear iteration scheme. The Jacobi-, Gauss-Seidel-, SOR- and Peaceman-Rachford-Newton, secant and Steffensen iterations are typical linear-nonlinear iteration schemes. By reversing the roles of the linear iterative method and the generalizedlinearized method it leads to the composite nonlinear-linear iteration procedure with the generalized-linearized method as the primary iteration and the linear iterative method as the secondary iteration. The Newton-, secant- and Steffesen-Jacobi, Gauss-Seidel, SOR, and Peaceman-Rachford iterations are typical nonlinear-linear iteration schemes. The composite nonlinear-linear iteration can be generalized to solve the generalized-linearized system not only by the linear iterations but also by certain other solvers such as the direct methods. The quasi-Newton methods, modified Newton-Raphson methods and accelerated modified Newton-Raphson methods are also generalized-linearized iteration schemes.26
Optimization in FE and DQE Techniques
3.3. Accelerated
modified Newton-Raphson
189
methods
In using the Newton-Raphson method to the equilibrium iteration, the stiffness matrix Krs and equivalent residual force vector have to be updated for each iteration step. The equivalent stiffness matrix can thus be constructed. Then the equivalent incremental stiffness equation for the iteration step n + 1 of a specific time stage o can be expressed by K?'nAU°
= R°'n
(41)
n+1
where K°'s is the equivalent stiffness matrix, AU°' the incremental displacement vector and R°'n the equivalent residual force vector. In using the standard modified Newton-Raphson scheme to the equilibrium iteration, only the stiffness matrix of the first iteration step, for each time stage, is necessary to be updated. The equivalent stiffness matrix can thus be constructed. Letting K°s denote the equivalent stiffness matrix of the first step of time stage o. Then the equivalent incremental stiffness equation can be expressed by k°sAU°'n+1
= R°'n.
(42)
In applying (42) to the equilibrium iteration, K°s can also be replaced by an equivalent stiffness matrix formed at a certain other iteration response stage in the incremental/iterative integration solution. The convergence of modified Newton-Raphson methods can be improved by using certain procedures to adjust the incremental response vector obtained by the modified Newton-Raphson prediction at each iteration step. The accelerated modified Newton-Raphson schemes using the GSR procedure are summarized. Through the introduction of an implicit equivalent secant stiffness matrix, denoted as K°'sn, the incremental displacement vector obtained by solving (42) can be used to construct a secant relation for hardening or softening response behaviours. This secant relation is shown to have the following form: R°'n = R°'n ± K°?AU°'n+l
(43)
n
in which R°' is an equivalent residual force vector after the modified NewtonRaphson prediction. The GSR method uses an accelerator defined by minimizing a certain defined system error to scale the incremental displacement vector. Let oj°'n+1 denote this accelerator, the updated displacement vector after acceleration can be expressed as [fo.n+1
=
Vo,n
+ uo,n+lAuo,n+l
(44)
and an equivalent residual force vector after acceleration can be expressed by the following form: R°'n+1 = R°'n ± u°'n+1K?snAU°'n+1.
(45)
This equivalent residual force vector provides important information for defining the system error.
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By minimizing t h e Euclidean norm of R°°'™+1, an accelerator denoted as uiZ' can be obtained which shows t o have t h e following form: o,n+l
f>o,n( fyo.n tir K"r
=
f
n
(R°s'
n
j}o,n\ r )
K
n
- R°s> )(R°s' - R°s'n)
( 4 6 )
'
T h e related acceleration scheme is GSR-MR. By using a diagonal matrix with all diagonal elements having the same value to replace t h e implicit equivalent secant stiffness matrix, (46) will result in t h e formula representing t h e Generalized Aitken accelerator proposed by Cahill. 2 7 It is worth mentioning t h a t no mathematical approximation is involved in deriving t h e accelerator u)Z'n+ . Therefore, in considering t h e exact description of t h e secant relation of (43) and t h e direction of the incremental response vector, t h e resulting iterative procedure is believed to be a highly consistent secant improvement-based iteration scheme. T h e energy norm defined as t h e inner product of residual force vector R°'n+1 and the incremental displacement vector caused by R°'n+1, along t h e linear deformation surface is also used t o evaluate the system error. This error can be expressed as Eo,n+l
=
^o,n+l^o,nyl^o,n+l
(4?)
By using t h e same procedures as those used in defining w j ' n + , another accelerator denoted as u>°'n+1 can be obtained: o,«+i
A r r o , n + l f>o:n
L =
±!±L
£V
AU°'n+1(R°'n-R°'n)' n+1
/48N
'
T h e improvement scheme using tu°' as t h e accelerator is G S R - M W . It should be noted t h a t the accelerator w ° , n + 1 is shown to have t h e same form as the single-parameter accelerator proposed by Crisfield 28 t h o u g h t h e fundamental concepts of acceleration, t h e mathematical formulations and t h e resulting iterative algorithms are actually different. In investigating t h e formulation procedures of these two accelerators, it is believed t h a t this similarity is caused by t h e fact t h a t both of these two approaches use the secant relation t o approximately describe the deformation behaviours of t h e two states used to construct t h e secant relation. It is also valuable t o investigate t h e difference of convergence performances between G S R - M R and G S R - M W theoretically. As already mentioned previously, t h e mathematical formulation of defining w°jn+1 is absolutely consistent. Therefore, t h e resulting iterative scheme is believed t o be highly reliable and t h e convergence performance should be good. On the other hand, inconsistency does exist in the formulation procedure in defining w°'™+1 since a mathematical approximation of using K°'sn to predict t h e incremental response vector is used t o construct t h e energy norm-based system error. This approximation will result in obtaining a less reliable evaluation of t h e system error, which will lead to a less reliable acceleration scheme with less promising convergence performance. And t h e inconsistency will be even
Optimization
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Techniques
191
severe for solving non-linear finite element problems if large time increments are used. The accelerated constant stiffness iteration is one of the accelerated modified Newton-Raphson methods. This scheme uses the linear elastic stiffness matrix to predict the initial incremental displacement vector for all incremental/iterative steps. The linear equation systems existing in the generalized-linearized FEM nonlinear iterations can be solved by using a certain direct or iterative solver. The most commonly used direct solvers are Gauss elimination, Cholesky decomposition and frontal method. Various techniques including the sparse implementation strategies, the domain decomposition and the parallel implementation was considered in implementing an efficient direct solution procedure into a FEM computer program. There are also many iterative solvers that can be used to solve a linear equation system. Among the indirect solvers the preconditioned conjugate gradient (PCG) methods have been attracting lots of the finite element programmers. In solving large linear equation systems, the PCG methods can offer promising performances due to the substantial reductions in computer memory requirements and the function of taking the advantage of vector and parallel processing strategies on computers that support these features. The iterative solvers possess a relatively high degree of natural concurrency, with the predominant operations in PCG algorithms being saxpy operations, inner products and matrix-vector multiplications. Among the PCG algorithms, the stabilized and accelerated version of the biconjugate gradient method, which is an extension of the conjugate gradient method to non-symmetric systems, is one of the most commonly used iterative solvers. The element-by-element solution procedure is also a useful algorithm which has considerable operation count and I/O advantages since no overall equivalent stiffness matrix is needed to be formed.
3.4. Explicit
predictor-corrector
iteration
The author has also proposed a diagonal stiffness-based predictor-corrector procedure for iteratively solving linear or non-linear finite element equation systems. 29 It is an explicit iteration procedure in the non-linear iteration. Instead of using an assembled overall equivalent stiffness matrix, this method only uses the diagonal elements of the overall equivalent stiffness matrix to predict the incremental displacement vector in carrying out the iterative solution. Consequently, only the diagonal elements of the element equivalent stiffness matrices are needed to be calculated. Thus the computer memory requirement can be minimized. Let K°'n denote the vector representing the set of the diagonal elements in the equivalent stiffness matrix K°'sn+1. Then, by using K°'n and referring (41), the following equation can be constructed. K°
= R°
(49)
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Chen
(49) and (44) represent the explicit predictor-corrector equilibrium iteration procedure. The discrete equation system can also be solved without updating the vector representing the set of the equivalent diagonal stiffness elements for each iteration step. Then, by modifying (42), the following equation can be constructed. k°{s)MJ°s'n+1
= R°'n
(50)
(50) and (44) represent a different version of the explicit predictor-corrector equilibrium iteration procedure. For solving a linear equation system such as the incremental equation of the step-by-step procedure, which might adopt the unbalance load correction instead of adopting the equilibrium iteration, the overall equivalent stiffness can be formed and used to calculate the equivalent residual force vectors required for the predictorcorrector iteration. However, it needs to save the overall equivalent stiffness matrix in the computer memory unit. Let AFS" denote the equivalent incremental load vector. The incremental equation of the step-by-step procedure for time stage o is expressed by K°SMJ°
= AFr°.
(51)
Then the equivalent residual force vectors used to define the scaling factor can be calculated by n n
R°' = AFr° - K°rs Y, At/,0'",
(52)
fc=l
R°
(53)
The predictor-corrector iterative procedure needs less computer storage space. It is also suitable for vector and parallel implementation. In applying this iterative procedure to solve a generic equation system, the amplification of longer period errors can be prevented by the introduction of a GSR correction which contributes to greater numerical stability. Thus all longer and shorter period errors can be effectively eliminated by this predictor-corrector solution. Numerical examples have proved that this iterative procedure has good numerical stability. It is also an efficient algorithm. This iterative procedure can also be used to the multi-grid solution in which the longer period errors, which may not be efficiently eliminated by the fine grid iteration, can be substantially reduced by the use of a coarse grid correction. The coarse grid can adopt either iterative or direct procedure. In implementing the FEM analysis program, various phases including preprocessing, calculation of elemental discrete equations, incorporation of boundary conditions, solution of system equations and postprocessing can be parallelized. However, the assembly of elemental discrete equations cannot take the advantage of parallel operation efficiently.
Optimization
3.5. FEM
in FE and DQE
193
Techniques
discretization
The total Lagrangian formulation is used for carrying out the finite element discretization. For load stage o, let u°'n(t,Xh), e°£(t,Xh) and a°kf{t,Xh) denote the updated displacement, strain and stress of iteration step n, respectively, in a finite element domain fle. Also let Au°'n+1(t,xh), Ae°£+1(t,xh) and Aa°k^+l{t,xh) denote the incremental displacement, strain and stress of iteration step n+1, respectively. Then the updated displacement u°'n+ (t,Xh), strain e°£+ (t,Xh) and stress n+1(t,xh)
= u°>n(t,xh) +
Au°>n+\t,xh),
e*r +1 (*.*fc) = C ( * . * f t ) + ^e^+1(t,xh), +
(54) +1
T°kP %xh) = *°k?(t,xh) + A*°£ (t,xh). In the above equation, the incremental strain can be expressed by Ae°£+1
= | ( A < > r l + Au-71
+ < > J f + 1 +
A^T+1=^mA<m"+1.
(56)
Let f° denote the body force of load stage o. Also let p and \i denote the mass density and damping coefficient, respectively. The dynamic equilibrium condition after updating the incremental information of iteration step n + 1 can be expressed as the following equation: • •o,n+l
.o.ro+1 , r o . n + l / r
,
o,n+l\i
ro
/m\
-put -M"i +Wk) (SH+ui'j )\,k = -fi(57) + Also let nk and T° denote the outward unit normal vector and traction force, respectively, on the element boundary. The traction condition is expressed by
TiO.71+1 i
O.n+1 / c
=nk<7kj
.
O.Tl + l \
(S*j+Ui,j
)•
Using the finite element interpolation function, denoted as ^a(xh), the incremental and updated displacements in a finite element can be discretized as follows: Au°'n+1(t,xh)
= 90(xh)Au°^+1(t),
u°'n+1(t,xh)
= 90(xh)u°^+1(t).
(58)
A Galerkin procedure can be constructed by using the dynamic equilibrium condition of (57) and the interpolation function ^fa(xh) which results in obtaining the following discretized equation:
{{-Pu°'n+l - K * + 1 + KP+1(Sa + *°) = ( - / ? , ¥ a ) - ((T°'n+1 - nka°£+\5ti
+ u^+1)),
*a>.
(59)
In the above equation (., .) and (., .) represent the integrations over the element and element boundary, respectively. Integrating by parts for (59) and considering
194
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Chen
the traction condition, then introducing (54), (55), (56) and (58) into the resulting equation the dynamic equilibrium equation for iteration step n + 1 of time stage o for a finite element can be obtained which is expressed as the following form: m
"0,n+l - „
ma0ui0
• o,n+l
+ ca0ui0
, i o.n
\
o,n+l
+k^h0Auh'0
where map = (p^a, *#) is the mass matrix, cap = (^a,
o,n
=r
/rn\
(60)
*/?) the damping matrix,
Kl0 = K?*f>,M, **,*) + ( ^ m ( % + u?j)(5lh + <;)* /3 , m , *Qlfc) the tangent stiffness matrix updated at the end of iteration step n, and " KTiSij + < ; ) , *a,*)
(6i) (62)
a force vector obtained by subtracting the internal force vector updated at end of iteration step n from the force vector due to body force and traction force. The damping can also be the Rayleigh or proportional damping with the damping matrix being the combination of a fraction of the stiffness matrix with a fraction of the mass matrix for certain structural problems. It should be mentioned that the concept of weak formulation is considered when assemble the incremental element stiffness equations into the overall incremental stiffness equation. It should also be mentioned that the residual force vector can be expressed by the incremental response quantities and external forces. For the elastic-plastic analysis a proper returning scheme must be used to set the final stress state on the yield surface if the resulting stress state defined by the elastic trial stress lies outside of the elastic region enclosed by the yield surface. 3.6. Numerical
results
The first problem is related to the predictor-corrector iteration of a static non-linear FEM system with geometric non-linearity. Two-dimensional problem was selected for the analysis. Dimensionless parameters were used to describe the geometry, material properties of plates, and external forces applied. A square plate, shown in Fig. 8, with side length and thickness being equal to 8 and 1, respectively, was considered. The Young's modulus of the material is 7000 and the Poisson's ratio is 0.33. Only concentrated point loads were applied to the structure. The arrow symbol is used to represent a load unit applied at the specified point in the arrow direction. The bilinear element was used to model the square plate and the ratio of residual norm to the norm of load vector was selected to be the indicator of convergence. Three analyses using pure diagonal stiffness iteration, GSR-MR-based and GSRMW-based predictor-corrector iterations were carried out. A certain high frequency loading condition was applied which could be seen in Fig. 8. In order to know the influence of degree of non-linearity to the convergence behavior of this model problem, three analyses using different load units were carried out. The resulting convergence informations were plotted in Figs. 9-11. From these figures, it is known that for the GSR-based predictor-corrector iteration method, no clear tendency
Optimization
in FE and DQE
Techniques
195
J, , '/.
\
V, /////////////////////////
Fig. 8.
A fixed plate subjected to concentrated inplane forces.
, Jacobi
'V\,
GSR-MW
-GSR-MR load unit = 10000. update diagonal stiffness [
I
I
I
I
I
I
I
1
5
10
15
20
25
30
35
ITERATION Fig. 9.
--_ I
1
I
1
40
45
50
55
NUMBER
Convergence of load 1.
^
60
C.-N. Chen
196
.Jacobi
1 0 -1
v«
^GSR-MW *" A
O U 03
-3
\ \ \
X
^—v__
\
cc
-4
en
-5
N_
\ \
/GSR-MR
-
o
-6 -7 _
-
\ \ \ \ load unit = 50000. \_ \ update diagonal stiffness 1
1 5
1 10
i
15 20 25 30 35 ITERATION NUMBER
i \
40
45
50
Fig. 10. Convergence of load 2.
of the influence of degree of non-linearity to the convergence rate can be seen. However, the convergence of pure diagonal stiffness iteration is largely dependent on the degree of non-linearity of the non-linear system analyzed. It also shows that the predictor-corrector iteration method has excellent numerical stability and convergence performance. Moreover, it shows that the GSR-MR corrector always performs the best. The second sample problem solved involves the static elastic-plastic analysis of a square plate with a square cutout, subjected to uniformly distributed axial load. The deformation is assumed to be linear. The description of the model problem can be seen in Fig. 12 in which the thickness of the plate is 1 mm. Elastic-perfectly plastic material with Young's modulus E being equal to 21,000kg/mm 2 , yield stress uy = 36.3 kg/mm 2 and Poisson's ratio v = 0.3, was considered. The modelling considers the symmetry property. Eight-node quadratic element adopting 3 x 3 quadrature rule was used to discretize one quarter of the plate domain. The mesh is shown in Fig. 13. The accelerated constant stiffness iteration was used to efficiently and reliably carry out the iterative computation in which Tresca's yield criterion was used to detect the plastification of integration points. The constant stiffness matrix was formed at the beginning of the incremental/iterative solution.
Optimization
in FE and DQE
197
Techniques
Jacobi
Vwwwwwwwwwwwvw
load unit = 100000. update diagonal stiffness -121
'
'
i
i
i
i
_l_
5
10
15
20
25
30
35
ITERATION Fig. 11.
J 40
I 45
L 50
55
NUMBER
Convergence of load 3.
The convergence indicator is defined as the ratio of residual norm to the norm of load vector. For accurate consideration, the value of convergence indicator for terminating the iterative computation is selected to be 10~ 6 . The response history was updated up to a near collapse load stage under which the structure showed globally unstable behavior due to the entire loss of elastic restoring capability caused by the formation of an unstable failure mechanism. It should be mentioned that the selective acceleration approach of which only when the calculated accelerator is larger than one the acceleration procedure is considered can also be adopted. The load-displacement curve with the displacement at point A for the plate model with a/W = 0.5 is shown in Fig. 14 in which the predicted value of the collapse load is 0.38ery. Convergency tests were carried out by using a single larger load increment of 0.3<7y to perform the equilibrium iteration. The convergence indicator is defined
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Chen
P i A A k A k k k ,
[tttttttt A
1 I
a
•*
*
u \
Fig. 12.
a
>•
w
\
\
\ \' 1 1 ? Y Y
'
I Y1 YJ '
A square plate with a square cutout subjected to a uniformly distributed load.
as the ratio of residual norm to the norm of load vector. The convergency curves of three different iteration algorithms were plotted. They can be seen in Fig. 15. The second problem been solved involves the elastic-plastic postbuckling analysis of a rectangular girder framework which can be seen in Fig. 16. Different structural models, which possess different plastic buckling behaviors, can be obtained by changing the material constants for horizontal or vertical members. Two dimensional plane stress theory was used to describe the mechanics behavior of the structure. Due to symmetry, one quarter of the structural domain was adopted for the finite element analysis. Eight-node quadratic finite element model adopting the standard 3 x 3 quadrature rule was used to discretize the analyzing domain. The finite element mesh is shown in Fig. 17. A uniformly distributed forced displacement is applied downward on line A-B. In detecting the plastification of integration points, Tresca's yield criterion was used. And the incremental/iterative procedure using the GSR-based accelerated constant stiffness iteration scheme was used to update the response history. Well converged results were used to study the plastic buckling behavior of the structure. The value of convergence indicator for terminating the iterative computation is selected to be 10~ 6 . Three model problems were analyzed. The results of response axial forces of the vertical members were plotted. They can be seen in Fig. 18. It shows that the model 1 problem, in which both horizontal and vertical members are elastic-perfectly plastic materials with significant difference of
Optimization
in FE and DQE
Fig. 13.
Techniques
199
The mesh.
6 ( x l O mm) Fig. 14.
Load-displacement curve of point A for a/W = 0.5.
material stiffnesses, possess a rather bifurcation type buckling behavior with which the buckling stress is very close to the yield stress. By using the same material properties for all structural members, a model 2 problem can be obtained. This model 2 problem shows limit point buckling behavior with which the buckling stress just
C.-N.
200
Chen
• constant stiffness • ui£ accelerating o w e accelerating
8o ¥ ' . o o • o = -2 on CM
-
3
o i-H
0
4
8
12
16
20
24
28
32
36
40
ITERATION NUMBER Fig. 15.
T h e convergence of equilibrium iteration. (Load increment A p 1 = 0.3
below the yield level. The results reflect the effects of two dimensional mechanics behavior to the two different buckling models. The two dimensional effects are even significant for model 2 problem. By considering that the structural material has a constant strain hardening, a model 3 problem can be obtained. The results of plastic buckling analysis for this model problem are also shown in Fig. 18. The last problem solved is the dynamic elastic-plastic analysis of a spherical cap subjected to a uniformly distributed step pressure with the effect of non-linear deformation considered. The problem is shown in Fig. 19 in which the geometrical quantities are: h = 0.104 x 10 2 mm, R = 0.566 x 10 3 mm and (3 = 26.7°, and the pressure p = 0.422kg/mm 2 . Eight-node axisymmetric finite element was used to discretize the model problem, which released certain deformation constraints imposed in developing classical shell theory. Twenty elements were used to model the cap. The standard quadrature rule was also used for carrying out the numerical integration. The material was assumed to have a constant strain hardening rate with material constants: Young's modulus E = 0.739 x 104 kg/mm 2 , Poisson's ratio v = 0.3, yield stress ay = 0.169 x 102 kg/mm 2 , strain hardening rate H' = 0.148 x 103 kg/mm 2 and mass density p = 0.262 x 10" 5 kg/mm 3 . The von Mises yield criterion was used for the yielding detection. The damping effect was neglected. The Newmark f3 scheme of constant acceleration with time increment At = 0.51 x 10~ 5 sec was used for the time integration. The accelerated constant
Optimization
- iA b=5mm
in FE and DQB
Techniques
201
u — /irirrm
*b*
B 1
i
t i
Thickness: b Material constants of vertical members — Young's modulus: E^ Poisson's ratio: v-^ Strain hardening rate: H[ Yielding stress: dyi Material constants of horizontal members — Young's modulus: E 2 Poisson' ratio: v 2 Strain hardening rate: Hj Yielding stress: a^2
L=50rrm
•
i
Fig. 16.
A rectangular girder framework.
stiffness iteration using GSR-MR was used for the equilibrium iteration. The convergence indicator is denned as the ratio of the norm of effective residual to the norm of boundary force vector. The value of convergence indicator for terminating the equilibrium iteration was selected to be 1 0 - 5 . The analyses were carried out for linear elastic model and non-linear elastic-plastic model, separately. Numerical results of the deflection at A were plotted in Fig. 20. It shows that after the yield occurs the restoring capability of the cap will decrease due to the loss of a certain ratio of the structural stiffness.
4. D Q E M for Solving Beam Problems The solution of non-prismatic beam problems is important since a lot of engineering structures are formed by non-prismatic beam members. A rather efficient method that can be used to develop solution algorithms for non-prismatic thin-walled beam problems is the DQEM. DQ approximates a partial derivative of a variable function
C.-N. Chen
202 A
E C
B
F D Fig. 17.
The mesh.
with respect to a coordinate at a node as a weighted linear sum of the function values at all nodes along that coordinate direction. 30 A generalized differential quadrature method has also been proposed by Shu and Richards. 31 In this method, a recurrence relationship is obtained by using Lagrange polynomials for calculating the weighting coefficients for the DQ discretization of any order derivatives. This method makes no difference with one fashion of the original DQ which uses polynomials or Lagrange polynomials to calculate the DQ weighting coefficients. The weighting coefficients calculated by these two methods are identical. Because only problems having simple regular domains and under simple external environment can be solved by using the DQ, the application of this method is very limited. The DQ has been generalized which leads to the GDQ. 32 ' 33 The weighting coefficients for a grid model defined by a coordinate system having arbitrary dimension can also be generated. The configuration of a grid model can be arbitrary. In the GDQ, a certain order derivative or partial derivative of the variable function with respect to the coordinate variables at a node is expressed as the weighted linear sum of the values of function and/or its possible derivatives or partial derivatives at all nodes.
Optimization
in FE and DQE
203
Techniques
X"
D
model 1
&
model 3
0)
u u
model 2
o
X
<
: £2 = 21000. k g / m m S v 1 = . 3 , Hj=0., ayi=82.4 kg/mm 2
01 d)
E 2 =21.kg/mm 2 ' V2=- 3 H 2 =0., a Y 2 =100.kg/mm 2 : E 1 =E2=21000.kg/mm 2 ,v 1 =v 2 =.3
C
o •H 0)
c
0)
e
^=H 2 =0.,
OYJ=OY2=82.4kg/mm2
: E 1 = E 2 = 21000.kg/mm 2 , v1 = v 2 =.3 H-i =H ? = 525kg/mm 2 , Oy, =0 Y ? = 8 2 . 4 kg/mm 2
I
0. 0.
.2
.4
.6
L__ I .8
^
1.
Dimensionless Forced Displacement 10 6/L Fig. 18.
Response history curves of the vertical members.
The DQ and GDQ have been extended which results in the EDQ. 3 4 , 3 5 In the EDQ discretization, the number of total degrees of freedom attached to the nodes are the same as the number of total discrete fundamental relations required for solving the problem. A discrete fundamental relation can be defined at a point which is not a node. The points for defining fundamental relations are discrete points. A node can also be a discrete point. Then a certain order derivative or partial derivative, of the variable function existing in a fundamental relation, with respect to the coordinate variables at an arbitrary discrete point can be expressed as the weighted linear sum of the values of function and/or its possible derivatives or partial derivatives at all nodes. Thus in solving a problem, a discrete fundamental relation can be defined at a discrete point which is not a node. If a point used for defining discrete fundamental relations is also a node, it is not necessary that the number of discrete fundamental relations at that node equals the number of degrees of freedom attached to it. This concept has been used to construct the discrete inter-element transition conditions and boundary conditions in the differential quadrature element analyses of beam bending problem and warping torsion bar problem. The author has proposed the DQEM. 36 Like the finite element method, in this method the domain of a problem is separated into a certain number of subdomains
204
C.-N.
Fig. 19.
Chen
A spherical cap subjected to a step pressure.
or elements. The DQ discretization is carried out on an element-basis. The governing differential or partial differential equations defined on the elements, the transition conditions on inter-element boundaries, and the boundary conditions on the boundary of the problem domain are in computable algebraic forms after the DQ discretization. All discretized governing equations, transition conditions, and boundary conditions are assembled to obtain an overall algebraic equation system. Since all relations governing a continuous problem are satisfied, the essence of this method is to find a rigorous solution numerically. Various techniques for easily defining discrete inter-element boundary conditions and boundary conditions of multidimensional problems have been proposed by the author. 3 7 - 4 0 It would be more flexible in treating the boundary conditions or transition conditions defined on the inter-element boundaries of two adjacent elements when they are applied to the DQEM. For analyzing bar, truss or two-dimensional potential problems, 36 ' 37 ' 40 no degree of freedom of the partial derivatives of the variable function can be assigned to the discrete points on the inter-element boundary on which certain external cause such as the concentrated force, fluid flow, conduction heat flux, etc. is applied if EDQ is not adopted. On the other hand, if the EDQ is adopted the DQEM can treat the natural transition condition and the boundary condition by assigning the degrees of freedom of the partial derivatives of the variable function to the element boundary nodes. The discrete governing equations can be defined on the inter-element boundaries as the average discrete governing equations of multiple elements. They can also be defined on the element boundaries
Optimization
,8
in FE and DQE
1.6 -0.25 -0.50
4-
-L—i 1
205
Techniques
3.2
4.0
5 0 A t
2.4.'
I \ :
f*—Linear elastic I analysis
-0.75 -1.00
i
-1.25 -1.50 Fig. 20.
Dynamic response of a spherical cap subjected to a step pressure.
without adopting the average treatment. Thus, elements having no interior nodes can also be used to the DQEM analysis. The philosophy of the proposed techniques for implementing the constraint conditions also hold good for solving beam or plate problems. The mapping technique can be used to develop irregular elements. Therefore this method has the same advantage as the finite element method of consistent boundary condition implementation and geometric flexibility. Hence a generic engineering or scientific problem can be converted into a numerical DQEM algorithm. And the related computer code can be systematically developed.
206
C.-N.
Chen
In generating the mesh, if the external cause is composed of various locally distributed causing functions in order to better approximate the true distribution of the external cause the mesh must be designed in such a way that the external cause in one element will not have two or more locally different distributed functions. The adoption of this adaptive discretization technique will lead to a better solution since that locally different causing functions will lead to locally different response functions. Without adopting this technique it will result in a poor approximation of the element-basis external cause if significantly different causing functions, in quantity or order of distribution, coexist in an element. Moreover, the analysis will be more efficient by adopting this adaptive discretization technique since in subdomains having locally higher order distributed causing functions, higher order elements can be used while in subdomains having locally lower order distributed causing functions, lower order elements can be used. In treating a concentrated external cause existing in the problem domain, the mesh can be designed in such a way that the concentrated external cause is located on some inter-element boundaries and included in the natural transition conditions or kinematic transition conditions. If the external cause is a force related quantity, it can also be located in some element domains and approximated by the composition of certain continuous functions based on the rule of force equivalence. However, the solution will not be able to reflect the locally transition response behavior. There is also a discrete element analysis technique QEM which also adopts the DQ. The original QEM was proposed to solve truss and frame structures. 41 In this method, the truss element is limited to a three-node second-order approximation, while a <5-grid arrangement is used to define the DQ discretization of the flexural deformation. The 5-grid is designed to approximately define certain boundary conditions at a point close to the boundary, and certain inter-element transition conditions at a certain point close to the inter-element boundary. Consequently, the definition of boundary conditions and inter-element boundary conditions is inconsistent. When developing the plane stress and plate bending QEM models, Striz et al. adopted a hybrid technique to incorporate the DQ discretization into a Galerkin finite element formulation and define a discrete element analysis procedure. 42 The theoretical basis of DQEM is rigorous since all fundamental relations are locally satisfied. Consequently, this method has excellent convergence property from the efficiency and reliability points of view. The FEM uses a certainly defined constraint law to constrain a certainly defined element-basis integral quantity and adopts the weak formulation which leads to inexact equilibriums of inter-element boundary forces and natural boundary conditions. Stationary principle or minimization of total potential energy, the principle of virtual work, weighted residual method, . . . , etc. are integral quantity-based constraint laws. It has been proved that the DQEM is efficient and reliable. 36 ' 43,44 In solving the free vibration of a beam on an elastic foundation, results obtained by the FEM cannot converge, while results obtained by the DQEM can converge very well.45 The DQEM has also been used to develop discrete space frame analysis models. 4 6 - 4 8
Optimization
in FE and DQE
Techniques
207
In solving the torsion problem of a solid bar by using extremely distorted twodimensional DQEM element, the results converged consistently by gradually increasing the element nodes from 3 x 3 up to 15 x 15. 39 In solving a two-dimensional potential flow problem, the results also converged consistently by gradually increasing the element nodes from 3 x 3 up to 17 x 17. 37 In solving the plate bending or shell problems, certain FEM elements may not converge well, while the DQEM plate analysis model can converge effectively. Consequently, the DQEM is an excellent numerical method for solving complex structures such as ship structures. The coefficient matrix of the overall algebraic equation system for DQEM analysis model is sparse. For a multidimensional DQEM analysis model, if interior elements and element grids are designed to be regular, not only mapping transformation is not necessary for defining an interior discrete equation but also the sparsest coefficient matrix can be obtained. The coefficient matrix would be much more sparse than that of the FEM. It is a great advantage in minimizing the computer memory and CPU time required.
4 . 1 . One-dimensional
EDQ
In using the EDQ to solve a problem, the number of total degrees of freedom attached to the nodes are the same as the number of total discrete fundamental relations required for solving the problem. A discrete fundamental relation can be defined at a point which is not a node. Then a certain order of derivative or partial derivative, of the variable function existing in a fundamental relation, at an arbitrary point with respect to the coordinate variables can be expressed as the weighted linear sum of the values of variable function and/or its possible derivatives at all nodes. Thus in solving a problem, a discrete fundamental relation can be defined at a point which is not a node. If a point used for defining discrete fundamental relations is also a node, it is not necessary that the number of discrete fundamental relations at that node equals the number of degrees of freedom attached to it. This concept has been used to construct the discrete inter-element transition conditions and boundary conditions in the differential quadrature element analyses of beam bending problem, warping torsion bar problem. Let 4>{C) denote the variable function associated with a one-dimensional problem. The EDQ discretization for a derivative of order m at discrete point a can be expressed by ^ = D i
t
^ ,
i = l,2,...,N
(63)
where N is the number of degrees of freedom and $ g the values of variable function and/or its possible derivatives at the N nodes. The variable function can be a set of appropriate analytical functions denoted by T p (£). The substitution of T p (£) in (63) leads to a linear algebraic system for determining the weighting coefficients D^ .
208
C.-N.
Chen
The variable function can also be approximated by 0(0 = *„«)*„,
P=l,2,...,N
(64)
where 'J'p(C) are the corresponding interpolation functions of $ p . Adopting ^ P (C) as the variable function (£) then substitute it in (63), a linear algebraic system for determining _D„a can be obtained. And the mth order differentiation of (64) at discrete point a also leads to the extended GDQ discretization equation (63) in which D^ is expressed by D^ = ^Z* \a • Using this equation, the weighting coefficients can easily be obtained by simple algebraic calculations. The variable function can also be approximated by 0(C) = T p (C)c p ,
p=l,2,...,JV
(65)
where T p (£) are appropriate analytical functions and cp are unknown coefficients. The constraint conditions at all nodes can be expressed as P = xPPcp, where XPP are composed of the values of T p (£) and/or their possible derivatives at all nodes. Then the variable function can be re-written as (£) = ^p(0xpp $p- Using this equation, the weighting coefficients can also be obtained D ^ = g(.mp \aXip1Various analytical functions such as sine functions, Lagrange polynomials, Chebyshev polynomials, Bernoulli polynomials, Euler polynomials, rational functions, . . . , etc. can be used to define the weighting coefficients. To solve problems having singularity properties, certain singular functions can be used for the EDQ discretization. The problems having infinite domains can also be treated. If only the values of the variable function at the nodes are used to define the EDQ discretization, the following Lagrange polynomials can be the interpolation functions used to define the weighting coefficients MC)=
n
fry--
P
= i,2,...,AT.
(66)
If two degrees of freedom used to represent and d(j)/dQ are assigned to a node, the EDQ can adopt the Hermite polynomials as the interpolation functions to define the weighting coefficients. For this model, the variable function is approximated by
0(o = E ^ ( c ) ^ + E ^ ( o ^ P=i
P=I
^
where tfp(C) and Hp(0
= (C - Cp)£p(C)
are
ydLp{Cp) i-2(C-CP)d(
Hermite polynomials.
L2P(0
(67)
Optimization
in FE and DQE
4.2. Dynamic equilibrium equations equilibrium equations
and
Techniques
209
Buckling
Figure 21 shows the definition of coordinate system. The fixed reference coordinate system on a cross section is xyz with O as the origin. A is a varying arbitrary point. The warping function can be defined by using Saint Venant's torsion theory. If the beam is thin-walled, the Leibnitz sectorial formula can also be used to define the warping function. Let A and 7W denote the cross section area and the first moment of sectorial area, respectively. Also let Co denote the normalized warping function. Then Co can be obtained by the following equation: Co = to - ^f. Consider that the length of the non-prismatic bar is I. By locating the origin of the coordinate system (x, y, z) at the centroid of the cross section and orienting the two axes on the cross section to the principal directions, various section constants can be defined. Then, the coordinates (xs,ys) defining the shear center S can be calculated by using certain section constants. Let us and vs denote the two lateral displacement components at the shear center. Also let w denote the average axial displacement. Assume that the length of the non-prismatic beam is large compared to the dimension of cross section. Thus, the transverse shear deformation is negligible. Since the cross section is rigid in its own plane, the displacements at an arbitrary point A can be
Fig. 21.
Coordinate system of non-prismatic beams.
210
C.-N. Chen
expressed as: U(x,y,z)
=us(z)
-(y-ya)6(z),
V(x,y,z)
dus(z) W(x, y, z) = w(z) — x dz where w(x,y) = cj(x,y) be obtained:
Izy —
dw dy
dz
d6(z)
xs)6(z), (68)
uj x
+ dz —n- ( >y)
— ysx + xsy. Using (68), t h e linear strain components can d2us dz2
dw dz
dvs(z) y—n— dz
= vs(z) + (x -
d2vs y
-
d26 ^ +
^
d6
2
dz~ ^
=
^
+ (x - xs)
Tz
-(y-
dx
ys) (69)
0.
Jxy
T h e stress components of a generic composite beam are expressed by &z =
+ <9347yz>
+ Qiilzy
Tzx = Qz$ez
+ Qlhlxz,
+ Qs5~fzx +
QiSlzy,
(70)
Ox — Oy = Txy = 0
where Q33, Q34, Q35, Q44, Q55 and Q45 are reduced stiffnesses. If t h e averaging stiffness approach is adopted, t h e reduced stiffnesses in (70) can b e replaced by the effective stiffnesses Q33, Q34, Q35, Q44, Q55 and Q45. T h e effective constitutive relations can t h u s b e obtained. T h e dynamic equilibrium equations have been derived by t h e author. 4 9 Let bx(x, y, z, t), by(x, y, z, t) and bz(x, y, z, t) denote t h e distributed forces in t h e structural domain in x, y and z directions, respectively; ix(x, y, t), ty(x, y, t) and tz(x, y, t) denote t h e b o u n d a r y traction forces on t h e two b o u n d a r y sections z = 0 and z = I; Qa/3, Qa0i a n d Qa0ij denote t h e effective stiffnesses with respect t o A, Ii and I j j , respectively; p, pi and pij denote t h e effective mass densities with respect t o A, Ii and Iij, respectively; and xs and ys denote t h e coordinates of shear center. T h e dynamic equilibrium equations are expressed as d2
d2
d2
\ )
2
+
2
'dz
~
d
33xxIxx
I^ T
83 dz d2 7^2 [
\
dt2dz (y.0)
d \ Qttyylyy dz2
K(
J*. dz2
>+
2
'0t dz
d3
+ dz
\
Pyyly
'dt2dz
(xs6)
dz\+pA
H(
d3e Pyujl]
y
qx
~
dz2 d2vs dt2
"dt2dz
o
Vs
d3B \ _ PXUJ*X
Q33yylyy-g^(xs8)
l ^ 3 5 r y - » r y ~r W34sy-*sy J
d29 dt2
_A (d'duus s pAl dt2
d2 ^ 2 1
d2^ [QMXOIXQ dz2
(-
f ~ Q~^2 22
\ d0\ —
d2
\
s
d~^2^y ^
QMSXISX)
pxx*xx
d2
f -
I
PXX*XX
dmy ~~dz~
(71)
Q33yu>lyuj
dH_ "dt2 dmx
qv + ~dY'
d3us dt2dz
„
2
d3vs vy 2 dz ' p y y I - ot dz (72)
211
Optimization in FE and DQE Techniques
^£)-£{(&*/ r + «3*.7.)s}+"£=» (73, d2us\
Q33XQIXO
_
d / / -
) ~*~ 7T~ | ( («35rx-'rx + Q34sx^sx)
Q2
9^2 I W33y<S-'ydi-^2- I + ^
~ 7T I l < * 3 5 r
r +
Qsisls)
+ Q33yu,Iyu^(xs0)>
a r -
o
[
2
j ^35r ! / -'ry + ^34sy^syJ
g^
~ ^
J
[^
-
s
2
~dz) " fa \ (^Sru-fru; + $343*1*0,) 7 ^ 2
-
-
o
s®) ~ (Q35rylry + Q34syIsy)-X-^(xs8)
2
z
d ((d ( d us \ T \d6\ d^\ \9^0,Ir* + Q34saIra) Q-zj+g-z \P^I^Qfid'z) 2 3 3
de
dr
+ Pvh dt2 = m z - ^ .
de_
yoaioo
dz
j
-~~ f + ~fl~2) Q33u>u^iJu)-K-^ ~ Q33xu>^xQ^-^(yst')
+ -X^-S (Q35rxlrx + Q34sxlsx)-^{v +
\ a2us\
=
o ,
1
P** **
m2dz
_
d
j
\ >
d3v: 2 \/Wl dt dz dz 3 a f \
d (_ +
W° ) + Py*Jy* dt2dz (x°u> J (74)
dt2dz
The boundary conditions are expressed as: 32v
d2
d20
-
Q33xxlxx p. 2 + Q33xxIxx-jr-2-\Ds6) — Q33xu>Ixui-jT-^
~
9 (*
[Q35rxlrx + Q34sxhxj
r d2us\
d [ (-
d r-
-g~ = My
T
\ 99}
1 ~X \ I ^s35rx-'rx ' ^c34sx-*sx I Tj
f
Or
5 ( -
d2 , „, d3us
_
' Pxx-^xx o.9£J
^ J = 0,
T
_
(75)
d2e d3
"r~ Pxx*-xx 012a
\Vs^)
Q3Q -
PXQIXQ-~T2~-=
92ws "7T~2
Vx + my
or
d2
<5MS = 0,
(76)
c>2#
-
Q33yylyy-Q—2\Xsv) + Q33yQ^yui-Q~^
(Q35rylry + Q s ^ ^ y ) 7 ^ = - M *
Or
5 ( - ^ j = 0,
(77)
212
C.-N.
Chen
d2v \ d (d2 d_ dz Qttyylyy-Q^ J + -g^{Q33yyIyy-g^(xs0)
d f /+
-
\P35rVIry
9~z\
+
~ d26} + Q33yQlyu>-g^ >
d3vs
\ d 6 \ _
QusyltyJ
-Q~Z | + Pyylyy
d3
_
Qfifc
~ Pyyhv
QfiQz
(Xs
Q3Q
+ Pyujlya dt2dz
=Vy-mx
Q^A-£
or
6vs = 0,
+ (Q35rIr + Q34SIs)
d2us
d2vs
-
(78) or
-Q-Z=P
d26
-
8w = 0, d2
-
(y.0) /B6\ S ( — j = 0,
2
d6 / \ d6 + Qttyulyw-jpz + [Q&ralra + QzisahaJ -jr- = MQ d
T d'uA
(o
(f>
I ~ \^35rx-trx o2 + Q34sylsy) ~ft~Y + (Qttrlr 2
d / -
de
~ -~~S Q33aQluiO-^-^
-
d
+ Q34su;Isui)
~ Pxu,Ixu,g^
7T~2
"T^" + ^ o Q34sls)
dz
r) ( / —
'A I
\ >+
Gj
m_ dz
\ d6 1
~fl~ \ [QzSrujIru
+ Q34su>IsQ ) 7T" f
d39
~ Pyulyutg^
d2
(80)
dv
I Q33yQ^yu,-Q^2
+ Q33yaIyu>-~-^{Xs9)
d3vs
_
T
-
— Q33xLjIxu>-~-2(ys8)
( d3us
+
2
\ d26 Q35ruiIrQ
°+d(h
T\ + ^Msx-lsx)
Q~Z I W33xalxQ-Qg2
or
d2u
T +n
(79)
+ POO^OQ^Q^
d3 -
}
Pxu,Ixu,Q^Qz-(yS0)j
d3 + Pyu,Iya-^g^(xs0)
= Mz+ma
or
86 = 0.
(81)
In (71) to (81), the following definitions involving forces, section constants and St. Venant's torsional rigidity are used. qx(z,t)
=
bxdxdy, A
mx(z,t)
qy(z,t)
=
bydxdy,
A
=
yb2dxdy,
my(z,t)
bzdxdy,
= - / / xbzdxdy,
A
mz(z,t)
p(z,t) =
A
(82)
A
—
(xby -ybx)dxdy,
mUJ(z,t) =
A
wbzdxdy,
Vx(t) =
A
Vyi*) = / / tydxdy,
P(t) = / / tzdxdy,
A
A
My{t) = -
xizdxdy, A
ixdxdy, A
Mx(t) = / /
yizdxdy,
A
Mz(t) =
(xty - ytx)dxdy, A
M^(t) =
u>tzdxdy. A
Optimization
A=
Ixx = / / x2dxdy,
dxdy, A
'xQ=
in FE and DQE
A
/ / xQdxdy,
213
Techniques
Iyy = / / y2dxdy,
IQS) = / / w2dxdy, A
A
IyQ = / / yuidxdy,
-y
Iaa = / / f ^
dxdy,
hb =
JJ {^ + X) dXdV' hb =II ( £ " ") ( S " X) dXdV' ^ =IJ^
A
A
Jl^xdxdy,
A
In, = JJ(jg-v)vdxdy,
9(D
/ / ^dxdy'
Isx =
A
Ir, = / / ( § £ - y) "dxdy,
/ / (^ + 7 l d x d l / '
Isy =
JJ ^ydxdy:
A
/ / I •»
A
h X j Qdxdy,
Gj = Qttaalaa
+ Qiibbhb
+
(83)
IQibablab-
It should also be mentioned that the coordinates of shear center are expressed by J
%s —
Us — "7
J
(84)
•
The buckling equilibrium equations have also been derived. 50 They are expressed as: US\ I V33x*i*x
dz2
^)^
d2 f -
d^\
dz2
A. dz
dQ\ d6 dz)dz)
s s
d2us \
-7-2 I Q33XLJIXQ
J dz] d9 (dMx x dz \ dz
Mxdlyy Iyy dz
d2
Jo
n\\
v
dz
, 0
dMy V dz
d2
°\
(rx
~ d~z~2
My dlxx Ixx dz
d
+
^
(85)
{^y^y^)
r
, 2 ) + ~~f~ 1 ( Q35rxlrx
Q33yui-iyui
d28\
~ d? {Q™** **^)
d^\Q33vyIyyd^{Xse)j
2
/dv . dvss V dz dz
i
1
C
j I l°c35ry-lry < *«c34sy*sy I .
d2 / -
dz2
^
d2 (-
\
{yse)
d^
d8\ s dz J d2
\ Qwyylyy
d2
T
Q33xxIxx
+
dz2 d_ 'pfdus_ dz \ dz d2 A~2 2 dz
+
~d~Z 1 \P35ryIry
+ +
= 0,
(86)
\ d2us \ Q3AsxIi J dz2 J Q^syhyJ -j-£- >
d
d2 f(A r A r \ d w \ J/S r d2° -j—\ \Q35rlr + V34s-'s 1 " J f + J ~ 2 1 Q33QolQQ-p2~
A ~
r d2° t as Q33xalxuj-T^2~{.ysV) 'dz2'
C.-N.
214
+
Chen
d. i(*. G j d£\_± — I —— <
Q33yoIyui^^(x 2 s8)
'dz '
dzj 2
d + ~T~\ (Q35rxlrx
+ Q34sxlsx)-j—^(ys^)
d2
)
f Q35ru>IrQ +
dz -
d
du.
d_ dz
XX HZ
>d$\
dv.
P s
^ -df-X°!z"+r;d-z)\
Mx dly yy
M^diif
( dMQ \ dz
Mydlxx
(xs&)
+ Q34syIsy)-j-2
d2i d2v. Mx dl-uv dus M,, L yy y dl wiXxxx dv + M + M, X v + Iyy dz dz + dz2 dz2 dz dz dMx dO d_ (ayMx - axMy + a^M^ dz dz dz dz
dMy dz
' dz2 j
2
-
~ (Q35rylry
d&
QZASQISQ) SUJ
T[S)
dz
0.
(87)
The boundary conditions are expressed as: Qz3xxLxx
d2u* A 2
d2
-
' ^33ix-»xx i 2 ^
s
dO ~ \Q35rxIrx
,
Q33 I 1 K°i33xx±xx
+ Q34sxlsx)
^ J — -j-\ 1 2
~T~
d2e f
W33xw-*xu; , 2
0
or 5 ( —— dz
Q33xxIxx-T^2~{ysQ)
—
d f f~ A T \ dO) ^/dus T [Q35rxlrx + Q34sxhx J "7" f + P {~~T~ dz MX dlyy dM, 0 or 5u, T dz dz
(88)
Q33XQIXU,-J-^
d0\
l-<
+ V* J ~ J
_
M
*
d9 dz (89)
yy
Q33yylyy -
d
,
[Q3brylry
(A T dz (^yytyy^T
,
+ Q34syhy)
d2y
2
\Xs")
+ Q33yu-lyu
~j- = 0
or
d
A
+ -J- \ [Qzbrylry
dMy dz
Q33yylyy
2
\f\ T & , + Tz\Q33yyIyyd^{Xs9) +
My dlxx XX HZ
Q34sylsy)
dz
2
5 f —— J = 0,
a\ dvs dz
= 0 or Svs = 0,
^
+
/S
Q
T
^ d6\
a I
(90)
d2
° ^ ^ „ dd
X
Uz)-MyTz (91)
Optimization
d2us
-
d2vs
-
+ QMy*Iy*-^
+ (QsSralru,
d2us\
d (h
in FE and DQE
-j- I V33xw-'xcZi ,
— [Qzhrylry
+ Q34su,Isd)
,
+ {Qsbrlr
2
J "J-J — J
+ Qzisuilsui
dz+rsdz)
Mydlyy\ Iyy dz J
,
(92)
d2vs\
al a
W~ I ^^y V ~TT
2
+ Q33yu>Iyu:-T-^(Xsd)
)
+ Qz4su)Isui)
>+
^JJ^
"j~ f
My
dz
dMx + ay ' \ dz (dMQ \dz
6 ( — J = 0,
Qz4sls)-j-
dz
u
OF
d (-
] ( Q^rGiIru
Mx
+ 2(ayMx - a x M y + a0M0)— 'dMy dz
-
d2
= Ma
+
~~ Q33xolxu>-pj;(ys0)
215
-
\d2us
-
Xs
\ dz
^
j ~ I V35rx^rx + WZAsxlsxj
2
~ ~T\ QwOuiIuiQ-pi
+ P ys
d29
-
/-
+ Qzisyhy)
+ [QzSruiIrQ
Techniques
Mx dl. Iyy dz
MQdI^ j{.s) dz
0 = 0 or 60 = 0. (93)
In (85) to (93), the following definition equations are also used: ax
*xxx
i ^xyy
27
*xxy
'
y
'
c
^,±xx *ujxx i *ujyy rW 2P--
2
' ^yyy
21 yy -*xx > Iyy
A
1
*X
Z1
"" 2
Ixyy = 11 xy dxdy, A
Ixxy = 11 x ydxdy,
IQJXX
=
y3dxdy,
Iyyy =
A 2
A
Iuyy = / / uiy2dxdy,
= / / x3dxdy,
A 2
ujx dxdy,
A
r2=r2+x2+y2,
I{^ = IQQ - (y2sIxx + x2sIyy).
A
The dynamic equilibrium equations of the composite beam considering the effects of shear deformations have also been derived. 51 The buckling equilibrium equations of the shear-deformable beam can straightly be written since only the mathematical terms related to elastic restoring forces are different from those of the shear-undeformable beam. 4.3. DQEM
static problem
analysis
Considering that the beam is isotropic and homogeneous with the Young's modulus E and the shear modulus G, and neglecting all inertia forces related terms, (71) to
216
C.-N. Chen
(74) lead to the following static equilibrium equations denned by the element-basis local coordinate variables x, y and z, and displacement parameters us, vs, w and 0 2 : „ d2 /
rf2M«\
drriy
,
E
94
d?{'"^-)""-^f'
s
<>
(75) to (81) lead to the following static boundary conditions: EIss^f
= Mv
(t)-*
or « 5 ( ^ ) = 0 ,
- £ — U s a - ^ - J = Vs+m5 E l
n
%
= -M-X
or <5us = 0,
(99)
or * ( § ) = < > ,
d ( d2vs\ -E-p[lyy-~\=Vy-ms EA^
08)
(100)
or 6vs=0,
(101)
=P or 5w = 0,
(102)
In (94) to (97) and (98) to (104), the following relations are used •is = -Ip ~r A(XS + ys), mis) =m,z +(ysqs sJ
- xsqy),
_ MiT/f + _i_ (ySVx /„-; T7. _ x ~ VT7_\ Mr(^ ) = z s v),
-IQQ = IQQ ~ Vs-'xx ~~ xs*yyi m%' = mQ + (xsms + ysmy), A!T(*) = M + (x M M^> a s s +
(105)
ysMv).
4.3.1. DQEM formulation The fundamental relations are defined by the physical coordinate system while the EDQ discretization is carried out on the natural coordinate system. Therefore, in using the EDQ technique to discretize the fundamental relations the transformation operations of coordinates and derivatives of displacements, between two different coordinate systems, have to be carried out. Let z\ and z^ e denote the global coordinates of node 1 and node Ne, which are two end nodes, respectively. The element
Optimization
in FE and DQE
217
Techniques
length le equals ze e — z\. Let ze be the coordinate variable of the local coordinate system with the origin located at node 1 of the element. Also let the range of the natural coordinate £ be 0 < C < 1- Then the coordinate transformation is expressed as ze = le£.
(106)
e
Using (106), the differential of z can be expressed as: dze = ledC
(107) e
e
Then the mth order derivative of the variable function (f> with respect to z can be written as dmche 1 dm
C.-N.
218
Chen
used to define the axial discretization, the nodes used to define the flexural discretizations and the nodes used to define the torsional discretization can be different. Let N^, iV|^, Ng^ and N% denote the numbers of nodes for defining the axial discretization, the flexural discretization in xe direction, the flexural discretization in ye direction and the torsional discretization, respectively; JV^, Ng , NB and TVf. denote the corresponding element degrees of freedom; and D^yk, D£ai, D c ^ and Dgsl denote the corresponding weighting coefficients. The number of discrete points need to be used to discretize equilibrium equations defined by the element displacements, w, us, vs and 9Z are N\ — 2, N% — 4, JVj, — 4 and N% - 4, respectively. Using (63) and (108), the discrete equation of (94) at a discrete point a in element e can be expressed as
(Pi:xx(a)
He)4
d(2
N%
D T, ™+2 »=i
1 dmya
= «!
I'
0 =
d(
f ,
dc
uai ~
/ j
II(O)
i=l
uai
t=l
(109)
4.
1,2,...,^
The discrete equation of (95) at a discrete point (3 can similarly be expressed as Nl "B
^je
Ee
u
ee U
%3 +
V
Z
Z^ Wi •+•
dC
(l )
Nf, "B
Jje
V
Zw uWi
At
£
+ 1
yy(0) 2 ^
3=1
1 !e
u
Wi
"SJ
j=i
rf
^Ig d( '
(110)
0 = 1,2,....N^-4.
The discrete equation of (96) at a discrete point 7 can be expressed as Ee >)
2
dA W\-n«C wyk + A ^ E ^ f c ) « f c = P ; . d( •£« fc=l fc=l
7 = 1 , 2 , . . . , ^ - 2 . (Ill)
And the discrete equation of (97) at a discrete point 5 can be expressed as d2j(sh
fi.
M>( , D
' E ir«+ J=l
2
0i
u.u>(<5) - S p
Z^
d(
1=1
n
eC3
, r(s)e
V^
neC
4
s 61 e5«i + io<s(«) Z ^ ^e.M ;=i
JV£ 7
1=1 0)e
f(
;=i
l,2,...,iV£-4.
(112)
In (109) to (112), the derivatives of section constants, m | , m§ and m^ at the related discrete points can also be calculated by the DQ. It is especially useful if the distribution function of a section constant, m%, m | or m% is not continuously differentiable up to the order of its derivative. The values of section constants, m§,
Optimization
in FE and DQE Techniques
219
m | and mf, at the two element boundary nodes and certain interior discrete points are used to define the DQ discretizations of the derivatives of these quantities. Let 4>(Q denote the distribution of these quantities. Then the DQ discretization for the mth order derivative of 4> at a discrete point a is expressed by ±J°=YDeF$l
(113)
i=l
where D-. are the weighting coefficients and N^ the number of points for defining the DQ discretization. Let N%, Nflf Nf2, Nf3, Nj, N§t and N§2 denote the numbers of nodes for defining the discretizations of the derivatives of A, Iss, lyy, IQQ, J, m,x(my) and rriQ , respectively. Then the derivatives at the related discrete points can be expressed by the following DQ discretization equations: NA dAe ' ,1Te dI Z 2 l _ V ^ rSeC Ae xxa A/2—i ~fk fc' A/-
2^,^03 Nf3
dC s
2-J
^
si
UQV
0j yyJ'
j=l
,2r(5)e
(=1
W d2Te " 2 J " i i c _ y ^ rSeC2 re Af2 2—/ ai xxV
' ST^ f) e C re 2—i aixxV
2 dC s
yyj>
j=l
,As)e
Nx
2
^.C s
Nf3
,
2—, ei aar r=i v
2slJsiJi
dC s
1=1
N^
e
rn
0t ~ 2-i Pt *=i
df
xf
(,2
= £%»». ^f- = E^« s )e .
^
di4)
Using (109), (110), (111) and (112), the following element matrix equation can be constructed: [ke}{~Se} = {fe}
(115)
where [ke] is an element coefficient matrix,
{n=[Ki
vii *>\ n.i Hsi nx 5§i ••• ^Nf,
V
1N%
W
»N-
"-«'
eC
ysNh
«IiV=JT
QIN«
(H6)
the element displacement vector, and
{Te}=[Pt
?xl Qn 4 f
••• P%%-2 Fx(Nh _4) ?1(iV« -4) 4 ? 4 - 4 ) J T (117)
d the element load vector. In (116), 9% d%si"•"" and" zaiel2ic Jrepresent ^ 1— -~~ F-1~5T' ^ si, "ysi 'xsi> " r a c " " ~~ -3F"
pr-^- and ^ e
( rn
1 ^ %
= p r - ^ , respectively, at node i. In (117), g | i ; g| 4 and m ^ p ,
1 dm|
j
(s)e
1 dm-
i
,
.
.
)e
Xr
d^ l ^1~~
represent . n
9* — F~Sc~' ^j/ + T'~df~ a n d m 2 — F dc ' r e s P e c t l v ely, at node 2. If no axially distributed force is applied, interior discrete points can be neglected and only two discrete points of the two element boundary nodes are necessary which are used to
C.-N.
220
Chen
define natural transition conditions or natural boundary conditions involving axial forces. Then no discrete axial equilibrium equation needs to be included in (115). If the element is axially rigid, w\ in {5e} and the related columns in [ke] can also be eliminated. Similarly, if no laterally distributed force in a certain direction is applied, the discrete flexural equilibrium equations in that direction can be neglected. If the element is flexurally rigid in a certain direction, the lateral displacements and their gradients in {5e} and the related columns in [ke] can also be eliminated. Furthermore, if no distributed torque is applied the discrete torsional equilibrium equations can be neglected. If the element is torsional rigid the angles of twist and the rates of change of twist in {6e} and the related columns in [ke] can be eliminated. For calculating the internal forces of a discrete element at a specific point, certain local derivatives of displacement components at that point have to be discretized. Using (63), (64), (108) and (116), the mth order derivative of ue with respect to ze at point a can be expressed as
^
=^u^J{
(us)
where
ID<:\ = [DS
0 0 ... 0 lr
o 0J. Bu
The mth order derivatives of v\, w% and 6\ with respect to ze at points /3, 7 and 5, respectively, can similarly be obtained: dmve 1 e m --i-[D (119) (le)m £\{5'} d(ze)-^ dmw* 1 [De£\{6e} (120) (le)m d{ze)m 1 dm0h d{Ee)m
-
{le)miKTjm
(121)
where LD2v"j = Lo 0 D * " ••• D^% 0 0 0 oj, LD&mJ = L0 ° ° ° ° Dtl •• 0 ° DSm-D
leD
l%^\-
The internal forces of a discrete element at an arbitrary point can be calculated. Denote F? the axial force in the element. F? is expressed as Ft = EeAe^.
(122) dze Using the EDQ discretization, the axial force at an arbitrary point 7 can be obtained:
Ft, = - T T ^ E ^ = V ^ J { * } fc=i
(m)
221
Optimization in FE and DQE Techniques
The distributions of bending moments in the element are
m^-E'II^,
MI = E^s0^.
(124)
Using the EDQ discretization, bending moments MJ at point (3 and M | at point a can be obtained: Jv
Tpeje M
x{{3) ~
^e _ V(«) _
M
fl„
(/e)2
2 ^ ^C«%' iVe V^ l_s
ra(a) He\2
j?e re (Je)2
L^e/S J l " ' '
(125) neC L>
2
J
2
-e _ ^ i i a i n e C I rxe-i uociUsi ~ ,ley \-Dua J \ d J"
The distributions of shear forces in the element are
~6% = -£ed^ {Ilsd(^r) - *» = - £e dF( J «d(F)0-
(126)
Using the EDQ discretization, shear forces V^e at point a and V^6 at point /3 can be obtained:
f pe
,„
NB
JA / J re
/
^ -^uai
e
(le)3 V dC
ES
NB
'
±
xx(a)
e
/
,
^uai
e
[D £\+I SSa[D^\y~6 }, (127)
' I frfC L<J+^L<jW>. @ yypi^vff •> I
(£e)3 V
v
The distributions of bimoment M?, warping torsion moment M"'e, Saint Venant torsion moment Mf ' e and the total torque M? in the element are Me -FeAs)e_fOt_
^-^dfFr M
| t ,e
=
joe Ge j e ^ z
^,e
_
™d
/rWe^i
M
* "-^dF^dW/
jjje
=
^ , e
+
M|t,e_
(128)
222
C.-N.
Chen
Using the EDQ discretization, M%, M?'e, M? 'e and Mf at point S can be obtained v-v e
_
,T>i5,e _
« M S ) V->
n
e C 2 he
1
^
aab
(s)e JV£ u>u>(<5) V - ^ neC
fie
<»>«
, .
•C'
/
£e
/d/i'-'
.,
I neC2 | rrei
iV° V"^
, r(»)e
., \
-
n
e C J 2e
(129)
^lD£\+l£slD£A){n L d£ 6sSJ wi^erf
(le)3 \ GeJ?5) ^ f
_
re Te
1=1
'
The DQEM requires that all condition equations at joints are satisfied. The condition equations have to be expressed as discrete forms. Let M J denote the number of elements connected to the inter-element boundary or analysis domain boundary, j . Also let Im denote the element node number of the m-?th element connected to the joint. Then I™3 is equal to 1 or the other element boundary node Nm of the m-'th element. The compatibility conditions, which are kinematic transition conditions, of joint j can be expressed as follows:
(13°)
{d)U = {<*?»} = &}
where {d™ j} and {d?} represent the global element nodal displacement vector of node Im of the m J 'th element and the global displacement vector of inter-element boundary j , respectively. The equilibrium conditions of external and internal forces at the inter-element boundary and natural boundary also have to be satisfied. Each equilibrium condition is either a natural transition condition or a natural boundary condition. Let vm denote an indicator defined by the local element node of an element at the joint. vm is defined as: +1
' -1,
if/ m
,
•
(131)
=1
And let v^3 and v^ denote the two lateral forces, F^3 denote the axial force, M™3 and M^3 denote the two bending moments, M™3 denote the bimoment, and Mp3 denote the axial moments, of the m^th element at inter-element boundary or natural boundary j . Also let Pi, P-j and P | denote the three concentrated forces, Ml, Mi and Ml denote the three concentrated moments, and M^x, M^y and M^z denote the three concentrated bimoments, applied at the inter-element boundary
223
Optimization in FE and DQE Techniques
or natural boundary, j . Then the equilibrium conditions of inter-element boundary or natural boundary, j , can be expressed as the following matrix equation:
Y, umi{Qmi}
= {Pj}
(132)
where
{QmJ}=[vf
vf
Ff
Mf
Mf
Mf\T
Mf
is the local element nodal force vector at the inter-element boundary or natural boundary, and {pi}
= [PI
pi
Pi
Mi
MI
Mi
MIX
MIZ\T
Mtv
the force vector of the forces applied at the inter-element boundary or natural boundary. Using (123), (125), (127) and (129), (132) can be rewritten as
Y^ „™> ([smJ}{6mJ} - {Pi}) = {Pi}
(133)
where 3
J
I cm I = L^i J
'dl™ m E m m _ »xxi ^ ml I nr)mm3 <>£2 \I + j - T ml . I n nmJ">t3I' L mj SSIml l (^)3 I dC vi ui™>J
dlf
E
L^ J = " ( ^ I cm3' i
/^± i nmJt n-^
j
0
L7 J -
mj
2
mi
m3 3
i
i cm j i —
yyIml
i nm'? 1 1
TTTt^
(dl^)™-3
pmi
] cm3 I _
7 3
n <^1 JJ . +i C -. " I n^ i •J - yy!f "- i L
•
•
\
Gm' Jml
3
T(s)m
£"" /'uiuilml nmiyi
I n 7 " 3or* ^" I
\-^SiImj}
are row vectors representing the seven rows of [Sm ], and {Pj} = [m-Im3
-m-Jmi
0 0 0 m^]m3
0jT.
A joint might have one or more prescribed displacement components. Let {dJp} and {dip} denote the vector formed by the prescribed displacement components and the vector formed by the corresponding displacement components of the joint. The
224
C.-N. Chen
condition equations of prescribed displacements can be obtained from the following vector equation: H)
(134)
= {%}•
With the kinematic transition conditions in mind then assemble all element matrix equations (115) and the equilibrium conditions (133) of the inter-element boundary and natural boundary, the overall algebraic equation system can be obtained. Another approach can be used to assemble all discrete fundamental relations. This approach includes the fourteen discrete equations for defining the internal element boundary forces in the element matrix equation (115) to form the element stiffness equation. In this element stiffness equation, each of the first seven component equations represents one equation for defining an internal force corresponding to each individual degree of freedom assigned to node 1 while each of the last seven component equations represents one equation for defining an internal force corresponding to each individual degree of freedom assigned to the other element boundary node. This element stiffness equation is expressed as [ke]{5e} = {rl
(135) e
where the first and last seven rows of the element stiffness matrix [fc ] are coefficients for calculating the fourteen internal element boundary forces, and {fe} = [Qh Q\N>
Qh
Fli
FIN*
Hi M£N<
% M?NC
Ml,
M*!
...
Q%N, T
M§NC
M*NC\ .
By using this approach, the equilibrium conditions of the inter-element boundary or natural boundary j can be expressed by Mj
J2 vmJ{QmJ} = {Pj}
(136)
where
{Qmi} = lQ?
Q?
Qf
Mf
Mf
Mf
M£
M£
Mt\T
is the element nodal force vector at the inter-element boundary or natural boundary, and is the column vector formed by the first or last seven components of the vector {re} for an element connected to the inter-element boundary or natural boundary j . 4.3.2. Implementing the DQEM computer program In implementing the DQEM analysis program, the effective use of computer memory units and computational efficiency have to be considered. The adoption of techniques for reducing computer memory requirements and the use of vector and parallel processing strategies on computers that support these features can raise the performance of the computer program. In implementing the DQEM analysis program,
Optimization
in FE and DQE
225
Techniques
various phases including preprocessing, calculation of elemental discrete equations, incorporation of boundary conditions, solution of system equations and postprocessing can be parallelized. However, the assembly of elemental discrete equations can not take the advantage of parallel operation efficiently. The linear equation systems existing in the DQEM analysis can also be solved by using a certain direct or iterative solver. Among the solvers implemented into the DQEM computer program, an algorithm employing elimination technique and various pivotal strategies can effectively solve the ill-conditioned Vandermonde equation system in calculating the weighting coefficients by adopting the maximum pivotal strategy. The algorithm was designed to minimize computer memory requirements and reduce arithmetic operations.
4.3.3. Numerical examples The first problem solved concerns a fixed-free I-bar, shown in Fig. 22, subjected to a uniformly distributed axial force.52 z axis is coincident with the centroid line. The cross sections at A and B are shown in Figs. 23 and 24, respectively. The web has the same thickness as the flange. The variation of width is b(z) = 60(1 — z/L + z2121?) with b0 = 40 mm and L = 1000 mm. The variation of depth is d(z) = d0(l — z/L + z2/2L2) with d0 = 80 mm. Only the degree of freedom of the axial displacement is assigned to each node which is also assumed to be a discrete point. The interior discrete points are used to define discrete equilibrium equations while the two element boundary nodes are used to define the joint condition equations. In the analysis the elements and discrete points in an element are equally spaced. The values of material constants are E = 206000 N/mm 2 and G = 82400 N/mm 2 . In discretizing the first order derivative of section area, the DQ is adopted. The numerical results are summarized and listed in Table 2. They are compared with exact solutions. By increasing either the number of elements or degrees of freedom per element, the results converge very fast to the exact solutions. It proves that the developed DQEM numerical model has excellent convergence rate.
p =l.N/mm
A V Fig. 22.
B 1000. mm
a
A non-prismatic I-bar subjected to a uniformly distributed axial load.
226
C.-N.
Chen
-40. mm
80. mm
2.mm
Fig. 23.
The cross section at A.
20. mm
H
H
40. mm
2.mm Fig. 24. Table 2. load.
T h e cross section at B.
Results of a non-prismatic I-bar subjected to a uniformly distributed axial
Element type 5-node
9-node Exact solution
Number of elements 2 4 6 2 4
w (mm) (at B) 0.1052838 0.1051554 0.1051508 0.1051500 0.1051498 0.1051498
x X x x x x
P(N) (at A) l
10" lO"1 10"1 10"1 10"1 10"1
0.9998271 0.9999883 0.9999977 0.1000000 0.1000000 0.1000000
x x x x x x
lc
|l«-U)exact|
(at B) 3
10 10 3 10 3 10 4 10 4 10 4
<
-2.894704 -4.273620 -5.021809 -5.720779 -7.000000
T h e other problem solved involves a fixed-free I-bar subjected t o a concentrated torque at the free end, which is shown in Fig. 25. It is a warping torsion problem. Two active D O F are assigned to each element b o u n d a r y node. However, only one D O F is assigned t o interior nodes. T h e element nodes and elements are equally spaced. Both five D O F element and nine D O F element are used to t h e numerical
Optimization
in FE and DQE
227
Techniques
Mz =1. Nmm |401
Z-
f\
/,
I
57
• * -
80.
1000. mm • Fig. 25. Table 3. Element type
A fixed-free I-bar subjected to a concentrated torque at the free end.
Results of a fixed-free I-bar subjected to a concentrated torque at the free end. Number of
9Z (rad)
elements
5-node
9-node
Exact solution
2 4 6 8 2 4 6
MQ
(at B) 0.1652444 0.1613057 0.1606309 0.1603994 0.1601066 0.1601054 0.1601054 0.1601054
x x x x x x x x
Mf (N • mm)
(N • mm) (at A)
10"- 4 10"- 4 10"- 4 10"- 4 10"- 4 10"- 4 10"- 4 10"- 4
0.4524914 0.4413221 0.4390149 0.4381887 0.4371161 0.4371121 0.4371120 0.4371120
x x x x x x x x
10 10 3 10 3 10 3 10 3 10 3 10 3 10 3
0.9598855 0.9897957 0.9954490 0.9974369 0.9999973 0.1000000 0.1000000 0.1000000
x x x x x x x x
\&z —®z, exact 1
(at B)
(at A) 3
log
10° 10° 10° 10° 10° 10 1 10 1 10 1
-1.493527 -2.125116 -2.483833 -2.736059 -5.125225 < -7.000000 < -7.000000
analyses. The five DOF element has one interior node located at the center of the element. One degree of freedom representing the angle of twist is assigned to the center node which is used to define a discrete equilibrium equation of torsion. The nine DOF element also has one interior node located at the center of the element. Five degrees of freedom of the angle of twist and its derivatives up to the fourth order are assigned to the center node. Nine equally spaced points are defined with the five points from the third one to the seventh one being used to define five discrete equilibrium equations. Numerical results are summarized and listed in Table 3. It also shows that the numerical results can converge very fast by either increasing the degrees of freedom per element or the number of elements. 4.4. DQEM
free vibration
AAA. DQEM
formulation
analysis
In the static analysis, the displacement parameters of the shear center are used to define the equilibrium equations, while the coordinate system used to define the section constants is located at the centroid. The reference coordinate system can also be arbitrarily selected. (71) to (74) and (75) to (81) can be used to solve problems by using the coordinate system with the origin located at the centroid to define
228
C.-N.
Chen
the section constants and by using the displacement parameters of an arbitrarilyselected point to define the equilibrium equations. In the present DQEM free vibration analysis model, the arbitrarily selected coordinate system is used to define the dynamic equilibrium equations and section constants of non-prismatic beams having isotropic material. 53 By neglecting the external forces and by considering the harmonic motion, the differential eigenvalue equations can be obtained. Let U, V, W and &z denote the modal displacements. Also let u denote the natural frequency. The EDQ discretizations of the differential eigenvalue equations lead to the following discrete equations: Ee
(d2Iema (a)
{le)A
m 7 , Dlai
d£2
\
+
2
N%
Ni
dlt ~H-
E Dlai t=l
d{
N hv V - r>e(3 / j "vaj
tre I ^ 2 re ^ B „ jje _ ^ _ [ a Jxj/(a) V eC 2 , 2aI*y(a) ~ / .e- 4N ^ I . *2i / j "vaj ' * jf
(l )
EB
(le)3
\
dC
d{
3=1
J
^i
rfC 1 ye vaj I rj
3=1 Nl
' ^^2 E D«ak + 2 ^ E D£k + Ika) £ D£k I Wl
\ dC,
I
d(
fc=i
d C
2
a
/ \
m
L_ f dJ ^(°)
Ixy(a) 5(Q dC
N'
"*
(/«)2
«=i
ivg
NB
7 , ^ f i a i + Isx(a) i=l
iVE
d/
fc=i
fc=i
* tr
^
4 A
— put
i e2 (l )
^B„ V ^ / ,
re ~ xxy(a)
N%
dl', (/e)4
+ ^li(a) E ^uai i=l
Uf
^uoa
E
fl„
"vaj
'
J
j= l
1 I dl%i \ d( ak
x y ( a ) / ,, "vaj J' = l
I
J
s +ii(a)J2Dtk)n
+ le
1
k= l
(dl[ujx(a)
•lyotOal
Ni-
NZ
E ^ai
+7
e;
^(«) E ^1,-. (=1
= 0,
(137)
a = l,2,...,JV|ti-4, iV=
£ e /rf2/;xy(0) E
fl
a e ) 4 ^c
N%
+ dC, - ^ tE <+^(« E D% \ut =l i=l
N ' V B„ V - neC2 j= l
2alxy(0)
J jre »B„ oaiOT(/3) V -
d(
3=1
ATS, DeC
_, re
V - r>er j= l
I
ye
Optimization
(
N%
yi0)
^2
( „e,,2
,2re
in FE and DQE
Are
.re
Techniques
N%
N\
e V ne< i 2? 5(/3) V i ? ^&k + / v{0) , V n e C 30k hy, e 2 ^ " a w + ~d(~^ * ~ )
#£
>re
JVT
\
#T
NeR
N%
+ 72 , 1 (dll ^ WV +^ 2n e^Cf ~ , re2_^ ^e./w SS(/5) V ^ n « {°»W r/e2-^ <W< I rfC2 I -2^ e 2
)
c ) l <*< ^ 1
Afl<5, |9<W
2i
i=l
N%
+
\
x
- ^w*< -
+
229
[ dI'yy(0) V - neC
, re
V
D eC 2
V?
i /<*'; fc=l
iv;
1 f^jfffl^^
+
—
0,
/3=l,2,...,iV^-4,
2 ^ ^e £ /3/
+ 7
N£
VDec2
«
^y(/5) 2; =^i
u
e,0i
e? (138)
C
N B E« I dll (7) N% D Ax7i Ui Ge)3 I ^C f7 , l~,i + ^I(7) E i=l WJ
^ f^fe E
+ (Z«)3
\
^
dC
N5
-°C7J
+ 7
D
5(7) E
«7j
^/
ldAti)
Jj7y2 1 " ^
Z_^ ^*7fc ^ ^(7) Z ^ ^«7fe I "'"fc
fc=i
fc=l
N%
iVJ
2
e p^ 1 ^ E
re
7
/ ^ 2 Te -^£(5)
(ie)4 \
de
^
= l,2,...,A^-2,
(139)
^ u ws V^ eC2 , o " J O x W V - n e C 3 , re V D e ^ 1 T? 2 + 2 ^ -^fi« + 2 ^ ^S<5» ^^(-5) 2 ^ ^ S5i | ^i dC i=l
[ d2lly(8) ^ 1^ j=i
eC2 DD
, g ^ ( v6j +2 At
B„
2^ J=I
U
VSJ
+^ m
2^ J=I
V
VSJ
| ^j
230
C.-N.
Ee
+ {i*y \
d'22LreLJUj(S)
dc2
M?
E^; jrC *
N'
D §Ui ~dT~ z1=1J hsi+ ^(8) z2 1= 1
6 bi
NZ
ATi
di
,eCa
2L [dJh^D<
Chen
D
NZ
+Je
TDe<
es
i=i
e P
LJ2
re
1
x
^ "
ax(S)
dn•V(S)
(i*)2
d<;
/ . Di5j
+ ^Qy(8) Z ^
= 0,
Uf
*
j=l
Ke
D
v5j
3=1
f*T
"T
(=i
i=i
e 2
C ) I dC
D
N%
N%
I dle
1
NZ D
*6i + ^2(«) z2 ~dT~ 2_, i=l i=l
(I*)
1
Nf,
dI
/ 2
e?
*(
(S = l,2,...,JVf.-4.
(140)
Using (137) to (140), the following local element eigenvalue equation can be constructed: ( [ f c e ] - W 2 [ m W } = {0}
(141)
where [ke] is an element coefficient matrix, [me] the local element mass matrix and
{r} = [Uf v{ w{ eii e^ e^ fa ••• °h„ vh wh 9*Nh ®hh e| w .
^.f
(142)
the local element modal displacement vector. The corresponding local element displacement vector is {5e} = \u\
v{ w\ l
N*
J
0%x 6^
N%
B\x
™N
a'n 7
vNi
In (142), GI,, e
GIN* ^
=
*%N-JT.
(143)
±^- and
ji-jf-, respectively, at node i. In (143), 9%i, 6^ and &% represent - ^
dei
= -p^r,
IHf = ]T^?- a n d -jj = j;~dC) respectively, at node i. If the element is axially rigid, W? in {fle} and the related columns in [ke] can also be eliminated. If the element is flexurally rigid in a certain direction, the lateral modal displacements and their gradients in {$ e } and the related columns in [ke] can also be eliminated. If the element is torsional rigid, the angles of twist and the rates of change of twist in {t?e} and the related columns in [ke] can be eliminated.
231
Optimization in FE and DQE Techniques
Denote F§ the axial force in the element. Ff is expressed as
F
S'-E''^-E,i;0r>+E'A'%-
<144»
Using the EDQ discretization, the axial force at an arbitrary point 7 can be obtained:
^7 = -(fja (4%L^?J +/57L^?J - ^ L ^ v J ) {^}- (145) The distributions of bending moments in the element are
We
d2Ue
r^ere xy
r,e re
e 2
yy
d{z )
„-„
r,e re^w"
e 2
y
d{z )
d2Ue
„e„
d2Ve
rp
XX
d{ze)2
d2Ue
Xy
d(ze)2
uv
' d{ze)2,
dz
,_,_ du) e _,. „ X
d2B\
T,e re
e
(146)
d 2 #§
"Xd(z e\2-
dze
Using the EDQ discretization, bending moments MJ at point j3 and M | at point a can be obtained:
m0 = M- =
Ee W
2
{iim[D^]+iiy,[D^\-i^y[D%\-ii,0[Df^
(le-- \D-^]+I--
|D!c2|-/e/e
|De-Cl-/5
cn,
in^l^Wr^
The distributions of shear forces in the element are
T/e
=
,el(fe
_^«M _ pe ±_ (re _f^_\
dz° \ x x d{z*)2 ) I pe d ( d2Q% \ J ux ^ A^e V l <52w^e\2 dz* d{ze)2)
+
,
pe^fre^
dz° \ x y d{z'Y J+ dz° \ x dz* dH _d*l__ d*W &§E Is xxs s2FM-lte + P Hxy 2F>^1P>~ P Hx1*2a*2 PIQHuJX 2 ^ V ^ dt dz dt dz dt dt dz'
V* = -E*—(p ^ ^ \ - F ^ f r ^ ^ \ + V + dz* V XVd(ze)2J dz* \yyd(ze)2J
pe^fre^l dze \ y dze
2 d (' d2d% \ d3u „ d3v , d36. r r d w + 1 2 Hxy 2 P yyy **-*^ 2 Hy 2 rWe\ Qy ^ s jTiifa + Phy-^^z - Ph-M - PiFOy 2
+ EB
dzt\ d(ze) J^ dt dz dt dz
dt
dt dz' (148)
C.-N. Chen
232
Using the EDQ discretization, shear forces V£ at point a and Vy at point (5 can be obtained: Ee
V-e -I
^ M a i r , e C 2 | + / ? _ \DeS3\\
dll^ | £ ) - c | + 7 ? I me 2 1 ^ _ (
+ !L (re V
V0
+Ie_
. r>eC31
<^° I D < 2 I + jg- Iz?- c3 1 0s a
I D ^ I + J 6 - I D - ^ l - / e / e l\I>e I - 7 e d/: fM l neC2
ID?C h
dt 2
{5e}
'
d/i;
dC L<J + ^ L < J J + ( " i f U>£ J + W > & J
(7e)3
d/j
d fL^j+^^
+
+ (d^L\Dee\
c
;j ;
*V0 I n e C 2 I , r e
v dc
IneC3 I
0,-/3 J
{*«}
( ' ^ L^J + I'm IDlU - ieih m0\ - lUe \-Dfj) ^p- (14Q)
where
ina\
= \Q o *!(Ca) ••• *^(C«) o o o oj.
The distributions of bimoment MQ, warping torsion moment M£'e, Saint Venant torsion moment M | ' , e and the total torque M§ in the element are d2u? d2e% M,5 = - £ / ; 2 *"VM~e\2 d{ze)2' " d(ze)2 d(f e ) 2 2 e d / d « d / d2v? Ee±(re Rlr^.e — r e " / r e " " z e uy dze\ dz \ d{z*)2 M £ * - dF(/**d(Fj* 3 dv , d39s • pla 2 - plan 2 PJQy 2 'dt dz' dt dz dt dz '2^e
e
jae
Ms2l'e = GeJe^1e,
d29% d{z*)2
QQ
(150)
Ml = Mf'e + Mf'e. z
dz
Using the EDQ discretization, M%, M ? ' e , Mf'e and Mf at point 5 can be obtained: Ee
M,u)5 M%e
(%*6lD$\ + ^ L ^ f J - W f l & j ) {^h
Ee
dI
kl\r><2\
W 'dl
^iDifsl+I'wslDgll)]
d{
~ -jj- {jassi^lsl e
Ms*'e = G ,Jf [D£M8°}, 9z$
+ItiyslDls\
. I n ' t 3 ^ 4-
(dI'°vS\neC-
{~Se} + lQu,slD§iS\) St,€
MI5 = M^ + Mil
(151) ^
2
'
Optimization
in FE and DQE
233
Techniques
The compatibility conditions, which are kinematic transition conditions, of interelement boundary j can be expressed by {¥>£} = { < " } = M
(152)
where { y ^ } and {
J2 umJ{QmJ} = {Pj)
(153)
7713 = 1
where
{Pi} = [Pi P> Pi Mi Ml Ml 0 0 0JT is the inertia force vector of inertia forces at the inter-element boundary or natural boundary. Let (xc,yc,zc) denote the global coordinates of the center C of a rigid body with the mass MK The mass is rigidly connected to the node O on z axis. The translational inertia forces can be expressed by P? = M^^^jp + (-ijk6JA0Xkoc), x x where e ^ is the permutation symbol and x^oc — kC ~ kO • Then the components of translational inertia forces are H = M ^
(u£ + zoceio
p
y = ^W2 (v° ~ zoce*°
H =M ^
- yocOio) , + Xoc9
Q
-
(w>0 + yocOio - xocOio)
.
Let Hfc denote the angular momentum of the mass M-7 with respect to C. The inertia moments can be expressed by M\0
= —^- + MHijkXjoc{8gt2°
0
tkim Qt2 XmOc)- Then the components of inertia moments are dt2 + Mj
xy
dt2
xz
2
dt2
2
d w>0 d v>0 dX0 2 2 dt2 ^ dt ' w » ^ dt yoc—^pi— zoc + yocyocQjl9 d2
°io
-yocxoc—^
d29{0 zocxoc o,2
+
C.-N. Chen
234
M
°y =
^o
h
-
+ Mj
d26;yO
+ 1:yy
dt2
**
dw0
d2ejx0
dt2
dt2
dt2
J
m2
zocVoc
dt2
d2(PyQ
-p
+ Mj xoc
dt2
y*
f d26j0
Mi
, d2e-zO
2 J
+x0cxoc^^
l
-I;
P
dt2
dt2
d26j0 ^
£S
gt2
2
d2vh
d u{
+(xocVoc
yoc- dt2
2
dt
+yocyoc)
d20jc dt2
where IL. is t h e inertia tensor of t h e mass Mj with respect t o t h e coordinate system (x, y, z) which has t h e origin located at C and t h e coordinate axes x, y and z oriented in x, y and z directions, respectively. Using (145), (147), (149) and (151), (153) can be rewritten as M1
>
Y^ vm'
[S m J ]{5 m '}+p m , [S m ']
where
Emi
[Sf\
dl
mj 3
(l )
dl
+
xxV
dt m jj<;22 syi7"3 |I nnm < I + lmJ 1U ] dt vl^
jmj (
dlm! U^L I Dmi< L
V dt
«/^
. ID m ^ 3 | ) xyI^^UvI^iJ
+I
I _i_ j
J
m j
5/m3L
Ajm? Qxl™j I n m ' C 2 I , rmj
In m ^ 2 II
*/ m i V
I r>m^3
dt dl
[Sf\
(I m.3\3 dl
+
dt
yyl™1
Inm
c
I +im
Inm ; I
[Dmi?\+Im!Jm,[D^^l
dt dl
xyl"
S/™J
dC df
uylm3
dC
| nmJC2
L^„ /mJ J
+1
cyi^
\-"§_Im,lJ
=
{Pj}
(154)
Optimization
F
/
J -
_;m
;l
F'
/
•
ul™3 >
"'x/"* 3 L
En (im3Y
+
TO/"*3
\
3
i _
3 2
rm *ylmj
J
l jyrn ( ^Sil"*1'J)
J
i\ ' i 2
. \r)m i 1 + r™J . I n m c I
«/">3 J
dr
3
^ v l ^
-2
(T™-3
I cm' I - _
[SE
[
rm inm'( yl™3' lJJwI™j
m J
"2
m \ + iflmi [D-iJ - r A & L ^ L J ) ,
3
1
'2
235
J
y'xyl™3
Qmiy
Techniques
•
L^J = -jpzy (^ ^ 4
in FE and DQE
J
u.x/™3 L " ^ - J ™ 3 J ; '
^ID^I+I^AD^]
d(
dlm\ ^
l
dC
vi™3 J
"S'mJ
L
c/-3
J
dl £m'
I Cm3 | _ L
7
__T (lm3
J
)
2
f J™3 \"3lm3
I n m J C 2 I , jmj + J \-UuI^J ut//m3
L
| r y n 3 C 2 I _ jm3 C/™ 3 J
| n m J ( ! |N QQlm3^UeiI^^j
1
are row vectors representing the seven rows of [Sm ],
| Cm3 I _
_
L Z
j^mJ y
J
(jmj
.\nm3(.]
mJ
XyI
L^J = " ^ ( ^
L
3
ulm
+ Tm3
'
yyl
[D&) + ^
.| D
L
m
vlm3
^ \-l
J
m 3
T
m 3
yImJ
L ^ J +^
, | *
L
3
m
'
u;/™'3
.I
J
L ^ L J)
are row vectors representing the three rows of [5 m 3 ]. Considering the harmonic motion, (154) results in the following equation: MJ
J2 v™' (\Sm1} - pmito2[Sm3^
{-dmi} - u2[Mj}{^}
=0
(155)
mi = l
where {^} is the global modal displacement vector of the inter-element boundary or natural boundary, j . It should be noted that the first six rows of [MJ] are
236
C.-N.
Chen
expressed by \M{\ = [-Mi
0 0 0 0 yocMi j
0 0 0J, j
[M|J = [0 -M
0 0 0 -xocM
3
j
j
0 0 0J,
[M Z\ = [0 0 -M
yocM
-x0cM^>
\M{\ = [0 zocW
-yOCMj
-P& -
0 xocMi
P.. +
-Ifo-xocxocM' L^eJ = [yocMj liy
yocVocM*
!L + zocxocMj
iit + yocxocMi [M3b\ = [-zocMi
0 0 0 0J,
0 0 0J,
xocVocMi
lli + zocyocM'
-xOCMj
0 0 Oj,
0 PAi + yocyoc)Mj
-Hi - (xocxoc
0 0 0J
and all elements in the last three rows of [MJ] are zero. With the kinematic transition conditions in mind then assemble all element eigenvalue equations and the dynamic equilibrium conditions of the inter-element boundary and natural boundary under free vibration, the overall eigenvalue system can be obtained. Another approach can be used to assemble all discrete fundamental relations. This approach includes the fourteen discrete equations for defining the internal element boundary forces due to modal displacements in the element eigenvalue equation to form another matrix equation. In this new matrix equation, each of the first seven component equations represents one equation for defining an internal force corresponding to each individual degree of freedom assigned to node 1 while each of the last seven component equations represents one equation for defining an internal force corresponding to each individual degree of freedom assigned to the other element boundary node. This matrix equation is expressed as ({ke}-w2[me}){&}
= {r}
(156)
where the first and last seven rows of [ke] and [rhe] are coefficients for calculating the fourteen internal element boundary forces due to modal displacement, and {7e}=LQIi QIN-
Qh
Hi
QW
Ml, F!N-
% M
IN*
Ml, M*N*
Mgx MINC
0 •••
0
M ^ \
T
is a force vector in which the first and last seven elements are internal element boundary forces due to modal displacements, while all other elements equal zero. By using this approach, the dynamic equilibrium conditions of the inter-element boundary or natural boundary, j , lead to the following equation J2 vmJ{Qm'}=uj2{Mj}{4P} m' = l
(157)
Optimization
in FE and DQE
237
Techniques
where
{QmJ} = [Qf
Qf
Ff
Mf
Mf
Mf
Mg
M%j
Mg\T
is the element nodal force vector at the inter-element boundary or natural boundary due to modal displacements, and is the column vector formed by the first or last seven components of the vector {r e } for an element connected to the inter-element boundary or natural boundary, j . 4.4.2. Numerical examples In the analyses, DOF per element is the same for all elements in solving a problem. The elements and discrete points in an element are equally spaced. The first problem solved concerns the longitudinal vibration of a fixed-free bar with a concentrated mass attached to the free end. The factor ^ j - is 1. In solving the bar problem, only one DOF representing the axial modal displacement is assigned to a node. The nodes in an element are also equally spaced. Thus nodes are also discrete points. The Lagrange interpolation functions are used to calculate the weighting coefficients. The natural frequency uin of the nth mode can be expressed a s u „ = ^f-\j— with C„ defined as the frequency factor. The frequency factors obtained are listed in Table 4. They are compared with exact solutions. It shows the numerical results converge fast by increasing either the number of elements or DOF per element. The second problem solved concerns the lateral vibration of a fixed-fixed beam. In solving this problem, the effect of rotational inertia is neglected and the elements are equally spaced. Consider a iV^-DOF element and define AC = l/(N^ — 1). The interior discrete points for defining the element-based eigenvalue equations are located at C, = (p — 1) AC, p = 3 , . . . , Ng — 4. The natural frequency w„ of the nth mode can be expressed as un = ^fr\l^\ with / the second moment of the cross section. Only one DOF representing the lateral modal displacement is assigned to an interior node. The nodes in an element are also equally spaced. The frequency factors obtained are listed in Table 5. They are compared with exact solutions. It shows that the DQEM solutions converge very fast by increasing either the number Table 4. Frequency factors of a fixed-free bar with a concentrated mass attached to the free end. DOF per element
Number of elements
5
2 4 2 4 6
7
solution
C\
C2
C3
0.86237 0.86059 0.86109 0.86044 0.86035 0.8602
3.54315 3.44639 3.47654 3.43372 3.42814 3.4267
6.91639 6.53406 6.60278 6.48289 6.45321 6.4373
238
C.-N. Table 5.
Frequency factors of a fixed-fixed beam.
DOF per element
Number of elements
5
2 4 6 8 2 4 6 8 2 4 6 8
7
9
Chen
Exact so lution
Ci
c2
c3
CA
c5
26.1279 22.7941 22.5337 22.4630 21.8060 22.3449 22.3677 22.3715 22.3883 22.3734 22.3733 22.3733 22.3733
55.4256 66.1269 63.3846 62.6119 61.9562 61.0706 61.5576 61.6364 61.6636 61.6851 61.6739 61.6730 61.6728
155.827 128.478 124.946 105.944 115.262 120.041 120.632 126.980 121.177 120.920 120.906 120.904
188.419 221.558 211.699 157.945 198.326 195.853 198.645 227.862 199.990 199.995 199.882 199.859
381.696 325.420 233.654 277.739 281.730 294.510 341.401 302.188 299.500 298.673 298.555
of elements or DOF per element. It proves that the DQEM vibration analysis model is excellent for analyzing the lateral vibration of a beam. 4.5. DQEM
Buckling
analysis
The Buckling equilibrium equations derived in the previous section can be used for the DQEM solution of non-prismatic bars under generic loading condition. Consider that the non-prismatic bar is subjected to a compressive end force P and buckles in x — z plane. Also assume that the shear center coincides with the centroid. Thus by using ys = 0, u = us, mx = 0 and Mx = 0 in (85), the governing equation can be obtained: d2 (d2u\ d2u T (158) dZ-2{Q^d^)-Pd^=°Bending moment My and shear force Vx can also be expressed by = r d2u My = Q^xx-2,
,r d (~ r d2u\ du v, = - - ^ 3 3 / « ^ ) + P ^ .
(159)
4.5.1. DQEM formulation For the buckling analysis, mapping is not used. The EDQ discretization of (158) at discrete point a in an element can be expressed as d2(Q33ixxy(a) 2 dz
^
, D
^ i=l
d(Q33ixxy
N
°
N
+2
—Tz
i=\
i=\
e
N
-P'£DZui
=0
i=i
where Ne is the DOF per element.
(160)
Optimization in FE and DQE Techniques
239
Let Nl and N^+1 denote the numbers of degrees of freedom assigned to the last node of element i and the first node of element i + 1, respectively. Then, at the interelement boundary of two adjacent elements i and i + 1, the kinematic transition conditions can be expressed by N
- u
e-
rliti+1
rlii*
i+1
' =
(161)
*
«Ar._Af.+i - «i > dz iibii dz % where N is the number of nodes of element i. Assuming that a concentrated elastic support with spring constant kl'l+1 existing at the inter-element boundary, the two natural transition conditions can be expressed by using (159), (pyi i ~ d2ui+1 d(Q33lxxYNi d2v?Ni dz dz^ +1
d(Q33lXx)\
+
<
d*u\
+1
i+1d*u\
h
(162) +l
du\
,M+1
J+1
_
If the stiffness Qzzlxx is continuous at the inter-element boundary, the compatibility of the derivatives of displacement at the inter-element boundary would be up to the second order. If the first order derivative of Q^Ixx is also continuous at the inter-element boundary and the support does not exist, the compatibility of the derivatives of displacement at the inter-element boundary would be up to the third order. For an elastic support free inter-element boundary, fcM+1 equals zero. In addition, if the inter-element boundary is a rigid support, the second of (162) must be replaced by u M + 1 = 0. In (161) and (162), ^ F l
replaced by u fti_Ni+k+v
while
for k = 1 , . . . , N£ - 1 can be
for m = 1 , . . . , A^ + 1 — 1 can be replaced by
d"^,
"m+i- ®n t f t e other hand, for k > Nl — 1 or m > N^+1 — 1, (63) must be used to discretize the related differential operations existing in (161) and (162). There are two boundary conditions at a boundary node. Assume that a specified boundary node is one of the two element boundary nodes of element n. Then the node number 7™ of the boundary node would be 1 or TV™. Considering that the boundary node has an elastic support with spring constant kjn, the two boundary conditions are f u ^ _
d ^ I x x Y ^ d ^
-
d?U^
dv!}
rn-
(163) 1
n
where u " is 1 for 7™ = 1 and - 1 for I = N". By using fc/» = 0 in (163), the boundary conditions of a free boundary can be obtained. By replacing the second of (163) with u™„ = 0, the boundary conditions of a simply supported boundary can be obtained. For a fixed boundary, the two boundary conditions are u"„ = 0 and d'z" = 0. In addition to the condition of zero slope, by using kjn = 0 in the second of (163), the second boundary condition of a guided boundary can be obtained.
240
C.-N.
Chen
The transition conditions and natural boundary condition can also be discretized by using EDQ. In (160), (162) and (163), the derivatives of the stiffness <533/Xx at the related discrete points can also be calculated by the DQ. It is especially useful if the distribution function of the stiffness is not continuously differentiable up to the order of its derivative. 4.5.2. Numerical examples The first problem solved concerns a compressed prismatic fixed-fixed bar. The length of the bar is L. The material is monoclinic with Q 3 3 being constant along the axis. The Hermite polynomials are used for calculating the weighting coefficients. In the analysis, elements and nodes in an element are equally spaced. The two DOF assigned to an element boundary node are used to define either the discrete transition conditions or the discrete boundary conditions. 2Ne - 4 equally spaced discrete points in each element are used to define discrete buckling equilibrium equations. These discrete points are defined by £ a = 4 ^ z r - Defining C = ?L\ as the load Q33I2
factor, numerical results of the first three critical load factors are summarized and listed in Table 6. It shows that the results of DQEM can converge fast by increasing either the DOF per element or the number of elements. It also shows that the approach of increasing the number of degrees of freedom performs better than the other one. The second problem solved involves the buckling analysis of a non-prismatic overhanging monoclinic bar with length L subjected to a compressive force P at the free end. At the middle point there is a roller support. The bar is a solid truncated cone with Ixx expressed as Ixx = ^-[2* — (2* — l)f;] 4 , where To is the value of Table 6.
Critical load factors of a compressed prismatic fixed-fixed bar.
D O F per element
Number of elements
6
1 3 5 1 3 5 1 3 5 1 3 5 1 3
8
10
12
14 Exact solution
Ci
0.4285714 0.3518793 0.3767991 0.2782830 0.3968114 0.3950997 0.3905148 0.3947501 0.3947821 0.3950413 0.3947844 0.3947842 0.3947742 0.3947842 0.3947842
x x x X x x X x x x x x x x x
10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2 10 2
c2
<53
0.6671324 X 10 2 0.7364972 X 10 2
0.1269444 x 10 3 0.1364942 x 10 3
0.8219256 X 10 2 0.8100730 X 10 2 2 0.7558177 X 10 0.8072120 X 10 2 0.8075981 X 10 2 0.8150056 X 10 2 0.8076367 X 10 2 0.8076293 X 10 2 2 0.8071510 X 10 2 0.8076299 X 10 0.8076292 X 10 2
0.1592622 X 10 3 0.1583756 x 10 3
0.1157340 x 103 0.1588826 0.1579597 0.1310206 0.1586241 0.1579374 0.1636877 0.1572701 0.1579140
x X x x x x x x
10 3 10 3 10 3 10 3 10 3 10 3 10 3 10 3
Optimization Table 7. DOF per element 5
7
9
in FE and DQE
Techniques
241
Critical load factors of a compressed non-prismatic overhanging bar. Number of elements 2 4 6 2 4 6 2 4 6
c2
X 10 x 10 x 10 x 10 x 10 X 10 x 10 x 10 x 10
0.2699958 0.3410679 0.3501234 0.3221310 0.3452757 0.3452336 0.3494950 0.3453623 0.3454465
c3 2
X 10 X 10 2 x 10 2 X 10 2 x 10 2 x 10 2 x 10 2 x 10 2 x 10 2
0.7358817 x 10 2 0.8847569 x 10 2 0.8546602 x 10 2 0.8351979 x 10 2 0.8418209 x 10 2 0.8431407 x 10 2
Ixx at the fixed end z = 0 is constant along the axis. The Lagrange polynomials are used for calculating the weighting coefficients. In the analysis, elements and nodes in an element are equally spaced. Each of the DOF assigned to nodes 3 , 4 , . . . , Ne — 2 in an element is used to define a discrete buckling equilibrium equation at that node. The other four DOF are used to define either the discrete transition condiPL' tions or the discrete boundary conditions. Defining C as the load factor, Q33J0 numerical results of the first three critical load factors are summarized and listed in Table 7. It shows that the convergence performance of the developed DQEM buckling analysis model is also excellent for solving the non-prismatic overhanging bar problem.
5. D Q E M Analysis of Potential Flow Problems The governing mathematical model of the incompressible inviscid fluid flow is a boundary value problem of partial differential equation. For the velocity potential formulation, the condition of irrotationality and the continuity equation of the twodimensional incompressible potential flow lead to the following Laplace equation:
+.
0
(164)
where <j> is the velocity potential. And the Neumann boundary condition on velocity is + ma
(165)
where v is the velocity in the outward normal direction, and I and m are the direction cosines of the outward unit normal, while on the Dirichlet boundary the velocity potential is assumed. Solution of the boundary value problem of Laplace equation provides the velocity components through the following definition vx =
(166)
242
5.1. Mapping
C.-N.
Chen
transformation
The element configuration would change from element to element in the mesh. By introducing an invertible transformation between a master element Q of regular shape and an arbitrary physical element fie, it should be possible to transform the partial differential operations on Sle so that they hold on fl. The mapping of Cl onto Cle is defined by the following coordinate transformations X~i = XI&)
(167)
e
where x^ are physical coordinates in Q and £ r are natural coordinates in Cl. Then the transformations of the first order partial derivatives of velocity potential are 4>,i = Zr-i,r
(168)
where £ r j is the inverse matrix of the Jacobian matrix xj r. And the transformations of the second order partial derivatives of the velocity potential are ,ij = f r , i £ a J < A , r s + €t,ij,t-
(169)
The outlined mapping transformations are generic which hold good for adopting any kinds of appropriate analytical functions. Thus various domain configurations and mapping techniques can be adopted. The simulation for transformation adopting the polynomial is carried out. The transformation relations are expressed by xj = N^r)xiy,
7 = 1,2,...,JV C
(170)
where xjy are xj and/or their possible partial derivatives with respective to £ r at nodes used to define the transformations, N7(£r) are the corresponding shape functions and Nc is the total degrees of freedom. Using (170), the first order partial derivatives of the physical coordinates with respect to the natural coordinates can be obtained. a
*,€j='ftW&.)ii7
(171)
And the second order partial derivatives of the physical coordinates with respect to the natural coordinates are x
*,«j€s = My.Cjfc (&•)**,•
(172)
The mapping transformations for serendipity C° triangular elements, serendipity triangular elements with incomplete first order derivatives, serendipity C° quadrilateral elements and serendipity Hermitian quadrilateral elements are illustrated. 5.1.1. Serendipity C° triangular elements In constructing the mapping transformation, the master triangular element in the natural space may be an arbitrary linear triangle. For convenience, the rectangular unit triangle is adopted. Let the natural coordinates be L\ and Li. A representative generic rectangular unit C° triangular element is shown in Fig. 26. Assume that the
Optimization
in FE and DQE
Techniques
l+m+n n+1 Fig. 26.
The serendipity rectangular unit C ° triangular element.
physical coordinates are n t h order on side 1 - n + 1; l-th. order on side n + 1 n + l + 1 and m - t h order on side n + i + 1 - 1 in t e r m s of t h e natural coordinates. T h e shape functions can be expressed by
Li
Ni
2(1
-L2)
^_i+1(l-L2)*?_i(i2)
L2
+ 2(1-Li) M =
*S-*+i(ii)*"-i(l-ii).
2 *{+n+i_i(£i + £2)*i_n_i(i - U 2(Li+L2) 1-Ll-L2,T; + 2(1-L2) *{+„+l-i(i2)*U-l(l-i2),
n + 2 < i
2
1-Li-L2 2(1-Li)
+ 2(£l
L2)
+l ^m+n+X-^l-il)^-/-^!^!)
+ L2)
* I + m + n + l - i ( l " Ll - L 2 ) * £ , - „ - l ( I l + £2),
/ + n + 2 < i < / + m + n, 2{LX+L2)
2(1 - L i ) +
£1 2(1-L2)
n{
2 , +
2(l-L1)
n{
lj
2
244
C.-N.
Chen
"™ = m£w*i{1 ~Ll ~u)+^r^r*<(1 -L2) 1 - Lx - L2 where
fc=l
V
= 1,
'
P= 0
(174)
in which h is the order of one side. Assume that the three sides have the same order n and define
Km-Ui—). = 1,
»^» p=l.
(175)
Then the shape functions can be expressed by
N =
* w^^)lK-»i{Li)*-i{1-Li)
+ A£_ i + 1 (l-L 2 )A?_ 1 (L 2 )], 2<*
i
1
- L2) n + 2 < i < 2n,
Li(l-Li-L2 1 T T T ! 2 ( i - 2KZ n - l TTT^Z. ) ( 3 n - ir T+ Tl T ) ' ^ n - i + i l - Li - L 2 ) x A?_ 2 „_!(Li + L 2 ) + A£„_ i+1 (l - Li)A?_ 2n _ 1 (L 1 )] ) 2n + 2 < i < 3rc,
M = Y M ( i i ) + A£(Li + L 2 ) + A£(l - L 2 ) - 1], AWi = Y [A^(L2) + A£(l - Lx) + A^(Li + L 2 ) - 1], A W i = ^ ^ " ^ [ A ^ l - Lx - L 2 ) + A^(1-L2) + A ^ ( 1 - L 1 ) - 1 ] .
(176)
Optimization
in FE and DQE
Techniques
245
5.1.2. Serendipity triangular element with incomplete first order derivatives Denote C the natural coordinate along a straight line having n nodes in the Li - L2 plane with Ci = 0,
+ {L2(n) - L2{1))x% (177)
i = 1,2,.. .,n where
f? = [i+vac - Ci)i b?(Oi 2 , g?(0 = « - co b?(0] 2 , 1
c=-2 n fc=l,fc^t
Ct -
„ , ..
Cfc'
<, _ Cfc
TT
tf(o= n
fc=l,/c#i C i - C k '
Equation (177) can be used to derive shape functions of serendipity triangular elements with incomplete first order derivatives. Serendipity Hermitian triangular elements with higher order derivatives can also be used for the mapping transformation. 5.1.3. Serendipity C° quadrilateral elements In constructing the mapping transformation, the master quadrilateral element in the natural space may be an arbitrary rectangle. For convenience, the unit square is adopted. A representative serendipity unit C° quadrilateral element is shown in Fig. 27. Assume that the physical coordinates are pth order on side 1 - p + 1;
(1,1) p+q+1
p+q+r+l
p+q+r+s «
(0,0) Fig. 27.
»
»
•
P+l
T h e serendipity unit C° quadrilateral element.
246
C.-N.
Chen
functions can be expressed by Wi = ( l - ^ + 1 ( 0 , Ni=^tp(v),
2
P + 2
Ni = Wptg+r+2-z(0>
+ q,
P + q + 2
1
Ni = (l-t)tp'p+
q+r+t+2_i(Ti),
+ q + r,
p + q + r + 2
+ q + r + s,
N, = (1 - »7)tf+1(0 + (1 - 0¥>i+1fa) ~ (1 - 0(1 - V), Np+1 = (1 - V)?+1fa) - £(1 - V), Np+q+r+1 = w i + 1 ( 0 + (l - o ^ £ - (i - OvAssume that the four sides have the same order p. Then the shape functions can be expressed by
iv^a-r,)^1^), Ni=Z
2
p+
^ = W3p+ 2 -i(0,
2
2p +
JVi = (1 - Ov&k-ifa).
2
+1
Nx = (1 - r?)^ (0 + (1 - OvT^fa) - (1 - 0(1 - r,), Np+1 = (1 - »,)^1(0 + ^+1(V) ~ €(1 - »?), iV2p+i=W^(0+^:i(r?)-^,
(179)
AT3p+i = W ? + 1 (0 + (1 - 0¥#lfa) " (1 - 0»75.1.4. Serendipity Hermitian quadrilateral elements Consider that the unit square in the natural space has an m x n serendipity grid. By using the two-dimensional node identification method, the following relations hold: DlD^Xap = D | £ > ^ a / 3 ,
0<s<m-l,0
(180)
where the following relation is used,
t
vn DlD^p =
dl'+M&l)
(181) a0
The physical coordinate x can be expressed by x(Z,r))=xs+xn-x.
(182)
Optimization
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247
Techniques
Let a(0,0), 6(1,0), c(l, 1) and d(0,1) be the four corner nodes. Then rn — 1
x
n-1
M
53 (/e- K£)£|) E ( # r i = aYl = l s=0 t=0
M
M ^ i«b
I%-1'\2,Ti)Dtrixa\*\
+
m— 1
n-1
1t M Yl 53(#£- ' (n,r,)Z?«) ^ f tfr (2,0£f^ U.
v
n=
s=0
3=1 U = 0
+ fr 2 ro - 1 ' s (2,0£>?a:/3|6 C ) and m—1n—1
^=EE EE s=0
t=0
+ ^"-1(2,OHr1(2,»?)r>II>^ where H™~1,s and HTp '* are Hermitian polynomials. 5.2. DQEM
formulation
Consider that the two-dimensional master element has two natural coordinates £ and rj. Then, the substitution of the transformation equations for <xx and
+ Fi& V)4>%n + ^ ( £ , i\Wm
+ F4(t, v)^
+ F 5 (e, v)
(183)
where *l(£. V) = C + &
F
^ V) = 2(C,x7?,x + iyV,y),
F3&V) = ^ * + V%, F
s(^,v)
Fi&V)
= £,** + tyy,
(184)
=V,xx +V,yy
And the substitution of (168) into (165) leads to the following transformed equation, v = (l£,x + m£,y)
(185)
Equation (166) can also be transformed by using (168), Vx=€,x,t
+ V,x,v>
V
V = t.,y,Z +V,y
(186) rs
The kinematic transition condition on the inter-element boundary d£l ' of two adjacent elements r and s is the continuity of velocity potential, which is expressed
248
C.-N.
Chen
as r = s, on
dW's
'
(187)
or the constraint of velocity potential which is expressed as r = ' = 4>r<',
o n <9fT's
(188)
rs
where 4> ' is the prescribed velocity potential. By using (185), the natural transition condition on the inter-element boundary can also be written as
0rC* + ™Tv) Vz + OX* + mT
( 189 )
where q is the rate of flow into the domain. Although the 2 x 2 quadrilateral element and the 3-node triangular element can be used to solve the two-dimensional potential flow problem, the convergence character is poor. Considering the quadrilateral element adopting the Lagrange family grid, by using (63) the discretization of (83) at node (a,/?) leads to the following equation:
+ F4(£a, V0)DeJm$em0 + F5(£a, V0)De£Kn
=0
(190)
where D^m and De^n are the weighting coefficients for the second order derivatives in £ and rj directions, respectively, and D^m and De^n are the corresponding weighting coefficients for the first order derivatives. Consider the inter-element boundary which is the £ = 1 side of element r and the £ = 0 side of element s. Then, the discrete continuity conditions of velocity potential are expressed by *Nifi = *if),
/?=l,2,...,JV„
(191)
and the constraint of velocity potential at a node on the inter-element boundary is
*N(f,=*u = *y-
(i92)
Using (189) the natural transition condition at a node on the inter-element boundary can also be obtained.
[lrNi{0)€Nl:(l3),x + miV5(/3)^iV€(/3),yJ ®' N^m^mp + [j^ (0)rlNi (0),x + mN4(/3)^V4(/3),2/)^3n$JV{n
+
[}i{0)€l(0),x + ml(0)£l(0),y) Dlm®m0
+ {ll(0)Vl{0hx + m{mVs1{0),y)Ds0mn
= q.
(193)
Letting element n be an element with £ = 1 side being the Neumann boundary, the discrete Neumann boundary condition at a node can be obtained by using (190) in
249
Optimization in FE and DQE Techniques
(185). v
- ^JV4(/3)?JV4(/3),x + mNi(0)^Ni(0),yJ
+ (l^WV^U
±J
NimVm0
+ ™nNd0)VnNiW,y) DZ^n-
(194)
The velocity components at discrete points in an element e can also be expressed by using (186). V
x,a0 - ^(a){0),xUam^m0
5.3. Mesh and element
+ V(a){0),x1J0n*ani
.
.
grid
Various techniques can be used to generate the mesh and element grids. The mapping technique is used. The selection of grid can be flexible. Considering the Lagrange family grid, denote N^ and Nv the numbers of levels in £ and r\ directions, respectively, in the master element of a physical element. The minimum value of N$ and Nv is 2. The convergence character of this 4-noded element is poor. If the element is a rectangle and the element sides are parallel to x or y axes, mapping results in scaling the lengths of the sides. It can reduce the arithmetic operations in calculating the element weighting coefficients. For solving a problem having irregular analysis domain boundary which has a curved line or a straight line not parallel to x or y axes, by designing the mesh in such a way that as many elements as possible are rectangles with the element sides parallel to x or y axes, the assembled overall algebraic system will have more zero elements in the coefficient matrix. It can reduce the computer storage and CPU time required for solving the problem. Mapping is also not necessary for triangles having three linear sides. Figure 28 shows an efficient mesh. The concept of generating efficient mesh can be used similarly to design efficient element grid of an irregular element with the inclined or curved element side attached to the analysis domain boundary. Figure 29 shows an efficient element grid model of an irregular element. For the GDQ discretization of this element grid model, only the calculations of weighting coefficients for partial derivatives at nodes on the curved element side need to solve linear algebraic systems or find the inverse matrix of a coefficient matrix. As regards the form of the assembled overall coefficient matrix, when assemble a discrete governing equation at an interior node, the number of data filled in is N^+Nv — 1 for the efficient element grid model as compared to N% x A^ for a fully irregular element grid model. When assemble a discrete natural boundary condition if the natural boundary is a straight line parallel to one of the coordinate axes, the adoption of efficient element grid model can also reduce the number of data filled in. Considering a boundary node at the intersection of a natural boundary line parallel to the y axis and an interior grid line parallel to the x axis, the number of data filled in is N$ for the efficient element
250
C.-N.
Fig. 28.
Chen
T h e efficient element.
I ' l l
I
I
I
I
I I I
I
I-+ + ++ + 4- + + -I- + + + + + + + + + + + + + -4- + + + ' -+ + + + + + + + + + + + + • 4- + + + + + + + + + + + 4 --t-H-+4- + + +- + + + + + + + + + + + + + +• + + +-++• i
i
i
i
' i
'
'
i
'
'
'
'
->- x Fig. 29.
The efficient element grid model.
grid model as compared to N^ + Nv — 1 for a fully irregular element grid model. When assemble the discrete natural boundary condition at a node on an inclined or curved natural boundary line the number of data filled in is also N% + Nv — 1. When assemble the discrete natural transition condition the number of data filled in can
Optimization
in FE and DQE
Techniques
251
also be reduced if the efficient mesh and efficient element grid model are used. It should be mentioned that the design of efficient element grid model for an element with inclined and/or curved sides attached to the analysis domain boundary might result in an extremely non-uniform distribution of element nodes. The DQEM and GDQEM analyses using elements having extremely non-uniform distributions of element nodes will also have excellent numerical performances. Sample analyses have been carried out which have proved the fact. 39 It should also be mentioned that in the DQEM and GDQEM analyses efficient mesh and efficient element grid can be designed so that in discretizing a fundamental relation defined at a discrete point in an element or on an element boundary parallel to one of the coordinate axes only the standard DQ has to be used which can significantly reduce the computer memory requirement and CPU time. For two adjacent elements having different numbers of nodes on the inter-element boundary, the number of kinematic transition conditions must be larger than the number of natural transition conditions. Let rid denote the difference between the two node numbers which equals the difference between the two numbers of transition conditions. To set up the kinematic transition conditions, the nodes on the interelement boundary must be arranged in such a way that only rid nodes in one element are not coincident with the nodes in the other element. The rid extra nodes are used to define rid extra kinematic transition conditions. In defining the rid extra kinematic conditions, the interpolation technique must be used. In addition, the transition conditions can easily be set up by designing the grids of the two adjacent elements in such a way that both elements have the same numbers of nodes on the inter-element boundary no matter what the orders of approximations and grid configurations are. The concept of efficient element grid can be similarly used to design the grids of two adjacent elements which have this type of connection. The direction cosines of the outward unit normal vector on the element boundary are necessary for defining discrete natural transition conditions and discrete Neumann boundary conditions. Various techniques can be used to calculate the direction cosines. The mapping technique used for generating the mesh and grid in conjunction with the tangent operation and the transformation operation of the first order Cartesian tensor are adopted. In the sample analysis, bilinear and quadratic elements are used. 5.4. Overall algebraic
system
For the DQEM potential flow analysis model, the total degrees of freedom must equal the number of discrete constraint equations. 37 ~ 39 The governing equation at a node is a governing equation constraint condition. An interior node can define only one discrete governing equation. The discrete Dirichlet and Neumann conditions are defined on the Dirichlet and Neumann boundaries, respectively. A node on the analysis domain boundary (ADB) but not an element corner node can define one boundary condition and one governing equation. At a node on the inter-element
252
C.-N.
Chen
boundary, if the node is not an element corner node, in addition to the continuity of velocity potential one discrete constraint equation of prescribed velocity potential or natural transition condition and up to two discrete constraint governing equations attached to the two adjacent elements can be denned. An element corner node might be able to define even more constraint equations. Consider that discrete governing equations are only defined at interior nodes. For an element corner node in the analysis domain which is the common node of iV/v natural inter-element boundaries (IEB) and NK kinematic inter-element boundaries, let NT denote the number of all constraint conditions. Then, for NK ^ 0, NT = NN + 1; for NK = 0 and <j> not prescribed at the node, NT = N^; for A ^ = 0 and cj> prescribed, NT = NN + 1 . The equation of setting as a prescribed value is also a kinematic constraint condition. Figure 30 is a typical element corner node in the analysis domain. For an element corner node on the analysis domain boundary (ADB) with two element-based segments of the analysis domain boundary being connected to it, if the two segments are Neumann boundaries, NT for various connections are: for NK / 0, NT = NN + 3; for NK = 0 and <j> not prescribed at the node, NT = NN+2; for NK = 0 and <j> prescribed, NT = NN + 3. For this type of element corner node, if only two elements are connected to the node with the inter-element boundary perpendicular to the analysis domain boundary which is a straight line within the two connected elements, then only one Neumann condition can be applied. If one of the analysis domain boundary segment is Neumann boundary while the other one is Dirichlet boundary, NT for various connections are: for NK ^ 0, NT = NN + 2; for NK = 0, NT = NN + 2. And if both of the two analysis domain boundary segments are Dirichlet boundaries, NT for various connections are: for NK ^ 0, NT = NN + 1; for NK = 0, NT = NN + 1. A representative of this type of element corner node is shown in Fig. 31 which is the type 1 element corner node on the analysis domain boundary. For an element corner node on the analysis domain boundary with one elementbased segment of the analysis domain boundary connected to it, if the segment is Neumann boundary, NT for various connections are: for NK ^ 0, NT = N^ + 2;
Fig. 30.
Typical element corner node in the analysis domain.
Optimization
in FE and DQE
Techniques
Fig. 31.
Type 1 element corner node on the analysis domain boundary.
Fig. 32.
Type 2 element corner node on the analysis domain boundary.
253
for Nfc = 0 and not prescribed at the node, NT = NN + 1; for NK = 0 and prescribed, NT = NN + 2. If the analysis domain boundary segment is Dirichlet boundary, NT for various connections are: for NK ^ 0, NT = NN + 1 ; for NK = 0, NT = NN + 1. A representative of this type of element corner node is shown in Fig. 32 which is the type 2 element corner node on the analysis domain boundary. Consider that only the values of velocity potential at nodes are used to define the GDQ discretization. Then in order to satisfy all constraint conditions at an element corner node, at the assemblage stage, one degree of freedom might not enough. However, we can use more than one constraint condition at that corner node by neglecting certain constraint conditions at interior nodes or at node on the inter-element boundary and other than corner nodes, and giving their degrees of freedom to the inclusion of extra constraint conditions other than the first one. For the DQEM analysis model, the NT constraint conditions can partially or fully be satisfied. We can also neglect all of the NT constraint conditions and give the degree of freedom of that node to the discrete governing equation at that node. The discrete governing equation at the element corner node can be defined as the average of the discrete governing equations of all elements connected to that node. The various techniques for selecting and implementing the constraint conditions at element corner nodes are flexible. Different approaches lead to different programming efforts. The overall algebraic system obtained by assembling all discrete constraint conditions is the discrete governing/transition/boundary equation system.
254
C.-N.
Chen
The GDQ can adopt the degrees of freedom used to represent the derivatives or partial derivatives of the variable function. In conjunction with the use of the EDQ with which the GDQ discretization can be defined at discrete points which are not the nodes, the DQEM can also assign the degrees of freedom of the partial derivatives of the variable function to the element boundary nodes. For analyzing the two-dimensional potential problems in order to automatically set the kinematic transition conditions by only using certain degrees of freedom assigned to the element boundary nodes, the degrees of freedom representing the variable function must be assigned to the element boundary nodes. The degrees of freedom representing the partial derivatives of the variable function can also be assigned to the nodes of all neighbor elements on the inter-element boundary and the compatibility conditions of the higher order partial derivatives can also be considered. However, if certain external cause such as the fluid flow, conduction heat flux, etc. is applied, no compatibility condition of partial derivatives can be considered. The discrete governing equations can be defined on the inter-element boundaries as the average discrete governing equations of multiple elements. They can also be defined on the element boundaries without adopting the average treatment. Thus, elements having no interior node can also be used to the DQEM analysis. For analyzing beam or plate problems in order to automatically set the kinematic transition conditions by only using certain degrees of freedom assigned to the element boundary nodes, the degrees of freedom representing the lateral displacement and first order derivative or partial derivatives of the lateral displacement must be assigned to the element boundary nodes. The degrees of freedom representing higher order derivatives or partial derivatives of the displacement can also be assigned to the nodes of all neighbor elements on the inter-element boundary and the compatibility conditions of the higher order derivatives or partial derivatives can also be considered. However, if the moment is applied the highest order of derivative or partial derivative that the compatibility condition can be considered is one. On the other hand, if the lateral force is applied on the inter-element boundary the highest order of derivative or partial derivative that the compatibility condition can be considered is two. The concept can also be used to treat the boundary conditions. It should be noted that if the highest order of derivative or partial derivative assigned to the element boundary nodes is larger than one, the EDQ has to be used. The philosophy inherent in the outlined techniques for defining discrete connection conditions on the inter-element boundaries, the discrete boundary conditions on the boundary and the discrete constraint conditions at the element corner nodes also holds good for other scientific or engineering problems. 5.5. Outward
unit normal
vector on element
boundary
Direction cosines of the outward unit normal vector on the element boundary are necessary for the natural transition conditions and the Neumann boundary conditions. They can be calculated by using the mapping technique. 37-39 Consider
Optimization
in FE and DQE
255
Techniques
the bilinear element. The calculation of the direction cosines at points on side r) = 0 is illustrated. The position vector of a point on the side is R(£) = xi + yj = [(1 - 0 * i + fr2]i + [(1 - Ovi + tV2]l
(196)
Then the unit tangent vector t can be expressed by f = - £ -
=
|^f |
^-x1)t
+ (y2-yi)3 2
[(x2-x1)
+
=
-
u
-pl
(197)
(y2-y1)^
Hence, the direction cosines of t are S2-S1
5
J/2-2/1
,10CA
a =
r , /? = (198) r. [(3:2 - * i ) 2 + (2/2 - 2/1)2]* K*2 - x i ) 2 + (2/2 - yi)T2 And the direction cosines of the outward unit normal vector n = li + mj can be obtained. 7T
—
7T
—
7T
/ = a cos — + /3 sin — = /?,
—
7T
m — —a sin — + j3 cos — = —a.
(199)
For the quadratic serendipity element, the position vector of a point on side r\ = 0 is R(0 =xi + yj = [(2£2 - 3£ + 1)1! - 4£(£ - l)x 2 + £(2£ - l)x 3 ] 1 + [m2 - 3£ + l)2/i - 4£(£ - 1)2/2 + ^(2? - 1)2/3] J-
(200)
Then the unit tangent vector t can be expressed by _
di
Ai + Bj
(201)
*^f-\ (A2+B2)* where A = (4£ - 3 ) n + 4(-2£ + l)x 2 + (4£ - l)x 3 , B = (4£ - 3 ) y i + 4(-2£ + 1)2/2 + (4£ - 1)2/3.
(202)
Hence, the direction cosines of t are A
a =
n
r,
/? =
B
,
r.
V (A2 + B2)i {A2 + B2)i And the direction cosines of the outward unit normal vector can be obtained.
/ = /3,
m
= -a.
s
(203) ;
(204)
C.-N.
256
5.6. Numerical
Chen
examples
The first problem being solved concerns the flow past a cylinder in a rectangular channel with a uniform inlet flow. The problem is shown in Fig. 33. By using centerline symmetry and midstream antisymmetry, one fourth of the domain shown in Fig. 34 is used for the analysis. The boundary of the analysis domain consists of four Neumann boundaries and one Dirichlet boundary. The Neumann boundary conditions involve zero normal velocity v = $,„ = 0 along AB, BC and DE, and a uniform inflow v = — 1 along AE. The antisymmetry on CD leads to vy = 0. Thus $ is constant along CD, and is set to be zero in the analysis. Five elements are used to model the analysis domain with the mesh being shown in Fig. 34. Two of the five elements are quadratic serendipity elements, one is the bilinear element and the other two are regular elements. In generating the coefficients for discretizing all relation equations, the numerical computations can be reduced for regular elements by using the scaling procedure instead of using the mapping technique. On AB, for all discrete points excepts A and B one Neumann condition per node is applied. The applied Neumann condition at G is attached to element 1. On BC, for all
—T
v=-l
ii ,1
Fig. 33.
E(0,4)
1
1 *
tV
1 1 4
\J
1
t
1 1 "" 1
1 .!3 v=l
,
Potential flow past a cylinder in a rectangular channel.
K(4,4)
H(6,4) 4 J(6,3)
4)
F(8,3) C(8,2)
1(6.5857864,1.4142136)
A(0,0) Fig. 34.
G(4,0)
B(6,0)
Mesh for analyzing the problem of flow past a cylinder.
Optimization
in FE and DQE
Techniques
257
discrete points except C one Neumann condition per node is applied. The applied Neumann condition at I is attached to element 3. On CD, each discrete point is set to have one Dirichlet condition. On DE, for all discrete points except D and E one Neumann condition per node is applied. On AE, each discrete point is set to have one Neumann condition. One constraint condition at J is considered in the analysis. Numerical tests adopting four different types of constraint, separately, are carried out. Type 1 constraint is the natural transition condition on JH. Type 2 constraint is the natural condition on JF. Type 3 constraint is the natural transition condition on JG. And type 4 constraint is the discrete governing equation at J. All five elements have the same type of element grid. Equally spaced discrete points in both £ and r\ directions are adopted for defining the element grid. The p refinement analyses were carried out. Numerical results are summarized and listed in Table 8. It shows that the velocity potentials and velocities at certain discrete points converge fast by gradually increasing the discrete points in an element for all of the four different types of constraint. The convergence characters of the four different types of constraint make no significant difference in the sense that all different types of constraint have the same numerical results up to five digits by only increasing the element discrete points up to 5 x 5. This problem can also be solved by only using two elements to model the analysis domain. By replacing the cylinder in the first problem with a square shaft having the length of the side equal to the diameter of the cylinder, another sample problem is solved. The analysis domain of the problem is also one fourth of the problem domain. Three elements are used to model the analysis domain to study the convergence by gradually increasing the discrete points in an element with the mesh being shown in Fig. 35. On AB, for all discrete points except A and B one Neumann condition per node is applied. On BN, for all discrete points except N one Neumann condition per node is applied. On CN, for all discrete points except C and N one Neumann condition per node is applied. On CD, each discrete point is set to have one Dirichlet condition. On DE, for all discrete points except D and E one Neumann condition per node is applied. On AE, each discrete point is set to have one Neumann condition. The constraint condition equation at TV is the discrete governing equation. In carrying out the p refinement analysis, equally spaced discrete points in both x and y directions are adopted for defining the element grid. All three elements have the same type of element grid. Numerical results of velocity potentials and velocities at certain discrete points converge fast by gradually increasing the discrete points in an element, which can be seen in Table 9. The convergence test for the h refinement procedure is also carried out by increasing the number of 3 x 3 grid elements. One constraint condition for each element corner node in the analysis domain is applied. It is the discrete governing equation at the corner node. Figures 36 and 37 show the 3 x 3 grid, 12-element mesh and 3 x 3 grid, 40-element mesh, respectively. The constraint condition equations at N and all interior element corner nodes are the average governing equations. The numerical results are listed in Table 10. They
258
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k S
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•
C.-N. Chen
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o o o o o o o o o o o o o o o o o o o o o o o o -s-" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ele ent grid ^H
s
211672 172663 178964 177848 178065 178007 210847 172633 178965 177848 178065 178007 211185 172633 178964 177848 178065 178007 212009 172631 178964 177848 178065 178007 230202 186001 192648 191459 191702 191642 231950 186022 192656 191462 191704 191643 231234 186014 192653 191461 191703 191642 229488 186086 192646 191460 191702 191642 317507 251048 261774 261106 262216 262727 316755 251087 261775 261106 262216 262727 317063 251086 261774 261106 262216 262727 317814 251093 261774 261106 262216 262727 582953 487465 502445 498822 499522 499344 582088 487469 502446 498822 499522 499344 582442 487467 502445 498822 499522 499344 583306 487480 502445 498822 499522 499344 105347 986284 100184 998552 999217 999053 105142 986281 100183 998552 999217 999053 105226 986282 100183 998552 999217 999053 105431 986274 100184 998552 999217 999053
Constraint type
Optimization
in FE and DQE
Techniques
259
1
M(7,<
E(0,4)
D(8,4) 3 F(8,3)
2
L*(7,3) K(7,2) C(8,2) N(6,2)
1
r- x
A(0,0) Fig. 35. Table 9.
B(6,0)
3-element mesh for analyzing the problem of flow past a square shaft.
Convergence of the p refinement analysis for the solution of flow past a square shaft.
Element §rid
3x3 5x5 7x7 9x9 11 X 11 13 x 13 15 x 15 17 x 17
A
<S>B
-0.159752 x 102 -0.112587 x 102 -0.111215 x 102 -0.110788 x 102 -0.110590 x 102 -0.110487 x 102 -0.110428 x 102 -0.110391 x 102
Fig. 36.
f^.Kr 2
-0.129752 x 10 -0.681504 x 101 -0.653784 x 101 -0.649073 x 101 -0.647264 x 101 -0.646098 x 101 -0.645328 x 101 -0.644843 x 101
0.382308 0.195833 0.209593 0.208901 0.210108 0.210427 0.210688 0.210825
v., 1
x 10 x 101 x 101 x 101 x 101 x 101 x 101 x 101
0.275326 0.199518 0.199353 0.199146 0.198983 0.198877 0.198815 0.198780
u 1
x 10 x 101 x 101 x 101 x 101 x 101 x 101 x 101
„
0.239665 x 101 0.196994 x 101 0.193430 x 101 0.192524 x 101 0.192076 x 101 0.191895 x 101 0.191796 x 101 0.191734 x 101
12-element mesh for analyzing the problem of flow past a square shaft.
also converge well by gradually increasing the number of elements. This problem can also be solved by using less elements to model the analysis domain. 6. G D Q E M analysis 6.1. Extended
GDQ
discretization
Consider an arbitrary M-coordinate GDQ model. 33 The dimension and grid configuration for defining the discrete point and node can be different. By adopting an
260
C.-N.
Fig. 37. Table 10.
40-element mesh for analyzing the problem of flow past a square shaft.
Convergence of the h refinement analysis for the solution of flow past a square shaft.
Mesh type A 3 12 40
3x3 3x3 3x3
Chen
V
B 2
-0.159752 x 10 -0.115443 x 10 2 -0.113064 x 10 2
x,C
2
-0.129752 x 10 -0.733397 x 10 1 -0.678379 x 10 1
V
"x,D
0.275362 x 10 1 0.190374 x 10 1 0.202098 x 10 1
0.310986 x 10 1 0.202478 x 10 1 0.198794 x 10 1
*,F
1
0.168343 x 10 0.203940 x 10 1 0.203537 x 10 1
one-dimensional node identification method to express both the discrete point and node, the extended GDQ discretization for a partial derivative of order m + ... at discrete point a can be expressed by 0(m+-),
3£"
^=D«"-$i)
i = l,2
JV.
(205)
The variable function can be a set of appropriate analytical functions denoted by Yj(£,...). The substitution of Tj(£,...) in (205) leads to a linear algebraic system for determining D ^ '". The set of analytical functions can also be expressed by a tensor having an order other than one. The variable function can also be approximated by ^ , . . . ) = *j(£.-..)*j.
J = 1,2,
,N
(206)
where $ j are the values of variable function and/or its possible partial derivatives at the iV/v nodes, and \?j(£,...) their corresponding interpolation functions. Adopting the set of ^j(^,...) as the variable function 4>(£,...), the same procedure can also be used to determine D^ ". And the (m + .. .)th order partial differentiation of (206) at discrete point a also leads to the extended GDQ discretization equation (205) in which D^ " i s expressed by 0(m+-)*.
Dt
(207)
d?
The variable function can also be approximated by (£,...) = Tj(t,...)cj,
j =
l,2,...,N.
(208)
Optimization
in FE and DQE Techniques
261
Then the weighting coefficients can also be obtained by Q(m+~)-£-. Xrt1-
(209)
In (208), the unknown coefficients and appropriate analytical functions can also be expressed by certain other tensors having orders other than one. By adopting a two-dimensional node identification method to express both the discrete point and node, the extended GDQ discretization for a partial derivative of order m + • • • at discrete point (a, S) can be expressed by ™« = £>«•••$«.
(210)
The variable function can be a set of appropriate analytical functions denoted by Tj S (£,...). The substitution of Tj S (£,...) in (210) leads to a linear algebraic system for determining D^Sil'. The set of analytical functions can also be expressed by a tensor having an order other than two. The variable function can also be approximated by ^ , . . . ) = *J-.(e,-.-)*j.
(2ii)
where $ J S are the values of variable function and/or its possible partial derivatives at the nodes, and ^fjS(^,...) are their corresponding interpolation functions. Adopting the set of *&ja(€, • • •) as the variable function (/>(£,...), the same procedure can also be used to determine £ ^ M . And the (m + • • • )th order partial differentiation of (211) at discrete point (a, 5) also leads to the extended GDQ discretization equation (210) in which D^a&il' is expressed by
aSil
Q£m . . .
(212) a5
Consider that the coordinate variables are more than one and that the interpolation functions can be expressed by forming products of two sets of functions with one set defined by three coordinate variables while the other set defined by the remaining coordinate variable. Then a representative of this type of interpolation functions can be ^fj(^, • • .)itfs('d). By using this type of interpolation functions, the weighting coefficients show to have the following form: D%u»P=D%-D%.
(213)
If the dimension of node identification and the number of coordinate variables are the same in expressing both the discrete point and node, the extended GDQ discretization for a partial derivative of order m + • • • + p at discrete point (a,..., 5)
262
C.-N.
Chen
can be expressed by
dt™ •••&>* =Dt:srA-i-
(214)
The variable function can be a set of appropriate analytical functions denoted by Yp...s(£, • • • >^)- The substitution of Y p ... s (£,... ,1?) in (214) leads to a linear algebraic system for determining D^''5i_v The set of analytical functions can also be expressed by a tensor having an order other than M. The variable function can also be approximated by 0(£,...,tf) = ¥,..•«(£.•--.tf)*,,-.
(215)
where $ p ... s are the values of variable function and/or its possible partial derivatives at the nodes, and \I/ P ... S (£,... ,$) their corresponding interpolation functions. Adopting the set of typ...s(£,..., -d) as the variable function (£,..., tf), the same procedure can also be used to determine D^,/'Si%l. And the (m + • • • + p)th order partial differentiation of (215) at discrete point (a,..., 5) also leads to the extended GDQ discretization equation (214) in which D^ "5i t is expressed by U
a-Si—l
'
*--*
...V
(216)
Consider that the interpolation functions can be expressed by the product of two sets of functions with one set denned by M — 1 coordinate variables while the other set defined by the remaining coordinate variable. Then a representative of this type of interplation functions can be ^ p ...(£,.. .)^ s (?9). By using this type of interpolation functions, the weighting coefficients show to have the following form:
^sZ^^-lM-
(217)
If the interpolation functions can be expressed by the product of P sets of functions with P larger than two and a coordinate variable only appearing in one set of functions, similar procedures can be used to determine the weighting coefficients which are components of the outer product of P tensors. The grid configuration of a two-coordinate grid model can be a triangle, a quadrilateral or a certain other configuration. There are three types of triangular grid: (a) Pascal triangular grid, (b) triangular grid having no interior discrete point, (c) triangular grid having interior discrete points but not being the Pascal triangular grid. And there are also three types of quadrilateral grid: (a) Lagrange family grid, (b) quadrilateral grid having no interior discrete point, (c) quadrilateral grid having interior discrete points but not being the Lagrange family grid.
Optimization
in FE and DQE
Techniques
263
The grid configuration of a three-coordinate grid model can be a triangle with the variable function defined by the area coordinates, a tetrahedron, a triangular prism, a hexahedron or a certain other configuration. There are three types of tetrahedral grid: (a) Pascal tetrahedral grid, (b) tetrahedral grid having no interior discrete point, (c) tetrahedral grid having interior discrete points but not being the Pascal tetrahedral grid. There are also three types of triangular prism grid: (a) Lagrange family grid having the Pascal triangular cross sections, (b) triangular prism grid having no interior discrete point, (c) triangular prism grid having interior discrete points but not being the Lagrange family grid. And there are also three types of hexahedral grid: (a) Lagrange family grid, (b) hexahedral grid having no interior discrete point, (c) hexahedral grid having interior discrete points but not being the Lagrange family grid. Consider the Pascal triangular grid with the coordinate variables being the area coordinates L\, L2 and L3 and let the variable function be approximated by the following equation which uses the three-dimensional node identification method, (L1,L2,L3) = * p ( L i ) * ? ( L 2 ) * f ( £ 3 ) * M f
(218)
where p, q and f are the numbers used to define a specified node x, with respect to L\, L2 and L3, respectively, and ^p(Li), ^ ( Z ^ ) and ^ ( L a ) can be defined by using the Lagrangian interpolation formula:
*f(L*)= ff .•r^k,t Trrrrr ~ -kfc.g
(219)
q=l,q&
where n is the order of the approximation. Then, the weighting coefficients for a partial derivative of order m + n + o at discrete point (a, (3,7) can be obtained by the following equation DLal%L°3=DtlD%D!;i.
(220)
The grid configuration of a four-coordinate grid model can be a triangular prism with three of the four coordinate variables being the three area coordinates for the triangular cross sections, the tetrahedron with the variable function defined by the volume coordinates or a certain other configuration.
264
C.-N.
Chen
Consider the triangular prism of Lagrange family grid having the Pascal triangular cross sections, and let the variable function be approximated by 4>{Li, L2, L3,0
= tfp(Li)*q-(L2)tf f(£3)*s-(C)$p
(221)
Then, the weighting coefficients for a partial derivative of order m+n+o+p at discrete point (Li, L2, Lz, C) can be obtained by the following equation: D$$£?
= D$D$D%D%.
(222)
Also consider the Pascal tetrahedral grid with the coordinate variables being the volume coordinates L\,Li,L% and L4 and let the variable function be approximated by the following equation which uses the four-dimensional node identification method, 0(Zi, L2, L 3 , U) = * p ( L i ) * 9 ( l 2 ) * r ( i 3 ) * a ( £ 4 ) $ M «
(223)
where p, q, f and s are the numbers used to define a specified discrete point with respect to L\, L2, L3 and L\, respectively, and ^p(Li), tyq{L2), ^IfiX'z) and ^s(Li) can be defined by using the Lagrangian interpolation formula (219). Then the weighting coefficients in a coordinate direction can be calculated by using the following equation: ap
~
(224)
dL\
For i larger than 1, Dap can also be calculated by using the following recurrent computation procedure: f2
n
k
n+1
= y^ DLkDLk q=l
1 ._ ^r1 —n +V^ nLk ap
2
nLk
9=1
n+1 L
]-) ±J
k ap
. j
- Y ^ DL'k
DLk
9=1
And the weighting coefficients for a partial derivative of order m + n + o + p at discrete point (Zi,Z2,Z<3,-£<4) can be obtained by the following equation:
DSSS^
= DLjD%D%D%.
(225)
It should be mentioned that if the Pascal triangular grid along with the area coordinates or the Pascal tetrahedral grid along with the volume coordinates are used to solve a problem coordinate transformations between the physical coordinates and the natural coordinates are necessary to be carried out. The mapping technique can
Optimization
in FE and DQE
Techniques
265
also be used to solve problems with irregular domain grids without the necessity of solving the linear algebraic system in order to calculate the weighting coefficients. It should also be mentioned that the appropriate analytical functions can be formed by using certain basis functions defined by the coordinate variables independently. The basis functions can be the polynomials, sine functions, Lagrange polynomials, Chebyshev polynomials, Bernoulli polynomials, Hermite polynomials, Euler polynomials, rational functions, . . . , etc. To solve problems having singularity properties, certain singular functions can be used. The problems having infinite domains can also be treated. 6.2. GDQEM 6.2.1. Numerical
potential
flow
analysis
simulation
In the GDQEM analysis, the analysis domain and element configuration can be arbitrary. The one-dimensional node identification method only using the velocity potentials at element nodes as the nodal variables is adopted for carrying out the GDQ discretization. The governing equation (164) at node i in element e can be discretized. Dff^
+ Df $*=(),
j = l,2,...,NN.
(226)
Letting element n be an element consisting of the Neumann boundary, the discrete Neumann boundary condition at node a can be obtained from (165). On the inter-element boundary dflr's of two adjacent elements r and s, let the /3th node of element r and /3th node of element s be a common node. Also let h and q denote the depth of flow region and the rate of flow into the domain, respectively. Then by using (227) the discrete natural transition condition can be defined. hTwD%*; + hmfaDft*; + hl^Df^
+ hmfaD'ffi = q.
(228)
The velocity components at node i in element e can also be obtained from (166). vlti = Dt?*ejt vl^D^'j.
(229)
6.2.2. Mesh and element grid Various techniques can be used to generate the mesh and element grids. The mapping technique is used. The appropriate analytical functions used to generate the weighting coefficients are the polynomials. The selection of polynomials can be flexible. The grid of an element can also be flexible. Let £ and r\ denote the non-dimensional one unit natural coordinates corresponding to the physical coordinates x and y, respectively. Also consider the Lagrange family grid and let N$ and Nv denote the numbers of levels in £ and r\ directions, respectively, in the master element of a physical element. The set of polynomials xp~1y9~1,
266 Table 11. cylinder.
C.-N.
Chen
Convergence of the adaptive p refinement analysis for the solution of flow past a
Refinement A
v
*,c
-0.10535 x 10 -0.98628 x 101 -0.10018 x 102 -0.99855 x 101 -0.99922 x 101 -0.99905 x 101
1
-0.58295 x 10 -0.48747 x 101 -0.50244 x 101 -0.49882 x 101 -0.49952 x 101 -0.49934 x 101
V
*,D
"x,F 1
0.31751 x 10 0.25105 x 101 0.26177 x 101 0.26110 x 101 0.26221 x 101 0.26272 x 101
0.23020 0.18600 0.19265 0.19146 0.19170 0.19164
1
x 10 x 101 x 101 x 101 x 101 x 101
0.21167 x 101 0.17266 x 101 0.17896 x 101 0.17785 x 101 0.17807 x 101 0.17801 x 101
p = 1,2,... ,N^, q = 1,2, ...,Nr! can be used to generate the weighting coefficients. It is equivalent to the approach of directly using the Lagrange polynomials to define the weighting coefficients. The element weighting coefficient matrices can be formed by using these smaller weighting coefficient matrices and the length of each side. It can reduce the arithmetic operations in calculating the element weighting coefficient matrices. Efficient mesh and efficient element grid model can also be used.
6.2.3. Numerical results The same problem model as that shown in Fig. 33 was solved. The mesh is also the same as that shown in Fig. 34. The constraint condition is the natural transition condition on JH. The adaptive p refinement procedure was used to analyze the problem. For the first stage, all elements have the same equally spaced 3 x 3 grid in the natural space. For the second stage, all elements have the same 5 x 5 grid in the natural space. After this refinement stage, the longer period errors have been reduced drastically. However, more refinement analyses are necessary since the shorter period errors are still significant. For the third stage, element 1 has the 5 x 5 grid while all other elements have the 7 x 7 grid. For the fourth stage, element 1 has the 5 x 5 grid while all other elements have the 9 x 9 grid. After this refinement stage the longer period errors become very small. In order to further reduce the shorter period errors, refinement process are continued. For the fifth stage, element 1 has the 5 x 5 grid, element 2 and 4 have the 9 x 9 grid and elements 3 and 5 have the 1 1 x 1 1 grid. For the sixth stage, element 1 has the 5 x 5 grid, elements 2 and 4 have the 9 x 9 grid and elements 3 and 5 have the 13 x 13 grid. After this refinement stage all longer period and shorter period errors are thought to be satisfactorily reduced. Numerical results are summarized and listed in Table 11. It shows that the velocity potentials and velocities at certain discrete points converge fast following the successive refinement. This problem can also be solved by using less elements to represent the analysis domain.
Optimization
6.3. Deflection
analysis
6.3.1. Numerical
simulation
in FE and DQE
267
Techniques
of a plate
The static deflection of the plate is also analyzed. The governing equation is expressed by d4w dx4
d4w dx2dy2
d4w ^ q(x, y) dy4 D
K
'
where w is the lateral displacement, D the flexural rigidity and q(x, y) the distributed load. Consider a clamped square plate having the side length a. By locating the origin of the coordinate system at the center of the plate, the boundary conditions are w = 0
,^=0, ox
for
x = ±^; 2
w
= 0,^-=0, ay
for
y = ±|. 2
(231)
In the analysis, one GDQEM element is used to represent the problem domain. Various grids can be used. The two-dimensional node identification method adopting the Lagrange family grid is used. For interior nodes only the degree of freedom representing the lateral displacement is assigned while multiple degrees of freedom are assigned to a node on the boundary to represent the lateral displacement and displacement gradient in the direction normal to the boundary line. Chebyshev polynomials are used to define the approximate displacement function. Thus, the problem domain is mapped onto the region fi = { — 1 < £ < 1 , — 1 < 77 < 1}. And (230) is transformed, d4w
W
d4w +
d£W
d4w +
W
a4q =
16#
where £ and 77 are natural coordinates. Let Tp(£i) denote the Chebyshev polynomials. They are expressed by Ti(£i) = l,
T2(&)=&,
T 3 (^) = - l + 2 ^ , . . . .
The displacement function can be defined by the two-dimensional expansion of the Chebyshev polynomials: w{Z,v)=Tp{£)Tq(n)cpq,
p = 1 , . . . , JV€+2;
q = 1 , . . . ,Nv+2.
(233)
Using Tj(&) = Tj(£i) in (208) and (209), the weighting coefficients D^ for mth order partial derivative in £, direction can be obtained. Equation (232) at an interior
268
C.-N. Table 12.
Chen
Results of the uniformly loaded plate.
Method EDQ
FEM
FDM
Mesh type
DOF
Dw/16qa 4
5x5 7x7 9x9 8x8 12 x 12 16 x 16 21 x 21 31 x 31 41 x 41
9 25 49 108 300 588 361 841 1521
0.8016 0.7909 0.7909 0.8148 0.8020 0.7969 0.8078 0.7984 0.7949 0.7909
Exact solution
x x x x x x x x x x
10"4 10~ 4 10"4 10"4 10-4 10"4 10-4 10"4 10-4 10"4
discrete point (a,/3) can be discretized, 4,
SSf <
+ 21%'Dpv
+ < < •
= ^gf •
(234)
The discrete boundary conditions are w
u = WN(J = °>
w
==w
n,i w
,i
=0
Nii,i '
=
w
iNr, = °.
w. JVTI.I
for
3 = 2, 3 , . . . , Nv - 1;
for
J = l,2,...,iV,;
f o r
i = 1 . 2 , • • •,
0, for
(235)
%
i = 1,2, ...,iV 5 .
In the above equation, A^ and Nv are the numbers of level of the Lagrange family grid in £ and rj directions, respectively. 6.3.2. Numerical results The Lagrange family grid is designed to have a form that at the discrete points the Chebyshev polynomials in both £ and rj directions are zeros. These Chebyshev sampling points are defined by f* |g= — cos ( tfa~_i ) ; <3 = 1,...,N^. Numerical results of the center displacements for the uniformly loaded plate are listed in Table 12. The ordinary finite difference method (FDM) and finite element method (FEM) Adini-Clough-Melosh element solutions are also included in the table. The exact value of y ^ r is 0.7909 x 10~ 4 . It shows that the current results converge fast by gradually increasing the Chebyshev sampling points. The EDQ needs much less degrees of freedom (DOF) than the FDM and FEM to converge.
6.4. Steady-state
heat conduction
analysis
The GDQEM steady-state heat conduction analysis is also carried out. The governing mathematical model of steady-state heat conduction is a boundary value
Optimization in FE and DQE Techniques
269
problem of partial differential equation. For the two-dimensional nonuniform problem with orthotropic medium, the governing equation is: (kxT,x),x
+ (kyTy),y + Q = 0
(236)
where T is the temperature, kx = kxh, ky = kyh, Q = Qh, h the thickness of the medium, kx and ky thermal conductivities and Q the energy generation rate. And the Neumann boundary condition is q = kJTtX + kymTy
(237)
where q is the conduction heat flux into the domain, and / and m are the direction cosines of the outward unit normal, while on the Dirichlet boundary the temperature is prescribed. Solution of the boundary value problem of the governing equation provides components of the heat flux through the following definition Qx
6.4.1. Numerical
=
™:r-*,£i
Qy =
"*y-*-,y
yZoof
simulation
In the GDQEM analysis, one element is used to represent the problem domain. The GDQ adopting the two-dimensional node identification method is used to discretize the fundamental relations. By only using the temperatures at nodes as the nodal variables, the discretized equation of (236) at node (a, /3) is
[net
t
.j~tex _ i L , ., . r)ex2
{"(aWaP^xaP^aPaP
^ ^MiP)^ap&P
- -L r>ey +
^(a){P)a0
l- -ney V"0 aP&P
+~ky(<*)(P)Kyp&p) T*0 + Q«0 = °-
(239)
The discrete Neumann boundary condition at node (a, J3) is 4aP = {kx(a)0)l(&)(P)DlXp&p + ky(&)(P)m{&)(P)De^&p)
T
(24°)
qViCC0 = -ky(a){p)Dey0BiST^.
(241)
&P-
And the two components of heat flux at node (a, /3) are Qx,ap = -kx(a)(p)Deax/}&0Ta^
The node arrangement in the problem domain can be arbitrary. The Pascal triangular grid is illustrated and used in the sample analysis. In calculating the weighting coefficients, the complete polynomials are used. Let n denote the order of the approximate complete polynomial. The temperature distribution can be
270
C.-N. Chen
expressed by T(x,y)
= Tpk(x,y)cpk,
0 < p < n,
l
(242)
where Tpk(x,y)=x"+1-kyk-1
(243)
and cpk are unknown coefficients. Then the weighting coefficients for the (m + n)th order partial derivative can be calculated from the following equation d(m+n)T
'5/3
X Da/3&0 J
where X-l-z
are
dxmdyn
a0
( 244 >
xlh*
obtained from the following nodal constraint relations Ta0 = Xa0&pc&0-
(245)
Direction cosines of the outward unit normal vector on the element boundary are necessary for the natural transition condition and the Neumann boundary condition. Various techniques can be used to calculate the direction cosines. The secant approximation is illustrated and adopted for the sample analyses. The unit tangent vector at a discrete point on the element boundary is constructed by the secant approximation which uses two or three consecutive discrete points on the element boundary. Then the direction cosines of the corresponding outward unit normal vector can be obtained by using the transformation law for first-order Cartesian tensors. Consider three counterclockwise consecutive nodes i, j , and k, viewed at a point inside the element, on one side of an element. Then the unit tangent vector tj can be approximated by (xk — Xi)i + (wfc — yAj -~ , tj = - i - = ^ ^ = — ^ 4 - = ai + /3j . (246) [{Xk - Xi)2 + (j/fc - Vi)2Y If i is the starting point of the element side, the unit tangent vector U can be expressed by (XJ — Xi)i + (j/9- — yAj -~ , . U =- ^ ^ ^ 4 =ai + 0j. (247) [(XJ -Xi)2 + (yj -yi)2}2 And if k is the end point of the element side, the unit tangent vector tk can be expressed by
ffc =
fo-xjff+fa-^J 2
=a.+
-pl
(24g)
2 2
[(xk - Xj) + {yk - yj) }
Using a and P, the direction cosines of the outward unit normal vector n = li + mj can be obtained. _
7
T
—
7
T
—
I = a cos — + P sin — = P,
7T
—
7T
m = —a sin — + P cos — = —a.
(249)
Optimization Table 13. Order of Pascal triangular grid 3 5 7 9 11 13 15
in FE and DQE
Techniques
271
Results of the steady-state heat conduction analysis.
T„ 0.277778 0.290720 0.290378 0.289937 0.289798 0.289733 0.289699
x x x x x x x
10 10 2 10 2 10 2 10 2 10 2 10 2
0.000000 0.187055 0.194590 0.195116 0.195707 0.195933 0.196049
Qx.B
%,A
1*,A 2
x 10° x 10 2 x 10 2 X 10 2 x 10 2 X 10 2 X 10 2
0.000000 0.187055 0.194590 0.195116 0.195707 0.195933 0.196049
x X X X X X X
10° 10 2 10 2 10 2 10 2 10 2 10 2
-0.187500 -0.210701 -0.203914 -0.203673 -0.204150 -0.203958 -0.203909
X 10 3 x 10 3 x 10 3 x 10 3 x 10 3 x 10 2 x 10 2
6.4.2. Numerical results The sample problem solved concerns the heat conduction of a triangular region. The medium is isotropic with the thermal conductivity k = 1. The sides of the triangle are the three straight lines x = 0, y = 0 and x + y = 1. The domain boundary is the Dirichlet boundary having a constant prescribed temperature T — 0. The interior region has a constant energy generation rate Q = 1000. The p refinement procedure adopting the Pascal triangular grid is used to analyze the problem. The Pascal triangular grid is designed by first using n + 1 horizontal lines to subdivide the domain into n subregions with n being the order of approximate polynomials. The horizontal lines define the levels of the Pascal triangular grid with the side y = 0 being the level zero. The level number will increase following the increase of the value of y. The highest level is level n+1. Let L denote the level number of a level. Then n + 2 — L equally spaced nodes are given to each level. Numerical results of the temperature at A(^, \) and components of the heat flux at A and B(0, ~) are listed in Table 13. It also shows that the results converge well by gradually increasing the order of the Pascal triangular grid. 7. D Q F D M Analysis of Composite Plate Problems The DQFDM has been proposed by the author. 5 4 - 5 7 The finite difference operators are derived by DQ. They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of GDQ. The derivation of higher order finite difference operators is also easy. The DQFDM is used to solve composite non-uniform plate problems. 7.1. Equations
of composite
non-uniform
plate
Let w denote the lateral displacement in z direction in a right-handed Cartesian rectangular coordinate system xyz with xy plane coincident with the neutral surface
272
C.-N. Chen
of the plate. By neglecting the transverse shear deformation, the relations between displacement w, and moments mx, my and mxy can be expressed by
D2e
D22 .-Die D26
A>6j
dx2 d*w
(250)
* dxdy
where Dij are plate rigidities which can be expressed by the reduced stiffnesses Qij and the thickness of the plate, D^ = JhL QijZ2dz. Considering that the plate is subjected to a distributed load p applied in the z direction, the equilibrium equation can be expressed by d4w =, d4w .= dAw „,„= . d4w + 4D:26 Dull - ^ + AD16^^- 3 + 2(£>i2 + 2D66)- 2 dx dy dx dy' W dxdy3 dD16\d3w - d4w „ / dDn 2 + D221TT + { dx + dy J dx3 dy* 3 d(D12 + 2D6e) | ^ A w d(D12 + Dee) d w 3aDi6 +2 2 dx dx dy + 2 dx dy dy 2 2 2 2 3 (d D m l 2d D16 i d D12 d w dD26 dD22\ dw + 2 dx + dx2 \ dx2 dxdy dy2 dy J dy 2 d2D16 d2Dee d2D26 d w +2 2 2 dx dxdy dy dxdy 2 2 d D22 d2w (d D12 2 ? ^ + + 2 + \ dx dxdy dy2 J dy2
d3 w
dxdy2
(251)
Consider that an edge parallel to the y axis. If the edge is clamped, the boundary conditions are dw(y) (252) w(y) = w(y), and = 0(y) dx where w(y) and 6(y) are forced displacement and rotation angle, respectively, applied on the edge. If the edge is simply supported, the boundary conditions are the first one of (252) and the following equation „
d\ „"iy dx
=
d2w 'dxdy
d2w * dy2
-
£11-^-0 2 + £ > 1 6 T ^ - + D L12-
'
-mx{y)
(253)
where fhx(y) is the bending moment applied on the edge. If the edge is free, the boundary conditions are (253) and the following equation -
d3w d3w + (D + 4D e) 12 6 dx2dy dxdy2 d2 d w dD16 dDi6 dDn 2dD6e + 2 dy ) dxdy dx dy dx 2 dDn dD2e d w dx + dy J dy
d3w dx3
+ +
=,
-
%
•
(254)
Optimization
in FE and DQE
273
Techniques
where Va is the edge force per unit length. For an edge parallel to x axis, the boundary conditions can be similarly expressed. 7.2. DQFDM
equations
of composite
non-uniform
plate
Let..., j — 3, j — 2, j — 1, j , j'• +1, j + 2, j'• + 3, . . . be consecutive discrete points on a grid line in the x direction, and l e t . . . , k — 3, k — 2, k — 1, k, k+1, k + 2, k + 3, . . . be consecutive discrete points on a grid line in the y direction. Assume that the discrete points in x direction have the same distance hx between two consecutive points, while the discrete points in y direction have the same distance hy between two consecutive points. Also assume that D^n and D^n are the weighting coefficients for the pth and qth order partial derivatives with respect to £ and r], respectively, with the range for both £ and rj being 1. Let the orders of Lagrange polynomials for defining D^n and D^n be M and N, respectively. Also let j and k be the r-th and s-th DQ discrete points, respectively. Then, the finite difference equation of (251) at the discrete point (j, k) can be derived by using the DQ discretization. Di ijk
M*ht
+
DrmwU+m-r)k
+
AD 16,jfe M3Nhxhy
2(-Pi2,jfc + 2-D66,jfc) ne M2N2h2h2 AD 26, jk
+ MN3hxhy 2
+ MN2hxh2y
1 +Mh
2 2 x
sn^{j-¥m—r){k+n
imDlnw{i+m-r)(k+n-s)
dDi6jk dy
+
D22,jk
N4hy
dy
9D26,jk
dx
dy
(d2Dlhjk \ dx2
jy\snwj(k+n-s)
DC rmw(j+m-r)k
d{Di2,jk + 2D66,jk)
dD 26,jk dx
s)
— s)
d(Dl2,Jk+ 2£>66,jfe)
dD-22,jfc dy
+ N3hy
nn
dDiejk dx
+ M2Nh2K
rmD^nWU+m-r)(k+r,
D
dDutjk dx
+ M3hx
D
U rm*J snw (j+m-r)(k+nimDlnw(j+m-r)(k+n-
s)
D
•s)
DV
d2D 16, jk + 2dxdy
snwi(k+r, d2Di2jk
dy2
Dirmw(j+m
2 d2D 16, jk r,d2D66,jk d2D26,jk 2 + MNhxhy dx dy2 dxdy d2D 12,jk d2D '26,jk 1 , d2D22,jk 2j ( — + 2 2 2 Dv + N h V dx dxdy dy2
Drm™
— r)k snw(j+m-r)(k+n-s)
•Wj(k+n-s)=Pjk-
(255)
In solving the plate problems, if the distance from the discrete point (j, k) to an x edge is larger than (M~23^ x , then r can be equal to ^~- which is a central difference in the x direction. Otherwise, r can not be equal to ^~ and the central difference cannot be used.
274
C.-N.
Chen
Similar concepts can be adopted for designing the finite difference model in y direction. In defining the finite difference equations of the boundary conditions, one auxiliary grid line just outside the plate and parallel to the boundary edge must be added. Let the two x edges be x = 0 and x = a. Then the finite difference equations of the boundary conditions of (252) for the edge x = 0 can be expressed by w2a=w2a
and j^-D^mw{2+m_r)a
= 62a.
(256)
The finite difference equations of (253) and (254) on the edge can be similarly expressed as -Dll,2a „ { 2 LJ 2mw(2+m-r)a M2h1 +
MNfr
ft
+
„-Dl2,2a r.-n'2 LJ w ^2^2 sn 2(a+n-s)
2
D
2mDlnw{2+m-r)(oc+n-s)
= -fhx,2a.
(257)
and D\l,2a Af3ft3
n£ L>
3
2mW(2+m-r)a
. -Dl2,2a + 4Z?66,2a MN2h h2 X
. D\6,2g + ^ M2Nh2h n
$
n
n
£2
n n
U
^mUsnw(2+m-r)(cc+n-s)
„2 L> 2mLJsnw(2+m-r)(a+n-s)
y
f 1 fdDn,2a , dD16,2a\ H Q- I L>2mW(2+m-r)a M2h2x \ dx +
MNhxhy V ^ ^ ~
+
~WJ
D2
rnDsnW(2+m-r)(a+n-s}
It should be mentioned that in real application, it is not necessary that the values of M and N are fixed in the sense that different terms in the finite difference equations can adopt different values of M and N. It should also be mentioned that the partial derivatives of plate rigidities at a discrete point can also be approximated by the finite difference discretization. 7.3. Numerical
examples
Various DQFDM algorithms can be constructed by adjusting the values of M and N for different terms in the finite difference equations. In carrying out the numerical tests, four different algorithms are used. In the algorithm Al, both M and N are four for discretizing a fourth order partial derivative with respect to a coordinate variable, while both M and N are two for discretizing a first or second order partial derivative with respect to a coordinate variable. In the algorithm A2, both M and N are four for discretizing a second or fourth order partial derivative, existing in the governing equation, with respect to a coordinate variable. However, in discretizing
Optimization
CO Q
1
o T—1
T -01 T -01
X X X b- <7> ^H to 1" •^1 CO CO o O o to to to b- b- b-
I
in FE and DQE Techniques
1
o
X X X X X
X X X X ai oo cs in N s « co CN CM cs cs o o o o to to to to b- t~ b- r-
o
I
o o o o o
X oo H co o to b-
I I l I l o oo o o X m oo in o to t-
d d o d dd I I I I I I
X X X X a> b- on to b- O) in • f o o o to to to t- b- b-
CO h i-l H S b-
CO Tf CO CO CD b_
Tt* ® CS CO (D b-
o o o x x x x x x x x
I I
o o o o o o o
(M in o bin b-
o o o o o o o
I
x -01
U
E
J3
S J3
CN Ol CO lO 00 O i-H co oo CM O) b- rH Oi S 00 GO N b ,
co co os i-t o o ao a o O H
o o o o o o o o o o o o o o o o bCl CD 00 t^
00 00 O 00 |>
lO CD CD I> t>
o
x x x x x x x x
CO 00 t-~ 00 O ) i-4 C D O CD C O H Oi 00 00 f-,
00 co oo in •^ o o CO <M i—l CO CT> t ffi M 01 00
O O O O O O O O
o
o
o
O o
o
o
1 1 1 1 1 1 1 1
X X X X X X X X X
o o o
to 00
T-H
o
CM ^f CO 00
o o o o o o o o o
• > *
o
o CN Tf CO 00 V) T-H
X X X X X X X X 1" to 00
lution Exact
3 Q fa
a a
760254 760263 760275 760283 760286 108127 921963 855878 823009 804471 793041 785516 780306 760279
276
C.-N.
Chen
Table 15. The non-dimensional displacement w x —Hj- of a simply supported anisotropic square plate subjected to a uniformly distributed lateral load. DQFDMsubd. 4x4 8x8 12 x 12 20 x 20 24 x 24 30 x 30 Exact solution
at(f,f) 0.194874 0.207903 0.211296 0.213156 0.213533
at(^,^) x x x x x
at (f, §)
2
10" 10"2 10~ 2 10"2 10~ 2
0.337379 x 10~
2
0.346140 x 1 0 " 2
0.363461 0.388439 0.394564 0.397777 0.398422 0.399458
x x x x x x
10~ 2 10~ 2 10~ 2 10"2 10"2 10"2
0.411 x 10~ 2
the boundary conditions, both M and N are two for discretizing a first or second order partial derivative with respect to a coordinate variable. In the algorithm A3, both M and N are four for discretizing a first, second or fourth order partial derivative, existing in the governing equation or boundary conditions, with respect to a coordinate variable. In the algorithm B3, both M and N are six for discretizing a first, second or fourth order partial derivative, existing in the governing equation or boundary conditions, with respect to a coordinate variable. In the algorithm C3, both M and N are eight for discretizing a first, second or fourth order partial derivative existing in the governing equation or boundary conditions, with respect to a coordinate variable. In the algorithm D3, both M and N are ten for discretizing a first, second or fourth order partial derivative, existing in the governing equation or boundary conditions, with respect to a coordinate variable. Two square plate problems were solved. The four boundary edges are x = 0, x = a, y = 0 and y = a. The first problem solved concerns a clamped isotropic uniform plate subjected to a uniformly distributed load PQ. Numerical results of the non-dimensional maximum displacement are summarized and listed in Table 14 in which subd. represents the subdivision of H interior domain. It shows that the algorithm A3 performs the best among the algorithms Al, A2 and A3 which have the same maximum order of approximation due to the consistent discretization of adopting the same M and N for all finite difference discretizations. It also shows that the performance of the algorithm will be improved by increasing the order of approximation. The second problem solved concerns the solution of a simply supported anisotropic uniform plate subjected to the same uniformly distributed load by using the algorithm A3. The plate rigidities have the following relations: D\2 = -Dn, Dn + 2D66 = 1.061£>ii and D16 = D26 = -0.174Dn. Numerical results are listed in Table 15. It also shows that the algorithm A3 performs well.
References 1. B. A. Szabo and A. K. Mehta, p-convergent finite element approximations in fracture mechanics, International Journal for Numerical Methods in Engineering 12 (1978) 551-560.
Optimization in FE and DQE Techniques
277
2. A. Peano, A. Pasini, R. Riccioni and L. Sardella, Adaptive approximations in finite element structural analysis, Computers & Structures 10 (1972) 333-342. 3. O. C. Zienkiewicz, J. P. de. S. R. Gago and D. W. Kelly, The hierarchical concept in finite element analysis, Computers & Structures 16 (1983) 53-65. 4. D. W. Kelly, J. P. de. S. R. Gago, O. C. Zienkiewicz and I. Babuska, A posteriori error analysis and adaptive processes in the finite element method: Part I — Error analysis, International Journal for Numerical Methods in Engineering 19 (1983) 15931619. 5. J. P. de. S. R. Gago, D. W. Kelly, O. C. Zienkiewicz and I. Babuska, A posteriori error analysis and adaptive processes in the finite element method: Part II — Adaptive mesh refinement, International Journal for Numerical Methods in Engineering 19 (1983) 1621-1656. 6. I. Babuska and W. C. Rheinboldt, Reliable error estimation and adaptation for the finite element method, Computational Methods in Nonlinear Mechanics (ed. J. T. Oden), North-Holland, Amsterdam, 1980. 7. W. C. Rheinboldt, Adaptive mesh refinement processes for finite element solutions, International Journal for Numerical Methods in Engineering 17 (1981) 649-662. 8. C. A. Felippa, Numerical experiments in finite element grid optimization by direct energy search, Applied Mathematical Modelling 1 (1977) 239-244. 9. A. R. Diaz, N. Kikuchi and J. E. Taylor, A method of grid optimization for finite element methods, Computer Methods in Applied Mechanical and Engineering 41 (1983) 29-45. 10. E. L. Washspress, Two-level finite element computation, Formulations and Computational Algorithms in Finite Element Analysis (eds. K. J. Bathe et al.), Cambridge, Massachusetts, 1977. 11. G. Loubignac, G. Cantin and G. Touzot, Continuous stress fields in finite element analysis, AIAA J 15 (1977) 1645-1647. 12. R. D. Cook, Loubignac's iterative method in finite element elastostatics, International Journal for Numerical Methods in Engineering 18 (1982) 67-75. 13. R. D. Cook and X. Huang, Continuous stress fields by the finite element-difference method, International Journal for Numerical Methods in Engineering 22 (1986) 229240. 14. C. N. Chen, A multiplicative and additive correction procedure for the iterative improvement of finite element solutions, PhD Thesis, 1986. 15. J. P. Ponthot and M. Hogge, On relative merits of implicit/explicit algorithms for transient problems in metal forming simulation, Proceedings of International Conference on Numerical Methods for Metal Forming in Industry, Baden-Baden, Germany, 2 (1994) 128-148. 16. C. N. Chen, Improved constant stiffness algorithms for the finite element analysis, Proceedings of NUMETA 90 (Swansea, UK, 1990) 623-628. 17. C. N. Chen, Efficient and reliable accelerated constant stiffness algorithms for the solution of non-linear problems, International Journal for Numerical Methods in Engineering 35 (1992) 481-490. 18. C. N. Chen, The differential quadrature finite element method, Applied Mechanics in the Americas (eds. D. Pamplona et al), American Academy of Mechanics 6 (1998) 309-312. 19. C. N. Chen, Efficient and reliable analysis of structural collapse by using a DQFEM along with a GSR-based accelerated equilibrium iteration procedure, The International Journal of Pressure Vessels and Piping 76 (1999) 411-420.
278
C.-N. Chen
20. C. N. Chen, DQFEM analyses of static and dynamic nonlinear elastic-plastic problems using a GSR-based accelerated constant stiffness equilibrium iteration technique, ASME, Journal of Pressure Vessel Technology 123 (2001) 310-317. 21. J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipations: the generalized-a method, Journal of Applied Mechanics 60 (1993) 371-375. 22. H. M. Hilber, T. J. R. Hughes and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, International Journal of Earthquake Engineering and Structural Dynamics 5 (1977) 283-292. 23. W. L. Wood, M. Bossak and O. C. Zienkiewicz, An alpha modification of Newmark's method, International Journal for Numerical Methods in Engineeing 15 (1980) 1562-1566. 24. N. M. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceedings of ASCE 85, EM3 (1959) 6794. 25. M. Hogge and J. P. Ponthot, Efficient implicit schemes for transient problems in metal forming simulation, Numerical and Physical Study of Material Forming Processes, Paris, France, 1996. 26. C. N. Chen, Newton-Raphson techniques in finite element methods for non-linear structural problems, one chapter in GORDON and BREACH International Series in Engineering, Technology and Applied Science, Volumes on Structural Dynamic Systems Computational Techniques and Optimization Finite Element Analysis (FEA) Techniques (ed. C. T. Leondes), Gordon and Breach, 1998, 149-244. 27. E. Cahill, Acceleration techniques for functional iteration of non-linear equations, IMACS Conference on Mathematical Modelling and Scientific Computing, Bangalore, India, 1992. 28. M. A. Crisfield, Accelerated and damping the modified Newton-Raphson method, Computers & Structures 18 (1984) 267-278. 29. C. N. Chen, A global secant relaxation (GSR) method-based predictor-corrector procedure for the iterative solution of finite element systems, Computers & Structures 54 (1995) 199-205. 30. R. E. Bellman and J. Casti, Differential quadrature and long-term integration, Journal of Computational Analysis and Applications 34 (1971) 234-238. 31. C. Shu and B. E. Richards, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stoke equation, International Journal for Numerical Methods in Fluids 15 (1992) 791-798. 32. C. N. Chen, A generalized differential quadrature element method, International Conference on Advanced Computational Methods in Engineering, Gent, Belgium, 1998, 721-728. 33. C. N. Chen, Generalization of differential quadrature discretization, Numerical Algorithms 22 (1999) 167-182. 34. C. N. Chen, Extended differential quadrature, Applied Mechanics in the Americas (eds. D. Pamplona et al.), American Academy of Mechanics 6 (1998) 389-392. 35. C. N. Chen, Differential quadrature element analysis using extended differential quadrature, Computers and Mathematics with Applications 30 (2000) 65-79. 36. C. N. Chen, A differential quadrature element method, Proceedings of the 1st International Conference on Engineering Computation and Computer Simulation, Changsha, China, 1 (1995) 25-34. 37. C. N. Chen, Potential flow analysis by using the irregular elements of the differential quadrature element method, ASME, Proceedings of the 17th International Conference
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38.
39. 40.
41.
42.
43. 44. 45. 46.
47.
48.
49.
50.
51.
52.
279
on Offshore Mechanics and Arctic Engineering, Lisbon, Portugal, 1998, Paper No. OMAE98-361, in CD ROM. C. N. Chen, The development of irregular elements for differential quadrature element method steady-state heat conduction analysis, Computational Methods in Applied Mechanics and Engineering 170 (1999) 1-14. C. N. Chen, The differential quadrature element method irregular element torsion analysis model, Applied Mathematical Modelling 23 (1999) 309-328. C. N. Chen, Adaptive differential quadrature element refinement analyses of fluid mechanics problems, Computational Methods in Applied Mechanics and Engineering 180 (1999) 47-63. A. G. Striz, W. L. Chen and C. W. Bert, Static analysis of structures by the quadrature element method, International Journal of Solids and Structures 31 (1994) 28072818. A. G. Striz, W. L. Chen and C. W. Bert, High accuracy plane stress and plate elements in the quadrature element method, Proceedings of the 36th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and AIAA/ASME Adaptive Structures Forum, New Orleans, USA (1995) 957-965. C. N. Chen, The warping torsion bar model of the differential quadrature element method, Computers & Structures 66 (1998) 249-257. C. N. Chen, Solution of beam on elastic foundation by DQEM, ASCE Journal of Engineering Mechanics 124 (1998) 1381-1384. C. N. Chen, Vibration of prismatic beam on an elastic foundation by the differential quadrature element method, Computers & Structures, 77 (2000) 1-9. C. N. Chen, The two-dimensional differential quadrature element method frame model, ASME, Proceedings of the 15th International Conference on Offshore Mechanics and Arctic Engineering, Florence, Italy, 1 (1996) 283-290. C. N. Chen, Differential quadrature element method frame analysis using extended differential quadrature, ASME, Proceedings of the 18th International Conference on Offshore Mechanics and Arctic Engineering, Newfoundland, Canada, 1999, Paper No. OMAE99-4209, in CD ROM. C. N. Chen, Vibrations of bar, beam and frame structures solved by differential quadrature element method using extended differential quadrature, ASME, Proceedings of the 18th International Conference on Offshore Mechanics and Arctic Engineering, Newfoundland, Canada, 1999, Paper No. OMAE99-5020, in CD ROM. C. N. Chen, Dynamic equilibrium equations of composite nonprismatic beams made of anisotropic materials, Proceedings of the 12th Conference on Naval Architecture and Marine Engineering, Taiwan, 1999, 524-537. C. N. Chen, The derivation of buckling equilibrium equations of arbitrarily loaded generic nonprismatic monotonic offshore/ocean bar structures and the analysis by DQEM using EDQ, ASME, Proceedings of the 19th International Conference on Offshore Mechanics and Arctic Engineering, New Orleans, USA, 2000, Paper No. OMAE 2000-2560, in CD-ROM. C. N. Chen, Dynamic equilibrium equations of composite anisotropic beams considering the effects of transverse shear deformations and structural damping, Journal of Composite Structures 48 (2000) 287-303. C. N. Chen, The development of differential quadrature element nonprismatic thinwalled beam model, Proceedings of the 4th National Conference on Structural Engineering, Taiwan, 1998, 381-388.
280
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53. C. N. Chen, Variational derivation of the dynamic equilibrium equations of nonprismatic thin-walled beams defined on an arbitrary coordinate system, Mechanics of Structures and Machines 26 (1998) 219-237. 54. C. N. Chen, A differential quadrature finite difference method, Proceedings of the International Conference on Advanced Computational Methods in Engineering, Gent, Belgium, 1998, 713-720. 55. C. N. Chen, Finite difference discretizations by difference quadrature techniques, Communications in Numerical Methods in Engineering 15 (1999) 823-833. 56. C. N. Che, Solution of composite nonuniform plate problems by the differential quadrature finite difference method, Computational Mechanics 26 (2000) 273-28. 57. C. N. Chen, Differential quadrature finite difference method for structural mechanics problems, Communications in Numerical Methods in Engineering 17 (2001) 423-441.
CHAPTER 6 C O M P U T E R TECHNIQUES A N D APPLICATIONS IN RAPID PROTOTYPING
GILL BAREQUET Department of Computer Science, The Technion—Israel Institute of Technology, Haifa 32000, Israel E-mail: [email protected] Rapid phototyping aims at fast and inexpensive production of three-dimensional models. It is necessarily accompanied by supporting software that allows data files to be input, processed, and prepared for manufacturing. Such software packages include converters between different formats of data, programs that repair flaws in the model descriptions, tools for editing and slicing the objects, etc. This chapter surveys computer techniques used for rapid prototyping. Keywords: Computer-aided manufacturing.
manufacturing;
geometric
computing;
layered
1. I n t r o d u c t i o n T h e Rapid Prototyping (RP) field has emerged in t h e last decade as means for fast and inexpensive production of three-dimensional models. Such models are needed in practically every industry for design verification, simulation, assembly, and integration of complex systems, etc. R P also has important applications in medicine (e.g. in surgery planning), chemistry (e.g. for verifying docking sites of macromolecules), in architecture, civil engineering, and more. T h e primary goal of R P is shortening t h e time-to-market of a product. Most currently available R P systems are based on layered manufacturing, t h a t is, on building t h e model slice by slice, usually from t h e b o t t o m up. T h e various existing systems differ in their physical fabrication processes and in t h e materials they use. T h e currently leading technologies are Stereolithography, Selective Laser Sintering (SLS), Laminated Object Manufacturing (LOM), Fused Deposition Modeling ( F D M ) , t o mention just a few. T h e interested reader may find a full listing of R P techniques elsewhere, e.g. in Ref. 1. T h e common for all layered-manufacturing techniques is the need t o input a model description (a computer file), process it, compute a series of cross-sections (slices) of it, and t r a n s m i t t h e slice d a t a t o the R P machine. In the next sections we review t h e various stages and features of this process. 281
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2. Input The major source of computer files input to RP is Computer-Aided Design (CAD) systems. There are numerous commercially available CAD systems, such as Catia (developed by IBM and Dassault), Pro-Engineer (by Parametric Technology Corp.), AutoCAD (by Autodesk), etc. CAD systems are used primarily to design smooth objects by using mathematical entities, such as Bezier surfaces, splines, NURBs (Non-Uniform Rational Bisplines), etc. The geometry of an object is kept by the CAD system in internal data structures, and can be saved in files whose formats are readable by RP systems. Up to date, all RP systems do not recognize input objects with smooth boundaries. They rather require a polyhedral approximation of the original design. The American-based firm 3D Systems, Inc., set in 1989 the de-facto standard file format in the RP industry. This format is called STL, an acronym for stereolithography, after the RP technology invented by the firm. It is now output by practically every CAD system and is recognized by almost all RP systems. This file format is an ASCII (or a binary) description of the triangulated boundary of a polyhedron. Each triangle is attributed by the 3-dimensional coordinates of its three vertices, and by the normalized vector perpendicular to the plane supporting the triangle. (A vector is normalized by scaling it so as to make its length 1.) The direction of the normal vector is outward the object, and the vertices of every triangle should appear in a counter-clockwise direction, when it is viewed from outside the object. Figure 1(a) shows a tetrahedron (a polyhedron whose boundary is made of four triangles), while Fig. 1(b) shows its ASCII STL-file description. The full specifications of this format are given in Ref. 2. The STL file format belongs to the class of single-list formats. Every format in this class contains only a polygon list, where each polygon consists of a sequence of three or more vertices, and each vertex is specified by its 3-dimensional coordinates. The other class contains two-list file formats. The first list consists of indexed points (usually but not necessarily, starting from index 0 or 1, and going up continuously), where each point is specified by its 3-dimensional coordinates. The second list consists of polygons, each specified as a cyclic sequence of indices that refer to the point list. The advantages of the second class are obvious. First, vertex coordinates need not be repeated. Second, adjacency information between two polygons that share an edge is implicitly specified by the existence of two successive vertex indices (i,j) in one polygon and a matching successive pair (j, i) in the other polygon. Note that the neighborhood information is not provided explicitly in either class. Nevertheless, the de-facto file-format standard in the RP industry belongs to the first class. As mentioned above, the CAD data are originally designed and stored in file formats that support smooth mathematical entities. Such formats include IGES (Initial Graphics Exchange Specification),3 VDA (Verband der Automobilindustrie), 4 STEP (STandard for Exchange of Product model data, ISO 10303),5 etc. The
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(a) A graphics display s o l i d Tetrahedron f a c e t normal 0.0000E+00 0.0OO0E+00 -1.0000E+00 o u t e r loop vertex 1.0000E+00 8.00001-01 0.0000E+00 vertex 0.0000E+00 0.0000E+00 0.00001+00 Yertex 4.0000E-01 3.0000E+00 0.00001+00 endloop endfacet f a c e t normal 6.2280E-01 ~7.7850E~0t 7.7850E-02 o u t e r loop Yertex 5.000OE-01 5.0000E-01 1.00001+00 vertex 0.0000E+00 0.0000E+00 0.00001+00 vertex 1.00001+00 8.00001-01 0.00001+00 endloop endfacet f a c e t normal -9.1077E-01 1.2144E-01 3.9466E-01 o u t e r loop Yertex 4.0000E-01 3.0000E+00 0.00001+00 Yertex 0.0000E+00 0.0000E+00 0.0000E+00 Yertex 5.0O00E-01 5.0000E-01 1.0000E+00 endloop endfacet f a c e t normal 8.4129E-01 2.2944E-01 4.89481-01 o u t e r loop vertex 5.OOOOE-Oi 5.0000E-01 1.00001+00 Yertex 1.O000E+00 8.0000E-01 0.0000E+00 Yertex 4.0000E-01 3.0O00E+00 0.0000E+O0 endloop endfacet endsolid
(b) An ASCII STL file Fig. 1.
A tetrahedron.
conversion of the data from these lies into an STL lie is done either by the CAD system (by an internal feature) or by an external program. Other sources of data for Rapid Prototyping include GIS (Geographic Information Systems), where a model of a three-dimensional terrain is to be constructed from topographic eleYation contours or from an xf/~grid of elevated points. Medical-imaging devices (CT, MRI, etc.) are yet another important source of data for rapid prototyping. Here the data are usually composed of a series of parallel
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slices, and the goal is to interpolate the boundary surface of a scanned organ and to manufacture a model of it. Thus, when the input data are not in the STL format (or any other format suitable for the specific RP system), one has to convert the file into another file readable by the system. Some translators merely change the syntax of a file by modifying keywords but not the geometric contents. Other translators apply some mathematical operations, such as tessellating (polygonalizing) smooth surfaces. Other translators interpolate a solid object from some partial data, e.g. from a set of points in space (a typical application in reverse engineering, see Ref. 6), or from a series of raster or polygonal cross sections (a typical application in medical imaging, see Refs. 7 and 8). See also Ref. 9 for a study of existing interfaces. When the boundary polygons of the input object may contain more than three vertices, but the specific RP system requires the data to contain only triangles, all the polygons need to be triangulated. A practical assumption is that each spatial polygon in the data is planar. Thus it can be rotated so as to become parallel to the xy-plane. There are a host of algorithms for triangulating a planar polygon [Ref. 10, §3]. The only known optimal algorithm for triangulating a planar polygon in ©(771) time, where m is the complexity of the polygon, is due to Chazelle. 11 However, this algorithm is extremely complex and difficult to implement. The running time of practical polygon-triangulation algorithms is typically 0(m log m). 3. Viewers and Manipulators All currently available RP systems are accompanied by supporting software products for visualizing the objects, measuring, modifying and repairing them, preparing them for production, and for controlling the production itself. In addition, various STL-file viewers are distributed by third-party firms. These viewers usually aim at the same goals except for controlling any specific production process. The more professional viewers are developed and distributed by commercial firms. Other viewers are the outcome of academic research. Morvan and Fadel, 12 Krause, Stiel, and Luddermann, 13 Barequet and Kaplan, 14 and Miiller, Joppe, and Meier 15 describe systems for processing and repairing of CAD data. Some viewers distinguish between a "model" mode, in which one object is visualized and/or edited; and a "scene" mode, in which the relative locations and orientations of multiple objects are edited. The first mode is used for repairing and enhancing a single object, while the second mode is used for designing a production job for the RP machine. 4. Repairing Objects 4.1.
Gaps
One of the main problems in polyhedral approximations of CAD objects is flaws ("gaps") in the boundary of the objects. Although many attempts were made to
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avoid these errors in the surface-fitting triangulation process (see, e.g. Refs. 16 and 17), many researchers (e.g. Sheng and Tucholke18) refer to these flaws as one of the most severe software problems in Rapid Prototyping. The problem usually appears in polyhedral approximations of CAD objects whose boundaries are described using curved entities. 16,18-20 In CSG (Constructive Solid Geometry) the object is described as the unions and intersections of cubes, spheres, cones, etc. whereas its B-Rep (Boundary Representation) may be described by NURBs Bezier surfaces, etc. Most flaws are caused by incorrect handling of trimming curves defined by intersections of surfaces.14 Other flaws are caused by missing surfaces or patches within a surface, or by incorrect handling of adjacencies between them. Flaws do not usually disturb graphics displays where the gaps between the surfaces are often too small to be seen or are handled in a straightforward manner. 21 However, it causes severe problems in applications which rely on the continuity of the boundary, such as rasterization algorithms 22 which play a crucial role in Rapid Prototyping. Consider flaws that are caused by improper approximation of trimming curves. The mesh points (i.e. the computed vertices of the polyhedral approximation) along an intersection curve between two surfaces are sometimes computed separately along each of its two sides, thereby creating two different copies of the same curve, causing a gap to appear between the copies. In the simple case, different point sets might be produced but according to the same curve equation; in the more complicated case, different equations of the curve, one for each surface containing it, are used for the mesh point evaluations. The first attempts to solve this problem, based only on the polyhedral description of an object, used only local information, and did not check for any global consistency violations. B0hn and Wozny19 treat only local flaws by iteratively triangulating them. They eliminate at each step the vertex which spans the smallest angle with its two neighboring vertices. Similarly, Makela and Dolenc20 apply a minimum-distance heuristic in order to locally fill cracks in the boundary of the polyhedron. Barequet and Sharir 23 were the first who attempted to solve the problem while considering the global consistency of the resulting polyhedron. Their algorithm uses a Geometric Hashing technique 24 for identifying similar portions of the flaws (which are then stitched) and an optimal minimum-area triangulation of 3-dimensional polygons for resolving the unmatched portions. The algorithm ensures that the resulting polyhedron is orientable, that is, that all the triangles can be oriented consistently. Sheng and Meier, 25 Morvan and Fadel, 12 and Barequet, Duncan, and Kumar 26 describe other systems that can repair defective STL files. 4.2. Normal
vectors
Another problem that sometimes occurs with STL files is inconsistent orientations of the triangles, or equivalently, wrong orientations of the vectors normal to the
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triangles. In STL lies, each triangle should be oriented counter-clockwise when it is viewed from outside the object. It is a common procedure to perform a depth-first search (DFS, see [Ref. 27, §23 J}) on the dual graph of the boundary of the object in order to find the connected components of the object. Then, it is possible to determine for each connected component whether it is orientable or not. (This is done by propagating the orientation information during the course of the DFS, and by checking consistency in the so-called "back edges".) If a component is orientable, it is then easy to orient all its triangles consistently, while the global orientation is set by finding the inside and the outside of the component by a standard ray-shooting technique.
4.3. Connected
components
The analysis of connected components also allows supporting software to split an object into several objects, each containing one connected component of the object, and/or to perform other operations selectively on only a subset of the components. Similarly, redundant components (e.g. remainders of supporting structures) can be identified and removed from the object. Figure 2 shows an object with two connected components: a ball trapped in a cage.
Fig. 2.
A 2-component object.
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5. Object Modifications 5.1.
Simplification
In practice, the resolution of an RP system is far coarser than the usual accuracy of an STL file (originated by a CAD system, other software systems, range sensors, etc.). Thus, the data in the file are in many cases an overdetailed approximation of the original object, in which a too small tolerance was allowed. The resulting tessellation of the object is indeed very similar to the original object, but it contains an unnecessarily large number of (close) vertices and very small triangles. There is a huge amount of literature on the surface-simplification problem. A comprehensive review is given by Heckbert and Garland. 28 Here are three typical strategies for reducing the complexity of an object: (1) Unifying sufficiently-close vertices into one vertex. The proximity threshold is usually supplied by the user. Since this process can degenerate edges into vertices and polygons into edges or points, the process should remove these degenerate entities. This feature is usually applied with threshold at most half of the minimum feature size of the RP technique. (2) Uniting coplanar (or almost coplanar) neighboring triangles into polygons of larger complexities. The planarity-tolerance threshold is usually supplied by the user. This feature is applicable only for RP system whose software supports polyhedral objects with boundary polygons, which are more complex than triangles. (3) Eliminating vertices from the boundary of the object (according to some priority function), and retriangulating the resulting "holes." Each such vertex-elimination operation reduces the number of triangles by two (according to Euler's formula). 5.2. Boolean
operations
Some computer programs that support RP systems perform boolean operations on solid objects. These operations include the computation of the union of two objects, their intersection, the difference between them, etc. Such operations are traditionally performed by CAD systems. The complexity of these computations is twofold. First, the exact locations of intersections between boundary triangles of the two objects need to be computed correctly and efficiently. (For example, it is inefficient to check every pair of triangles, one from each object, for possible intersection.) Second, the information on local intersections between triangles have to be considered very accurately for computing the result of the boolean operation on the two objects. A typical operation involves the computation of the intersection of the object with a plane. As we discuss below, this is the basic operation performed in layered manufacturing for computing a slice. This operation is also useful for cutting an object into two or more pieces. This tool is usually used when the model is physically too large for a specific RP system. For cutting an object into two parts, the software characterizes each triangle according to which side of the plane it is in. A triangle
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which intersects the plane is split (by the plane) into a triangle and a quadrangle. The quadrangle is then split into two triangles, and each of the three new triangles is characterized as above. Each of the two sets of triangles is finally closed to form a solid object by additional triangles, which lie within the cutting plane. 5.3.
Offsetting
Sometimes the object is described as an open polyhedral surface instead of as a solid polyhedron. In this case the surface should be offset (by some user-defined amount) so as to form a solid shell. Offsetting a single triangle is a rather simple operation, although the computation of the offset vertices should be performed with care. Two important issues need to be taken care of: First, the offset surface may intersect itself or the original surface (especially if the original surface contains sharp bends). These intersections need to be identified efficiently. (For example, it is inefficient to check every pair of triangles for possible intersection.) Second, these intersections need to be eliminated by performing boolean operations between portions of the object. The nature of these operations is usually subject to the design of the offsetting tool. 5.4. Parting
line/surface
Rapid Prototyping is essentially different from molding. However, RP can be used for generating the molds themselves. An important step in mold design is the computation of a parting line (or a parting surface) along which the mold is split into two parts, which can be taken apart to reveal the molded object. For convex objects parting lines/surfaces always exist. One would then like to design the mold so as to achieve the flattest parting surface, to avoid undercuts (overhanging portions) of the model, and/or to optimize other production parameters. Numerous algorithms and strategies for mold design have been suggested in the literature. See, e.g. Ref. 29. 6. Slicing 6.1. Intersecting
with
planes
The most important software component of rapid-prototyping systems is the "sheer," that is, a module that computes the cross-sections of the produced model (or models). To compute the intersection between a polyhedron and a plane, we first need to know which triangles of the polyhedron boundary are intersected by the cutting plane. A simple approach is to compute this information from scratch for every slice. A more efficient approach is to use the fact that slices are produced one after the other in increasing (or decreasing) heights above the bottom of the model. Thus we maintain a list of "active" triangles (that are intersected by the current cutting
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plane). Every triangle becomes active at some slice, remains active at some continuous range of slices, then ceases to be active at some other slice and never becomes active again. Maintaining the active triangles along the sequence of slices is implemented, e.g. by two priority queues, whose keys are the minimum and maximum Z coordinates of the triangles. Given the active triangles at some height, computing the slice is rather straightforward. For each triangle, we compute the segment that is the intersection of the triangle and the cutting plane. (Horizontal triangles are handled separately as a special case.) Since every triangle is oriented, that is, its inside and outside are well defined, we can also determine within which side of the segment (in the cutting plane) the material is. Each slice segment is thus directed such that the material is on, say, its right side. The collection of all these segments forms the polygonal representation of the slice: a collection of closed oriented polygonal contours whose interiors (exteriors in case of holes) are the slice material. Note the effect of "non-watertight" (unclosed) objects on this stage of the slicing. Small gaps in the 3-dimensional boundary of the polyhedron are reflected by small 2-dimensional gaps in the slice polygons. In other words, the polygonal contours can be broken into polygonal chains. Under the assumption that these gaps are very small in practice, it is rather easy to locate matching endpoints of such chains and unite them "head-to-tail" (see Fig. 3). These and more slicing recipes are given by Dolenc and Makela.30 Rock and Wozny31 show how to use topological information (that is, information about adjacencies between triangles) to speed up the slicing process. Direct slicing of smooth CAD data was suggested by Vuyyuru et al.32 They showed that it was less time-consuming, and that it produced more accurate slices that occupied less memory resources. However, direct slicing of smooth freeform surfaces is much more computationally involved (and error-prone) task than slicing of polyhedral data. 6.2.
Pen-plotting
Some rapid-prototyping systems behave like "pen-plotters": they need only the set of closed polygons per slice, so the slicing software needs only pack them in the appropriate format and send to the target machine. Other systems (e.g. the SLA
(a) Before Fig. 3.
(b) After Fixing broken slice contours.
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series of 3D Systems) make some hatching process, where a laser beam first solidifies liquid along the slice polygons, and then makes a hatching of the interiors of these polygons. 6.3.
Rasterizing
Some rapid-prototyping systems require slice data as pixel bitmaps (raster data). The rasterization is usually based on a well-known scan-conversion algorithm. 22 The slice polygons are scanned along the Y axis from top to bottom, and for each particular value of Y, the intersections of the scan-line with the contours are either entry or exit points of the scan-line to/from the "material" area. The type of the point (entry or exit) is determined according to whether the scan-line crosses the edge from its "left" side to its "right" side (recall that the segments are directed) or vice versa. Note that for this purpose the segments within a slice need not be collected into closed polygons: they can appear in random order but still the algorithm performs correctly. After all the crossing points (intersections of the scanline with slice segment) are computed (for one specific scan-line), they are sorted according to their X coordinates, and the pixel values along the scan-line are readily available. When the sliced object contains one component or several nonoverlapping components, every slice contains a set of simple nonintersecting polygons. In this case all the crossing points along a scan-line appear in alternating order, starting with an entry point and ending with an exit point, and, as mentioned above, the material lies between each point with odd index to its successive point with even index. In more complex objects, solid components may overlap. The polygonal slices of overlapping objects can contain overlapping polygons. In the scan-line level, this is reflected by some permutation of the crossing points: the scan-line may enter two components in a row (the second entry-point is the entry to the overlapping area) and then exit from both (the first exit-point occurs when the scan-line leaves the overlapping area). More generally, any valid parentheses-sequence of entry and exit points is legal; the permutation of the points reflects the overlapping pattern of the components. We have to count then the "level" of material along the scan-line. Initially (at —oo along the X axis) it starts with level 0 (outside any component), then every entry point increases the level by 1, and every exit point decreases the level by 1. Finally (at +oo along the X axis) the scan-line should return to level 0. The simple observation is that the material lies wherever the level is greater than 0; level k means that k components overlap at this portion of the scan-line.
6.4. Adaptive
slicing
Given a specific orientation of a model, the factor that affects the most the production time is the slice resolution, that is, the vertical height of every built layer.
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The finer the resolution is, the more layers are built and the larger the build time is. In order not to sacrifice quality for speed, several methods have been suggested for designing a production job with a varying slice resolution so as to minimize the number of slices while locally always admitting a user-defined tolerance (deviation of production from the original design). Sabourin, House, and B0hn 33 propose a 2-step adaptive slicing algorithm. The object is sliced in the first step into thick slabs, whose height is the maximumpossible production layer thickness. In the second step, each slab is resliced uniformly with higher resolution which suffices for meeting the desired model accuracy. Hope, Roth, and Jacobs 34 use layers with tilted boundaries, so that their adaptive-slicing algorithm closely matches the required surface and also reduces the stair-case effect. The algorithm was implemented in software which slices bispline data directly. Tata et al.35 vary the layer thickness according to the local geometry. Their algorithm adaptively slices the object so that some tolerance parameter is not violated. They provide four such parameters: cusp height, maximum deviation, chord length, and volumetric error per unit length. 7. Machine Features 7.1. Support
structures
Some rapid-prototyping processes, e.g. Stereolithography, require that support structures anchor components to the building platform, to prevent sagging or distortion of the model during the production (see Ref. 36). Majhi et al.37 minimize the volume of the support structures needed for manufacturing a model by finding its appropriate orientation. Some RP techniques introduce internal grid pattern in the support structure. The pattern typically depends on the specific technique, but occasionally the operator has some control of it. Support structures are removed in a postprocessing step after the job is completed. Figure 4(a) shows a mold cavity insert for an optical block amplifier. The slice shown in Fig. 4(b) was taken at about 2/3 of the model height. The model structure is shown in light and dark green, while the support structure is shown in red. a 7.2.
Hatching
As mentioned above, some systems require hatching of the polygonal slices instead of rasterizing it. This is typical for rapid-prototyping techniques that use a moving laser beam. Since the beam has a fixed width, beam-width compensation should be applied to the contour lines. This is done by offsetting inward the contours by an amount equal to the radius of a laser spot. Arkin, Held, and Smith 38 describe a few related optimization problems and solutions. a
T h i s figure is courtesy of Sanders Prototype, Inc. of New Hampshire, USA.
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(a) 3-dimeEsional display
(b) Slice Fig. 4. Model and support structures.
8* Job Optimization 8.1. Model
orientation
There are several factors that affect the planning of a production job. 39 Model orientation is the most important factor for production quality. It directly influences the building time, the manufacturing cost, the surface finish, and the dimensional accuracy of the model. 15 Majhi et al37 present an algorithm for finding the model orientation that minimizes the stair-case error caused by the layer thickness, the volume of the support structures, or the contact area between the support and the model. In a follow-up
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work, Majhi et a/.40 suggest methods for optimizing combinations of these criteria. These algorithms are basically built upon constructing and searching arrangements of curves on the unit sphere. The caveat of some of the variants of the algorithms is that they are appropriate for convex objects only. 8.2.
Packing
In most cases the capacity of the rapid-prototyping system is larger than the size of a single built model. It is therefore more economical to produce a batch of models, packed in the physical machine volume, in one run. Supporting software often offer tools for editing such a production job, which allow the user to position, rotate and mirror, translate, and scale multiple models in a production job. Some systems offer automatic on-line collision detectors, which notify the user about the positioning of two overlapping or too-close models. Some other systems offer automatic tools that pack efficiently multiple models in a given volume. The result is obviously an approximation of the optimum packing, as this problem is known to be AfVcomplete. b References 1. M. Burns, Automated Fabrication-Improving Productivity in Manufacturing (Prentice Hall, New Jersey, 1993). 2. 3D Systems, Inc. (Valencia, CA), Stereolithography Interface Specification, P / N 50065-S01-00, 1989. 3. National Institute of Standards and Technology (NIST), The Initial Graphics Exchange Specification (IGES), Version 5.1, MD, 1991. 4. Verband der Automobilindustrie e.V. (VDA), VDA Surface Interface, Version 2.0, Germany, 1987. 5. M. S. Bloor and J. Owen, Product Data Exchange, UCL Press, ISBN 1-85728-279-5, 1995. 6. H. Hoppe, T. de Rose, T. Duchamp, J. McDonald and W. Stuetzle, Surface reconstruction from unorganized points, Computer Graphics (Proc. SIGGRAPH), 1992, 295-302. 7. G. Barequet and M. Sharir, Piecewise-linear interpolation between polygonal slices, Computer Vision and Image Understanding 6 3 (1996) 251-272. 8. W. E. Lorensen and H. E. Cline, Marching cubes: A high resolution 3D surface construction algorithm, Computer Graphics 21 (1987) 163-169. 9. C. C. Kai, G. K. Jacob and T. Mai, Interface between CAD and rapid prototyping, Part 1: A study of existing interfaces, International Journal of Advanced Manufacturing Technology 13 (1997) 566-570. 10. M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry: Algorithms and Applications (Springer-Verlag, Germany, 1997). jVP-complete is the class of all problems for which the existence of a polynomial-time algorithm implies the existence of such algorithm for all NV problems. The latter class contains all the problems that can be solved nondeterministically in polynomial time. Whether or not MV problems can be solved deterministically in polynomial time is one of the major open problems in computer science.
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11. B. Chazelle, Triangulating a simple polygon in linear time, Discrete & Computational Geometry 6 (1991) 485-524. 12. S. M. Morvan and G. M. Fadel, IVECS, Interactive correction of .STL files in a virtual environment, Proceedings Symposium on Solid Preeform Fabrication, Department of Mechanical Engineering, University of Texas at Austin, 1996, 491-498. 13. F.-L. Krause, C. Stiel and J. Luddermann, Processing of CAD-data—conversion, verification and repair, Proceedings 4th ACM Symposium on Solid Modeling, Atlanta, GA, 1997, 248-254. 14. G. Barequet and Y. Kaplan, A data front-end for layered manufacturing, ComputerAided Design 30 (1998) 231-243. 15. D. H. Miiller, M. Joppe and I. R. Meier, RP-workbench—Continuous visual and interactive data preparation for Rapid Prototyping processes, Proceedings IASTED International Conference on Computer Graphics and Imaging, Palm Springs, CA, 1999, 17-22. 16. A. Dolenc and I. Makela, Optimized triangulation of parametric surfaces, ComputerAided Surface Geometry and Design (Mathematics of Surfaces IV) (A. Bowyer, ed.), The Institute of Mathematics and its Applications Conference Series, 48 (Clarendon Press, Oxford, 1994) 169-183. 17. X. Sheng and B. E. Hirsch, Triangulation of trimmed surfaces in parametric space, Computer-Aided Design 24 (1992) 437-444. 18. X. Sheng and U. Tucholke, On triangulating surface model for SLA, Proceedings 2nd International Conference on Rapid Prototyping, Dayton, OH, 1991, 236-239. 19. J. H. B0hn and M. J. Wozny, Automatic CAD-model repair: Shell-closure, Proceedings Symposium on Solid Preeform Fabrication, Department of Mechanical Engineering, University of Texas at Austin, 1992, 86-94. 20. I. Makela and A. Dolenc, Some efficient procedures for correcting triangulated models, Proceedings Symposium on Solid Freeform Fabrication, Department of Mechanical Engineering, University of Texas at Austin, 1993, 126-134. 21. H. Samet and R. E. Webber, Hierarchical data structures and algorithms for computer graphics; Part II: Applications, IEEE Computer Graphics & Applications 8 (1988) 5975. 22. J. D. Foley, A. van Dam, S. K. Feiner and J. F. Hughes, Computer Graphics: Principles and Practice, 2nd ed. (Addison Wesley, 1990). 23. G. Barequet and M. Sharir, Filling gaps in the boundary of a polyhedron, ComputerAided Geometric Design 12 (1995) 207-229. 24. H. J. Wolfson and I. Rigoutsos, Geometric hashing: An overview, IEEE Computational Science & Engineering 4 (1997) 10-21. 25. X. Sheng and I. R. Meier, Generating topological structures for surface models, IEEE Computer Graphics & Applications 15 (1995) 35-41. 26. G. Barequet, C. A. Duncan and S. Kumar, RSVP: A geometric toolkit for controlled repair of solid models, IEEE Trans, on Visualization and Computer Graphics 4 (1998) 162-177. 27. T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, 5th ed. (The MIT Press, McGraw-Hill Book Company, 1991). 28. P. S. Heckbert and M. Garland, Survey of polygonal surface simplification algorithms, Multiresolution Surface Modeling Course (course 25 of SIGGRAPH '97), Los Angeles, CA, 1997. 29. J. Majhi, P. Gupta and R. Janardan, Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design, Proceedings 1st ACM Workshop on Applied Computational Geometry, Philadelphia, PA, Lecture Notes in Computer Science, 1148 (Springer-Verlag, 1996) 39-47.
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INDEX
assembly relationship reasoning system, 153 assembly structure, 93, 94, 99 AutoCAD, 282 automated fixture configuration design (AFCD) system, 105, 139, 140, 157, 158 automated fixture design system, 168 automated fixture design technique, 168 automated fixture planning, 168 automated modular fixture configuration design system, 131 average discrete governing equations, 204
AfP-complete, 293 2D assembly drawings, 37, 54, 68, 70 3D systems, Inc., 282 3D part drawings, 37, 54, 68 3D relief, 7 accelerated constant stiffness iteration, 191 accelerated modified Newton-Raphson methods, 189 acceleration scheme, 190 accessibility of fixturing surfaces, 114 adaptation of shape, 81 adaptive discretization technique, 206 adaptive slicing, 290 adaptive slicing algorithm, 291 adaptive-growth-type, 75 adaptive-growth-type 3D representations, 99 adaptive-growth-type assembly structure representation, 92, 95 adaptive-growth-type shape representation, 76, 81 additive correction constants, 176 additive correction functions, 176 additive correction procedures, 175 additive correction solution, 177, 178 architecture of assembly relationship reasoning., 154 Art-to-Part process, 2, 3, 11 Art-to-Part technique, 26 ArtCAM environment, 2, 7 ASCII STL-file description, 282 assembly criteria, 156 assembly criteria between fixture elements, 151 assembly drawings, 55 assembly features, 147, 148 assembly knowledge, 59, 70 assembly relationship, 155
Bezier surfaces, 282 backward setup planning, 106, 129 bispline data, 291 block based correction, 182 block based interpolation functions, 183 block interpolation functions, 176, 178 boundary conditions of multidimensional problems, 204 boundary representation, 285 Buckling equilibrium equations, 209, 213 CAD data, 282 CAD systems, 287 CAD/CAM, 2 CAD/CAM software, 20 CAD/CAM system, 11 CAD/CAM techniques, 27 candidate fixturing surfaces, 113, 124 candidate locating surfaces, 121 Catia, 282 cell division model, 78, 79 clamping planning, 119 CNC machining, 29 coining processes, 30 composite beam, 215 composite nonlinear-linear iteration, 188 297
298
Computer Aided Design, 2, 35, 73, 106, 282 computer aided design and engineering, 171 Computer Aided Manufacturing, 2, 35 Computer-Aided Fixture Design, 104 computer-aided modular fixture design, 168 computer-aided process planning, 104, 105 computer-integrated manufacturing systems, 104 conditions of solid elements, 48 configuration design, 73-77, 81, 84, 87, 95 configuration design constraints, 76 configuration design in satellite design, 87 configuration design of a satellite, 88 configuration design process, 96 configuration design solution, 95, 99 configuration design solutions, 87, 90, 99 configuration design unit, 82, 85 configuration of the system, 87 connected components, 286 consistent boundary condition implementation, 205 constraint conditions, 176, 253 constraint of configuration design, 79 constraint optimization problem, 74 constraint satisfaction, 76 constraint satisfaction process, 82 constraints in configuration design, 76 constructive solid geometry, 285 convergence indicator, 197, 201 convergence performance, 190 convergency curves, 198 conversion algorithm, 17 correction procedure, 178 cross-sectional views, 54, 59 cubic element equations, 38, 42 cubic elements, 40, 42, 44 design knowledge, 60, 70 design process, 74 detailed embodiment design, 87 determination of spatial positions of fixture elements, 163 development process, 75 developmental rules, 76 differential quadrature (DQ) based discrete numerical methods, 172
Index differential quadrature element analyses, 171, 203 differential quadrature element method, 172 differential quadrature finite difference method, 172 differential quadrature finite element method, 172, 187 direct method, 173 direct slicing, 289 discrete analysis techniques, 171 discrete constraint conditions, 253 discrete element analysis technique, 206 discrete fundamental relations, 203, 224, 236 discrete governing equations, 204 discrete inter-element boundary conditions, 204 discrete inter-element transition conditions, 203 discrete space frame analysis models, 206 discretization, 171, 172 distributions of bending moments, 221 distributions of shear forces, 221 Dotted Edge Conditions, 49, 52 DQEM analysis of potential flow problems, 241 DQEM Buckling analysis, 238 DQEM formulation, 216, 247 DQEM frame analysis model, 217 DQEM free vibration analysis, 227 DQEM potential flow analysis model, 251 DQEM static problem analysis, 215 DQFDM analysis of composite plate problems, 271 dynamic equilibrium conditions, 233, 236 dynamic equilibrium equations, 209, 210 dynamic finite element problems, 185 dynamic incremental equilibrium equations, 187 dynamic nonlinear finite element system, 185 EDQ discretization, 216, 221, 238 effective fixturing function, 156 efficient element grid model, 251 element coefficient matrix, 230 element eigenvalue equations, 236 element matrix equations, 224 epoxy mould, 28
299
Index equations of composite non-uniform plate, 271 equilibrium equation, 180 equilibrium iteration, 185, 188 equilibrium iteration technique, 180 Euclidean norm, 190 evaluation function, 79 evolutionary algorithms, 73, 74 example fixture elements, 155 Existence Conditions, 42, 49 Existence Conditions In Assembly, 61, 64 explicit predictor-corrector equilibrium iteration procedure, 192 explicit predictor-corrector iteration, 191 extended differential quadrature, 172 extended GDQ discretization, 259, 261 extended GDQ discretization equation, 208, 261, 262 feasible fixture configuration design, 129 feasible fixture design, 114 feasible regions of clamping points, 122 feature accuracy, 109, 126 feature accuracy calculations, 128 feature based design, 106 feature combination types, 121 feature recognition, 62 PEM linear element equation, 217 finite difference approximations, 173 finite difference method, 171 finite difference representations, 173 finite element discretization, 193 finite element analysis, 198 finite element approximation, 173 finite element mesh, 198 finite element method, 172, 203, 205 finite element solution, 173-175, 181 fixture components, 115 fixture configuration, 104, 105, 119 fixture configuration design, 105, 116, 125, 128, 133, 138, 148, 166, 168 fixture design, 103, 113, 116 fixture design activities, 104 fixture design in manufacturing systems, 104 fixture element information, 147 fixture element library, 138 fixture elements, 105, 136, 137, 148 fixture functional unit, 140
fixture planning, 105, 111-117, 119, 123,
124 fixture planning systems, 123 fixture structure, 136, 137 fixture structure tree, 137 fixture unit generation algorithm, 163 fixture unit generation module, 159 fixture unit mounting algorithm, 163 fixture units, 137 fixturing features, 112, 114 fixturing features of a workpiece, 134 fixturing points, 114 fixturing requirements, 112 Fixturing stability, 114 fixturing stability consideration, 122 fixturing surfaces, 112, 113 flexible fixturing, 104 flexible fixturing methodology, 168 flexible manufacturing systems, 103 form tolerance, 109 format of STL file, 13 function carriers, 80, 86, 96, 97 function carriers shape, 86 Fused Deposition Modeling, 281 Galerkin procedures, 181 Galerkin residual equation, 180 Gauss elimination, 191 GDQEM potential flow analysis, 265 Generalized Aitken accelerator, 190 generalized differential quadrature element method, 172 generalized differential quadrature method, 202 generalized non-linear iteration techniques, 188 generalized-linearized iteration schemes, 188 generalized-linearized methods, 188 generation of 3D relief, 4 generation of assembly structure, 94 generation of surfaces, 3 generic composite beam, 210 generic differential quadrature, 172 generic non-linear equation system, 188 genetic algorithms, 82 genetic information, 75 genetic programming, 94 genetic information, 77 genotype, 82
300
Geographic Information Systems, 283 geometric constraint, 80 geometric constraint solver, 82 geometric constraints, 88 geometric constraints solver, 84 geometric elements, 37 Geometric Hashing technique, 285 geometric information, 107 geometrical constraints, 91 global modal displacement vector, 233, 235 graphics displays, 285 grid model, 202 GSR-based accelerated constant stiffness iteration scheme, 198 GSR-based predictor-corrector iteration method, 194 hatching, 291 heat treatment, 126 Hermite polynomials, 208 high resolution relief, 16 high speed dynamic problems, 187 implementing the DQEM computer program, 224 improper approximation, 285 incremental response vector, 190 Initial Graphics Exchange Specification, 282 integration of complex systems, 281 integration of dynamic equilibrium equation, 187 integration of product design, 103 inter-element boundaries, 204 inter-element boundary, 204, 251 inter-feature accuracy relationships, 108 interference checking algorithm, 163 interference checking module, 164 interference of assemblies, 93 internal data structures, 282 interpolation functions, 175, 178 iteration step, 181 iterative correction procedure, 181 iterative method, 173 iterative solution procedure, 187 Jacobian matrix, 242 job optimization, 292
Index kinematic transition conditions, 206, 224, 233, 236, 239, 251, 254 Lagrange polynomials, 202, 208 Lagrangian formulation, 179, 193 Laminated Object Manufacturing, 281 lattice faces, 38, 42 layered manufacturing, 281, 287 layered-manufacturing techniques, 281 layout constraint solver, 82 layout constraints, 89, 91 layout constraints solver, 85, 90 layout of function carrier, 76 Legendre polynomials, 176 linear elastic analysis, 205 local element eigenvalue equation, 230 local element modal displacement vector, 230 longer period errors, 174, 182 machine features, 291 machining accuracy specifications, 113 machining tool database, 131 manipulators, 284 manufacturing accuracy, 126 manufacturing accuracy requirement, 128 manufacturing errors, 110 manufacturing feature analysis, 168 manufacturing feature classification, 107, 108 manufacturing feature database, 131 manufacturing feature description, 109 manufacturing feature information description, 109 manufacturing feature-base, 107 manufacturing feature-base needs, 106 manufacturing features, 103, 106-109 manufacturing information, 106 manufacturing planning, 103, 106 manufacturing planning applications, 106 manufacturing process database, 131 manufacturing processes, 110, 126 mapping transformation, 242 Mass Condition In Assembly, 61 mathematical model of the incompressible inviscid fluid flow, 241 measuring errors, 110 Medical-imaging devices, 283 mesh generation procedure, 174 model building, 3
Index model description, 281 model of assembly structure, 93 model orientation, 292 modified Newton-Raphson methods, 191 modular fixture design, 103 modular fixture design with workpiece, 165 modular fixture element assembly, 143, 144 modular fixture element assembly relationship graph, 106 modular fixture element modeling, 144 modular fixture elements, 140-142, 145, 148 modular fixture structures, 131, 134, 138 modular fixture systems, 116,142, 145, 147 modular fixtures, 104 multi-dimensional block, 178 multi-level finite element procedure, 179 multi-level solution procedures, 174 multiplicative correction distribution function, 175 natural boundary condition, 180 natural transition conditions, 206 Newton-Raphson methods, 188, 189 Newton-Raphson procedure, 182 non-geometric information, 106, 107 non-linear elastic-plastic analysis, 205 non-linear elastic-plastic model, 201 non-linear finite element problems, 191 nonlinear elasticity problem, 179 Numerical Control, 36 NURBs, 282, 285 object modifications, 287 offsetting, 288 operation planning, 104 operational information, 107 optimal minimum-area triangulation, 285 orientation tolerances, 111 orthographic view, 38, 49, 50, 52, 59, 62, 64 orthographic views, 36-38, 44, 54, 70 phenotype, 82 physical prototype, 74 pixel bitmaps, 290 planarity-tolerance threshold, 287 polygon-triangulation algorithms, 284
301 polygonal slices, 290 polyhedral approximations, 285 polyhedral description of an object, 285 postprocessing phase, 173 predictor-corrector iteration, 194 Pro-Engineer, 282 product design, 1 product specifications, 74 production cycle, 103 production cycle time, 1 production model, 106 production of three-dimensional models, 281 production quality, 292 programmable fixtures, 104 prototype system, 87 prototyping process, 11 protrusion features, 107 rapid prototyping, 12, 281, 285 rapid prototyping systems, 12, 288, 290, 293 Rapid Prototyping technology, 8 rapid-prototyping processes, 291 rasterization algorithms, 285 rasterizing, 290 reconstruction techniques, 62 relief detail, 7 relief image, 15 repairing objects, 284 resin prototype, 26 resolution of the relief image, 15 RP techniques, 291 rubber mould, 28 scan-conversion algorithm, 290 scanning of artwork, 2 scanning software, 2 search algorithm, 56 secant improvement-based iteration scheme, 190 sectional view, 64 selective Laser Sintering, 281 sequence of slices, 289 setup planning, 103, 107, 112, 125, 126, 130-132, 134 setup planning algorithm, 130 setup planning decision making, 131 setup planning information, 114 shape adaptation, 82
302
shape adaptation process, 96 shape feature generating process model, 77 shape feature generation process, 78 shape generation, 77 shape generation process, 78 shape generation unit, 97 shape of function carrier, 76 shape profiles, 7 shear-deformable beam, 215 shorter period errors, 174 side clamping surfaces, 122 side locating planning, 122 simulation of configuration design, 97 slice resolution, 290 sliced object, 290 slicing, 288 small batch production, 104 solid element equation, 53 solid element equations, 44 solid elements, 37, 55 solid models, 37-39, 44 spatial co-ordinate, 175 STandard for Exchange of Product model data, 282 standard interpolation functions, 176 standard quadrature rule, 200
Index static non-linear FEM system, 194 Stereolithography, 281, 282 291 Stereolithography Apparatus, 2, 9, 20 Stereolithography technology, 9 STL file format, 282 STL files, 15, 285 STL-file viewers, 284 strain gradient expressions, 173 surface models, 37 surgery planning, 281 system configuration, 88 time to market, 11 transformation relations, 242 triangle reduction, 17 triangular mesh file, 3, 7-9 Turbo-C language, 8 virtual geometric elements, 36 wireframe model, 36, 37, 40, 46, 50, 52, 59, 62, 63, 65 working die, 29 workpiece accuracy, 116 workpiece blank, 103 workpiece design, 112 workpiece model, 103, 116, 122
C o m p u t e r H i d e d and Integrated Manufacturing Systems This is an invaluable five-volume reference on the very broad and highly significant subject of computer aided and integrated manufacturing systems. It is a set of distinctly titled and well-harmonized volumes by leading experts on the international scene.
The techniques and technologies used in computer aided and integrated manufacturing systems have produced, and will no doubt continue to produce, major annual improvements in productivity, which is defined as the goods and services produced from each hour of work. This publication deals particularly with more effective utilization of labor and capital, especially information technology systems. Together the five volumes treat comprehensively the major techniques and technologies that are involved.
ISBN 981-238-980-6
World Scientific www. worldscientific.com 5249 he
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