MATHEMATICS FOR INDUSTRY: CHALLENGES AND FRONTIERS
SIAM PROCEEDINGS SERIES LIST Bermúdez, Alfredo, Gomez, Dolores, Hazard, Christophe, Joly, Patrick, and Roberts, Jean E., Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation (2000) Kosaraju, S. Rao, Bellare, Mihir, Buchsbaum, Adam, Chazelle, Bernard, Graham, Fan Chung, Karp, Richard, Lovász, László, Motwani, Rajeev, Myrvold, Wendy, Pruhs, Kirk, Sinclair, Alistair, Spencer, Joel, Stein, Cliff, Tardos, Eva, Vempala, Santosh, Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (2001) Koelbel, Charles and Meza, Juan, Proceedings of the Tenth SIAM Conference on Parallel Processing for Scientific Computing (2001) Berry, Michael, Computational Information Retrieval (2001) Estep, Donald and Tavener, Simon, Collected Lectures on the Preservation of Stability under Discretization (2002) Achlioptas, Dimitris, Bender, Michael, Chakrabarti, Soumen, Charikar, Moses, Dey, Tamal, Erickson, Jeff, Graham, Ron, Griggs, Jerry, Kenyon, Claire, Krivelevich, Michael, Leonardi, Stefano, Matousek, Jiri, Mihail, Milena, Rajaraman, Rajmohan, Ravi, R., Sahinalp, Cenk, Seidel, Raimund, Vigoda, Eric, Woeginger, Gerhard, and Zwick, Uri, Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2003) Ladner, Richard E., Proceedings of the Fifth Workshop on Algorithm Engineering and Experiments (2003) Barbara, Daniel and Kamath, Chandrika, Proceedings of the Third SIAM International Conference on Data Mining (2003) Olshevsky, Vadim, Fast Algorithms for Structured Matrices: Theory and Applications (2003) Munro, Ian, Albers, Susanne, Arge, Lars, Brodal, Gerth, Buchsbaum, Adarn, Cowen, Lenore, Farach-Colton, Martin, Frieze, Alan, Goldberg, Andrew, Hershberger, John, Jerrum, Mark, Johnson, David, Kosaraju, Rao, Lopez-Ortiz, Alejandro, Mosca, Michele, Muthukrishnan, S., Rote, Gunter, Ruskey, Frank, Spinrad, Jeremy, Stein, Cliff, and Suri, Subhash, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2004) Arge, Lars, Italiano, Giuseppe F., and Sedgewick, Robert, Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithmics and Combinatorics (2004) Hill, James M. and Moore, Ross, Applied Mathematics Entering the 21st Century: Invited Talks from the ICIAM 2003 Congress (2004) Berry, Michael W., Dayal, Umeshwar, Kamath, Chandrika, and Skillicorn, David, Proceedings of the Fourth SIAM International Conference on Data Mining (2004) Azar, Yossi, Buchsbaum, Adam, Chazelle, Bernard, Cole, Richard, Fleischer, Lisa, Golin, Mordecai, Goodrich, Michael, Grossi, Roberto, Guha, Sudipto, Halldorsson, Magnus M., Indyk, Piotr, Italiano, Giuseppe F., Kaplan, Haim, Myrvold, Wendy, Pruhs, Kirk, Randall, Dana, Rao, Satish, Shepherd, Bruce, Torng, Eric, Vempala, Santosh, Venkatasubramanian, Suresh, Vu, Van, and Wormald, Nick, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2005) Kargupta, Hillol, Srivastava, Jaideep, Kamath, Chandrika, and Goodman, Arnold, Proceedings of the Fifth SIAM International Conference on Data Mining (2005)
Demetrescu, Camil, Sedgewick, Robert, and Tamassia, Roberto, Proceedings of the Seventh Workshop on Algorithm Engineering and Experiments and the Second Workshop on Analytic Algorithmics and Combinatorics (2005) Ferguson, David R. and Peters, Thomas J., Mathematics for Industry: Challenges and Frontiers. A Process View: Practice and Theory (2005)
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MATHEMATICS FOR INDUSTRY: CHALLENGES AND FRONTIERS A PROCESS VIEW; PRACTICE AND THEORY
Edited by
David R. Ferguson The Boeing Company (retired) Seattle, Washington
Thomas J. Peters University of Connecticut Storrs, Connecticut
Society for Industrial and Applied Mathematics Philadelphia
MATHEMATICS FOR INDUSTRY: CHALLENGES AND FRONTIERS A PROCESS VIEW: PRACTICE AND THEORY
Proceedings of the SIAM Conference on Mathematics for Industry: Challenges and Frontiers, Toronto, Ontario, October 13-15, 2003. Copyright © 2005 by the Society for Industrial and Applied Mathematics. 1098765432 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Control Number: 2005931140 ISBN 0-89871-598-9
is a registered trademark.
CONTENTS
1
Introduction
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Part I: Industrial Problems
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Paradigm-Shifting Capabilities for Army Transformation John A. Parmentolo
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Computational Simulation in Aerospace Design Raymond R. Cosner and David R. Ferguson
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LEAPS and Product Modeling R. Ames
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3D Modelling of Material Property Variation for Computer Aided Design and Manufacture M. J. Pratt
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Part II: Mathematical Responses to Industrial Problems
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A Framework for Validation of Computer Models James C. Cavendish
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Numerical Investigation of the Validity of the Quasi-Static Approximation in the Modelling of Catalytic Converters Brian J, McCartin and Paul D. Young
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A Framework Linking Military Missions and Means P. J. Tanenbaum and W. P. Yeakel
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Computational Topology for Geometric Design and Molecular Design Edward L F. Moore and Thomas J. Peters
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Discretize then Optimize John T. Betts and Stephen L Campbell
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Transferring Analyses onto Different Meshes D. A. Field
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Bivariate Quadratic B-splines Used as Basis Functions for Collocation Benjamin Dembart, Daniel Gonsor, and Marian Neamtu
VII
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Part III: The Process of Mathematical Modeling, Practice & Education
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A Complex Systems Approach to Understanding the HIV/AIDS Epidemic Carl P. Simon and James S. Koopman
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A Combined Industrial/Academic Perspective on Fiber and Film Process Modeling C. David Carlson, Jr. and Christopher L. Cox
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Helpful Hints for Establishing Professional Science Master's Programs Charles R. MacCluer and Leon H. Seitelman
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Author Index
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Introduction This collection of papers is a novel publication for SIAM. It is first, and foremost, a forum for leading industrial and government scientists and engineers to describe their work and what they hope to accomplish over the next decade. The goal is to focus the mathematical community on research that will both advance mathematics and provide immediate benefits to industry. So, this is not a book on mathematics in the strictest sense. It is not organized around a single or even multiple mathematical disciplines. The papers contain few, if any, statements and proofs of theorems. Rather, they focus on describing the needs of industry and government and on highlighting mathematics that may play a role in providing solutions. The papers were elicited primarily from the conference, Mathematics for Industry: Challenges and Frontiers, sponsored by SIAM in October, 2003. That conference itself was different from other SIAM conferences in that it was a forum for scientific experts from industry and government to meet with members of the mathematical community to discuss the visions they had for their industries and the obstacles they saw to reaching those goals. The conference was the result of over a decade of conversations among academics and industrialists as to why SIAM was not having a greater impact on industry and why there weren't more industrial scientists participating in SIAM. There are probably many answers to those questions but one struck us as particularly relevant - the traditional mathematical format expected by SIAM audiences is not a comfortable means for communicating the engineering concerns of industry and government. Thus, it is difficult for the two communities to talk effectively with each other and to look to each other for help and solutions. It occurred to us that a partial solution to this problem might be to hold a conference where invited representatives from industry and government, senior scientists, engineers, and managers, would be asked to talk simply about their industrial directions and perceived obstacles without the restriction that their talks fit a narrow mathematical model. It would then be left to the mathematicians to extract relevant issues where mathematical research could be helpful. In June, 2001 a group of industrial and academic mathematicians met at a workshop at the Fields Institute in Toronto to consider organizing such a conference1. The rough outline of the conference emerged from the workshop along with another interesting idea. The workshop participants began to realize that there was another recurrent theme that arose, namely that the traditional role of mathematicians as valuable, but narrowly focused, contributors to engineering projects 1
See http://www.fields.utoronto.ca/programs/cim/00-01/productdev/
1
was quickly being replaced by the notion that mathematics is itself a fundamental technology needed to advance many of the concepts of industrial interest. One particular example that came out of the workshop was that of relying on virtual prototyping methods and then the modeling and simulation technologies needed to validate virtual prototypes: an area where mathematicians naturally lead. This led to considering that a new mathematical discipline may be emerging whose research is centered on mathematics in industry and is based upon a broad synthesis of using existing mathematics to solve industrial problems while discovering further extensions and unifications of the mathematics that are essential to practice - thus enriching both the application of interest and the discipline of mathematics. A corrollary to this realization was that there is a need for a different kind of industrial-academic collaboration. In the past, industrialists often seemed to believe that the right mathematics for their problems existed in some academic closet and all they had to do was find it, while academics treated industrial problems only as a source for test cases and funding, with no intrinsic value for the development of mathematical theories and methods. In our minds a change was needed. For the benefit of both, industrialists needed to make a serious attempt to communicate to the mathematical community what they were trying to do, while the academic community needed to understand that the solution processes for industrial problems serves as a rich nurturing ground for new mathematics and demands close collaboration of industrialists and academics. All of this discussion on the relevance of SIAM to industry and the desire for a different kind of industrial-academic collaboration resulted in first, the Mathematics for Industry: Challenges and Frontiers Conference and then, eventually, in this book. The book itself has three parts: Industrial Problems in which leading government and industry scientists and engineers discuss their industries and the problems they face; Mathematical Responses to Industrial Problems where illustrative examples of mathematical responses to industrial problems are presented; and finally, The Process of Mathematical Modeling, Practice & Education where the actual process of addressing important problems by mathematical methods is examined and illustrated and the role of education is discussed. Summary and Acknowledgements
This book covers a broad range of mathematical problems that arise in industry. It contains some of the novel mathematical methods that are developed to attack the complexities of industrial problems. The intent is to bring both the beauty and sophistication of such mathematics to the attention of the broader mathematical community. This should simultaneously stimulate new intellectual directions in mathematics and accelerate timely technology transfer to industry. The editors thank all the authors for their contributions. The editors speak on behalf of all the members of the conference organizing committee in acknowledging, with appreciation, the financial support of SIAM.
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Part I: Industrial Problems In Part I, leading industrial and government scientists, engineers, and mathematicians present problems relevant to their industries. These are not articles boasting triumphant solutions. Rather, through these essays, the authors challenge the mathematical research community to help with some of the big issues facing industrial and government scientists. They present their problems in order to provide context for developing new mathematical theories and discovering novel applications for traditional mathematics. They leave it to the mathematical research community to figure out just what those theories and applications should be. The problems presented also provide a novel glimpse into how industrial and government practice actually works and the need to reach out to other disciplines for ideas and solutions. There are four articles in this section. The first: Paradigm-Shifting Capabilities for Army Transformation, by John Parmentola, Director for Research and Laboratory Management, U.S. Army, describes fundamental challenges facing the army as it transforms into a lighter and more flexible force, the Future Combat System (FCS), all the while retaining needed military capabilities. The FCS is constructed around the Brigade Combat Team, envisioned to be a self- organizing, self-configuring, and self-healing network of over 3,000 platforms maintaining secure communications while moving. Among the challenges that FCS will need to meet are: reducing equipment weight, improving soldier protection, developing better armor, and providing secure and mobile command and control. Parmentola outlines how the military is looking to technologies such as High Performance Computing, miniaturization, new materials, nanotechnologies, and network science to help meet those challenges. He concludes with a call for Army partnerships with academia, industry, and U.S. allies to realize the needed advances and changes. The next two articles focus on computer simulation and the verification and validation of virtual prototypes. In Issues in Relying upon Computational Simulation for Aerospace Design, Raymond Cosner and David Ferguson, both from Boeing, describe changes that intense competition is bringing to design in the aerospace industry. They argue that computational simulation will become an even more pervasive tool in the design of complex systems. But challenges to achieving the full potential of simulation remain, among which are issues of fidelity and accuracy, throughput and cost, and confidence in computed simulations. Using their experiences in computational fluid dynamics and geometry as representative technologies for computational simulation, they describe how things stand now in industry and
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identify some major barriers to advancement. Robert Ames of the Naval Surface Weapons Center, Carderock Division focuses on the entire ship design, manufacture, and assembly processes and the requirements for Smart Product Models for virtual prototyping and operational simulation to support testing, evaluation, and acquisition. His article, LEAPS and Product Modeling examines the challenges of bridging modeling solutions with simulation federations while addressing validation, verification and analysis issues of large acquisition programs. In his paper he describes an integrated architecture to support such complete simulations and the geometry, topology, and application frameworks needed to implement such an integrated architecture. The final article of Part I looks at manufacturing and focuses on the emerging technology of layered manufacturing. In 3D Modelling of Material Property Variation for Computer Aided Design and Manufacture, Michael Pratt of LMR Systems surveys the state of layered manufacturing and the challenges remaining. Layered manufacturing methods are ideal for interfacing directly with CAD design systems and can be used to build very complicated artifacts with intricate geometry and internal voids. However, there are a number of issues yet to be settled.
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Paradigm-Shifting Capabilities for Army Transformation John A. Parmentola * Abstract The tragic events of 9/11 and our nation's response through the war in Afghanistan and the current conflict in Iraq reflect major changes in our national security strategy. Our current adversary, global terrorism, has no boundaries. We no longer know where, when and how U.S. vital interests will be threatened. As a part of the Joint Team, the Army is undergoing Transformation to ensure that it is relevant and ready to defend U.S. interests anywhere on the globe. Through Transformation, the Army is focusing on advancing capabilities of the Current Force and planning for the exploitation of future technology trends to ensure that the Army continues to be the most relevant, ready and powerful land force on earth well into the future. This article will describe some of the ways the Army is meeting these future challenges through its strategic investments in research and technology for the Future Force. These investments are likely to result in paradigm-shifting capabilities which will save soldiers lives while ensuring that future conflicts are won quickly and decisively.
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Transformation to a New Force
The U.S. Department of Defense (DoD) has embarked on an extraordinary process of change called Transformation, that is, the creation of a highly responsive, networked, joint force capable of making swift decisions at all levels and maintaining overwhelming superiority in any battlespace. In support of this process, the Army is developing the Future Combat System (FCS), a major element of its Future Force, which will be smaller, lighter, faster, more lethal, survivable and smarter than its predecessor. Transformation will require that the Army make significant reductions in the size and weight of major warfighting systems, at the same time ensuring that * Director for Research and Laboratory Management, U.S. Army.
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U.S. troops have unmatched lethal force and survivability. It also means that the Army and other military services (as well as coalition forces) will be interdependent. To better appreciate the process and nature of Transformation, it is important to understand the motivation behind the concept. From a historical perspective, the Army's Current Force was largely designed during a period in time when the U.S. had a single superpower adversary contained in a geographic battlespace that was thoroughly analyzed and well understood. Their tactics, techniques and procedures were also well understood, and large-scale U.S. materiel and forces were pre-positioned. The Cold War was characterized by a fragile and dangerous detente between two competing superpowers, the U.S. and the Former Soviet Union. This situation is in stark contrast to our current security environment where we are a nation at war against an elusive enemy called terrorism that can strike anywhere at anytime through unanticipated means. We have experienced this emerging form of warfare through attacks by terrorists on our homeland and across the globe. In this era of threat uncertainty, we cannot be sure where, when and how our adversaries will threaten our citizens, our allies and U.S. interests. Consequently, our military forces must be more responsive, more deployable, our Soldiers multi-skilled and able to transition quickly from a direct ground combat action to stability and support operations in short periods of time. Unlike the threat-based approach of the Cold War, the Army is transforming to a capabilities-based, modular, flexible and rapidly employable force as part of a Joint Team, and our current warfighting strategy now reflects this new approach. Important Transformational characteristics of new information-age operations are speed of planning, speed of decision making, speed of physical movement, speed with which physical barriers are overcome, and effects-based precision fires, as demonstrated in the mountains of Afghanistan and the urban and desert scenarios of Iraq. Operation Enduring Freedom (OEF) in Afghanistan, initiated less than a month after the attacks of September 11, was carried out with great speed and precision. In Operation Iraqi Freedom (OIF), we went from direct ground combat action to stability and support operations in a matter of three weeks. A recent addition to the Current Force is the Stryker Brigade Combat Team, a smaller, lighter and faster combat system that embodies several of the desired Transformational warfighting capabilities of the Future Force (Figure 1).
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Brigade Combat Teams
To meet the goals of Transformation, the Army is developing Brigade Combat Teams (BCTs). The FCS-equipped BCT is comprised of about 2550 Soldiers and FCS equipment that together comprise approximately 3300 platforms (Figure 2). The challenge is to ensure that when 3300 platforms in the BCT hit the ground, all are networked and ready to fight. The key characteristics of the network are the ability to self-organize, self-configure, self-heal and provide assured and invulnerable communications while the BCT is moving, which means everything is moving together - command-and-control, information, logistics, etc. When links break, other platforms will be made available to restore them. All communications will be se-
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Figure 1. A Stryker Brigade Combat Team on patrol in Iraq. Source: U.S. Army Photo cure through the use of various techniques such as encryption, low probability of detection and low probability of intercept as well as directional antennas so as to limit the ability of the adversary to detect and intercept signals. Of utmost importance on the battlefield is the knowledge of where everyone is - where I am, where my buddy is, and where the enemy is. For the BCT, each of its 3300 platforms senses a portion of the battlespace and then transmits the most salient aspects of that portion back to a command-and-control vehicle that is assembling a common operating picture (COP) of the battlefield. As the BCT moves, it is constantly updating and reconfiguring the 3300 platforms to fill out this COP. The ultimate goal is a COP that minimizes latency so as to enable our Soldiers to execute complex operations with great speed and precision to devastate any adversary.
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Future Combat Systems (FCS)
FCS is a paradigm shift in land combat capability that will be as significant as the introduction of the tank and the helicopter. FCS is comprised of a family of advanced, networked air-based and ground-based maneuver, maneuver support, and sustainment elements that will include both manned and unmanned platforms.
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Figure 2. Elements of the Future Combat Systems (PCS) Brigade Combat Team. For rapid and efficient deployability using ClSOs, the maximum weight of each ground platform will not exceed 20 tons. The FCS system-of-systems family currently consists of 18 vehicles that range from combat vehicles to command, control, communications, computers, intelligence, surveillance, and reconnaissance (C4ISR) command centers and medical vehicles. In addition, there are unmanned aerial vehicles (UAVs) that include larger platforms for the more robust sensor and communication packages, as well as smaller UAVs that carry small payloads for local reconnaissance missions. FCS also includes remote launch systems capable of carrying rockets and mortars, a robotic mule to aid the Soldier with respect to sustainment and logistics, as well as numerous types of unattended ground sensors for unprecedented situational awareness.
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Transformation Challenges
To meet the Army goals for "strategic responsiveness", that is, the ability to deploy a brigade combat team in 96 hours, a division in 120 hours, five divisions in 30 days, and to fight immediately upon arrival, the Army must overcome a number of technical challenges. These include: reducing the weight of soldier equipment while improving soldier protection; making lightweight combat systems survivable;
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and ensuring that command-and-control centers are mobile and much more capable (Figure 3).
Figure 3. Army Transformation from the Current to the Future Force.
4.1
The Weight of Equipment
Today, soldiers must carry as much as 100 pounds of equipment, which has a dramatic effect on their agility and endurance (Figure 4). The Army goal is to reduce the effective fighting load to 40 pounds, while improving protection against threats from the enemy and the environment. As a first step, the Army is developing robotic "mules" that can follow soldiers into battle and carry a good part of the load. 4.2
Improved Soldier Protection
The Army is also pursuing novel ways to use nanomaterials to protect against ballistic projectiles and chemical and biological attacks and to enable the soldier ensemble to perform triage through active-control materials and diagnostic sensors. An immediate challenge is to protect against injuries to the extremities, the most prevalent injuries on the battlefield (Figure 5). The Army Research Laboratory, in collaboration with the Army Center of Excellence in Materials at the University of Delaware, has developed a new Kevlarbased garment by applying shear thickening liquids to the material. These substances are composed of nanoparticles of silica suspended in a liquid, such as
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Figure 4. Impact of the Soldiers load on performance. Source: Dr. James Sampson, Natick Soldier Center polyethylene glycol. When a high-speed projectile impacts into these liquids, the nanoparticles are compressed into a rigid mass that resists penetration. At slow speeds, the nanoparticles are able to move around the projectile, offering little or no resistance to a slow-moving projectile. The result is a garment with normal flexibility that is completely stab resistant. The garment is currently being assessed to determine its effectiveness with respect to other types of injuries to the extremities. Recently, the Army Institute for Soldier Nanotechnology at the Massachusetts Institute of Technology (MIT) discovered a novel, active-control material, dubbed "exomuscle", that might be used as a prosthesis to help soldiers handle and lift heavy objects. Exomuscle might also be embedded in the soldier ensemble, along with physiological monitoring and diagnostic sensors. The soldiers uniform could then act as a tourniquet to limit blood loss or perform CPR, as needed on the battlefield. 4.3
Stronger, Lighter Weight Armor
Currently, the most advanced combat system is the Abrams tank, which weighs more than 70 tons and can only be deployed either by C-5 aircraft (two per aircraft) using special runways, C-17 aircraft (one per aircraft), or ship and rail. The Abrams tank has a remarkable record of limiting casualties (only three in combat since its deployment nearly 20 years ago). To meet the new deployment goals, how-
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Figure 5. In Operation Iraq Freedom (OIF), majority juries/casualties are due to lack of protection to extremities.
of in-
ever, the Army must use C-130-like intratheater cargo aircraft to transport troops and equipment. Traditional approaches to survivability have relied heavily on armor, which has driven up the weight of ground combat systems. Because of the weight limits of FCS, the Army must develop a new survivability paradigm that relies on speed, agility, situational understanding, active protection systems, lighter weight armor, signature management, robotic systems, indirect precision fire, terrain masking, and various forms of deception rather than heavy armor. Realizing this new paradigm will require sophisticated research tools. For example, suppose for each of the 10 parameters listed above there are 10 points to explore. This means there are 10 billion points representing varying degrees of survivability. So where in this 10-dimensional volume are the acceptable levels of survivability for light combat systems in desert terrain, rugged terrain, urban terrain, and jungle terrain, taking into account the environmental conditions associated with them? Analyzing this complex 10-dimensional volume experimentally is both unaffordable and impractical. Therefore, we must rely on modeling and simulation. Fortunately, with focused research, emerging technological developments, and advances in high-performance computing (HPC), it is anticipated that the Army will be able to conduct trade-off analyses to help resolve this critical issue (Figure 6). Armor will undoubtedly remain an important aspect of survivability, and many innovative approaches are under development, including advanced lighter
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Figure 6. High performance computing trends in computation and affordability. Source: Dr. Rich Linderman, RADC weight composite armors and ceramic armor that can sustain greater loading for longer periods of time, thus increasing its ability to dissipate energy. At the Army Research Laboratory, scientists and engineers have modified the surface roughness of glass fibers that go into composite ceramic armor at the nanoscale. By fine tuning this roughness at the nanoscale to better match the characteristics of the epoxy resin that holds these fibers in place, it has been possible to increase the frictional forces during debonding when a kinetic energy penetrator impacts the armor. This has enabled the armor to sustain greater loading for longer periods of time, hence, increasing its ability to dissipate energy. These novel materials have enabled engineers to trade levels of protection for reductions in armor weight. 4.4
Mobile, More Capable Command-and-Control Centers
Another challenge is making command-and-control centers mobile and capable of maintaining the momentum of the fighting force. Currently, these centers are massive and relatively immobile. They move at less than 10 miles per hour, not as slow as the air traffic control center at a major airport. One of DODs top five goals is network-centric warfare, the central element in fully realizing Transformation in this century. The network must include individual soldiers on point, operations centers
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in the theater of operation, and the home station, which can be anywhere in the world. Communications and the network are the key foundational elements of PCS and the Future Force.
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Trends in Science and Technology
Because the Army strategy for transformation is strongly dependent on the continuous infusion of new technologies, trends in technology are continually monitored and assessed to determine their applicability to meeting the Armys needs. Certain trends are expected to persist well into this century. These trends include: time compression; miniaturization; and the understanding and control of increasingly complex systems. 5.1
Time Compression
Time compression involves the conveyance of information at the speed of light, and, more importantly, the ubiquitous availability of HPC that can process information very rapidly. Knowledge management, data processing, data interpretation, information routing, and link restoration for assured communications will be essential to situational awareness. Real-time, multisensor, data-fusion processing will be possible with embedded HPC capabilities. This technology will also be important for autonomous unmanned systems and reliable autonomous seekers for smart munitions. Advances in silicon-based HPC are likely to be overtaken by rapid developments in molecular electronics, and possibly DNA and quantum computing, with speeds that will make current supercomputers seem like ordinary pocket calculators. According to futurist and inventor Dr. Ray Kurzweil, we can expect a steady, exponential progress in computing power. At that rate of advance, we could have embedded HPC with remarkable speeds within the next decade. If Dr. Kurzweil is correct, computing ability will exceed the ability of all human brains on the planet by 2050 (Figure 7).
5.2
Miniaturization
Space continues to be "compactified", as more and more functions are performed by devices that take up smaller and smaller spaces. Golf-ball-size systems on the horizon include advances in microelectromechanical systems (MEMS). These systems will improve sensor systems and lead to low-cost inertial-navigation systems, diagnostics, prognostics, microcontrol systems, and so forth. Miniaturization will also improve logistics. Maintenance of warfighting systems on the battlefield will be managed in real time through predictive capabilities involving sophisticated prognostic and diagnostic systems, all connected and communicating on the FCS mobile wireless network. Further advances in miniaturization will result in inexpensive, self-contained, disposable sensors, such as Smart Dust (Figure 8). These small, inexpensive sensors will be dispersed by soldiers on the battlefield in handfuls over an area where they will self-organize and self-configure to suit the particular situation.
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Figure 7. Over time, exponential growth in computing power will exceed that of all human brains. Source: Dr. Ray Kurzweil, Kurzweil Technologies Miniaturization will also have a major impact on flexible display technology, conformal displays that can be placed on a soldiers face plate or wrapped around the arm. The Armys Flexible Display Center at Arizona State University leads the field in research in this area. Within this decade, we expect to realize a wireless device contained in a six-inch long, one-inch diameter tube (Figure 9). Anticipated advances in miniaturization, computer memory, computational speed, and speech recognition should lead to a compact device capable of video recording, speech recognition, embedded mission-rehearsal exercises, stored illustrative manuals, wireless communications, and real-time situational awareness through a flexible display, all in a compact form that will easily fit into the pocket of a soldier. We will also be working on the development of very small complex machines, such as nanobots that can perform microsurgery, prostheses that can enhance soldier capabilities, and machines that can go into places that are dangerous to humans. The development of micro unmanned aerial vehicles (UAVs) the size of a human hand, or even smaller, is within our grasp (Figure 10). Micro UAVs will enable soldiers to gather information about threats and provide both lethal and nonlethal capabilities, while keeping soldiers out of harms way. Our inspiration for this system is the common bumblebee (Figure 11). This small creature, with a body weight that is essentially all nectar, has a horizontal thrust of five times its weight and is capable of flying at a speed of 50 km per hour with a range of 16 km. Recently, researchers have discovered that the bumblebee
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Figure 8. Smart dust, a miniaturized, inexpensive, self-contained disposable sensor. Source: Dr. Kenneth Pister, University of California at Berkeley
navigates by balancing information flow from its left and right optical systems. Our current challenge is to understand the control system that enables this small creature to land precisely and exquisitely with zero velocity under turbulent conditions. Achieving this capability in a micro UAV will require extensive research on small- scale instabilities at low Reynolds numbers, the development of lightweight, durable materials, and sophisticated control systems that can work in turbulent environments. We will also have to develop highly efficient active-control materials and low-noise propulsion systems with compact power and energy sources that can operate reliably and for extended periods of time. Through biotechnology, we have a real opportunity to take advantage of four billion years of evolution. Biotechnology could lead to the engineering and manufacturing of new materials for sensors and other electronic devices for ultra-rapid, ultra-smart information processing for targeting and threat avoidance. Dr. Angela Belcher of MIT has tapped into the biological self-assembly capabilities of phages
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Figure 9. Wireless flexible display for use by the Soldier on the battlefield. Source: Eric Forsythe and David Morton, Army Research Laboratory
Figure 10. Micro-unmanned aerial vehicles (UAVs). Source: M. J. Tarascio and I. Chopra, University of Maryland (viruses that infect bacteria) that could potentially enable precise, functioning electrical circuits with nanometer-scale dimensions. By allowing genetically engineered phages to self-replicate in bacteria cultures over several generations (Figure 12), Dr. Belcher has identified and isolated the phages that can bind with particular semiconductor crystals with high affinity and high specificity. These phages can then self-assemble on a substrate into a network forming exquisitely precise arrays. The ultimate goal of this research is to replace the arduous fabrication of electronic, magnetic, and optical materials with genetically engineered microbes that can selfassemble exquisitely precise nanoscale materials based on codes implanted in their DNA. By exploiting living organisms as sensors, we are making advances in detection and identification. After all, why invent a sensor when evolution has already done it for you? The U.S. Army Medical Research and Materiel Command has developed
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Figure 11. The bumblebee a highly agile and efficient navigator; Source: M. V. Srinivansan, M. Poteser and K. Krai, Australian National University a technique for using common freshwater Bluegill sunfish to monitor water quality in several towns around the country. The system successfully detected in real-time a diesel fuel spill from a leaking fuel line at a New York City reservoir. Fortunately, the reservoir intake was off line at the time of the incident, and no contaminated water reached consumers. A Belgian research organization, APOPO, has developed a way to detect land mines using giant African pouched rats. In Tanzania, these rats have been trained to detect land mines with extraordinarily high detection rates. Research is ongoing on the detection of explosives by parasitic wasps, the early diagnosis of pulmonary tuberculosis by rats, and the detection of certain types of cancers in humans by dogs. 5.3
Control of Increasingly Complex Systems
Our understanding and control of increasingly complex human-engineered and biologically evolved systems continues to improve. Besides creating new materials from the atom up and managing these new configurations through breakthroughs in nanotechnology and biotechnology as described above, we are improving our control of the communications network to support the Future Force. The FCS network
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Figure 12. Self-assembly characteristics of genetically modified phages. Source: Dr. Angela M. Belcher, MIT will be a network of humans collaborating through a system of C4ISR (command, control, communications, computers, intelligence, surveillance, and reconnaissance) technologies. Humans process sensory information and respond through an ad hoc communication network, which affects network performance and, in turn, feeds back into human behavior. For the network to meet the Armys goals, we need a better understanding of the best way for humans to behave and collaborate on such a network. Although multi-hop mesh networks hold out the promise of self-organizing, self-configuring, self-healing, and higher bandwidth performance, we still need considerable research to understand network performance in a wide range of conditions to optimize protocols for military operations. We especially need to identify network instabilities to ensure that the network remains invulnerable to attack.
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Network Science
The network is the centerpiece of network-centric warfare and the Armys transformation to the Future Force. There are networks in all aspects of our daily lives
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and throughout the environment, such as the Internet (we are still trying to understand how it works), power grids (we could have used a common operating picture in the Northeast in 2002 to avoid a blackout); and transportation (cars, trains, and airplanes). There are also social networks comprised of people and organizations. Studies of social networks focus on understanding how interactions among individuals give rise to organizational behaviors. Social insects, such as bees, ants, wasps, and other swarming insects, also operate as networks. There are networks in ecosystems as well as in cellular (the human brain) and molecular (e.g., metabolic) systems. We are learning how information is processed throughout the prefrontal cortex of the brain and where various types of events occur in this region of the brain (Figure 13). There are about 100 billion neurons in the brain, approximately half of them in the cerebellum. Modeling and simulation at the University of Pennsylvania has resulted in a depiction of the dynamic activity of approximately 10,000 neurons in the cerebellum . Although this is only a small fraction of the total, we continue to advance our understanding of how neuronal networks function and affect human behavior. One goal is to understand how the brain and cognition work to learn about the software of the brain and its application to artificial intelligence. This knowledge will significantly affect virtual reality, robotics, human-factors behavioral science, and smart munitions, all of which are likely to be important to the Armys transformation to the Future Force. However, we currently lack a fundamental understanding of how networks operate in general.
Figure 13. Information flow between regions of the prefrontal cortex. Source: Adapted from Laura Helmuth, Science, Vol 302, pg 1133 The network of network-centric warfare will be a system of connections be-
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tween humans organized and interacting through a system of technologies. This network will be a highly nonlinear sense-and-response system about which little is known and for which there are few research tools to enable us to predict performance. Although the main focus is on C4ISR technologies and associated concepts, they are only part of the picture. At a very basic level, the rules or principles governing the behavior of this complex system are not well understood. Consequently, we do not have a language appropriate for describing the dynamics or a systematic mathematical formalism to make predictions of network performance for comparison with experimental data. We will need a multidisciplinary approach to advance our knowledge. This network is an example of an entire class of complex systems that exhibit network behavior. Therefore, rather than focusing research on the network of network-centric warfare, there may be an opportunity to advance knowledge and develop synergies along a broader front that will improve many complex systems and processes that exhibit network behavior. This new front could be called "network science", and progress in network science could have significant impacts on many fields, including economics and sociology. Research in network science could address a number of intriguing and important questions. Do seemingly diverse systems that exhibit network behavior have the same or similar underlying rules and principles? Is there a common language that can give us insight into the behaviors of these systems? Is there a general mathematical formalism for a systematic study of these systems? What should the Army focus on in the near term (010 years), midterm (1020 years), and long term (beyond 20 years) to advance Future Force capabilities?
7
Conclusions
The Army faces formidable technical challenges on its path to Transformation. We are already seeing the emergence of a paradigm-shift in capabilities that will save the lives of soldiers and lead to a smaller, lighter, faster, and smarter force. The Army partnerships with other Services and government agencies, academia, industry, and U.S. allies are essential to advancing science and engineering to realize the vision of the Future Force. Our investments in science and technology will enable us to overcome the many technical challenges associated with Transformation, but more importantly, to ensure that when our soldiers are called upon to defend freedom and liberty anywhere in the world, they come home safe and victorious.
8
Acknowledgement
The author is deeply indebted to Irena D. Szkrybalo for her creative comments and careful editing of the original transcript associated with my SIAM conference briefing. She also made several important suggestions, which significantly improved this paper. A version of this paper was presented at the National Academy of Engineering Symposium on Biotechnology in February 2004.
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T [1] P. A. ANQUETIL,H-H Yu, J.D. MADDEN, P.G. MADDEN, T.M. SWAGER, AND I.W. HUNTER, Thiophene-based conducting polymer molecular actuators, in Smart Structures and Materials 2002: Electroactive Polymer Actuators and Devices, Proceedings of SPIE, Vol. 4695, edited by Y. Bar-Cohen. Bellingham, Wash.: International Society for Optical Engineering (SPIE), 424434. [2] G. BUGLIARELLO, Bioengineering: An Engineering View, San Francisco: San Francisco Press, Inc., 1968. [3] P. FAIRLEY, 2003. Germs that build circuits. IEEE Spectrum Online. Available online at: http://www.spectrum.ieee.org/WEBONLY/publicfeature/nov03/1103bio.html. [4] L. HELMUTH, 2003. Brain model puts most sophisticated regions front and center. Science 302(5648): 1133. [5] W.E. HOWARD, 2004. Better displays with organic films. Scientific American 290(2): 7681. [6] R.E. JENSEN AND S.H. MCKNIGHT. 2004. Inorganic-organic fiber sizings for enhanced impact energy absorption in glass reinforced composites. Submitted to Composites Science and Technology. [7] R. KURZWEIL, 1999. The Age of Spiritual Machines: When Computers Exceed Human Intelligence. East Rutherford, N.J., Viking Penguin Group. [8] R. KURZWEIL, 2003. Future Technology Vision Document. Report to U.S. Army Leadership and Army Science Assessment Group. [9] R. KURZWEIL,2003. Societal Implications of Nanotechnology, 2003. Testimony to the Committee on Science, U.S. House of Representatives. [10] S.-W. LEE, S.K. LEE, AND A.M. BELCHER. 2003. Virus-based alignment of inorganic, organic, and biological nanosized materials. Advanced Materials 15(9): 689-692. [11] Y.S. LEE, R.G. EGRES, AND N.J. WAGNER. 2003. The ballistic impact characteristics ofKevlar woven fabrics impregnated with a colloidal shear thickening fluid. Journal of Materials Science 38(13): 2825-2833.
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[12] S. LLOYD, 1995. Quantum-mechanical computers. Scientific American 273(4): 140145. Scent Detection Workshop. 2004. Unpublished papers delivered at the Scent Detection Workshop, Sokoine University for Agriculture, Morogoro, Tanzania, July 2730, 2004. [13] T.R. SHEDD, W.H. VAN DER SCHALIE, M.W. WIDDER, D.T. BURTON, AND E.P. BURROWS. 2001. Long-term operation of an automated fish biomonitoring system for continuous effluent acute toxicity surveillance. Bulletin of Environmental Contamination and Toxicology 66(3): 3928 [14] M.V. SRINIVANSAN, M. POTESER, AND K. KRAL. 1999. Motion detection in insect orientation and navigation. Vision Research 39(16): 27492766. [15] P. STONE, 2003. Cebrowski Sketches the Face of Transformation. Available online at: http://www.defenselink.mil/news/Dec2003/nl2292003 _200312291.html. [16] M.J TARASCIO AND I. CHOPRA. 2003. Design and development of a thrust augmented entomopter: an advanced flapping wing micro hovering air vehicle. Pp. 86-103 in Proceedings of 59th Annual Forum of the American Helicopter Society. Alexandria, Va.: AHS International. [17] T.P. TRAPPENBERG, 2002. Fundamentals of Computational Neuroscience, Oxford, U.K.: Oxford [18] UNIVERSITY PRESS. University of Pennsylvania. 2004. Modeling and Simulation of the Cerebellum. Available online at: www.neuroengineering.upenn.edu/finkel/. [19] U.S. ARMY. 2004. Department of Army Executive Summary. LTC R.K. Martin, U.S. Army. [20] U.S. ARMY. 2004. The 2004 Army Posture Statement. Available online at: www.army.mil/aps/04/index.html. [21] U.S. ARMY. 2004. The Way Ahead: Our Army at WarRelevant and Ready. Available online at: www.army.mil/thewayahead/relevant.html. [22] U.S. ARMY, OFFICE OF FORCE TRANSFORMATION. 2004. Top Five Goals of the Director, Force Transformation. Available online at: www.oft.osd.mil/top_five_goals.cfm, July 16, 2004. [23] U.S. DEPARTMENT OF DEFENSE. 2004. Transformation. Available online at: www. defenselink. mil/ Transformation. [24] B. WARNEKE, M. LAST, B. LEIBOWITZ, AND K. PISTER. 2001. Smart dustcommunicating with a cubic-millimeter computer. IEEE Computer 34(1): 4451.
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[25] H.-H. Yu AND T.M. SWAGER. In press. Molecular actuators: designing actuating materials at the molecular level. IEEE Journal of Oceanic Engineering 29(2).
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Computational Simulation in Aerospace Design Raymond R. Cosner* and David R. Ferguson^ Abstract Driven by pressures to improve both cost and cycle time in the design process, computational simulation has become a pervasive tool in the engineering development of complex systems such as commercial and military aircraft and space launch vehicles. Fueled by the tremendous rate of advance in engineering computing power, acceptance of computational design tools and the results from using those tools has accelerated in the last few years. We anticipate this pace for improving our capabilities likely will continue to accelerate. Based on this view of enabling technologies, we present a vision for the future of aerospace design, ten to twenty years from now. We will then identify some of the major barriers that will hinder the technical community in realizing that vision, and discuss the advances that are needed to overcome those barriers.
1
Introduction
Modeling and simulation are tools that allow companies to design new products both more quickly and of higher quality than before. Computational simulations such as Computational Fluid Dynamics (CFD) are key to modern modeling and simulation efforts. However, even while these tools continue to gain acceptance as a primary tool of aerospace product design, they have yet to realize their full potential in the design process. In our view, in order to achieve its full potential, the entire aero CFD process must become more highly automated over a wide variety of product types, geometry representations, and user skills as encountered in a *Senior Technical Fellow, Director for Technology, Boeing Integrated Defense Systems, St. Louis, MO. t Boeing Technical Fellow - retired.
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typical industrial environment. Despite considerable improvements and successes, especially in academic institutions and government labs (see for example the work described in [9]), this kind of productivity has yet to arrive at the typical industrial setting. Even while there are a number of factors that are accelerating the acceptance of these tools, factors such as: • Improving maturity and capability of the computational tools and the supporting infrastructure, • Clear benefits gained from computational simulation in previous applications, which encourages design teams to be more aggressive in the next application, and • Customer and market pressure which dictates ever-faster design cycles, often to a degree impossible without computational simulation. There are other factors that continue to limit acceptance, factors such as: • Insufficient accuracy in various applications • Throughput and cost issues including — Lack of versatility of the tools in addressing complex engineering challenges over widely varying product classes — Inability of the tools to meet the data generation rates required to support a large engineering project — Computational failures requiring human intervention and — Total cost of the simulation based process in a large engineering project. • Lack of confidence in computational simulation. This article discusses these factors, with emphasis on those which limit acceptance of computational simulation in the aerospace design process. The authors experiences are mainly in geometry and Computational Fluid Dynamics (CFD), and this article is illustrated with examples from that domain. However, the key issues transcend individual analysis domains.
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Factors Favoring Computational Simulation
Computational simulation has made great strides in the last decade. Tools are becoming more useful in that they are beginning to have the capability of producing required accuracy at an acceptable cost, over a wide range of design space. There have been clear successes in engineering development which have been made possible by effective use of computational simulation. These successes have motivated others to adopt computational simulation in their design processes. Several factors are promoting an increased reliance on computational simulation in the design process. Increased market pressure and globalization are both
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making traditional experimentation based design more difficult while, at the same time, computing itself is becoming more capable. The aerospace market is intensely competitive at all levels, from major manufacturers such as Boeing down to the level of smaller manufacturers creating an imperative to use design processes which define practical aerospace products quickly and at low cost. But aerospace product development is a long process, typically spanning a decade or more from the initial concept to the point where the product enters service. The long time span is based on the need to develop a vehicle that can fully meet a large, aggressive, complex set of requirements. The conflicting needs of quick design cycles versus supporting complex requirements has resulted in portions of the US Department of Defense adopting a spiral development process aimed at accelerating bringing new capabilities into service. This is done by defining a sequence of products with increasing capabilities. The initial product from design spiral one might have 70% of the ultimate capabilities. The products of subsequent spirals will have greater capabilities. This approach is intended to bring products into service quickly with the capabilities that are ready for use, without waiting for the remaining capabilities to reach acceptable maturity. Further, the spiral development approach provides the opportunity to gain operational experience in the earlier products, thus leading to a more capable product later. One key to success of the spiral development process lies in the ability to execute a single design cycle very quickly, a few months, or perhaps a year instead of multiple years in a traditional process. This means that a traditional experimental design approach such as one based on wind tunnel testing may be inappropriate and so, spiral development will encourage a trend toward an increased focus on computational simulation to obtain key data. Globalization and mergers are also driving increased reliance on computational simulation. Suppliers from around the world contribute to the development of new aerospace vehicles. Further, a design team likely is now dispersed over many more geographic locations than previously. Both of these factors a global supplier base and a geographically dispersed organization mean that it is impractical to rely on a single physical source of design data, e.g., a wind tunnel, based in a specific location. It is imperative that the designers have access to tools, no matter where the designer is located, in which they have a high degree of confidence in the data generated and which are suitable to a wide variety of applications and products and are not tuned to specific products. This again points to an increased role for computational simulations which have the capability of generating high quality data and are not tied to a specific location. Computational tools and their required computing infrastructure also are becoming substantially more capable each year. As they become more capable, there is a tendency to employ them more in the design process. Based on these observations, it seems reasonable to anticipate that computation simulation will be the primary data source for aerospace product development in the next few years. The key advantage of computational simulation will lie in its extension to multidiscipline analysis and optimization, for which there is simply not an affordable competing capability from any other technology or design process. However, even as tools become more useful and better adapted to todays markets, they still fall
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short of having the full impact we want and need. To get to that full impact several challenges must be met. We turn now to discussing some of those challenges.
3
Challenges
To realize this vision of the future, computational simulation must demonstrate its readiness for a larger role in three key areas. Fidelity and Accuracy "Is the data good enough?" Computational analysis must demonstrate it is capable of achieving the quality of data required. Thus, data must be at an acceptable level of fidelity meaning that it is complete in its representation of geometry, flight conditions, and the physics of the flowfields. At acceptable fidelity, analysis must demonstrate that it achieves reliably the required level of accuracy to meet engineering needs as the primary data source. Throughput and Cost "Does CFD provide data quickly enough and at an affordable cost?" Engineering design is based on large amounts of data used to assess the effects of each vehicle parameter that can be altered by the design team. Vehicle performance must be estimated at a range of flight conditions spanning the complete flight envelope including unusual or emergency conditions. And, it must span the parametric range of the vehicle state, e.g., deflection of each control surface, other factors such as landing gear up or down, various bays or cavities open or closed, and so on. These requirements mean that large number of cases must be analyzed, and the entire cycle of analysis will be repeated at various points in the design process when significant changes are made to the baseline configuration. Thus, the analysis process must be capable of sustaining a high throughput of data, at an acceptable cost in analysis, man-hours, computing resource requirements, and other metrics. Confidence "Can we rely on computed data in making critical decisions? Will we know if something is wrong?" Unless the program team has sufficient confidence in the simulation data to make key design decisions based solely on the computed data, the computational simulation process will add only modest value to the design program. If the design team finds it necessary to hedge their bets by acquiring independent data from a different source such as wind tunnel testing, then they are apt to question why they needed to obtain the computational data at all. One key aspect, therefore, is to be able to assure the statistical accuracy of the data based on the actual characteristics of individual elements of data, rather than based on a universal statement that all the data is accurate to, say, ±1%. For example, we need to get to the point where we can say that a specific element of data, with 95% confidence, is accurate within ±1%, while another data element, due to specific cause, is less accurate, say, ±5%. Each of these factors accuracy, throughput, and confidence will be considered in turn.
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3.1
Accuracy
Throughout most of the history of CFD, the research focus has been on achieving ever-improving accuracy over a range of conditions. This accuracy improvement has been achieved through research on numerical algorithms, methods for modeling common physics such as turbulent boundary layers and shear layers, and methods for modeling special areas of physics such as combustion or multi-phase flow. In our experience, for aerospace design applications, useful accuracy often means at most ±1% uncertainty in force and moment predictions CONSISTENTLY. To achieve this level of accuracy, four factors must exist: • It must be possible to employ dense grids where critical to the analysis without incurring penalties by adding high density where it is not needed. • The analysis must converge to tight criteria. Shortfalls in accuracy are generally blamed on deficiencies in turbulence models. We were amazed how much better our turbulence models performed, once we developed the practice of running on dense grids with tight convergence criteria. • Skilled users must be available. The most significant factor in quality of CFD predictions is the users ability to make optimal decisions in user-controlled modeling parameters such as grid construction, selection of algorithms and physical models among several available options, and management of the solution from initialization to convergence. • Engineers must anticipate key physical interactions likely to exist in a new analysis flowfield, and assess realistically the modeling requirements and the accuracy that can be expected before the engineering program commits to using CFD analysis to provide the required data. This requires experience and judgment in both physical and computational fluid dynamics. There are several areas of common flow physics where accuracy remains troublesome for engineering CFD analysis. Where these physical interactions are important to the engineering problem, it is more difficult to achieve the level of accuracy required to meet engineering design needs. Two of the most critical areas are: • Boundary Layer Transition Current transition models are based on gross correlations of limited experimental data to modeling parameters such as momentum thickness Reynolds number. These models generally are not adequate for high-accuracy predictions in complex (e.g., realistic) 3-D flowfields. Improved transition models are needed which take into account specific localized details of the flowfield, and which add only a modest increment to the total computing cost of the analysis. • Heat Transfer Heat transfer (or adiabatic wall temperature) continues to be a difficult parameter to predict at high accuracy. There are several reasons for this difficulty. First, fluid temperature or heat transfer rate is a difficult quantity to measure experimentally much more difficult to measure than fluid velocity. So, there is only a small amount of detailed experimental data
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available to guide development of models and solution methods. Second, the asymptotic temperature profile near the wall is non-linear, whereas the asymptotic velocity profile (either laminar or turbulent) is linear. A much higher grid density is required to capture accurately the nonlinear temperature profile adjacent to the wall. This near-wall profile determines the wall temperature or heat transfer rate. Third, the wall temperature or heat transfer rate is very strongly affected by the state of the boundary layer laminar, transitional, or turbulent. The transport properties of air are well known for laminar flow, but they are modeled approximately by turbulence models for turbulent flow. We will not recite the limitations of turbulence models here. The transport properties are even less well known for transitional flows, where the fluid is changing from a laminar state to a fully turbulent state. So, transitional and turbulent flow prediction is subject to the familiar limitations of current models for those regions of a flowfield. There is an additional issue. Models for transition and turbulence predict (with recognized deficiencies) the local eddy or kinematic viscosity of the fluid. The effective heat transfer coefficient must be determined for thermal predictions. This heat transfer coefficient can be determined from the kinematic viscosity using a proportionality constant, the Prandtl number. The Prandtl number for laminar flow is a known property of the fluid, but for turbulent or transitional flow it is subject to significant uncertainty. Generally, a constant value of 0.9 is assumed but this is a very weak assumption. Thus, calculation of temperature or heat transfer in transitional or turbulent flow is subject to all the uncertainties of predicting mean velocity in those flow regimes, plus the uncertainty associated with the turbulent Prandtl number. And, the grid requirements are more stringent due to the nonlinear asymptotic behavior of the temperature profile. All in all, it is a very difficult calculation to execute with high accuracy.
Also important but beyond the scope of this paper are analyses involving unsteady flows, chemistry, and turbulence as these are beyond our particular areas of expertise. However, we suggest that turbulence and chemistry modeling are areas where basic research is needed to capture the physics of interest and to develop numerical algorithms which capture unsteady flow physics to the required accuracy remembering all the while that levels of accuracy that were acceptable ten years ago are not considered sufficient today. As accuracy increases in steady-state applications, we become less comfortable with inaccuracies that exist in unsteady calculations. As experience is gained using CFD for systems with chemical reactions (combustion, for example), we see more clearly areas where further improvements would lead to significant gains in the engineering product development process. In any metric, once we can regularly achieve satisfactory results at a given level in some metric, the engineering community begins calling for a higher level of performance (accuracy, cost, etc).
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3.2
Throughput and Cost
The three primary sources of aerodynamic data design are computational analysis, wind tunnel testing, and flight testing. Of course, the product must already exist before data can be obtained from flight testing. So, the incumbent source of aerodynamic data for design (prior to building the vehicle) is the wind tunnel. The wind tunnel has been the primary data source since before the flights of the Wright brothers. A strong and well-developed body of practice has evolved based on wind tunnel testing. While wind tunnel testing is subject to various limitations and deficiencies, these issues are well understood throughout the aerospace community. Thus, there is high confidence that in most situations the uncertainties in wind tunnel data are known and understood, and that situations where unacceptable uncertainty exists can be identified. Based on this degree of understanding of data uncertainty, critical design decisions based on data from wind tunnel testing can and are made routinely. Consider the factors involved in using CFD data as a direct replacement for wind tunnel data. Significant time and money are needed to initiate a wind tunnel test, but once testing is underway the marginal cost per data point (a single combination of model geometry and test conditions) is minimal. Thus, while CFD analysis may enjoy a cost and cycle time advantage in situations where only a small number of data points are required, the advantage vanishes as the number of needed data points is increased1. Eventually, a crossover point is reached where wind tunnel testing becomes the faster and cheaper source of data. The crossover point in the number of cases where analysis is faster or cheaper is currently a few hundred cases (data points). And it is rising every year. There are many segments of the design process, such as analysis of components rather than the whole vehicle, where data is required in quantities where analysis is the faster and cheaper data acquisition path. For example, propulsion system design inlets and nozzles was one of the earliest areas where high-end Navier-Stokes CFD gained a prominent role. For this segment of design, testing is particularly expensive and computational simulation is very cost effective. Data requirements in an airplane development cycle may change as the design goes through the cycle. Generally, data is required in smaller quantities in the initial stages. As the design gains maturity, large parametric database are required to support trade studies and to verify design performance in a variety of measures throughout the flight envelope. In the detailed (final) design stage, databases of 30,000 points or more are common and oftentimes over a half million points will be needed. As noted above, a wind tunnel is very efficient at generating large databases. Test data can be acquired to fill a large database at a cost in the general range of $30 to $150 per point, for a full airplane geometry including all movable control surfaces, protuberances, etc.2 ^n this context, a data point consists of a set of data for one combination of vehicle geometry and flight condition. This single data point may consist of thousands of discrete measured values in a test, or tens of millions of calculated values in an analysis. One CFD analysis run generates one data point in this terminology, though that data point may consist of millions of discrete predicted values on a grid. 2 An important factor in assessing the cost of a wind tunnel test is whether a suitable wind
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To compete with testing for the task of building large databases, CFD analysis must at least be competitive with testing that is, execute 30,000 cases in less than 12 months at a cost of less than $150 per case. And, the CFD cost per case must include all elements of cost (as does the wind tunnel cost): labor, computing facilities, software licenses and support, facilities (buildings and floor space), and overhead. This is a demanding requirement, and CFD analysis is not yet competitive in the role of building large databases although progress is being made on a number of fronts. See for example, [9] where they describe a distributed parallel computing environment and a software system capable of generating large data bases for parametric studies. One area where industry continues to struggle despite advances in technology is geometry where improvements will lead to significant gains in throughput is in geometry. The geometry representation of a vehicle is the most basic representation of all and is the starting point for all downstream applications including especially the engineering analysis applications of fluid dynamics, electro-magnetics, and structural analysis. It also forms the basis for other important applications such as numerical controlled machining and even the development of installation and maintenance guides. Most would assume that in todays world the geometry representation is brought forward directly from the CAD system used by the designer into the analysis package and that all the information is present to enable the application. Sadly, this is not so at present. At Boeing we estimate that upwards of 30% of engineering time for analysis is in fact spent on preparing the designers geometry for analysis. This is a tremendous drain on resources and our vision for the future is to eliminate as much as possible this activity. Geometry preparation consists of several tasks, any or all of which may be present in any one application. These tasks include importing the geometry from a source such as a CAD system, modifying the geometry by simplifying if needed and eliminating unnecessary or intractable features, adding pieces of geometry needed to enable the analysis, and, most obnoxious, going through a quality assessment process and repairing the geometry, and lastly exporting the geometry to the next user. The distribution of effort over the various tasks for the typical CFD process3 is captured in the following table.
tunnel model is available. A wind tunnel model is a precision scientific instrument, with extensive sophisticated instrumentation. The cost of designing and building a model from scratch can get into multiple millions of dollars. Therefore, if a suitable model is available already (or an existing model can be easily modified to meet the new need), then the cost of testing will be at the low end of the range mentioned above and the desired data can be acquired within 1-2 months. If a new model must be built, then the cost will be at the high end of the range, and it could easily take a year to obtain the needed data. While the accuracy of test data depends on many factors, as a general statement most data can be obtained with uncertainties of ±1%. 3 This table is, of course, only qualitative. These estimates will vary according to the particular problem. They are, however, representative of a typical CFD process.
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Average CFD Process Average % of total CFD effort Activity Geometry Preparation 25% Mesh Generation 30% Flow Solution 25% Post Processing 20% What this table shows is that after the designers work is completed, geometry preparation still constitutes a significant portion of the CFD task and, indeed, of other analyses such as computational electro- magnetics. The situation is such that in many instances geometry preparation issues make an analysis infeasible or too expensive to perform [5]. Our goal is to remove upwards of 50% of CFD processing cost by significantly reducing geometry preparation with the ultimate goal of having geometry moved directly from design to automatic mesh generation procedures. The affect of this would be to reduce CFD time by at least 50% but, even more, it would give the designer an on-demand capability for flow analysis. This in turn would significantly increase opportunities for analysis in the design loop. For example, being able to move directly from design through mesh generation is one of the key enablers for multi-disciplinary design and optimization. Achieving these goals will require answers to several key questions and issues. Among these are: • Is it possible to have a master geometry model that is immediately usable by all disciplines? Or will every master geometric model have to be modified to meet the requirements of specific downstream applications? If the answer is yes, a master geometric model is possible, then it is imperative that the geometry and analysis research communities begin work on understanding what such a master model might look like and what, if any, advances in analysis need to be made to make use of such a master model. If the answer is no, there can never be a master geometric model then research must be undertaken to understand how at least to limit the amount of individual modifications needed to enable all relevant analyses. In either case, research is needed to understand what exactly should be in a master or core geometry model and how should analysis codes be configured to take full advantage of the core geometry model. • How can the transfer of geometry between different parts of the engineering process with varying requirements for geometry quality, fidelity of details, included features, and mathematical representations be improved? Often it is suggested that the answer to this question is to impose a single CAD system with a single requirement for quality, fidelity, etc. from the beginning. However, such a simple answer does not seem feasible [7]. At best, it assumes that the final requirements for accuracy, fidelity, etc. are known at the beginning of what can be a very long design process. At worst it imposes a degree of fidelity on all components for the process that may simple be too burdensome for the overall process. Do the initial design concepts really have to have the accuracy and fidelity of detail design?
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• How should geometry from many sources be handled considering that there are multiple sources within any company and where suppliers and partners use different CAD systems? This will remain a key issue whether a master geometry model can be made available or not. Our experience within a large corporation certainly suggests that uniformity of geometry sources within the same corporation may be a practical impossibility. But certainly, uniformity across a broad spectrum of customers, partners, and vendors is a certain impossibility. Thus, even if we can obtain a seamless transfer of geometry from design, through mesh generation and into analysis within a single corporation, the need for handling geometry from multiple sources will continue. We have discussed some of the overarching issues to be addressed when considering how geometry, meshing, and analysis systems should work together. We now turn our attention to some of the specific problems that arise with geometry and cause meshing and, in particular, automatic meshing to be very difficult and often to fail all together. Geometry problems include gaps where surface patches fail to meet correctly, twisted patches where normals are reversed, missing geometry pieces, overlapped surface boundaries, and an excessive number of surface patches being used to define a specific geometry. The sources for these problems can be in the CAD system itself, in the design process, or in the underlying mathematics of the process. For example, gaps in the geometry although not likely could be caused by the designer not doing a proper job. Gaps are more likely caused by a failure in a mathematical algorithm designed to compute the curve of intersection between two surfaces. The algorithm could fail by simply not having a tight enough tolerance4 thus allowing gaps that are too large or it could fail to find all components of an intersection, or simply fail to return an intersection at all. Surface overlaps on the other hand are usually the result of the designer not paying careful enough attention to the details. Thus, the gap problem often can be traced directly to the mathematics of the CAD system while the overlap and missing geometry problems are more likely to be process problems. Excessive numbers of patches used to define geometry undoubtedly result from the designer needing very precise control but not having a rich enough assortment of tools within the CAD system to obtain that control resulting in the only recourse being to continue to refine the geometry into finer and finer pieces. Difficulties in finding and fixing geometry problems continue to vex designers and engineers and remain a major cause of wasting both engineering labor and time to complete the CFD process. Although some tools exist for detecting and repairing flaws in geometry, most critical flaws still are detected visually and repaired in an ad hoc manner. Improving and perhaps automating CFD specific geometry diagnostics and repair tools would significantly reduce cycle time. Even with the advent of some promising tools, see [2], repairing geometry continues to be difficult. There is a huge 4
Caution: The temptation here is to attempt to set a tolerance that is tight enough to handle all geometries and all applications. We do not know how to do this[6]. The accuracy requirements on surface intersections vary by orders of magnitude from very loose to satisfy simply graphical display requirements to moderately tight for CFD applications, to extremely tight for some electromagnetic applications.
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process bottleneck in availability of people with skills to do a geometry task and this is compounded by the fact that effective repair often requires users proficiency in two complex skill areas, e.g., CAD system and CFD. Given the complexity and persistence of these problems, we dont believe we should expect immediately a tool with all the required functionality that would remove all problems. Rather, borrowing from the spiral development process in design we suggest that a similar approach could work here and that there are a number of areas where results could lead to immediate improvements. We mention a few. The first phase5, spiral one, would be to develop metrics for geometry quality appropriate to each user group and then develop tools to evaluate quality against the metrics. Too often the required quality is a vague notion of what works that is developed by users of specific software for specific applications. Thus, it is almost impossible for a new comer to understand what makes one instance of geometry acceptable and another not. This means it is impossible to automate these processes. Such metrics and tools should be generated both for internally developed geometry and for geometry received outside the organization. Further, the metrics should be communicated to all providers of geometry and most especially to the CAD vendors. The second spiral, building on results from the first spiral, would begin work on developing robust geometry and analysis tools. The sensitivity to levels of precision should be eliminated if at all possible. This may require substantial changes to both geometry and analysis tools in use today. A background level of geometry quality should be identified. This would also help in the question of core or master models. The next phase, spiral three, would examine the question of how healed or fixed geometry gets absorbed back into a CAD system. This involves obtaining a deep understanding of the flow of geometry data through the entire product life cycle including the need to provide feedback upstream through the process on geometry problems, to address design changes, to preserve geometry fitness to understand tradeoffs between process cost/time and geometry quality. 3.3
Confidence
Weve covered increasing accuracy and throughput and reducing cost as important topics in promoting the role of computational simulation in engineering design. We now turn to perhaps a more fundamental question: Can we rely on computed data in making critical decisions? Will we know if something is wrong? This is the crux of the matter in gaining acceptance of computational simulation in the engineering process. One way of viewing this issue is to think in terms of having a useful understanding of the level of uncertainty in each piece of data produced by the computational simulation process. This uncertainty estimate must be sound if the estimated uncertainty is in fact too low, then the users are exposed to a potentially dangerous failure. If the estimated uncertainty is too high, it may unnecessarily discourage users from gaining the value in the data. 5 There would actually be no need to hold off on future spirals until the first ones are done. Rather, there are possibilities for concurrent development. We only divide them into cycles as a convenience for understanding the work.
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The difficulty in gaining confidence in fluid flow results is compounded by the fact that both physical fluid dynamics and computational fluid dynamics exhibit highly nonlinear behavior where a small change in the conditions defining the flowfield in question (physical or computational) can produce important changes in the flowfield and where the response of the physics and the computational simulation to a change in the flow conditions generally is different. We assert, without offering proof, that it is impossible to run enough validation cases to establish complete confidence throughout a needed design space, when the physics are nonlinear. In the rest of the paper, we will discuss both todays essentially a priori methodology and possible improvements and a new, a posteriori methodology that may result in dramatic increases in confidence. Todays Validation Process
Today, validation is an a priori process described as follows where a significant body of literature exists on practices for verification and validation of computational simulations [1]. Recall that • Verification is the process of establishing the degree to which the computational simulation provides accurate solutions of the selected mathematical model. • Validation is the process of establishing the degree to which the computational simulation provides accurate representations of a selected physical flowfield, or a defined range of flowfields. In todays practice, validation is the primary method used to establish confidence in computational predictions. The general approach is: 1. Select a high-confidence set of experimental data, as "close" as possible to the analysis application which it is desired to validate. Due to the issue of differing nonlinear response (physics vs. computation) determining the best experimental data close to the desired validated analysis objective can be a very subjective and judgmental process. 2. Conduct repeated analyses of the validation problem(s), adjusting the computation model to determine the operational procedures required to attain the best possible accuracy and cost. This is a trial-and-error process. 3. When satisfied that the optimal combination of accuracy and cost has been identified, document the key lessons, expected accuracy, and expected cost from the validation results. The key lessons will include user-controlled modeling parameters that were adjusted (optimized) to achieve the best results, such as grid topology, density, and structure, flow solver options, guidance on simplifying the representation of the physical flowfield while maintaining acceptable accuracy and cost, etc. 4. Apply these lessons from the validation process in new applications. The new applications by definition are different in some significant way from the
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existing data used to guide the validation. Therefore, the user must exercise judgment in applying the knowledge from the validation process in the new application. At several points in this sequence, the user must apply judgment to achieve good results. Thus, one can see that the skill, experience, and insight of the user are critical factors in achieving satisfactory results in a "validated" application. There is an important consequence of this process. The person performing the validation typically repeats the analysis a number of times, in the process of learning the optimal combination of modeling techniques and user-controllable modeling parameters. Without having such intent, the person conducting the validation has in fact identified the best-case outcome of the analysis methodology, in the validated application. Thus, the validation process actually winds up validating the best accuracy that the method can obtain rather than validating a level of accuracy that can be expected of most users in most situations. For engineering applications, it would be far better to conduct validation based on an approach which sets expectations at a level which can be achieved most of the time in a first-case solution, rather than an expectation that can only be achieved rarely in the first analysis. This iterative approach to validation has an additional implication. The use of an iterative trial-and-error process for setting expectations on CFD accuracy leads to reliance on highly tuned analysis models that often are very sensitive to specific users. Such a process is non-robust. The preferred analysis process is robust in that the quality of results is relatively insensitive to usage practices, over a meaningful range of practice. With a robust process, the natural range in variations due to differences in user practices should have little impact on the quality of results. In such a process, the rate of first-try success (acceptable quality) is high. A nonrobust process, on the other hand, is one that requires careful adjustment of usercontrolled parameters to produce acceptable quality. Todays CFD process exhibits several non-robust characteristics, particularly with regard to geometry and grid issues. This non-robustness ultimately is related to the selection of algorithms used for CFD analysis. Algorithm development has long been directed based on considerations of spatial and temporal order of accuracy, suitability for high-rate parallel processing, and other issues. The ability of the algorithm to provide acceptable results over a range of grid characteristics has generally not been a factor in the development and implementation of algorithms in a computational code. The lack of robustness is illustrated by the iterative nature of most validation exercises. In turn, the expectations set by publishing validation data based on careful tuning of the computational analysis tend to perpetuate highly tuned, non- robust processes. It is a chicken-and-egg problem. To summarize: the primary approach to CFD quality assurance today is validation. However, the validation process has several weaknesses and falls well short of providing a complete solution to the problem of assuring quality of CFD data in the engineering design process. Key areas of concern are: • Validation cases almost always are simpler than the real applications simpler in terms of both the geometry of the problem, and the flowfield physics exhib-
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T
data, essential for value in validation, for simplified geometries. Frequently, validation also is based on integrated flowfield parameters such as drag. If validation is performed on that basis, without additional data describing the details of the fluid interactions in the validation cases, then there is a real risk of false optimism in the validation case due to offsetting errors and those errors wont be offsetting in a new application, in the subsequent design process. • CFD data quality is strongly dependent on small details of the modeling approach. For example, accuracy can be strongly affected by small details of the computational grid and its interaction with the numherical algorithms. Fluid dynamics (physical or computational) is a domain where small, localized features of a flowfield can have far-reaching impact, e.g., formation of a vortex, or details of a shock wave boundary layer interaction. Thus, a localized modeling deficiency can destroy all value in the overall solution. • As discussed above, it is very difficult to model the real application while exactly preserving the operational approach that was successfully validated. Success in this extension from the validated outcome is heavily dependent on the judgment of the user, which in turn is based on the users skill, experience, and insight. To mitigate the impact of these issues, a design team relying on CFD predictions often adds an additional step after the analysis results have been provided: a careful review and assessment of those results by independent experts, that is, by knowledgeable people who had no role in producing the CFD predictions. To succeed in this role, the reviewing expert must be skilled in the pertinent aspects of both physical and computational fluid dynamics. People with skills in both areas can be hard to find. In any event, adding a process of human inspection greatly reduces the throughput of the CFD analysis process, thus reducing the value of the CFD analysis even if no deficiencies are found in the CFD data. For all these reasons, it does not appear that the current approach to quality assurance in computational simulation data can be retained as we seek to increase substantially the throughput of the computational simulation process. A new approach is needed. Issues that Degrade Quality
There are many factors that contribute to uncertainty in CFD predictions, or in other computational simulation disciplines. Each factor must be addressed to improve the overall analysis quality. Thus, there cannot be a single silver bullet approach to improving confidence in CFD simulation data each factor affecting uncertainty must be dealt with, appropriately. It can be helpful to consider the contributors to uncertainty in two categories: those factors which are inherent in our tools and processes, and those which can be controlled by the user. Clearly, the approach to improvement is very different in these two categories. Uncertainties due to fundamental limitations in the analysis codes include:
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• Algorithm errors truncation, roundoff, phase error, etc. • Limitations of the physical models for turbulence, transition, chemical reactions, etc. Uncertainties due to user-controllable factors in usage of the computational tools include: • Imperfect grids, • Inadequate convergence spatial or temporal, • Inappropriate turbulence model locally or globally, • Omitted physical interactions, e.g., boundary layer transition, and • Boundary condition simplification or ambiguity. Many additional examples could be listed in these two categories. As discussed to this point, a priori validation is a valuable tool for improving the quality of CFD predictions. However, the most careful and complete validation process cannot provide full assurance that acceptable results will be realized when CFD analysis is conducted for a new problem, in the engineering design process. This is due to several reasons. Chief among them is that user-controlled factors are the chief factor in establishing solution quality, and user practices inevitably are different in small but important details between the validated cases and the real cases analyzed in engineering design activities. This sensitivity to small details of usage practices suggests the CFD analysis process is not robust and that new processes are needed. Potential New Approach
We must establish methods for verifying solution quality that take into account specific details of usage practices, in producing specific sets of data. This means that a capable quality assurance process must include a posteriori methods of evaluating data quality inspection methods. At least two different approaches to solution assessment can be identified, and both can be useful. • Physics-Based Assessment Test the full solution, or regions of the solution, for conformance to known physical laws. For example: discontinuities across shock waves should be in accord with the Rankine-Hugoniot laws, and the net circulation along a steady-state vortex should be constant. Local solution quality can be assessed based on the degree to which the solution exhibits this expected (required) behavior. An example of a physics-based local assessment is seen in Figure 1, below. This figure illustrates a prototype tool which identifies shock wave locations, and assesses the local modeling accuracy based on conformance to the Rankine-Hugoniot laws of shock wave jump conditions.
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• Rule-Based Assessment Test localized subdomains of the solution for their conformance to sound modeling practices for example, test the actual grid density against the density known to be required for high accuracy, given the flow interactions which are seen to be present in the selected subdomain. Various metrics are used that correlate with accuracy, e.g., grid spacing normal to the wall.
Figure 1. Physics-Based Assessment of Shock Wave Modeling Accuracy Both of these approaches (physics-based and rule-based) begin with an assessment of solution quality or uncertainty in localized regions of the solution domain. These localized uncertainty estimates must then be combined to build uncertainty estimates for global parameters of engineering interest, e.g., lift, drag, pitching moment. Several methods can be conceived to do this. Perhaps the most rigorous approach, and also the most computationally expensive, would be to use adjoint methods to combine a set of local uncertainties into global uncertainties. Simpler and more approximate methods could be conceived based on Greens functions or influence matrices. One approach which has received some attention is that of calculus of belief functions [8]. This provides a mathematically formal process for combining various component uncertainty assessments into a composite assessment of a global measure of uncertainty. An initial prototype has been developed using a rule-based approach and belief function calculus to establish confidence boundaries
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on CFD predictions of store separation trajectories6. The prototype approach for this application used a set of metrics which correlated with various uncertainties associated with the store separation analysis uncertainties for example in each of the three components of net aerodynamic force and net aerodynamic moment on the store. Evaluation of each of theses metrics at various points in a solution leads to estimates of components of uncertainty affecting the motion of the store. These component uncertainties are rolled up to overall uncertainties in the forces and moments on the store using belief function calculus, and a Monte Carlo method is then used to build an envelope of the confidence limits of the store separation trajectory [4]. Another interesting approach to uncertainty estimation was developed by Childs [3]. His approach enables estimates of local truncation error, that is, one type of error that results from using a too- coarse computational grid. Users would prefer to use the coarsest grid consistent with the required levels of accuracy, because solutions will run faster on a coarse grid. The penalty for using too coarse a grid is unacceptable uncertainty in the solution, and in the absence of suitable guidance most users will err on the side of caution and accept the resulting penalties in CFD throughput. Childs used neural networks to identify the truncation error in a CFD solution. The network was trained using CFD data on a high-density high-quality grid as truth data, and then it was used to assess local errors in other solutions. Key Elements of New Approach
Three examples have been presented to show the reader that an a posteriori uncertainty estimation approach can be developed for CFD analyses. Mimicking the DoD spiral development process, any such assessment approach probably would be brought into service incrementally. Initial capabilities would be relative simple, and the process would gain sophistication based on experience. Probably, the rulebased approach would be the most likely starting point. The knowledge required to implement this approach would be captured in three stages of research: Identify metrics relating to local or global solution quality which can be evaluated by a posteriori evaluation of a CFD dataset, conduct validation studies to quantify the relationship between each metric and the related sources of uncertainty and build a prototype to combine the discrete uncertainty measures into global uncertainty estimates relevant to the aerospace design process. The second step suggests that an entirely different approach to validation is needed. Current validation processes, as discussed above, generally are based on finding the modeling approach that yields the best accuracy on a (hopefully) representative problem. Then, the process relies on the judgment of the user in replicating the validated modeling approach in a new analysis problem. In the proposed new approach, validation instead is focused on quantifying the accuracy obtained in a range of modeling methodologies, with 6
A store is any object, weapon, fuel tank, sensor, or some other object, that is intentionally separated from an airplane in flight. A key issue is to ensure the store separates cleanly from the aircraft at all flight conditions where it might be released. In particular, it is important to ensure that the store does not follow a trajectory which might cause it to re-contact the airplane which could result in loss of the airplane and crew. Therefore, it is vitally important to understand the degree to which confidence can be placed in CFD predictions of a safe store separation trajectory.
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specific attention to the quantitative uncertainty resulting from non-optimal modeling. One key concept is to build this quantitative understanding of uncertainty prior to the engineering application of the analysis methods. A user cannot be conducting grid density studies during the live application of CFD in the design process; the required knowledge of uncertainties must be gained prior to launching the engineering design analyses. The required uncertainty rule base of course should cover a number of metrics that relate to solution quality. These metrics include (but are not limited to) grid density, grid imperfections, turbulence model usage, and spatial and temporal convergence. The uncertainty estimation process must lead to estimates of uncertainty that are quantitative, well founded in validation data, mathematically and statistically valid, and are predictive in nature. 3.4
Statistical Process Control
With such a process for a posteriori assessment of simulation data quality, many paths open up for adding value to simulation data. One of the most immediate benefits would be to enable statistical process control (SPC) as a means of assuring data quality in high-rate computational simulation processes. Statistical Process Control (SPC) is a method of monitoring, controlling and, ideally, improving a process through statistical analysis. Its four basic steps include measuring the process, eliminating variances in the process to make it consistent, monitoring the process, and improving the process to its best target value. This would be SPC in its familiar form as taught, for example, in industrial engineering curricula except that SPC would be applied to datasets coming out of a computational engine rather than widgets coming out of an assembly line. Statistical Process Control, enabled by a posteriori quality assessment tools, is an effective and proven tool for assuring quality in high-rate production processes. It does not require a quantum leap to apply these concepts to the production of computational data, instead of the traditional applications of production of mechanical components. An automated quality assurance process such as this is essential to achieving high-rate computational simulations required for engineering product development, since human inspection of the computational data is impractical (as discussed above). For an introduction to SPC see [10]. 3.5
Additional Added Value
With valid quality assessment techniques in place, statistical process control is only the first step. Once these assessment methods are in routine use, we will naturally begin building a database of historical data on quality assessments of data emerging from the high-rate analysis process. Analysis of this database will identify key factors that correlate with uncertainty in the computational data. These key factors potentially would include: • Geometric features that lead to high uncertainty. • Flowfield conditions or combinations of flowfield features that lead to high uncertainty.
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• Modeling selections or modeling practices that lead to high uncertainty. • Specific codes or code options that lead to high uncertainty. This information, of course, can be used immediately to guide tool and process improvement studies, and to justify and guide basic research initiatives. Analysis of the historical quality database also, of course, can be performed to identify the users that are having difficulty achieving high data quality in specific situations. This information can be used to improve training materials and user support capabilities in the organization. Of course, the database also will be invaluable in identifying the modeling practices that consistently provide superior data quality. With a process in place for quality assessment, capable of supporting high- rate computational simulations, the value of CFD data (or other domains of computational simulation) inevitably will increase greatly. This process of collecting and analyzing data will lead to establishing the capability to predict uncertainty in future analyses, based on the characteristics of the future analysis. This in turn will have a magnifying factor on the value of the computational data, since this capability will help ensure that the computational simulations are used wherever they are the most effective tool in providing the desired data. A quantitative understanding of uncertainty in CFD data (and/or other domains of computational simulation data) is needed to enable design teams to rely on the predictions. Without this understanding, engineering programs must ensure that they err on the side of conservatism, by acquiring additional data from independent sources to confirm predictions, by adding safety margins, and by other means. These defensive requirements, undertaken because the uncertainties in the data are not understood, lead to added cost and time in developing new products. Quantitative understanding of the sources of uncertainty also will be a powerful tool to focus our efforts to improve the CFD process. It will make it easier to justify these improvement efforts, by allowing the CFD team to state clearly the benefits in risk, cost, and cycle time that will be obtained through specific improvements. Finally, quality assessment will lead to uncertainty prognostics. With this ability to forecast data uncertainty in various applications, we can optimize the design data flow to achieve the best possible cost or cycle time at a predetermined level of risk that the aerospace product will meet its design objectives. And, we can use this process to understand the tradeoff between risk, cost, and cycle time in planning future product development programs. These benefits will lead to another round of dramatic growth in the value of computational simulation in the aerospace product development process.
4
Summary
Aerospace industries need improved design processes for both cost reduction, quality improvement, and break-through new vehicle concepts. In our view computational simulations, CFD and, for another example, computational electro-magnetics among others will play key roles in these new design processes but there are significant challenges that must be overcome, challenges related to accuracy of simulation, throughput and costs, and confidence in simulation results. Among specific areas where work is needed are:
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• Faster CFD analyses with greater accuracy, • Automated griding, • Improved handling of geometry inputs, and • Better quantitative understanding of uncertainty in CFD data. Our stretch goal for CFD that would indicate that the discipline is beginning to have the capabilities discussed in this paper would be the ability to generate 30,000 test cases/year at an average cost of $150 and with a computable error of less than 1% for all force and moment predictions for full airplane geometry including all control surfaces and perturbances.
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Bibliography [1] AIAA Committee on Standards for CFD, Guide for the Verification and Validation of Computational Fluid Dynamics Simulations, AIAA G-077-1998 [2] url: www.transcendata.com/products-cadfix.htm [3] R.E. CHILDS, W.E. FALLER, G.A. CEDAR, J.P. SLOTNICK, AND C.-J. WOAN, Error and uncertainty estimation in CFD, NEAR TR 574 (Nielsen Engineering and Research), 7 June 2002 [4] A. W. GARY, L. P. WESLEY, Uncertainty management for store separation using the belief function calculus, 9th ASCE EMD/SEI/GI/AD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability (PMC2004), Albuquerque, NM, July 2004 [5] R. T. FAROUKI, Closing the gap between CAD models and downstream applications, SIAM News, 32(5), June 1999 [6] T. J. PETERS, N. F. STEWART, D. R. FERGUSON, AND P. S. FUSSELL, Algorithmic tolerances and semantics in data exchange, Proceedings of the ACM Symposium on Computational Geometry, Nice, France, June 4 - 6 , 1997, 403 - 405. [7] D. R. FERGUSON, L. L. MIRIAM, AND L. H. SEITELMAN, PDES Inc. Geometric Accuracy Team, Interim Report, July 24, 1996. [8] W.L. OBERKAMPF, T.G. TRUCANO, AND C. HIRSCH, Verification, Validation, and Predictive Capability in Computational Engineering and Physics, presented at Foundations for Verification and Validation in the 21st Century Workshop, Johns Hopkins University, Applied Physics Laboratory, Laurel, MD, October 22-23, 2002. [9] S. ROGERS, M. AFTOSMIS, S. PANDYA AND N. CHADERJIAN, NASA Ames Research Center, Moffett Field, CA; E. Tejnil and J. Ahmad, Automated CFD parameter studies on distributed parallel computers, Eloret Institute, Moffett Field, CA 16th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2003-4229, June 2000 [10] http: //reliability.sandia.gov/index.html
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LEAPS and Product Modeling R. Ames*
Abstract The requirements for product model integration via virtual prototyping suggest that the underlying mathematical framework must be robust, extensive, and accurate. The challenges of integrating modeling solutions with simulation federations while addressing Validation, Verification, and Analysis issues of large acquisition programs will be discussed.
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Introduction
Ships are arguably the most complex of inventions. Dan Billingsley, of the Naval Sea Systems Command writes, The typical ship is comprised of hundreds of times as many parts as the typical aircraft, thousands of times as many parts as the typical power plant, ten thousands of times as many parts as the typical vehicle. Indeed, our more complex ships fly aircraft off the roof, have vehicles running around inside, and have a power plant in the basement all incorporated in a floating city capable of moving at highway speeds around the oceans of the world. The process of ship development is likewise complex, particularly for naval warships. It involves thousands of individuals in hundreds of corporations, governmental and regulatory bodies operating throughout the world. Each ship is in some ways unique. A ship may have a conception-to-retirement lifespan of 50 years involving both those notyet-born when it was launched and those who will retire before it retires. Certainly today's ship will outlive several generations of information technology applied to its development, construction, and service life support [3]. The estimated design cost of the US Navy's new destroyer is $3-5B. Detail design will add another $500M. The production costs for the first ship are estimated at $2.7B. These costs rule out building operational prototypes as a means of test, evaluation, and selection. As a result, ship programs are moving toward * Naval Surface Warfare Center, Carderock, MD
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virtual prototyping and operational simulations within an integrated architecture or framework that is, toward the development of Smart product models. Further, with the new emphasis on the war on terror, system acquisition decisions are being framed in light of a larger context. Specifically, DOD Instruction 5000.2R is being reformulated in favor of a broader Joint Capabilities Integration and Development System JCIDS 3170.01, which focuses more on system-of-system requirement and upfront systems engineering [8]. Just as in the development of mechanical products, software development must follow a comprehensive process involving requirements, design, development, testing, and deployment. If a naval ship acquisition program must ensure that operational requirements can be met by a particular design before metal is cut, then one can conclude that a virtual prototype of this product must pass the test of validation and verification. An essential component of any virtual prototyping simulation will be a credible and comprehensive smart product model that encompasses all system dependencies and system performance. Since virtual prototypeing simulations operate on smart product models, requirements on smart product models are extensive and complex. This paper describes software development work being done in support of emerging System-of-System and Smart product model requirements. The ability to automatically conduct analyses in a multitude of disciplines is a cornerstone for validation and verification of Smart product models. Consequently, the emphasis here is to develop techniques that support multidisciplinary analysis. Three related software development efforts are described. The first, Geometry and Engineering Mathematics Library (GEML), provides a mathematical framework for geometry, grid data structures, analysis characterization, and behavior modeling. The second, Geometry Object Structure (GOBS), is a framework that provides a unique modeling capability for geometry, as well as more complete functionality for discretization, evaluation, modeling of complex solids of different material elements, logical view construction, and bi-directional communication of topology between geometry systems, modelers, and analysts. The third, Leading Edge Architecture for Prototyping Systems (LEAPS), provides the product information representation methodology required to facilitate the work of an integrated product team. Finally, underlying this development is the idea that all shared or integrated product information must follow an ontology of machine-processable data.
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Approach
The approach assumes that the requirements for simulation include an architecture that is integrated, extensible, supports life-cycle product knowledge, and is physically valid. Unlike arcade games whose simulations may lack a certain reality when it comes to performance, simulation based design systems must ensure that the physics of the product is sufficiently accurate to support acquisition decisions. This paper will describe an approach to smart product modeling using new representational methods. It starts with the assumption that all disciplines need smart product model data; that the need for a common representation exists; and
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that the smart product model must contain sufficient information to convey design intent and detail for all disciplines. The fundamental premise is made that smart product model data is composed of geometric, topological, performance, and process data and that product and process integration occurs through a common representation. In essence, a common smart product model supports the application of information technology to all aspects of product development, manufacturing, and operation. To understand how the smart product model concept with its emphasis on integrated design and analyses domains changes the traditional approach to design software, consider Figure 1 where a notional architecture supporting a smart product model with relationships between functional domains including design and analysis as well as utilities and tools common to all users is shown. Traditional tool development has focused on the boxes, not the arrows, with the consequence that the requirements for the individual boxes have dominated. However, the requirements implied by the arrows change the picture considerably. For example, when requirements for design and analysis are added, the architecture becomes like Figure 2, a much more complex concept.
Figure 1. Notional Smart Product Model Architecture A cursory review of the requirements imposed on a smart product model shows the complex and extensive need for geometry and topology, domain decompositions of product space, performance and property data, materials, and many other data. As product life cycles evolve from concept to manufacturing to maintenance, it is clear that for a smart product model to characterize reality it must be flexible, extensible, and incorporate a superset of all requirements imposed on it. If the domains we call design and analysis use applications such as CAD, FEA, and CFD to fulfill these functions then the smart product model must be capable of representing
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and communicating the required data to these applications. This means, for example, that if a particular application needs discretized data of a certain form, then the smart product model must be capable of generating the data in the required form.
Figure 2. Notional Architecture Data Requirements It is evident that one of the major roadblocks to integration is confusion over the form of representation of geometry. Geometry is the common element in the design and analysis of complex systems and the form of the representation determines the level of effort involved in generation of analysis data, provides the means to communicate the design for product review and mockup, and describes the product for manufacturing. But often the form will change as particular applications change. Further, different organizations and disciplines may require different forms or views of the geometric representation. Thus there are conflicting requirements on the geometric form. On the one hand, there is a desire for a single, common form to aid in overall product data integration while on the other hand there are requirements, both technical and historical, for different forms supporting different disciplines and organizations. It is not surprising then that when complex systems are designed, the integration of performance data with geometry becomes critical and problematic. In this paper, data is viewed from two perspectives: Topological Views and Common Views emphasizing the way designers and engineers view design, performance, process, and product data (see Section 4). Smart product models start with methods and algorithms used to characterize shape, behavior, and content. These methods are fundamental to the modeling process. Likewise, the use of these mathematical methods and the associations made among them, topological and common views, provide the building blocks for system design. Through proper model topology we provide a framework for mul-
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tidisciplinary integration necessary for a fully capable design environment. This framework is implemented using three component technologies: GEML, GOBS, and LEAPS. GEML provides the mathematical foundation necessary for characterization of geometry, analysis, and their coupling. GOBS takes this mathematical foundation and builds unique topology models that allow for multiple disciplines to access to the design without duplication and redundancy. Finally, LEAPS combines the modeling of GOBS with an application framework providing an environment for designing and assessing complex systems. The following three sections describe these component technologies and the relationships among them.
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Mathematical Methods: GEML
The Geometry and Engineering Mathematical Library, GEML, provides a mathematical basis for modeling geometry. GEML is a C++ implementation of the DTJNTURBS [9] subroutine library. The objective of GEML is to provide a suite of mathematical objects for design and analysis. This object library includes functions to: 1) import, export, modify, and evaluate geometry data; 2) develop, modify, adapt, grid or mesh data with associated spline functions; and 3) store resultant geometry, grid, and analysis data in a common framework with dependencies maintained. Using these objects, a GEML based framework can contain and control all information relating to geometry, grids, and analysis. Since analysis is dependent on grids, and grids on geometry, the coupling of these elements in an object-oriented data structure is logical. GEML provides for flexible representation and object dependency. It includes such features as geometry varying with time; mapped geometry (composition of functions); grids denned in the parametric domain, and n-dimensional functions for analysis and behavior modeling. In order to model geometry, grids, and analysis data it is necessary to have a mathematical basis which facilitates representation of all three and allows for both implicit and explicit coupling of the data. In GEML, splines and composition of functions provide the essential building blocks for geometry and analysis description and coupling. The splines used are unrestricted in the number of independent and dependent variables allowed and this permits the modeling and interpolation of any number of geometric and analysis variables such as Cartesian coordinates, velocities, pressure and temperature; and with any number of independent parameters such as time, frequency, and iteration. For instance, traditional geometric entities like parametric spline surfaces (x,y,z) — f(u,v) are extended to n-dimensional mappings as, for example, (x, ?/, z, Vx, Vy, Vz, Cp, temp,) = g(u, v, time). The removal of dimensional constraints affords tremendous flexibility in the modeling of objects. Composition of functions permits straight forward and timely reparameterization of surface elements along with attached behavior models. We illustrate the concepts of n-dimensional splines and composition of functions as follows. Begin with a surface S parameterized as F(u, v, time] with a single time slice shown in Figure 3. The function F is an example of an n-dimensional spline. Now suppose there is a subsurface S* that is of particular interest. To obtain a parameterization of S* proceed by finding a function g mapping a second
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domain D<2 into the original domain D\ and covering the preimage of S*. Then obtain a parameterization of <S* by forming the composite function G(s, t, time) = F(g(s,t),time)) . We can include the results of any analysis on S* by exploiting the n-dimensional character of splines. If, for instance, p(s, t) is the pressure at a particular time at the point F(g(s,t),time)^ then we can construct a model of S* and pressure by forming the composite function (x,y,z,p) = H(s,t,time) = (F(g(s,t),time),p(s,t)). See [1] for more details.
Figure 3. Composition Function
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Model Topology: Geometry Object Stucture (GOBS)
GEML provides a framework for representation of geometry and analysis data, but design integration across multiple disciplines requires that the representation of the design model be sufficiently rich to support a variety of analyses including differing discretization requirements. Simply having the mathematical and geometrical objects and methods is not sufficient. Further, the idea of supporting multiple analyses and discretization requirements means that the model must explicitly contain all required information needed to support those requirements in a machine-processable form. In particular, a smart product model is not to rely on inferring information from a rendered image. To organize all this data, we have implemented concepts of views, topological and common. Topological Views: Designers and engineers rarely, if ever, view an object as abstract mathematics. Rather, they view objects, quite naturally, as physical things
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having properties or characteristics important to their design goals. Thus, for example, while a solid model may provide a sufficient abstract mathematical description of a ship's hull, a designer or engineer would never be satisfied with such a description. Rather, they would also want to know performance and other data about that hull so they could make decisions as to its suitability for whatever project was in hand. To support designers and engineers while maintaining connections to the abstract mathematics needed in software implementations, we devised the term Topological View to encompass shape data in the form of surfaces, trimmed and untrimmed, data describing the connectivity relationships among the shape entities, i.e., the topological structure, and data capturing performance and behavior knowledge. Common Views: Common Views are the primary vehicle by which domain analysts and designers will view or interrogate a smart product model. A Common View is a way of organizing a collection of views for convenience in analysis and is simply a logical grouping of Topological Views and/or other Common Views. One example of a Common View could be Habitability Spaces on Deck 3 containing Topological Views of all the habitable compartments on Deck 3. Another Common View, Ship Habitability Spaces might contain the Common View Habitability Spaces on Deck 3 as a member. Similar uses of Common Views could include Exterior Surfaces, Compartments, Machinery Spaces, or Mast or a Weather Exposed Structures and Components. To further understand the idea behind views, consider an example of three compartments within a ship as depicted in Figure 4. This case, while simple in appearance, poses a number of challenges to a Smart product model. One of the elements in this model is the transverse bulkhead 2, (Trans-2). Consider the roles that Trans-2 plays in the overall product and the information that must be available for the use of designers and analysts. Certainly, the topological view of Trans-2, shape, topology, and properties like permissible loads or manufacturing tolerances must be available for design and analysis. Information about the roles Trans-2 plays in other topological views, front face of the bulkhead forms part of the topological definition of the front compartment while sections of the back face of the bulkhead form faces for the other two compartments, must be preserved and made available as needed. Structural analysts will want to know what structural components are connected to Trans-2 so the information that Trans-2 is connected to the hull, port and starboard, the longitudinal, and decks above and below must be maintained. While these roles and connections may be self evident from the graphical image, nevertheless they must be explicitly defined in the Smart product model. In addition, there are locations on this bulkhead that may be of interest to analysts such as the corner points at intersections with other surfaces (longitudinal, hull, deck, etc.). Further, the bulkhead could be a watertight bulkhead bounding a zone on the ship and may be grouped together in a common view of all the watertight bulkheads. The Geometry Object Structure (GOBS) [2] has been developed to sort out and manage these different views and connectivity relationships. As previously discussed, GOBS supports two views: topological and common. It also provides objects, CoEdge and CoPoint, to maintain connections among various entities. A
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Figure 4. Three-Compartment Test Case CoEdge object contains all the information about a connection, the surfaces that are connected and the edges in each surface where the connection is made, i.e., edges that are to be declared equivalent in Cartesian space. An example of such a connection is the intersection at a deck edge and the hull defining a common boundary (Figure 5). Similarly, a CoPoint maintains information on points of connection. Note that the CoEdge and CoPoint objects provide the ability to traverse boundaries, logically, explicitly, and with information on the relationships of surface parameter spaces. This affords many advantages including the ability to grid or mesh across trimmed surface boundaries with the gridding nodes changing continuously. The GOBS structure allows for the implementation of differing views into a single smart product model using topological and common views. It also allows for future extensions, which will include additional n-dimensional objects and composition of functions. These extensions will allow more flexibility in discretization for physics based analysis, variational or time-dependent analysis mappings, and allow more complete representations for engineering analysis in general. The objective is to provide the means to construct views of complex objects that contain many entities and whose representation cannot be reduced to any single entity type and to provide the means to communicate this representation bi-directionally between existing systems, other geometry modelers, and selected analyses. The GOBS technology allows geometry and attributes to be presented to engineering modelers and analysts in a form which allows for convenient discretization according to the requirements of their models and analyses.
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Application Framework: LEAPS
A brief overview of the GOBS topology built on GEML mathematics has been presented as the foundation for development of a new suite of tools. An environment for integration, however, requires a complete framework for development not just geometric objects. The Naval Surface Warfare Center, Carderock Division (NSWCCD) developed a framework called Leading Edge Architecture for Prototyping Systems, LEAPS. The LEAPS framework supports smart product modeling for the generation of virtual prototypes. As mentioned earlier LEAPS implements the GOBS technology along with other framework elements.
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Figure 5. GOBS Topology The LEAPS framework was designed to support smart product modeling in general and to date has demonstrated an ability to support conceptual and preliminary ship design and analysis. The complexity and diversity of naval ship design and analysis deals with information that is complex, extensive, ambiguous, and dynamic. To support these realities, the LEAPS architecture takes a meta-model approach to smart product model development. To understand the LEAPS architecture it is necessary to have an ontology for concepts such as Application Programming Interface, smart product model, Meta Model, and other terms used today in discussing integrated environments and their computational framework. A brief description of LEAPS elements is presented here to help with the understanding of this framework. However, a detailed report on the LEAPS architecture is beyond the scope of this paper. See [7] for more details. LEAPS processes and applications communicate to a smart product model database, LEAPS/PM, using an Application Program Interface, LEAPS/API. These applications write objects that are instantiations of a set of generic classes, LEAPS/MM. These objects and their associations are named and instantiated in compliance with a formalized Product Meta-Model, LEAPS/PMM, where the Product Meta-Model is specific to a class of products, e.g., cars, ships, planes. The LEAPS Meta-Model (LEAPS/MM) is a set of generic classes used in defin-
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Figure 6. Sample Application: Gridding across Boundaries
ing a Smart smart product model. Fundamental to product development is the notion that a design progresses through various concepts and that these concepts must perform in some condition or scenario. Similarly, product development centers on a given set of functional requirements against which a concept's design is measured. The LEAPS/MM supports the notion of Studies that contain Scenarios and Concepts, and Concepts contain smart product model data. The LEAPS Application Programmers Interface (LEAPS/API) is a set of C++ classes available for software development that implements the LEAPS/MM. The LEAPS Product Meta-model (LEAPS/PMM) is the object oriented schema or product characterization of a category of products, (e.g. combatant surface ships). The LEAPS/PMM defines a specific category of products as objects and/or templates of objects. The LEAPS Product Model (LEAPS/PM) is the instantiation of the LEAPS/MM. The LEAPS Data Base (LEAPS/DB) is the persistent store for any
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LEAPS/PM. LEAPS Applications (LEAPS/APP) are individual applications that communicate with the LEAPS/DB through the LEAPS/API. These individual applications come in many forms and can be implemented to meet individual needs. To summarize the LEAPS processes, applications communicate to a smart product model database, LEAPS/PM, using an application program interface, LEAPS/API. These applications write objects that are instantiations of a set of generic classes, LEAPS/MM. These objects and their associations are named and instantiated in compliance with a formalized Product Meta-Model, LEAPS/PMM. LEAPS provides support for information sharing and exchange requirements of scientists and engineers through an Application Programmers Interface, API. Through this interface, LEAPS has the capability to integrate the assessment efforts of multiple organizations. The classification of tools as Design vs. Analysis is more about process than architecture. In the LEAPS architecture everything is an application communicating with the smart product model. Data requirements for each application suggest that computational analysis will require a design before any analysis can be performed. The benefits of using LEAPS processes have been demonstrated on two analysis applications and are note worthy. Sample metrics for Data Preparation (Extraction and Transformation) show a decrease from 13.5 work-days to .5 workdays for Ship Infrared Signature Prediction and 20 work-days to 1 work-day for Radar Target Strength predictions. LEAPS smart product model data supports the initialization of HLA federates allowing multiple assets to be fully described at the product level while publishing performance data to an HLA simulation. Thus, performance data for each asset is available at the required fidelity of the simulation enabling the use of simulations requiring intimate knowledge and detail about a particular asset's performance and behavior. A sample architecture is shown in Figure 7.
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Summary
In summary, the requirements for modeling complex systems involve three components; a mathematical core, a methodology for managing geometry, topology, and performance data, and an application framework. These three components are the basis for a specific design environment implementation. The GEML library provides the necessary mathematical tools to compose advanced structures and models. The GOBS classes take advantage of GEML's technology to provide explicit geometry relations and analysis views of the design. LEAPS integrates these classes into a development environment designed for information-sharing in a teaming situation. The LEAPS architecture also provides for the inclusion of legacy software into the smart product model via the LEAPS API. Finally, LEAPS enables HLA Federate Object Model development by way of dynamic object and property creation, and using common taxonomy for common attributes; thus allowing products like ships, planes, and tanks to participate in a battlespace HLA simulation.
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Figure 7. HLA Simulation Infrastructure
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Glossary of Terms
Virtual Prototype (VP) Functionality that enables users to visualize, interrogate, and analyze a smart product model. Model A physical, mathematical, or otherwise logical representation of a system, entity, phenomenon, or process [5]. Simulation A method for implementing a model over time [5]. Integrated Architecture The structure of components, their interrelationaships, and the principles and guidelines governing their design and evolution over time [6].
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Bibliography [1] Bob Ames and Dave Ferguson, Applications To engineering design of the general geometry, grid & analysis object In DTJVURBS, 5th International Conference on Numerical grid Generation in CFD and Related Fields, Mississippi State University, April 1-5, 1996. [2] Bob Ames and Richard Van Eseltine, A framework for modeling complex structures, Hydrodynamic Technology Center 1st Symposium on Marine Applications of Computational Fluid Dynamics, March 10, 1991. [3] Dan Billingsley, Naval Sea Systems Command,, An activity/transaction model for ship design development, ICC AS 2002. [4] Alan K. Crawford, Bob Ames, Greg K. Harrington, and Rich Van Eseltine, Neutral format data exchanges between ship product models and analysis interfaces, International Conference on Computer Applications for Shipbuilding, ICCAS 1999 [5] DoD Directive 5000.59, DoD Modeling and Simulation (M&S) Management, January 4, 1994 [6] DoD Integrated Architecture Panel, 1995 [7] Myles M. Hurwitz, Leading edge advanced prototyping for ships (LEAPS): an integrating architecture for early stage ship concept assessment software, 2nd ASNE Modeling, Simulation, and Virtual Prototyping Conference, Arlington, VA, Nov 24-25, 1997, 135-141. [8] Chairman of the Joint Chiefs of Staff, Operations of the Joint Capabilities and Development Systems, CJCSM 3170.01A. [9] Naval Surface Warfare Center-Carderock Division, The Geometry and Engineering Mathematical Library User ManualS, Version 2.1, by Boeing Mathematics and Computing Technology Group
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3D Modelling of Material Property Variation for Computer Aided Design and Manufacture M.J. Pratt
Abstract Layered manufacturing (LM), also known as solid freeform fabrication or rapid prototyping, is an important emerging manufacturing technique that creates an artefact by depositing successive layers of material under computer control. Originally, objects manufactured by LM methods were composed of homogeneous material. However, new methods of optimal design specify 'functionally graded' or 'heterogeneous' material distributions, and developments in layered manufacturing technology provide the means for manufacturing them. The problem of representing variable material distributions for design and manufacturing purposes essentially requires the parameterization of the interior of a general solid model, and the paper surveys published methods aimed at achieving this. It is likely that other applications will emerge in future for the modelling of properties in the interior of volumes with complex boundaries.
1 Introduction The term layered manufacturing (LM) is used for a family of additive manufacturing processes in which objects are constructed layer by layer, usually in a series of parallel planar laminae. Alternative names for these processes include solid freeform fabrication (SFF) and rapid prototyping (RP). The last term stems from the fact that such processes were initially used primarily for the creation of non-functional prototype parts used for 'look, feel and fit' evaluation of component shape as part of 'LMR Systems, 21 High Street, Carlton, Bedford, MK43 TLA, UK.
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the design process. However, the use of LM is now being extended into other areas of manufacturing, in particular the production of moulds and dies, and it is likely that it will soon become used as a method for small batch production of functional components of engineering products. Many LM processes currently exist, using different materials and layering methods [6]. The following classes of methods may be identified: Photopolymer solidification (e.g., stereolithography, solid ground curing), in which a liquid resin is hardened layer by layer with a laser or ultraviolet lamp. Material deposition (e.g., fused deposition modelling), in which drops or filaments of molten plastic or wax are deposited to construct each layer. Powder solidification (e.g., selective laser sintering and three-dimensional printing) , in which powdered material layers are solidified by adding a binder or by sintering with a laser. Parts can be built from ceramics, nylon, polycarbonate, wax or metal composites. Lamination (laminated object manufacturing and solid ground curing). The first of these methods uses lasers to cut layers from sheets of paper, cardboard, foil or plastic, stacks them and bonds them together. The second uses a cut mask to expose regions of resin to be solidified by an ultraviolet lamp. Weld-based approaches. Now emerging from the research stage, these use welding and cladding techniques to build metal parts [28]. LM processes have several advantages over other manufacturing methods. They operate largely automatically on input data generated directly by a CAD system. Their process planning requirements are comparatively low, the primary requirement being that of determining an optimal orientation in which to build the product. They can be used to build very complex artefacts with intricate geometry and internal voids, or to create multiple ready-assembled components simultaneously, without the use of special tooling. On the debit side, the layer by layer mode of building LM objects generally leads to surfaces that have a staircase appearance, and it is at present difficult to achieve good tolerance or surface finish specifications. Poor surface quality is sometimes overcome by performing finishing operations, such as grinding or polishing, after an object is built. LM products may also have inferior material properties when compared with objects manufactured by other means. Additionally, some of the chemicals used in LM processes present health hazards. Much research is currently directed towards overcoming these and other problems.
2 Material distribution in LM objects Although objects manufactured using LM were originally composed of single materials, the manufacture of multi-material objects is now becoming possible. By
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contrast with older manufacturing processes, LM in principle permits complete 3D control over interior material constitution, and the following possibilities are emerging: • Entire pre-existing components may be embedded as the object is built; • Discrete 3D regions of different materials may be deposited through the use of multiple deposition heads; • The simultaneous use of multiple deposition heads also makes it possible to deposit continuously varying combinations of different materials. The term functionally graded material (FGM) has been used to describe such material distributions. • Reinforcing fibres may be laid down with the deposited material to obtain directional strength characteristics in the built object. The first two of these present little difficulty from the point of view of mathematical representation, but the third and fourth raise some interesting problems. Both of them are also potentially important in practice, as explained in the following paragraphs. New design methods such as homogenization for structural topology design [2] specify varying material properties in the interior of a designed object. As we have seen, LM provides a means for producing such artefacts. If such design practices come into productive use some means will be needed of representing the material distribution throughout the volume of the designed object, to serve as output from design and input to LM manufacturing. Many applications have been identified for objects composed of functionally graded materials. Single components may be manufactured that exhibit local control of surface and interior properties including corrosion resistance, electrical or thermal conductivity, hardness, magnetic properties, malleability and refractive index [26]. Some of the specific applications mentioned in the papers surveyed in the following sections are as follows (see also [10, 30, 54]): • Cutting tools for machining, with hard but brittle materials for the tip, grading to stronger metal for the shank. • Drug delivery pills or implants, in which the active chemical is embedded in a capsule that breaks down gradually in the human body and releases it according to a specified schedule. • Gradient index lenses, in which the optical properties result from gradients in the refractive index of the material used rather than (conventionally) through the external geometry of a homogeneous lens. • Medical implants, using exterior materials chosen for wear resistance and good bonding to bone, grading to interior materials chosen for strength. • Turbine blades, with ceramic exterior for thermal properties grading to metal in the interior for strength.
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Further, recent developments in LM have demonstrated the deposition of reinforcing fibres as an object is built [12]. This leads to obvious possibilities for design in terms of directional strength characteristics. Now the design representation will need to capture not only an isotropic distribution of material composition but also additional anisotropic directional information. Although this paper is primarily concerned with the representation of isotropic inhomogeneity, it is believed that all of the methods reviewed can be extended easily for the capture of material anisotropy. A related but slightly different material representation problem arises because in most LM processes (with the exception of those classed above as lamination methods) material is deposited or hardened in a strand or filament, using some area-fill strategy within the boundaries of each layer. This being so, the structural properties of an LM object depend on the particular area-fill strategy employed. For a given method they will be affected by the lengthwise strength of each filament, and by the strengths of its bonds to neighbouring filaments in the same layer and to the underlying layer on which it is deposited. It is therefore highly desirable to be able to represent the microstructure resulting from the deposition patterns within layers, so that analysis programs tailored to particular physical choices of material and process can be used to calculate strength characteristics of manufactured parts. However, this aspect of the material representation problem has so far been given less attention by the research community, and no attempt has been made in this paper to survey work on the representation of the material microstructure. Neither is there any attempt to deal with microstructures generated by the embedding of reinforcing materials (e.g., carbon fibres) in a matrix. Although this method is widely used, it is not an LM method in the sense defined here (except for the technique mentioned in the previous paragraph, which is still experimental). With present technology, it is the scan paths generated by LM systems (analogous to NC cutter paths in numerically controlled machining) that give rise to the particular microstructure in a specific LM object, and they will provide the basis for the capture of the associated microstructure. But the design of the original geometry/material model of the type that is the focus of this paper must be completed before scan paths can be generated. Thus the incorporation of microstructure modelling may be regarded as a separate problem, though ideally it should be treated in a manner compatible with the macrostructure representation. Another type of microstructure is porosity. Schroeder, Regli et al. [45] have developed a representation for solid objects with material, density and porosity variation using stochastic methods. They suggest that this is the first use of stochastic geometry in the CAD modelling area, and provide experimental data to validate the approach. An important application is modelling the porosity of bio-materials, which is critical in the creation of replacement bone tissues. The topic lies outside the main stream of this survey, however, and will not be pursued further here. This paper provides a review of methods that have been proposed for the mathematical representation of solids with variable or functionally graded material properties, on the macroscopic scale. There are two known previous surveys of the same kind [17, 37], one by the present author. Since their publication much further work has been done, as evidenced by the fact that the only specific methods
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for the representation of material property distributions cited in the author's own previous paper were those of references [18, 23, 24, 43] and [47] of the present paper. Furthermore, there are signs that the field is beginning to mature. The list of publications surveyed here is not completely exhaustive, but it is hoped that all the major approaches have been covered. The author offers his apologies in advance to any group that may feel that its work has been unfairly neglected in the paper.
3 Representations for material distributions — Initial considerations For the representation of material macrostructure, what is ideally needed is some means of defining a function whose domain is strictly limited to the interior and boundary of a solid model of a 3D object. This is easy if the object is bounded by constant values of the coordinates in some 3D coordinate system, e.g., in the case of a rectangular block or a circular cylinder. More generally, it is also easy for a triparametric volume resulting from a mapping of such a coordinate system from some parameter space into Euclidean space. However, for most realistic mechanical engineering objects there is no such simple parameterization. If the boundary of the defined material distribution coincides only approximately with the boundary of the object model, then there may be interior regions near the model boundary where no distribution is defined. This problem could be overcome by ensuring that the domain of the distribution is larger than that of the entire CAD model, but that is also non-ideal — it leads to difficulty in the precise specification of the distribution on the object boundary itself, which is likely to be a frequent requirement from the design point of view, as can be seen from some of the application examples given earlier. Several well-known methods, including the finite element method, appear at first sight to be well suited to the problem described. However, these methods generally make use of an approximated or simplified object boundary, and therefore do not meet the requirement on accuracy of the boundary of the material representation domain. The representational methods so far proposed fall into two broad classes. The first, based on decomposition of the product shape into simpler geometric elements, has several subclasses, as indicated in the following list: 1. Decomposition methods — these discretize the object's volume into geometric elements of various kinds that individually have simple parameterizations. The possibilities include • • • •
Regular or semi-regular decomposition into cells of uniform type; Irregular mesh-type decomposition; Irregular decomposition into geometric primitives; Decomposition into 'material features';
2. Global methods — these provide means for the construction of a single specific parameterization for the interior and boundary of the entire object.
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Examples of all these approaches are given in what follows. The type of representation that eventually becomes dominant may depend upon the nature of the practical design systems that emerge for the definition of objects with continuously variable material properties. Conversely, of course, it is possible that the discovery of an 'ideal' representation for functionally graded material objects may influence the development of systems for the design of such objects.
4 Decomposition methods This section describes several decomposition methods that have been suggested in the literature of functionally graded material modelling. 4.1 Voxel-based modelling The most elementary form of cellular decomposition uses voxels, small cuboidal cells forming a regular 3D grid. Chandru et al. [7], among others, have advocated the use of voxel modelling for LM purposes in this manner. Any 3D solid model may easily be converted into a voxel-based representation. Typically, a grid is imposed over the object and all cells more than 50% occupied by material are considered to be part of the object and the rest to be exterior to it. Voxels are usually small in relation to the overall object size, and almost invariably the physical properties associated with each voxel are taken to be homogeneous in its interior. For the present application, therefore, material variation will be modelled in terms of discontinuous changes across voxel boundaries. In one sense such a representation is well suited to layered manufacturing, since each layer in the fabrication process can be represented as a planar layer of voxels whose vertical dimension is the layer thickness. On the other hand, voxel models suffer from the same disadvantage as objects manufactured by LM, in that they exhibit surfaces with staircase effects. All faces of the voxel model are parallel to the three coordinate planes, and in general the faces of a real object can only be approximated. In general, any reorientation of the model requires recalculation of its voxel representation. Some further important limitations of the voxel approach are that • A voxel model is not suitable for analytical purposes, e.g. for finite element analysis. There are far too many elements, if each voxel is regarded as an element; it is only possible to modify the mesh by changing its grid characteristics, and the grid-based mesh is far from ideal for most types of analysis. • It is not easy to associate attribute information (e.g., material conditions on the object boundary) with entire faces of an object model if those faces are discretized into assemblies of voxel facets. • If voxel height is equated to layer thickness, some of the simplicity of the method is lost when adaptive slicing is used. This is the calculation of a different thickness for each individual layer in the deposition process, based on considerations of (for example) surface finish of the manufactured object.
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Octree decomposition [16, 29] may be cited as an example of what is referred to in this paper as a semi-regular approach to decomposition. It is more sophisticated than pure voxel decomposition, in that while all cells have the same cuboidal form they vary in size by factors of two in their linear dimensions. In its basic version the octree method uses small cells near the boundary of an object model to achieve acceptable resolution there, but larger ones in the interior in order to keep the overall number of cells within bounds. Since the cells are all cuboidal it is easy in principle to model variable material distributions inside all cells in terms of local Euclidean coordinate systems. However, such an approach still suffers from most of the disadvantages listed above for the pure voxel method except that the number of cells is smaller. Bajaj [1] has combined the use of octrees with Cl-continuous triparametric polynomial spline functions to model spatial distributions for visualization purposes. This is essentially an interpolation approach. The boundary of the interpolated data set emerges from the process in a triangulated representation, whereas what is needed for LM is a fit of the spatial distribution to the pre-specified boundary of a CAD model. A further approach that is akin to the use of a regular mesh is based on the modelling of solid shapes as triparametric volumes [27]. If this is done the material distribution in the interior may be represented as a function of the parametric coordinates. If a B-spline formulation is used, for example, and the material composition is specified at the knots defining the mesh, then it may be interpolated in exactly the same way that the geometry is interpolated, in terms of the B-spline basis functions. It is noted in [40] that time dependency (of material composition or of shape) may easily be included in such a representation. Unfortunately, many engineering shapes do not lend themselves to being modelled by a single trivariate volume, and to build a complex shape as a combination of such volumes leads to continuity problems in the material distribution where the different volumes meet or intersect. 4.2 Mesh-based decomposition. Irregular decompositions of the finite element type provide alternative possibilities for the modelling of material distributions. One such proposal is made by Jackson et al. at MIT [18], who mesh a solid model into tetrahedral finite elements. In general, the material composition in the model may be represented in terms of a vector-valued function M n (x) = (mi (x), 7712 (x),... ,m n (x)), where • x is the position vector of a point in the domain of the object, • n is the number of different materials to be combined in the LM process, and • Y^i=i m i( x ) — 1? so that rai(x) > 0, i £ [l,ft], expresses the relative concentration of material i at the point x. In [18] the designer is allowed to specify the material composition in terms of distance from some geometric datum (for example, the axis of a rotational object) or
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from the object surface. This fixes the values of the components of M n (x) at the nodes of the mesh. The cited reference appears to be the first in which distance fields have been advocated for the specific purpose of calculating material distributions, though related approaches have recently become prevalent, as will be seen in what follows. The material composition at the mesh nodes having been determined, the values in cell interiors are then interpolated in [18] by trivariate Bezier polynomials using the nodes as control points. Only piecewise constant and piecewise linear variations were considered in this work. However, additional smoothness in the material distribution between cells could in principle be attained by adding further nodes to the mesh and using Bezier representations of degree two or higher. One advantage of this type of approach is that the model can be directly used for finite element analysis [21, 22, 35]. Against this, it may be questioned whether a mesh that is optimal for design is equally suitable for subsequent analysis. Other drawbacks of the finite element mesh approach are as follows: • Although this method can specify the material composition exactly at the mesh vertices, elsewhere the composition is computed by interpolation, and is therefore only approximate. • As in the voxel approach the object boundary is also approximated, and consequently the material definition domain does not fully coincide with the object domain. Clearly it would also be possible to decompose the object model into the hexahedral elements that are preferred by many finite element analysts. However, the same advantages and disadvantages apply in this case also. Deformation of a regular mesh.
Fadel and Morvan [8] propose a method for modelling the shape of a heterogeneous material artefact based on a deformation of a regular mesh, either in 2D or 3D. While the nominal topology of the mesh remains unchanged in the deformation, it is suggested that coincidence of nodes and edges should be allowable in the deformed mesh to give greater flexibility. The authors recommend that geometric interpolation within the mesh should initially be linear (though generalization is of course possible), because most LM processing software only handles polygonal slice (layer) shapes. However, the interpolation of material properties in the mesh may use some alternative method permitting the modelling of smooth distributions. An example is given of a design for a multi-material flywheel, in which Bezier polynomials are used for modelling both the shape and the material distribution. The approach to the modelling of geometry appears to restrict the deformation method described in [8] to the handling of comparatively simple shapes. It is suggested that such shapes may be combined through the use of the Boolean combinations (union, intersection, difference) of constructive solid geometry (CSG) [13]. However, these operations are not interpreted traditionally in terms of point sets, but in terms of refinements of the mesh. These will allow, for example, included
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regions with different material properties, and (if desired) transitional regions over which blending occurs between outer and inner regions. The examples provided for such modifications are rather simple ones, and it is not clear whether this approach will permit the modelling of completely general shapes. It is worth noting that modern commercial CAD systems have abandoned the use of pure CSG because of limitations in this respect. 4.3 Decomposition into volumetric primitives A decomposition method known as heterogeneous solid modeling (HSM) was developed at the University of Michigan for the representation of multi-material objects [24, 34]. Although the mathematical formulation is slightly different, the mode of representing material properties at a point is equivalent to that described in the section dealing with the finite element approach. The notion of an rm-set is introduced as a subset of the product space of E3 with M n as defined earlier. The corresponding subset of E3 is an r-set in the usual solid modelling sense [41], and that of M" assigns a material composition to each Euclidean point of the r-set. Within the r-set, therefore, a vector-valued function is required to specify the material distribution. This is assumed to be C°° continuous to ensure that material discontinuities can only occur at r-set boundaries. The domain of the material distribution function should ideally correspond to the r-set. This is easy to achieve when the boundary of the r-set has a simple geometric configuration, but to cater for more complex cases the rm-set is said to be undefined for points in the exterior of the r-set. The case where an r-set is occupied by a single pure material can obviously be handled by a material distribution function with ra^ = 1, rrij = 0 for j ^ i. A material object (rm-object) is now defined as a finite collection of rm-sets having the following properties: • No pair of rm sets in the rm-object share any interior points. • The material distribution function applies only to the interior of each rm-set. • At boundary points shared by adjacent r-sets the material composition is defined by a specialized combination operator. For the representation of material variation it is proposed in [24] that a coordinate system be associated with each rm-set for use in defining the material composition function. The best choice of coordinate system will depend on the basic shape of the underlying r-set. For example, regions that are approximately cuboidal or approximately cylindrical will most appropriately have their material distributions represented in cartesian or cylindrical coordinates respectively. The HSM method provides material distributions that are continuous in each r-set, but not in general between the r-sets defining the shape and internal configuration of an rm-object. It also suffers from the disadvantages that • Except for very simple objects the domain of the material distribution function will not coincide with the domain of the object it is defined for. As
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an example, consider a gear-wheel. Its shape is basically cylindrical, and a cylindrical coordinate system is therefore appropriate for defining the material distribution function. However, at the periphery of the gear-wheel the teeth represent departures from the cylindricity of the configuration, and there is no flexibility in refining the material distribution function there unless further rm-sets are defined for the teeth and used in the specification of a more complex rm-object. • The basic method provides no means for achieving smoothness of the material distribution function throughout the interior of an rm-object. Further work at the University of Michigan, to be described later [49, 50], has extended the HSM method to overcome the problems listed above. In particular, a blending technique is provided to ensure a smooth continuous overall distribution of material properties across boundaries between adjacent rm-sets. A theoretical extension of the HSM approach in a different direction was described by Kumar et al. [23], who pointed out that the same basic approach can be used for modelling a variety of spatially-related product properties beyond material distribution. The cited paper notes also that the association of one or more fields of properties with a spatial model constitutes what is known to modern differential geometers as a fibre bundle [31]. The use of fibre bundles for related purposes has also been studied by Zagajac [56]. However, Biswas et al. [4] observe that the fibre bundle formulation 'does not translate directly into concrete computational solutions'. Wang and Wang [55] adopt an approach with similarities to HSM, defining a heterogeneous object as a finite collection of non-overlapping regions in which material properties may exhibit continuous variation. As with HSM, these properties are in general subject to discontinuous changes at the interfaces between material regions. The authors propose a variational method for the design of such objects that gives rise to a system of coupled partial differential equations (PDEs). The material distribution functions within individual material regions emerge as solutions of those PDEs. Designs may be optimized in terms of dimensions, geometry, topology and material properties. The cited paper gives examples of optimal design of 2D multi-material structures. 4.4 Decomposition into material features The primary distinction between decomposition into volumetric primitives and into material features is analogous to the distinction between the use of constructive solid geometry (CSG) and feature-based constructional methods in CAD systems. The use of pure CSG in commercial CAD systems died out more than ten years ago, but it has left an important legacy in the feature-based methods that characterize today's CAD systems — in both cases the primary model representation is in terms of the sequence of operations used to construct the model. CSG provided mathematical rigour [41] but lacked geometric flexibility and engineering semantics. Feature-based systems make no pretence at rigour, but are more flexible in use and (at least in principle) capture engineering semantics during the design process.
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The notion of the form feature (often referred to simply as a feature) has been growing in importance in computer aided design and manufacturing for some twenty years [36, 46]. The concept is that of a local shape configuration on an object that has significance in some lifecycle phase of that object. Different engineering applications in general decompose an object into its constituent features in different ways, and many feature viewpoints may therefore exist for the same artefact. The following example illustrates some basic ideas regarding form features. Consider a cooling fin on the cylinder head of an air-cooled internal combustion engine. This is a form feature from the design viewpoint, because it provides functionality (specifically, cooling). The fin may also be regarded as a form feature in the manufacturing phase, because it somehow has to be generated during the manufacturing process. But a method-specific manufacturing viewpoint of the cylinder head may not regard the fin as a primary form feature; if it is to be manufactured by machining, for example, material will have to be removed to reveal the shape of the fin, and from this viewpoint it is the removed material that constitutes the primary form feature. Features may also be defined for other life-cycle activities including structural analysis, post-manufacture inspection, assembly with other parts, and so on. The importance of the feature concept in computer aided design and manufacture is that a feature-oriented representation of a part shape is easier for application software to comprehend than a purely geometric representation, because the features are semantically related to the nature of the application, and provide it with information in the form it needs for its operation. There is no generally agreed method for the representation of a form feature; sometimes it is convenient to regard a feature as a (usually connected) set of elements belonging to a boundary representation model, and at other times it makes more sense to think of a feature as a volume (for example, a volume of material to be machined away). The idea of form features has been applied in LM as in other manufacturing methods. In the context of functionally graded materials, a research group at the University of Michigan has defined the concept of the material feature as a volumetric element that has some specified material distribution within it [38, 39, 40]. An important distinction between this and the earlier HSM method is that material features do not in general correspond with the simple geometric primitives that were the focus there. It should be noted that some authors use the term 'material feature' to denote a rather different concept, a reference element with respect to which a material distribution is defined. Thus, for example, gradation may be required to be linear with distance from a reference line or a reference surface. In this paper the term material datum will be used for such a reference element, by analogy with the datum concept in geometric dimensioning and tolerancing [53]. This allows the use of material feature to be reserved for the volume over which the distribution is defined. The approach taken in [39] is unusual in that it considers two feature viewpoints simultaneously. The part shape is considered firstly as a Boolean combination [13] of design feature volumes, in which the operations concerned are either union or subtraction. Secondly, it is given an alternative representation as the union of a set
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of material feature volumes. In this second representation the set intersection between any pair of material features is required to be null, and the union of all these volumes is required to correspond to the overall part volume — the union operations for material features are therefore all of the specialized 'glue' type. From either point of view the overall model is nonmanifold, with internal feature boundaries. In the work reported on, the various feature volumes and their interior material distribution are modelled in terms of triparametric B-splines. The material distribution in features containing non-uniform material is determined in terms of specified distributions on the exterior of each feature volume and the use of a finite element discretization of the diffusion equation to calculate the distribution in the interior. Samanta and Koc [44] also use B-splines to model the geometry of parts and their material features. As usual with feature-based models, it is possible to make modifications subject to constraints on inter-feature relationships, and this permits the formulation and solution of an optimization problem based on a part's functional requirements. Material variation is defined in terms of functions of parametric distance from one or more material datums. Another feature-based approach is that of Kou and Tan [19, 20]. Here the primary concern is modelling the material distribution within an individual feature, less attention being given to the construction of more complex volumes. Geometrically, the features dealt with correspond to the 'swept' volumes that can be generated by CAD systems by linear extrusion or rotation of planar 2D profiles. However, Kou and Tan associate material distribution functions with the lower-dimensional entities involved in the generation of the volume (e.g., elements of the original profile, and the directrix curve along which it is swept). These entities are material datums in the terminology introduced earlier in this paper. Then, when the feature volume is generated, the material composition at interior points is calculated as a inverse-distance-weighted average of the compositions at the closest points on each material datum. The method of Kou and Tan has the definite advantage that it is closely aligned with an existing and widely-used CAD system method for generating feature volumes. The method of Siu and Tan [51, 52] has also been classified as feature-based for the purposes of this survey. The approach is in principle quite general, but it is not clear whether what has been implemented is applicable to complex objects — most of the examples given by these authors are geometrically rather simple. The method can handle any number of materials, and regions of material property variation are defined in terms of what are termed 'grading sources'. A grading source contains information regarding • the combination of materials related to that grading source, • a material datum, and • a material grading function. The material datum may be a point, curve or surface. The use of an axial line allows for axisymmetric distributions, for example. It is stated that it is also possible
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to use the outer surface of a solid as a material datum to generate a material gradient in the surface region. This is easy enough to achieve if that surface is composed of planar, cylindrical or spherical faces, but it is not explained how it can be done for objects with more general geometry. That is the main reason why the method of Siu and Tan is classified here as a feature-based rather than a global method. A material grading function is a user-specified function /(cf), where d is distance from the grading source. As such, it is defined throughout Euclidean space, but its effect is limited, firstly to the interior region of the solid concerned (its effect is disregarded for exterior regions) and secondly to the effective grading region of the grading function, which is defined by f(d) € [0,1]. Material composition is assumed to be constant in the interior of an object at points that are not in the effective grading region. Inside such regions, however, the components of the material composition vector Mn (x) defined earlier are defined as functions of the value of f(d) at the point x, subject to the condition that those components sum to 1. More than one grading source may be associated with a single solid, and in this case there may be overlap between the effective grading regions of the grading sources involved. In these regions of intersection the material composition is determined in a rather complicated manner through the use of a dominance factor ri and a weighting ratio Wi that are both associated with the ith grading source. A further feature-based approach is that of Shin and Dutta [49, 50]. This is based ultimately on the HSM work of Kumar and Dutta described earlier [24], but extends it considerably. The primary similarity is that it is a constructive technique based on the CSG philosophy; the primitive volumes used may include the swept volumes that are termed 'features' by most CAD system vendors1. The method introduces the notions of the heterogeneous primitive set (hp-set) and generalized heterogeneous object (h-object) as generalizations of the rm-set and rm-object defined for the earlier method. Several different techniques are provided for the specification of the material distribution function in an hp-set, which corresponds to a volumentric primitive as used in CSG modelling: 1. coordinate system based and independent of the geometry of the hp-set, as in the original method of Kumar and Dutta [24]; 2. based on a distance function, and similar to the approach used by Siu and Tan [51, 52]; 3. based on material blending functions, using one of two methods that are described in [49] — • Blending with respect to angle. This may be used for blending in a 2D region bounded by two convex profiles, one contained by the other. It requires a reference point to be defined within the inner profile. The material composition at a point on any radius vector through the reference point is determined in terms of the ratio of the distance from the lr This is strictly an incorrect use of the term 'feature', because CAD systems generate these volumes as pure geometric shapes and do not endow them with the application semantics that characterize a true feature.
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inner boundary to the point in question and the distance along the radius vector between the two boundary profiles. The implication is that the 2D region can then be extruded or rotated to generate a solid with the specified inhomogeneous material distribution. The intention is apparently to use this method for cases where the material distribution on each boundary profile is constant, but there is an obvious generalization to the case where variable distributions are specified on each profile, as in the method of Kou and Tan [19, 20]. • Inverse distance blending and (more specifically) R-function blending. R-function blending was first used in the present context by Shapiro et al., and it is described in the next section under global methods. In the method of Shin and Dutta it is apparently used within hp-sets rather than globally for the object as a whole. 4. based on sweeping functions, for which three cases are defined: • The swept volume is considered to have an embedded coordinate system in which x and y are cartesian coordinates in the plane of the swept profile and z is distance along the sweep path or directrix. The material distribution is specified as a function of these coordinates. • As for the previous case except that the coordinates in the plane of the profile are polar. • The third option is not very clearly described, but it appears to provide a blend in the sense of the directrix between material conditions specified at the intial and final positions of the profile being swept. In [49] it is stated that more than one of the above methods may be combined in the definition of the material distribution function in a single hp-set, and rules are given for calculating the result. Although more complex models are built from hp-sets by the use of Boolean operations, the geometry of the hp-sets themselves is actually represented in boundary representation (B-rep) form [13]. The material composition at internal points is however computed by traversing the associated Boolean tree structure. The primary problem in computing Boolean results arises in cases of union or intersection between volumes for which the point-set intersection is non-null. In such a case both volumes will originally have their own defined internal material distribution, and the question is, how is the distribution calculated in their region of overlap after they are combined? Shin and Dutta give a procedure that uses a weight factor a G [0,1]. Where two volumes intersect and one of them has an associated weight factor a = 1, the material distribution function in that volume remains unaffected in the intersection region, and a distance-based blend is defined between the distribution on that part of the boundary that is included in the second volume and the original distribution defined in the second volume. The situation is reversed if the weight factor for the first volume is a = 0; then it is a — 1 for the second volume, whose original material distribution therefore remains unchanged. When a lies between 0 and 1 the situation is intermediate, and a blending effect also occurs
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in the intersection region itself, presumably (though this is not clearly stated) between the distributions on those parts of the boundary of the intersection volume that originally belonged to each of the two initial volumes. It might have been thought that no blending would be needed in the case of the Boolean subtraction operation, but in fact this operation is defined in such a way that the material distribution in the subtracted volume may have an influence on the distribution in the volume remaining after its removal. The Boolean 'join' or 'glue' operation is separately dealt with; it is the case of union for which the (non-regularized) set intersection consists only of boundary points. This allows the creation of a blended distribution between hp-sets that before their union have only surface contact. As has been shown, although the approach of Shin and Dutta is based upon the earlier work of Kumar and Dutta, it includes a range of capabilities similar to those devised by other workers in the field. This provides flexibility, but at the inevitable expense of a complex user interface. The most significant contribution of the method, in this author's opinion, is the blending capability providing smooth overall material distributions as simpler volumes are combined into more complex ones. The lack of this was a severe disadvantage of previous constructive methods. The final feature-based method that will be described in this section is a further contribution from MIT [26], where the team concerned has moved away from its initial approach based on the finite element philosophy because it suffers from the disadvantages mentioned earlier. The new method is feature-based, and designed specifically for compatibility with the methodology used by modern CAD systems. In particular, this work adopts the now standard approach of representing a model of a product by a dual structure, an unevaluated procedural representation expressed primarily in terms of the sequence of operations used in building the model, plus an evaluated model of the boundary representation or some closely related type [13]. The procedural model is robust and very easy to modify, simply by changing values of some of its inputs. However, it provides none of the explicit geometrical information needed by most applications downstream of design. The evaluated model, on the other hand, provides all the detail needed by applications. It is also used to generate the screen display on a CAD system, for user interaction with the system and for geometrical computations. This is a more delicate model than the procedural one, and rather than try to update it in line with the procedural model as new operations are performed it is easier to discard it at each step and generate a new one — the power of modern computers now allows that to be done very quickly, although it would have been a major and time-consuming task for the hardware of twenty years ago. With most modern CAD systems, the operations used in building a model are feature creation operations, which as interpreted by CAD vendors are little more than shape macros, creating local geometric configurations on the model — any associated application semantics exist only in the mind of the designer. What is proposed in [26] is the use of this standard CAD methodology of design in terms of geometric features, with the extension of the feature notion to include a material distribution. With such an approach it is not only the geometry of the feature that is easy to edit (in terms of the defining parameters of the
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feature) but also its material aspect. For each feature a composition function is defined, providing a mapping, in terms of one or more variables, from the geometric shape of the feature to the material composition at points within the feature. In some cases, editing the feature geometry may lead to an automatic reevaluation of the composition function; in others, the user will be prompted to change it appropriately. Constraints may be imposed on the material composition to ensure (for example) that no non-manufacturable distributions are specified. The types of features provided include the volume features commonly implemented in CAD systems, user-defined 'surface features' (combinations of specified faces that play the effect of material datums according to the terminology of the present paper), pattern features (typically rectangular or circular arrays of simple features) and transition features (blends, fillets, chamfers). A composition function is defined in terms of the materials concerned, a parameterization method chosen from a methods library, the material datums on which the distribution is based, the parameters concerned and any subfunctions used. The latter are of several types, but in particular they include the functions used to parameterize particular geometric feature configurations and the functions used to map geometric parameters to material composition ratios. Parameterization is in terms of distance functions from one or more datums, a concept that is explored more fully in the following section. An efficient method of calculating minimum distance to a general surface has been developed [25], based on a digital distance transform algorithm [5]. The region of influence of each distance function is limited by upper and lower distance bounds imposed by the user; outside these limits the value of the function is taken to be constant. The determination of material composition in terms of distance from multiple material datums uses multiple applications of the single datum method, applied in a specified sequence. This allows the domain of a single volumetric material feature to be expressed as a binary subdivision tree whose leaves represent subdomains that are disjoint except for boundary points. The root of the tree is the entire volume of the feature. A weighting method is defined that apportions the effects of the various contributions to the overall composition. In cases where material distributions are defined in terms of compositions specified on one or more surface datums composing a user-defined 'surface feature' then Laplace's equation is used to compute a smoothly blended material distribution, using the specified compositions as boundary conditions. A boundary element formulation is used for the solution.
5 Global approaches The global methods differ from those described earlier in not being decompositionbased, and all methods of this type so far described are based on the use of distance fields. These were mentioned earlier, but more detail will be given in this section. The basic principle, as described in Biswas et al. [4], may be expressed in terms of a generalization of the classical Taylor series. Using similar notation to the cited reference, we may write the distance field as
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where //. denotes a derivative in the usual manner and the term in $ is a remainder term. In the classical interpretation, the series represents the behaviour of a univariate function / as the scalar d moves away from the point corresponding to d = 0. The difference in interpretation here is that /(O) is taken to represent not a point but rather a point set, and d to represent distance from that point set. Recall that one of the standard ways of representing a curve or surface is by the use of an implicit equation /(x) = 0, in which x represents a point as previously. With the given interpretation, the equation displayed above provides a representation of the value of / in terms of (in some sense) distance from the point set /(x) = 0. The point set in question may represent the closed boundary of a region (the volume of an entire object or a material feature in the sense discussed earlier). More generally, it may represent a material datum that does not form part of such a boundary, and in fact (as shown in [4] and discussed further below) it is possible to use a combination of boundary and other datum elements if desired. The practical use of this approach amounts to the determination of the coefficients in the generalized Taylor series so that the material distribution based on it satisfies specified constraints imposed by design, analysis or manufacturing considerations. Often, a small number of terms in the series may be sufficient to give satisfactory results. In fact an exactly computed distance field will have discontinuous derivatives at any point equidistant from more than one point on a point set being interpolated, but it has proved possible to define approximate distance fields with desirable smoothness properties. Once f(d) has been determined, it may be used to specify a material distribution as in the method of Siu and Tan described above. The Taylor-series representation of a general distance field with respect to a single element, and methods for building combinations of such distance fields, were first described by Rvachev in Russian publications (see the references in [4] and [43]). Other methods have subsequently been developed [3, 9, 11]. One way of controlling material distributions with respect to more than one element is through the use of R-functions as originally suggested by Rvachev. One of several approaches of this kind, described in [43], is based on two key ideas: Shepard interpolation. This is a well-known method for the interpolation of values defined at sets of randomly distributed points in any number of dimensions. It is based on the use of functions that are radially symmetric about a central point and decrease in value with increasing distance from that point, usually according to some inverse power of the distance. The overall interpolant is a linear combination of such basis functions, one centred at each interpolated point. Shepard [48] (see also [14]) popularized the use of such methods in the 1960s, though in fact they had been used for isolated applications before that time. Essentially, they interpolate function values fi at a corresponding set of points x$ by a function ^V Wi(x)fi, where the
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Wj(x) are the Shepard basis functions. These must be positive and continuous, must form a partition of unity and also satisfy the interpolation condition Wj (xj) = Sij. Subject to these conditions various formulations are available. There also exists a wider class of related radial basis function methods [14]. If inverse distance weighting is used, the functions Wi(x) are constructed by normalizing the distance function according to Wi(x) — d~ M t (x)/EjC^ ^ (x), where the summation is over all the interpolated points and di (x) is the Euclidean distance between the points x and Xi. This ensures that the values of each weight function are restricted to the interval [0,1]. In practice, an equivalent but numerically more stable form of this relation is used [43]. The exponents /^ control the smoothness of the interpolant at Xj; provided //j > 1 it will be differentiate pi — 1 times at x^ [14]. It is also possible, by an extension of the method, to fit prescribed derivative values at the interpolation points x,. A further possibility is to replace the inverse distance weighting functions by more general influence functions; these must be subject to the conditions stated above and they permit the definition of other types of material property distributions. Examples are given in [4]. R-functions. These should not be confused with the r-sets described earlier. An R-function is a real-valued function whose sign is completely determined by the signs of its arguments. Such functions provide analogies to the Boolean logical functions. Simple examples are provided by min(x 1,0:2) and max(x 1,0:2), which are analogous to Boolean 'and' and 'or', respectively, if we take + and — values of the arguments to correspond to the logical values of TRUE and FALSE. Many other R-functions are known [43], generally proving more suitable than those given above for practical applications since they give better continuity in the generated interpolation functions. The analogy with Boolean functions allows any shape element of a solid model, expressed in terms of Boolean combinations of half-spaces, to be defined in terms of a single implicit function, by composition of those half-spaces using R-functions. The composition process may be performed automatically. It is shown in [47] how this may also be done for individual elements of a boundary representation model, which may be used in the generation, through many compositions, into an implicit function representing a full B-rep model. Then a generalization of Shepard's interpolation method can be used for interpolating continuous distributions (e.g., of material properties) specified on the boundary elements or any other defined material datums. In [43] it is shown how boundary derivative distributions may also be interpolated. An important feature of the R-function method (RFM) is that it defines transfinite interpolants, which interpolate continuous functions rather than sets of discrete values. The use of RFM and other methods for computing smooth approximate distance fields enables the representation of continuous material property distributions, without the use of any meshing or other form of decomposition, inside any volume bounded by implicitly defined surfaces. Furthermore, such approximations may be normalized up to the rath order, which is to say that the approximating function
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and its derivatives of orders 2...m in the direction normal to the point set being interpolated are all zero, while the first-order normal derivative has value 1. This ensures that absolute distances are smoothly approximated with good accuracy near the point sets concerned. Since constraints on the distribution may also be defined in the interior of the volume, as mentioned above, there is potential for local control anywhere it is required. RFM therefore provides a unified means for the calculation and representation of material properties throughout the interior of a B-rep model. However, it is not a perfect fit for material property variation in LM, for the reasons that • RFM may only be used in its basic form for models whose boundary surfaces are defined by implicit surfaces. Its extension for models involving parametric surfaces such as NURBS is not trivial, though it is claimed in [4] that it is always possible. • The implicit function corresponding to a complex B-rep model will be extremely complex. It is possible to use the method without computing it explicitly, by traversing the equivalent of the Boolean tree defining the function in terms of the individual elements involved and the R-functions used to combine them, but the computational requirements are nevertheless heavy. As mentioned earlier, other approaches also exist for the generation of approximate distance fields, but space does not permit them all to be described here. Two related global method methods have been described by Park et al. [32, 33]. In the first of the cited papers, solids are modelled in generalized CSG terms [13] using implicitly defined shapes (e.g., ellipsoids) and a blending method originally due to Ricci [42]. This enables the determination of a single implicit equation f ( x , y, z) — 0 representing the surface of the entire overall shape. Variable material composition is specified in a 'boundary layer' between the surface of the object and its homogeneous interior. The function f ( x , y, z) in the implicit surface equation is used to define a measure of distance from the surface, and the thickness of the boundary layer is defined in terms of this measure. As described in [32], the method handles binary material combinations. The relative concentration profile across the boundary layer is controlled by 'bias' and 'gain' parameters that affect the symmetry and steepness of the profile. This method is referred to as 'hypertexturing'. Hypertexturing is a distance field method that is global because distances are measured with respect to a unified representation of the object boundary. The second of the papers by Park et al. [33] describes a method referred to as 'volumetric multi-texturing'. As with the hypertexturing method, it is aimed primarily at representing material distributions near the surface of a solid. However, it is designed to operate with boundary representation models represented in terms of closed shells of faces. It is said that face representations are combined using the Ricci blending technique mentioned earlier — though little detail is given on precisely how this is done — which provides an implicit representation that can be used as a the basis of a distance function. This method also allows 'global' material
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gradients to be defined; these appear not to be defined in terms of boundary elements of the object, and apply throughout its interior. The following comments apply to the methods of Park et al.: • The hypertexturing paper [32] contains little detail, but the method seems straightforward in principle. It can presumably be extended to handle more than two materials. It gives a limited form of control, restricted to the region of the object boundary — though this covers the most important requirement in the present state of functionally graded material technology. As with RFM, hypertexturing in its current form cannot directly handle models with parametrically defined surfaces. The method is not compatible with commercial CAD systems or standards for the representation of shape, because the blending technique used to generate smooth shapes differs fundamentally from the blending methods used in current CAD systems. • By contrast, volumetric multi-texturing is more complex but better suited to modern CAD practice in that it is based on a boundary representation approach. Otherwise the aims and principles are similar to the hypertexturing method, though the availability of global material gradients in addition to surface gradients gives more flexibility. Like the previous method, this one does not work directly for objects with parametric surfaces. • The paper [33] states that the volumetric multi-texturing method has been implemented with the solid modeller ACIS, and describes the methods for defining face-specific and global material gradients.
6 Discussion and conclusions This paper has surveyed published methods for the representation of material distributions in CAD models. The first methods proposed for the purpose date from no earlier than 1995, and this is therefore a relatively new field of research. Nevertheless, it already exhibits significant evolution and some degree of convergence of approach. Emphasis seems to be moving away from earlier cell-based and finite element-type decompositions of the objects to be modelled, and most recent methods are feature-based approaches, in increasingly many cases using some form of distance field technique for the representation of material distributions. A trend towards the design of representations and algorithms that are compatible with the functionality of modern CAD systems is also observable. That is very encouraging, as it points to the possibility of providing the definition of inhomogenous material distributions as an add-on application to such systems. The alternative, of starting from scratch and developing new commercial systems with both solid modelling and material distribution capabilities, would be much more onerous. With regard to the distinction between decomposition and global methods, it will be important that practical implementations of global methods provide localization techniques allowing design capabilities at the feature level. In the light of the above comments, the present writer has reservations about two of the approaches shared by some of the methods surveyed:
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1. The use of trivariate B-splines for the modelling of solids. This is not compatible with practice in most major commercial CAD systems, which represent 'simple' surfaces in simple ways. Thus, typically, a plane can be fully defined in three dimensions by a point and a (normal) direction, and a cylinder by a point, a direction and a radius value, where the first two entities define the axis. These are representations of unbounded surfaces, but bounded regions may then be defined on them by topological means [13] (the specification of edges and vertices forming the boundaries of a face, for example). It is perfectly possible to use B-splines to represent simple surfaces (e.g., the plane, circular cylinder, cone, sphere and torus) and bounded regions on them, but such representations are unduly verbose, the use of general B-spline algorithms for computations involving simple surfaces is inefficient and the engineering semantics of simple surfaces tend to be lost if they are represented in a generalized way. 2. The use of partial differential equations (PDEs) for determining material distributions. Here the objection is less on technical grounds and more a matter of gut feeling. Such an approach may of course be used appropriately to model the result of some physical process. However, in other cases it seems better to define material distribution in terms of an explicit functional relationship, giving the system user more direct control over it than is available indirectly through the specification of boundary values and (possibly) adjustable parameters present in the PDE concerned. Further, the necessity to compute the solution of the PDE adds a computational overhead. As regards the choice of equation, Laplace's equation is well known to have smooth solutions, which is a point in its favour. However, the justification for using the diffusion equation is harder to see, because the material distributions that can be generated using LM as described earlier certainly do not result from a diffusion process. An additional practical consideration stems from the industrial requirement for the exchange of models between dissimilar CAD systems. This typically uses a neutral exchange format such as the international standard ISO 10303, informally known as STEP ('STandard for the Exchange of Product model data') [15, 34]. Some of the methods described earlier, in particular those using mesh-based decompositions of the finite element type, are compatible with STEP as it currently exists. However, when commercial systems emerge that provide the capability for representing functionally graded material distributions it may be necessary to extend the standard appropriately in order to cover that capability, depending on the representational technique used. Finally, it should be emphasized that, although this paper has concentrated primarily on scalar distributions, the methods described can all be used, in principle, for the componentwise modelling of material non-isotropy and other vector-valued properties in domains with geometrically complex boundaries. Some difficulties may arise because of the occurrence of additional constraints in such cases [4] — for example, in modelling the distribution of a unit vector with variable direction
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it is necessary that the squares of the components sum to 1 - but these appear not to be insuperable.
7 Acknowledgment The author gratefully acknowledges assistance from Deba Dutta, Georges Fadel, Nick Patrikalakis, Vadim Shapiro, S. T. Tan and three anonymous referees, who supplied information and references used in compiling this survey.
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Bibliography [1] C. L. Bajaj. Modelling physical fields for interrogative visualization. In T. N. T. Goodman and R. R. Martin, editors, The Mathematics of Surfaces VII. Information Geometers Ltd., Winchester, UK, 1997. [2] M. P. Bendsoe, A. Diaz, and N. Kikuchi. Topology and generalized layout optimization of elastic structures. In M.P. Bendsoe and C.A. MotaSoares, editors, Topology Design of Structures. Kluwer Academic Publishers, Amsterdam, 1993. [3] A. Biswas and V. Shapiro. Approximate distance fields with non-vanishing gradients. Graphical Models, 66, 3, 133 - 159, 2004. [4] A. Biswas, V. Shapiro, and I. Tsukanov. Heterogeneous material modelling with distance fields. Computer Aided Geometric Design, 21, 3, 215-242, 2004. [5] G. Borgefors. Distance transformations in arbitrary dimensions. Vision, Graphics and Image Processing, 27, 321-345, 1984.
Computer
[6] M. Burns. Automated Fabrication. Prentice-Hall, Englewood Cliffs, New Jersey, 1992. [7] V. Chandru, S. Manohar, and C. E. Prakash. Voxel-based modelling for layered manufacturing. IEEE Computer Graphics & Applications, 15, 6, 42 - 47, 2004. [8] G. Fadel and S. Morvan. Multi-material representation and design issues. In I. Gibson, editor, Software Solutions for Rapid Prototyping, pages 155 - 224. Professional Engineering Publishing, 2002. [9] S. F. Frisken, R. N. Perry, A. P. Rockwood, and T. R. Jones. Adaptively sampled distance fields: A general representation of shape for computer graphics. In Proc. 27th Annual Conf. on Computer Graphics & Interactive Techniques, pages 249 - 254. ACM Press/Addison-Wesley Publishing Co., 2000. [10] A. Ghosh, Y. Miyamoto, I. Reimanis, and J. Lannutti, editors. Functionally Graded Materials: Manufacture, Properties and Applications, volume 76, 1997. [11] S. F. F. Gibson. Using distance maps for accurate surface representation in sampled volumes. In Proc. 1998 IEEE Symposium on Volume Visualization, pages 23 - 30. IEEE Press, 1998.
80
[12] R. W. Gray IV, D. G. Baird, and J. H. B0hn. Effects of processing conditions on short TLCP fiber reinforced FDM parts. Rapid Prototyping Journal, 4, 1, 14-25, 1998. [13] C. M. Hoffmann. Geometric and Solid Modeling. Morgan Kaufmann, San Mateo, CA, 1989. [14] J. Hoschek and D. Lasser. Computer Aided Geometric Design. A. K. Peters, Wellesley, MA, 1993. [15] International Organization for Standardization, Geneva, Switzerland. ISO 10303:1994 - Industrial Automation Systems and Integration - Product Data Representation and Exchange, 1994. (The ISO catalogue is at http://www.iso.ch/cate/cat.html; search on 10303 for a listing of parts of the standard). [16] C. L. Jackins and S. L. Tanimoto. Oct-trees and their use in representing three-dimensional objects. Computer Graphics & Image Processing, 14, 249 270, 1983. [17] T. R. Jackson, W. Cho, N. M. Patrikalakis, and E. M. Sachs. Memory analysis of solid model representations for heterogeneous objects. ASME J. Computing & Information Science in Engineering, 2, 1, 1-10, 2002. [18] T. R. Jackson, N. M. Patrikalakis, E. M. Sachs, and M. J. Cima. Modeling and designing components with locally controlled composition. In Proceedings of the Solid Freeform Fabrication Symposium, Austin, TX, 1998. [19] X. Y. Kou and S. T. Tan. Representation and visualization for heterogeneous cad models. In Proceedings of the 2004 International CAD Conference, Pattaya, Thailand, 24 - 28 May 2004, 2004. [20] X. Y. Kou and S. T. Tan. A hierarchical representation for heterogeneous object modelling. Computer Aided Design, 37, 3, 307 - 319, 2005. [21] A. V. Kumar and J. Lee. Designing and slicing heterogeneous components for rapid prototyping. In Proceedings of the 2000 Solid Freeform Fabrication Symposium, Austin, TX, Aug. 2000, 2000. [22] A. V. Kumar and A. Wood. Representation and design of heterogeneous components. In Proceedings of the 1999 Solid Freeform Fabrication Symposium, Austin, TX, 9-11 Aug. 1999, 1999. [23] V. Kumar, D. Burns, D. Dutta, and C. M. Hoffmann. A framework for object modeling. Computer Aided Design, 31, 9, 541 - 556, 1999. [24] V. Kumar and D. Dutta. An approach to modeling and representation of heterogeneous objects. ASME J. Mechanical Design, 120, 4, 659 - 667, 1997.
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[25] H. Liu, W. Cho, T. R. Jackson, N. M. Patrikalakis, and E. M Sachs. Algorithms for design and interrogation of functionally gradient material objects. In CDROM Procs. 26th ASME Design Automation Conf., Baltimore, MD, September 2000, page 141 (of abstracts). ASME Press, 2000. [26] H. Liu, T. Maekawa, N. M. Patrikalakis, E. M. Sachs, and W. Cho. Methods for feature-based design of heterogeneous solids. Computer Aided Design, 36, 12, 1141 - 1159, 2004. [27] A. Marsan and D. Dutta. On the application of tensor product solids in heterogeneous solid modeling. In Proceedings of the 1998 ASME Design Engineering Technical Conference, Atlanta, GA, Sept. 1998, 1998. [28] J. Mazumder, J. Koch, K. Nagarathnam, and J. Choi. Rapid manufacturing by laser aided direct deposition of metals. Technical report, University of Illinois, Department of Mechanical Engineering, 1996. [29] D. Meagher. Geometric modelling using octree encoding. Computer Graphics & Image Processing, 19, 129 - 147, 1982. [30] Y. Miyamoto, W. A. Kaysser, and B. H. Rabin, editors. Functionally Graded Materials: Design, Processing and Applications. Kluwer Academic, 1999. [31] C. Nash and S. Sen. Topology and Differential demic Press, New York, NY, 1983.
Geometry for Physicists. Aca-
[32] S.-M. Park, R. H. Crawford, and J. J. Beaman. Functionally gradient material design and modeling using hypertexture for solid freeform fabrication. In Proc. 1999 Solid Freeform Fabrication Conference, Austin, TX, USA, 1999. [33] S.-M. Park, R. H. Crawford, and J. J. Beaman. Functionally gradient material representation by volumetric multi-texturing for solid freeform fabrication. In Proc. 2000 Solid Freeform Fabrication Conference, Austin, TX, USA, 2000. [34] L. Patil, D. Dutta, A. D. Bhatt, K. K. Jurrens, K. W. Lyons, M. J. Pratt, and R. D.Sriram. Representation of heterogeneous objects in ISO 10303 (STEP). In Proc. ASME International Mechanical Engineering Congress and Exposition, Orlando, FL, November 2000. ASME Press, 2000. [35] J. Pegna and A. San. CAD modeling of multi-modal structures for freeform fabrication. In Proc. 1998 Solid Freeform Fabrication Conf., Austin, TX, USA, 1998. [36] M. J. Pratt. Solid modeling and the interface between design and manufacture. IEEE Computer Graphics & Applications, 4, 7, 52 - 59, 1984. [37] M. J. Pratt. Modelling of material property variation for layered manufacturing. In The Mathematics of Surfaces IX. Springer-Verlag, 2000. (Proc. IMA Conference in Cambridge, UK, 4 - 7 September 2000).
82
[38] X. Qian and D. Dutta. Feature methodologies for heterogeneous object realization. In I. Gibson, editor, Software Solutions for Rapid Prototyping, pages 225 - 261. Professional Engineering Publishing, 2002. [39] X. Qian and D. Dutta. Feature based design for heterogeneous objects. Computer Aided Design, 35, 3, 319 - 329, 2003. [40] X. Qian and D. Dutta. Physics-based modeling of heterogeneous objects. ASME J. Mechanical Design, 125, 416 - 427, 2003. [41] A. A. G. Requicha. Representations of rigid solids — Theory, methods and systems. ACM Computing Surveys, 12, 437 - 464, 1980. [42] A. Ricci. A constructive geometry for computer graphics. Comput. J., 16, 2, 157 - 160, 1973. [43] V. L. Rvachev, T. I. Sheiko, V. Shapiro, and I. Tsukanov. Transfinite interpolation over implicitly defined sets. Technical Report SAL 2000 - 1, Spatial Automation Laboratory, University of Wisconsin - Madison, 2000. [44] K. Samanta and B. Koc. Feature-based design and material blending for freeform heterogeneous object modeling. Computer Aided Design, 37, 3, 287 305, 2005. [45] C. Schroeder, W. C. Regli, A. Shokoufandeh, and W. Sun. Computer-aided design of porous artifacts. Computer Aided Design, 37, 3, 339 - 353, 2005. [46] J. J. Shah and M. Mantyla. Parametric and Feature-based CAD/CAM. Wiley, 1995. [47] V. Shapiro and I. Tsukanov. Implicit functions with guaranteed differential properties. In W. F. Bronsvoort and D. C. Anderson, editors, Proc. 5th ACM Symposium on Solid Modeling and Applications. Association for Computing Machinery, New York, NY, 1999. [48] D. Shepard. A two-dimensional interpolation function for irregularly spaced data. In Proceedings of 23rd ACM National Conference, pages 517 - 524. Association for Computing Machinery, New York, NY, 1968. [49] K.-H. Shin and D. Dutta. Constructive representations for heterogeneous objects. ASME J. Computers & Information Science in Engineering, 1, 3, 205 217, 2001. [50] K.-H. Shin, H. Natu, D. Dutta, and J. Mazumder. A method for the design and fabrication of heterogeneous objects. J. Materials & Design, 24, 5, 339 353, 2003. [51] Y. K. Siu and S. T. Tan. CAD modelling and slicing of heterogeneous objects for layered manufacturing. In I. Gibson, editor, Software Solutions for Rapid Prototyping, pages 263 - 282. Professional Engineering Publishing, 2002.
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[52] Y. K. Siu and S. T. Tan. Source-based heterogeneous solid modelling. Computer Aided Design, 34, 1, 41 - 55, 2002. [53] V. Srinivasan. Theory of Dimensioning. Marcel Dekker, 2004. [54] S. Suresh and A. Mortensen. Fundamentals of Functionally Graded Materials. Cambridge University Press, 1997. [55] M. Y. Wang and X. Wang. A level-set based variational method for design and optimization of heterogeneous objects. Computer Aided Design, 37, 3, 321 337, 2005. [56] J. Zagajac. Engineering Analysis over Subsets. PhD thesis, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, 1997.
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Part II: Mathematical Responses to Industrial Problems The writers of the papers in Part II provide insights and examples into how mathematical thought and practice are being applied to industrial problems. The articles also support the idea that mathematicians and mathematics are undergoing a change from a narrowly focused but important group and technology to being critical components in today's increasingly simulation-driven industrial environment. The articles further support the idea that industrial mathematics is emerging as an important discipline within the mathematical community. Seven essays make up this section, with subject matter from a variety of mathematical disciplines and with applications ranging from how to validate computer models to how to link military operations to resources. All seven articles discuss mathematics needed to support modeling and simulation, each with a different emphasis. The first two articles investigate validation of computer models. The question of whether or not the predictive power of any computer model is sufficient for the intended use of the model is the theme of A Framework for Validation of Computer Models by James Cavendish of General Motors. In this paper, Cavendish describes a six-step model for validation based on a Bayesian statistical strategy for developing error bounds on model predictions. This paper portrays the complexity of mathematical issues confronting industrial mathematicians, scientists, and engineers who are charged with finding any reasonable and applicable solution. Brian J. McCartin and Paul D.Young of Kettering University in Numerical Investigation of the Validity of the Quasi-Static Approximation in the Modelling of Catalytic Converters, show how to determine, using a fully dynamic model, values of physical parameters for which the quasi-static approximation is valid. One of their important numerical tools is exponential fitting. They also use the fully dynamic model to investigate inherently transient phenomena that are not approachable by the quasi-static approximation. The next two articles explore novel applications of mathematical theories. In A Framework Linking Military Missions and Means, Paul Tanenbaum of the Army Research Laboratory, and William Yeakel of ORSA Corporation, consider the problem of coordinating material requirements with actual military mission
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needs. They use elements of the theory of Boolean lattices to develop a framework that links military missions with the means to accomplish the mission. Next, potential applications of computational topology to both geometric design and molecular design are examined by Edward L. F. Moore of Pratt &; Whitney, and Thomas J. Peters of the University of Connecticut, in Computational Topology for Geometric Design and Molecular Design. They explore connections between automation of geometric design and robust, topological processes. In computational molecular design they examine the potential impact of knot theory on the design process, illustrating how mathematicians expand theory to meet problems. The final three papers in Part II focus on computational aspects of modeling and simulation. In the first of these, Discretize then Optimize, John Betts of Boeing, and Stephen Campbell of North Carolina State University, investigate numerical solutions of high-index differential algebraic equations. They compare the specific solution strategies of the direct transcription method, discretizing first and then optimizing, versus the indirect approach of optimizing and then discretizing. A single geometric model may generate different meshes for different analyses and may require that results that depend on one mesh be transferred to the second mesh. Stamping and structural meshes derived from a single CAD model provide an example. David Field of General Motors looks at these problems in Transferring Analyses onto Different Meshes. Data fitting is the topic of the last paper of this section, Bivariate Quadratic B-splines Used as Basis Functions for Data Fitting. Benjamin Dembart and Daniel Gonsor, Boeing, and Marian Neamtu, Vanderbilt, present the mathematical theory of bivariate B-splines and show that they have excellent spectral properties making them useful for, among other things, data fitting and scattering problems in computational electromagnetics.
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A Framework for Validation of Computer Models James C. Cavendish* Abstract Computer simulations and math models are used in industry to design, understand and control various physical systems and processes and to predict the performance of these systems and processes under various operating conditions. The two main approaches to establishing credible confidence statements about the predictive capabilities of computer models are verification and validation. Verification is the process of determining that a computer model (that is, the operational computer program or "code") is a correct (i.e. bug-free) and accurate implementation of a conceptual model of a physical system or process. Model validation is the process of determining the degree to which output from a computerized model accurately represents reality. In this paper we present a framework that enables the implementation of the validation process. This framework is being developed as part of a Research Agreement between the National Institute of Statistical Sciences (NISS) and General Motors. Underlying the proposed validation framework is a systematic six-step procedure based upon a Bayesian statistical methodology. The Bayesian methodology is particularly suited to treating the major issues associated with the validation process: quantifying multiple sources of error and uncertainty in math models; combining multiple sources of information; and updating validation assessments as new information is acquired. Moreover, it allows inferential statements to be made about predictive error associated with model predictions in untested situations (that is, prediction beyond the validation comparisons between calculations and experimental outcomes).
* General Motors Research and Development Center, Warren, Michigan
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1
Introduction
We view the most important question in evaluation of the predictive capability of a computer model to be the following: Does the computer model provide predictions that are accurate enough for the intended use of the model? The purpose of this paper is to document an invited presentation made at the SIAM Conference on Mathematics for Industry: Challenges and Frontiers, October 13 - 15, 2003, where we presented a systematic six-step math model validation process intended to help answer that question. This was done by sketching out a Bayesian statistical strategy for developing error bounds on model predictions with the interpretation that there is a specified confidence (say, 80%) that the corresponding true process value being modeled will lie within the range of those error bounds. Although seldom done in practice, such error bounds a'nd confidence estimates should routinely be provided whenever model predictions are made. A Caveat: The process of model validation is inherently a hard statistical problem. In fact, the statistical problem is so hard that one rarely sees model validation approaches that actually produce error bounds and confidence estimates on computer model predictions. The intent of the SIAM presentation was essentially to provide a 'proof of concept' that it is possible to provide such bounds and estimates for predictions of computer models, while taking into account all of the uncertainties present in the problem. However, the computations required in the methodology we presented can be intensive, especially when there are large numbers of (uncertain) model inputs, large numbers of unknown parameters, or a large amount of data (model-run data or field data.) We call the reader's attention to a reference of particular importance: Bayarri et. al. [1]. This reference provides a down-loadable PDF file that contains a technical report presenting all of the technical details associated with our proposed validation strategy developed as part of a research agreement between General Motors and the National Institute for Statistical Sciences. This report also provides practical applications of the strategy to several math models of interest to the automobile industry. The paper is structured as follows. In Section 2 we provide some general comments on the role and use of mathematical models. These models include finite element models of structures, computational fluid dynamic models, as well as simulation models not derived from basic first principal physics. Also in Section 2 we pay particular attention to the roles of uncertainty and error quantification in mathematical models. In Section 3 we fix these issues of uncertainty and error quantification by considering two very simple and similar math models; one with very little associated uncertainty and high predictive power, and one with only moderate uncertainty and yet very low predictive capability. In Section 4 we provide a specific definition of math model validation and a sketch of a six-step validation process that is based upon a Bayesian statistical methodology. This paper is deliberately short on technical detail. Complete technical detail is provided in Bayarri et. al. [1] and Higdon [5]. Section 5 is used to summarize the paper.
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2
Kinds of Math Models
Computer models and simulations are used in many industries to design, understand and control various physical systems and processes and to predict the performance of these systems and processes under various operating conditions. However, there is presently little agreement on systematic procedures for assessing the credibility of their predictive capabilities. When using the term, "mathematical model" it is useful to distinguish between two types: a conceptual model and a computerized implementation of the conceptual model (that is, the operational computer program or "code"). Often, the conceptual model is derived from continuum laws of nature believed to describe the behavior of the system or process being modeled. For example, in computational fluid dynamics the conceptual model is dominated by partial differential equations accounting for conservation of mass, momentum and energy. Other math models do not have their basis rooted in physical laws of nature and as such do not involve a computerized model that implements a set of governing differential equations. For example, industrial economists postulate economic pricing models which involve equations that relate product demand, supply and revenue to product price, and which can be used to aid multi-product pricing decisions aimed at optimizing corporate profits. Other similar examples are to be found in industrial production models, planning models and marketing models. In principal such models can be validated in essentially the same way as the physics-based models. However, it may not be feasible to conduct controlled field experiments for meaningful validation comparison with model predictions.
2.1
Math Model Usage
In the engineering design of a device, system or process, we are most interested in the performance of the system as a function of its design (that is, its geometric configuration, operating conditions, material properties, etc. ) In principal we could simply make an educated guess at performance, however, for a complex process such as resistance spot welding of automobile sheet metal panels, or the process of crashing a vehicle into a fixed barrier, our guess would most likely be quite inaccurate. Through the use of a well-validated math model of the system, however, we hope to considerably reduce our uncertainty in such system performance estimates. Math models, particularly those derived from known laws of nature, require us to estimate various model variables (e.g. conductivities, densities, heat transfer coefficients, etc.) for which our uncertainties are much lower than the uncertainty that we have in our original guess of the entire system performance itself - and then combine these more accurate estimates into an estimate of performance that contains considerably more certainty than our educated guess. A major problem with the use of math models and computational simulations in support of product and process design is that all models are only abstractions of reality, and insight and understanding they can provide is usually limited. It is important that model builders and code users understand the limitations of their models. It's unfortunate, however, that almost all computational models are used in
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a fashion that is deterministic; that is to say, they are seldom exercised extensively to explicitly account for error and uncertainty, and they do not provide boundaries on the range of valid model application. In this paper we will focus on a model validation strategy that explicitly accounts for model error and uncertainty and which establishes confidence levels that statistically measure the predictive accuracy of mathematical models and simulations.
3
Two Very Simple Math Models and Their Uncertainty Characteristics
In order to get a feel for the role that uncertainty plays in evaluating the predictive capabilities of a math model, we consider two very simple and similar math models. One model is characterized by very low uncertainty in its model input parameters and it is a model that possesses high predictive accuracy. The second model is one for which we can obtain only moderately accurate input data and yet has very poor predictive power. The lessons to be learned from these two models is that we will never know the magnitude of the error in a prediction unless we quantify the uncertainty associated with the model, and propagate that uncertainty through the computations explicitly. The question of how many rigid particles can be packed into a given container is a very old one dating back to the 16th century when mathematician, Johannes Kepler posed it when the particles are identical perfect spheres. There is a simple mathematical model that can be postulated to help address the issue: Np = Number of particles Vp = Volume of a single particle Then,
Vc — Volume of the container.
and
In Equation 1, n is called the packing density, a number between 0 and 1 which denotes the fraction of the container that is occupied by the collection of rigid particles instead of air. This is a simple mathematical model; one capable of producing the exact number of particles, Np, ifVp, Vc, and /j, are the correct inputs to the model. Spheres: If the particles are identical spheres of diameter D, then the number of spheres in Equation 1 becomes:
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To test the predictive capability of the model, consider packing an empty bottle of drinking water with BB's (see Figure 1 and Figure 2) 1 . As BB's and water bottles are manufactured items, they are subject to some manufacturing variation, however small. The bottle that I've chosen to fill with BB's has its volume written on it (Figure 3).
Figure 1. A 12 ounce bottle of drinking water.
Figure 2. The bottle packed with BB's. It's not clear that the actual volume of the bottle is exactly 354.75 cc; does that figure include the small amount of air in the neck of the bottle when it's filled 1 When making my SI AM presentation during the spring of 2003 at a colloquium hosted by Oakland University, I circulated the bottle packed with BB's and asked for "educated guesses" as to the number of BB's. I received the following guesses: 4,000 5,700 6,300 7,500 9,000 10,000 12,000 20,000 and 40,000 - an enormous spread in the "probability density function" on original educated guesses before modeling takes place.
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Figure 3. Determination of the bottle's volume. with water? It would not be unreasonable for the volume of the bottle to be between 350cc and 360 cc or about 355cc±1.5%. Now, what about the diameter of the BB's? Considering the box from which the BB's came (Figure 4) we can conclude that these are 4.5 mm caliber BB's, so D — .45cm. To estimate the variation in this number, I selected 10 BB's at random and used a micrometer to measure each diameter (Figure 5). Sure enough!! To within negligible measurement error, the BB's do have diameters of .45cm. Finally, we need to estimate the value of the packing factor, p. Entering the words, "packing density of spheres" into Google's search engine produces Figure 6, among many others. When identical spheres are packed face-centered-cubic, or hexagonal into an infinite volume, then /i is about 74% (proven in 1998 by a mathematician named Thomas Hales). When packed randomly, p, is about 64%. An eyeball comparison of Figure 6 and Figure 7 seems to indicate that the BB's pack in non-random fashion - that is, they appear to pack in a more or less regular repeating nested pattern. Therefore, it's not unreasonable to estimate /j, to be 70% ± 2%. Summarizing, for D = .45cm, we have
and
Plugging these values into Equation 2 produces a model prediction for the number of BB's in the bottle, NBB, that satisfies
The true number of BB's in Figure 2 is 5280. If we pick the midpoint of (4986,5430), or 5208, then our model prediction is within 1.5% of the true value. This high accuracy is, of course, caused by the low uncertainty attendant to the inputs of the model. Oblate Spheroids: An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis (rotating the ellipse about its major axis
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Figure 4. Box from which the BB 's came.
Figure 5. Measuring the diameter, D, of a BB. produces a prolate spheroid.) Consider next filling the empty bottle in Figure 1 with M&M candies (Figure 8). M&Ms are objects of fairly uniform shape and they are reasonably well approximated by oblate spheroids (see Figure 9 where I have cut an M&M in half). Whereas the volume of a sphere of radius D is 7r£)3/6, the volume of an oblate spheroid is 7rD2T/6 where D is the diameter of the spheroid and T is its thickness. Thus, under the assumption that M&M's are identical spheroids (which of course they are not), we have that the number of M&M's in the bottle in Figure 8, NM&M-, is predicted to be
Unlike BB's which have very low manufacturing variation, M&M's have considerably more manufacturing variation. Selecting 10 M&M's at random, and using the micrometer again (Figure 10 and Figure 11), I found that
and
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Figure 6. Examples of non-random sphere packing & their densities.
Figure 7. A closeup of the BB 's in Figure 2. It is not clear whether oblate spheroids pack more or less densely than spheres2. However, it is obvious from Figure 8 that there is a higher degree of randomness in the packing of spheroids than spheres. Thus, there is considerable uncertainty in our estimate of the packing density //, to use for M&M's in equation (3). Any guess we make could be in error by as much as, say 20%. For our purposes we will assume that
To summarize:
2 In a February 13, 2004 issue of Science magazine, Paul Chaikin, Salvatore Torquato and colleagues [3] proved that when packed randomly, identical oblate spheroids achieve a packing density of about 68% - a surprising result given that randomly packed spheres fill only 64% of space. See also http://www.princeton.edu/pr/pwb/0223
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Figure 8. The bottle packed with M&M's.
Figure 9. M&M's can be well-approximated by oblate spheroids. The simple math model we are using in (3) contains only the four input variables listed above. The only error in the model is in the estimation of the inputs to the model and any computational error we make in evaluating (3). For ease of computation, I assumed that the four input variables are uniformly distributed random variables bounded by the percentage errors indicated above. Taken by themselves, these are fairly modest uncertainties and yet they result in evaluations of (3) that range over a factor of nearly 2.5 to 1. That is to say, the maximum predicted number of M&M's contained in the bottle shown in Figure 8 is 450; the minimum number of M&M's is predicted to be 182. The true number of M&M's contained in the bottle is 301. This is a huge amount of uncertainty, considering the simplicity of the model, the relatively small variation in the input data, and the reasonableness of the assumption that M&M's are well approximated by oblate spheroids.
4 Uncertainty Quantification in Math Models The input data that engineers provide as input to their math models can often only be estimated, and hence their models always include uncertainty. In addition
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Figure 10. Measuring the diameter, D, of an M&M.
Figure 11. Measuring the thickness of an M&M. to input data uncertainty, there is often model form error (the model doesn't adequately capture the appropriate physical phenomenon), uncertainties in loads, initial/boundary conditions, uncertain material properties, etc. And yet they almost always fail to include an uncertainty analysis with their predicted model output results. We will never know the magnitude of the prediction error produced by an uncertain math model unless we account for and propagate uncertainty through the computations explicitly. Because math models never produce exact and perfect predictions, they cannot be used with impunity. It's the uncertainty that accompanies model predictions that is the main reason that decision makers don't rely much on math model results; it's this uncertainty that gives us the greatest grief.
5 A Proposed Strategy for Math Model Validation Math model validation is the process of determining the degree to which a model's output matches the "real world" as determined by comparing carefully selected and conducted field experiments with model calculations. The process must provide tolerance bounds and confidence bounds on model predictions. (For example, 5.17±
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.44 for a model prediction of 5.17, with the interpretation that there is a specified chance (say, 80%) that the corresponding true process value would lie in the specified range, (4.73, 5.61) ). Model validation can be thought of as a series of activities or steps, roughly ordered by the sequence in which they are usually performed. The completion of some or all in the series of steps will typically lead to new issues and questions, requiring revision and revisiting of some or all of the activities, even if the model is unchanged. New demands placed on the model and changes in the model through new developments make validation a continuing process. The six-step validation strategy outlined below allows for such dynamics. Step 1. Specify model inputs and parameters with associated uncertainties or ranges — the "Input/Uncertainty (I/U) map" This step requires considerable expertise to help set priorities among a (possibly) vast number of inputs. As information is acquired through undertaking further steps of the validation process, the I/U map is revisited and updated.
Step 2. Determine the evaluation criteria Determine the physical quantities that are to be used for comparison between model calculations and field data during validation experiments. Defining criteria must account for the context in which the model is used, the feasibility of acquiring computer-run and field data, and the methodology to permit an evaluation. In turn, the data collection and analyses will be critically affected by the criteria. Moreover, initially stated criteria will typically be revisited in light of constraints and results from later analyses.
Step 3. Design validation experiments. Both computer and field experiments are part of the validation (and model development) process; multiple stages of experimentation and replicate experimentation will be common. The need to design the computer runs along with field experiments can pose non-standard issues. As noted above, any stage of design must interact with the other parts of the validation framework, especially the evaluation criteria.
Step 4. Computer model approximation (if necessary) Fast running model approximations (for example, response surfaces) are often key for such analyses, as well as for design, optimization and Step 5.
Step 5. Perform output analysis, sensitivity analysis and compare computer model output with field data Uncertainty in model inputs will propagate to uncertainty in model output and estimating the resulting output distributions is often required. The related "sensitivity analysis" focuses on ascertaining which inputs most strongly affect outputs, a key tool in refining the I/U map. Comparing model output with field data has several aspects. . The relation of reality to the computer model
("reality = model + bias") . Statistical modeling of the data (computer runs and field data where "field
data = reality + random measurement error")
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• Tuning/calibrating model input parameters based on the field data • Updating uncertainties in parameters (given the data) • Accuracy of prediction given the data. The methods used here rely on a Bayesian formulation. The fundamental goal of assessing model accuracy is determined by this formulation. Step 6. Feedback information into the current validation exercise and feed forward information into future validation activities
Feedback refers to use of results from Step 5 to improve aspects of the model, as well as to refine aspects of the validation process. Feed-forward refers to the process of utilizing validations of current models to predict the validity of related future models, for which data may be lacking. The above framework is elaborated in detail in Bayarri et. al. [1].
6
Summary
Improved computational math models offer the potential to reduce design, development and manufacturing time and cost by providing better predictions and a more complete exploration of system and process design space. However, these models and simulations will not achieve this potential unless credible uncertainty statements about their predictions accompany them. Key to attaining this goal is a model validation process that permits comparison of computational predictions to experimental outcomes over a meaningful set of validation experiments - a process that will provide confidence in the predictive capability of the computational model. In this paper we have outlined a six-step approach based upon a Bayesian statistical methodology for validating mathematical models. Developed with researchers from the National Institute of Statistical Sciences, the six-step process is discussed in technical detail in Bayarri et. al. [l], where it is also applied to two model problems of interest to the automobile industry. The validation process allows inferential statements to be made about predictive error associated with model predictions made in untested situations.
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Bibliography [1] M. Bayarri, J.O. Berger, D. Higdon, M. Kennedy, A. Kottas, R. Paulo, J. Sacks, J. Cafeo, J. Cavendish and J. Tu, A framework for the validation of computer models, in Proceedings of the Workshop on Foundations for V&V in the 21st Century, D. Pace and S. Stevenson, eds., Society for Modeling and Simulation International, 2002. Also available as a downloadable Tech report #128 at http://www.niss.org/downloadabletechreports.html. [2] J. Cavendish and J. Cafeo, Framework for validation of computer models applied to a resistance spot weld model, Proceedings of the 2003 ASME International Mechanical Engineering Congress, Washington, B.C., 2003 [3] A. Donev, I. Cisse, D Sachs, E. A. Variano, F.H. Stillinger, R. Connelly, S. Torquato and P. M. Chaikin, Improving the density of jammed disordered packings using ellipsoids, Science, Vol 303, February 13, 2004, pp. 990 - 993. [4] G. A. Hazelrigg, On the roll and use of mathematical models in engineering design, Transactions of the ASME, Vol. 121, Sept 1999. pp. 336 - 341. [5] D. Higdon, M. Kennedy, J. Cavendish, J. Cafeo, and R. Ryne, Combining field data and computer simulations for calibration and prediction, SI AM Journal on Scientific Computing, Vol 26, 2004, pp. 448-466.
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Numerical Investigation of the Validity of the Quasi-Static Approximation in the Modelling of Catalytic Converters* Brian J. McCartirtf 1
and Paul D. Young^
Abstract
The modelling of automotive catalytic converters entails the numerical solution of a system of partial differential equations for chemical species concentrations as well as gas and solid temperatures. In the model of Oh and Cavendish (1982), four oxidation reactions are included while the simplified model of Pease, Hageman and Schwendeman (1994) accounts for only a single oxidand (CO). However, both models make a quasi-static approximation in the heat and mass transfer equations. In this paper, we employ the fully dynamic single oxidand model and approximate it using, among other numerical tools, exponential fitting. In this way, we are able to determine those values of the physical parameters for which the quasi-static approximation is valid. Furthermore, we are able to investigate inherently transient phenomena such as engine stall and pulsating flow.
*SIAM Conference on Mathematics for Industry: Challenges and Frontiers t Applied Mathematics, Kettering University •f General Motors Tech Center
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2
Introduction
In response to more stringent federal regulation of automobile exhaust emissions, 1975 and later model cars have incorporated a catalytic converter into their exhaust system [1]. The most common variety today is the three-way monolith catalytic converter (Fig. 1). It is usually placed on the underfloor of the vehicle (Fig. 2) but is sometimes found attached directly to the exhaust manifold (Fig. 3). Its function is to chemically convert pollutants such as carbon monoxide (CO), unburned hydrocarbons (HC), and nitrogen oxides (NOX) into less noxious substances such as carbon dioxide (CO?), water (H^O}, and nitrogen (A^).
Figure 1. Monolith Catalytic Converter
This objective is achieved by passing the hot exhaust gases through a honeycomb ceramic substrate. This ceramic structure is painted with a metallic washcoat containing precious metals such as platinum, rhodium, and palladium. As the converter heats up, these materials catalyze the requisite chemical reactions. Thus, in order to perform efficiently, the catalytic converter must be heated as quickly as possible [2]. The warm-up behaviour of catalytic converters first became a subject of mathematical modelling in the landmark paper of Young and Finlayson [3] where a basic model was developed and approximated by orthogonal collocation. There followed the extremely influential paper of Oh and Cavendish [4] wherein the model of the chemistry was significantly refined and a numerical procedure was developed specifically tailored to the stiff nature of the resulting differential equations.
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Figure 2. Underfloor Catalytic Converter However, the sheer complexity of these pioneering models made mathematical analysis of the warm-up behaviour of catalytic converters well-nigh impossible. Thus, the beautiful paper of Pease, Hagan, and Schwendeman [5] introduced a simplified version of the Oh and Cavendish model wherein only a single oxidand (CO) is considered. This permitted them to perform a refined mathematical analysis of the formation of the so-called light-off point and its subsequent propagation upstream to the inlet region thereby completing the warm-up process. Yet, all of these models employed a quasi-static approximation [6] whereby the transient terms in the heat and mass transfer equations are ignored while the transient term in the diffusion equation is retained. As very eloquently pointed out by J. Rauch [7], such an approximation requires both an estimate of the resulting residual at each time step as well as an assessment of the accumulation of error resulting from such residuals for its complete justification. In the present paper, we present a numerical procedure for the fully dynamic single oxidand model. This procedure is based upon Heun's method for the diffusion equation and an exponentially fitted method of characteristics for the heat and mass transfer equations. Numerical results from this fully dynamic model are then compared with those resulting from making the quasi-static assumption. The parameter range over which this approximation is valid is thus firmly established. Additionally, the fully dynamic model is employed to study the effects of engine stall and pulsating inlet flow upon the warm-up behavior of catalytic converters. Both of these phenomena are excluded by the quasi-static assumption.
3
Single Oxidand Model
In this section, we present a summary of the single oxidand model [8]. With reference to Fig. 4, we will focus on a single converter channel and average over the cross-section of the monolith. In this model, only a single oxidand (CO) is accounted for since it is the most abundant pollutant and the dominant heat source.
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Figure 3. Close-Coupled Catalytic Converter The governing initial-boundary value problem is specified by [9]:
The first two equations account for mass and heat transfer, respectively, between the gas and the solid converter. The third equation accounts for the chemical reaction taking place at the surface of the converter. The fourth equation accounts for heat diffusion along the length of the converter. The dependent variables are Cg (concentration of CO in the gas), Cs (concentration of CO on the surface of the converter), Tg (temperature of the gas stream), and Ts (temperature of the solid converter). The physical parameters are // (void fraction of the monolith), v(t) (velocity of the gas stream), a (mass transfer coefficient), /3 (heat transfer coefficient), 7 (chemical activation energy), a(x) (catalyst surface area), A (reaction rate constant), and 6 (thermal diffusivity). The remaining quantities pertain to the initial and boundary conditions prevailing in the system.
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Figure 4. Single Converter Channel These equations have been nondimensionalized whereby the scaled length of the converter is unity, a scaled time unit is approximately 14 seconds, and a scaled velocity of 1 is 10 meters per second. For further details of the scalings employed, the reader is referred to [8]. The quasi-static approximation is the result of setting // = 0 and jettisoning the initial conditions on Cg and Tg. This is typically justified by the disparity in size between /j. and a, /?, v (see Table 1). Note that /j. has been scaled and thus the physical value of the void fraction is .7.
4 4.1
Numerical Approximations Approximation of Heat Diffusion Equation
Referring to Fig. 5, an open circle represents a predicted mesh quantity while a solid dot represents a corrected mesh quantity. After semidiscretization [10] of the heat diffusion equation, we employ Heun's method which is an explicit second order accurate predictor(Euler)-corrector(Trapezoidal Rule) method. The resulting fully discrete equations are:
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Figure 5. Heun's Method
These equations are modified at the endpoints in the usual fashion in order to account for no-flux boundary conditions. This method is conditionally stable and requires that r := 7^2 < i- We next describe how to evaluate C™+1 and T£+1.
4.2
Approximation of Mass and Heat Transfer Equations
Both the mass and heat transfer equations are hyperbolic with identical characteristics of slope fj,/v [11]. Thus, they may be reduced to ordinary differential equations and numerically integrated along a characteristic by exponential fitting [12]. We will call this the exponential method of characteristics (see Fig. 6). Defining the Courant number p = ^^. (see Fig. 6), if p < 1 then we use second order upstream differencing supplemented by the Lax-Wendroff scheme [2] at the inlet while if p > I then we employ the angled derivative method [10]. These schemes coincide when p—\. With v(t) evaluated at time £ n +i/2, the exponential method of characteristics is second order accurate and unconditionally stable. Below, the expression p(u,v,w; z] represents the parabolic interpolant to the values u,v,w evaluated at z. The function M(z) :~ Ign-er*) was introduced in the context of exponential fitting in [12]. In order to avoid catastrophic cancellation in the ensuing exponentially fitted approximations, we utilize the following Maclaurin expansions for \z\
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Figure 6. Exponential Method of Characteristics Approximation of Plug Flow Reactor Equation
The mass transfer equation may be rearranged in the form of the plug flow reactor equation of chemical engineering [11]:
Referring to Fig. 6, it may be approximated via:
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Table 1. Numerical Simulation Parameters Parameter a ft 7 6 V A a V s~i in
L
'a
r-T'lU
.9
rncoid s
Ax At
Numerical Value 2.0 2.0 10.0 0.005 0.001 0.035 1.0 0.5 1.0 -0.05 -1.0 .01 .001
Approximation of Heat Exchanger Equation
The heat transfer equation may be rearranged in the form of the heat exchanger equation of chemical engineering [11]:
Referring to Fig. 6, it may be approximated via:
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Figure 7. Warm-up: v/p = 500
5
Numerical Examples
We next apply the preceding numerical approximations to the study of three fundamental questions: • When is the quasi-static approximation justified? • What is the effect of engine stall on warm-up? • When may a pulsating inlet flow be replaced by a mean-flow approximation? Unless otherwise specified, numerical parameter values are those of Table 1. More refined discretizations both in time and space have been employed with no appreciable difference in the numerical results. 5.1
Quasi-Static Approximation
Our first simulation, the results of which are displayed in Fig. 7, has v//j, = 500 with p := -^| = 50 and r := 7^72 = -05. The 21 solution curves shown are successively separated by 800 time units (19.2 seconds). Thus, the entire simulation takes place over a span of 384 seconds. There are, in fact, two sets of curves plotted, a solid set for the fully dynamic model and a dotted set for the quasi-static approximation. Their coincidence is a testament to the effectiveness of the quasistatic approximation in this parameter range. We observe in this figure the three
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Figure 8. Warm-up: v/ n — 9 characteristic regimes of the warm-up process: a gradual (first couple of minutes) increase in temperature along the converter, followed by the rapid (less than a minute) formation of a sharp light-off profile downstream, completed by a slow (more than two minutes) propagation of the light-off point upstream to the inlet. This effectiveness is pushed to the limit in Fig. 8 where v = .009 and, consequently, v/jj, = 9. Here we begin to see small differences between the two models as the arrows point (here and subsequently) to solution curves at identical times. Fig. 9 goes even further by using v = .005 and, consequently, v/jj, = 5. What is surprising is the rapid degeneration of the accuracy of the quasi-static approximation evident upon comparing Figs. 8 and 9. Incidentally, in these two figures only, the number of time steps between successive curves has been increased to 8000 so as to permit the observation of more of the warm-up process. 5.2
Engine Stall
As indicated in Fig. 10, our next sequence of simulations involve engine stall for 7 seconds. Once again, the solid curves will represent the numerical results from the fully dynamic model with engine stall. However, the dotted curves now represent the fully dynamic model without stall. The quasi-static model is singular for zero velocity and, hence, is inapplicable. In Fig. 11, engine stall occurs 28 seconds after ignition and prior to light-off. This is seen to produce a retarding effect on the warm-up process. In Fig. 12,
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Figure 9. Warm-up: v/p, = 5 engine stall occurs 98 seconds after ignition and during light-off. This is seen to produce a surprising advancing effect on the warm-up process. In Fig. 13, engine stall occurs 154 seconds after ignition and after light-off. This is seen to produce a negligible effect on the warm-up process. Similar trends would be expected if the engine was turned off during the warm-up process and restarted before cooling.
5.3
Pulsating Flow
For the close-coupled catalytic converter shown in Fig. 3, its proximity to the exhaust manifold calls into question the appropriateness of assuming a steady, uniform flow of exhaust gas through the converter. Instead, the flow conditions are more likely to be pulsatile [13]. Such pulsating flow conditions have been studied experimentally [14] and our final sequence of simulations do the same numerically. Conventional wisdom in fluid dynamics maintains that, when a flow is pulsating rapidly, a mean-flow approximation may be made whereby the unsteady velocity profile is replaced by a steady one with the same average velocity. Thus, Fig. 14 displays the velocity profile v(t) = .5 + .5sin(27r/£) pulsating between 0 and 10 meters per second at a frequency of / Hz. Once again, the solid curves will represent the numerical results from the fully dynamic model with pulsating flow. However, the dotted curves now represent the fully dynamic model with a steady flow of v = .5. The quasi-static model is singular for zero velocity and, hence, is again inapplicable. Moreover, the time derivatives
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which // multiplies will be substantial for large pulsation frequencies even if the flow does not stagnate thus necessitating the use of the fully dynamic model. In Fig. 15 (At — .001), a pulsation frequency of 10 Hz unsurprisingly produces significant differences. However, the pulsation frequency of 100 Hz of Fig. 16 (Ai = .0001) is much closer to typical engine firing rates yet we still find observable differences. Upon increasing the pulsation frequency to I KHz we observe in Fig. 17 (Ai = .00001) a coincidence of the two model results and a consequent validation of the conventional mean-flow approximation. What is surprising is that it requires such a high pulsation frequency to bring about this coincidence. This is symptomatic of the need to account for pulsating flow for such close-coupled catalytic converters and points once again to the inadequacy of the quasi-static approximation when analyzing or simulating such configurations.
6
Conclusion
In the foregoing, we have presented a comprehensive numerical approach to the fully dynamic single oxidand model of automotive catalytic converters. This has been compared to the corresponding model with the quasi-static assumption appended. This comparison shows that for v > 10/i the quasi-static approximation is excellent. While this result might be anticipated on intuitive grounds, the nonlinearity of the mathematical model necessitates either theoretical or, as provided herein, numerical justification. However, once the inlet velocity drops below this threshold, the accuracy of the quasi-static approximation degenerates with startling rapidity. This precludes its use in numerical studies of transient phenomena such as engine stall and pulsating flow. In contrast, the fully dynamic model can be used to numerically study such interesting engine operating conditions. As was revealed above, engine stall prior to light-off has a retarding effect on the warm-up process while its occurrence during light-off has a surprising advancing effect. The effect of stalling after light-off was seen to have negligible effect. For the case of pulsating flow, our results indicate that a surprisingly high pulsation frequency of 1 KHz is necessary in order to fully justify the invocation of a meanflow approximation. In closing, it should be pointed out that the computational overhead of using the fully dynamic model is not significantly greater than that for the quasi-static approximation. Thus, while the quasi-static approximation has much to speak for it in a purely theoretical investigation, in fully numerical investigations its utility is negligible. Finally, while the present study has focused on the single oxidand model, it is anticipated that similar qualitative trends would be observed with the inclusion of more comprehensive chemical reactions.
7
Acknowledgements
The authors would like to express their sincere gratitude to Mrs. Barbara A. McCartin for her dedicated assistance in the production of this paper. Figure 1 is by Bryan Christie (http://www.sciam.com), Figure 3 is from http://www.gillet.com, and Figure 2 is from http://customimport.simplisticmedia.com.
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Figure 10. Engine Stall: Velocity Profile
Figure 11. Engine Stall: Early Time
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Figure 12. Engine Stall: Intermediate Time
Figure 13. Engine Stall: Late Time
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Figure 14. Pulsating Flow: Velocity Profile
Figure 15. Pulsating Flow: f — 10 Hz
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Figure 16. Pulsating Flow: f = 100 Hz
Figure 17. Pulsating Flow: / = 1 KHz
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Bibliography [I] K. C. TAYLOR, Automobile Catalytic Converters, Springer-Verlag, 1984. [2] B. A. FlNLAYSON, Numerical Methods for Problems with Moving Fronts, Ravenna Park, 1992. [3] L. C. YOUNG AND B. A. FlNLAYSON, Mathematical Models of the Monolith Catalytic Converter, AIChE J., Vol. 22, No. 2, pp. 331-353, March 1976. [4] S. H. OH AND J. C. CAVENDISH, Transients of Monolithic Catalytic Converters: Response to Step Changes in Feedstream Temperature as Related to Controlling Automobile Emissions, Ind. Eng. Chem. Prod. Res. Dev., Vol. 21, No. 1, pp. 29-37, 1982.
[5] C. P. PLEASE, P. S. HAGAN, AND D. W. SCHWENDEMAN, Light-off Behavior of Catalytic Converters, SIAM J. Appl. Math., Vol. 54, No. 1, pp. 72-92, 1994. [6] L. A. SEGEL AND M. SLEMROD, The Quasi-Steady-State Assumption: A Case Study in Perturbation, SIAM Review, Vol. 31, pp. 446-477, Sept. 1989. [7] J. RAUCH, Review of "Wave Motion", Amer. Math. Monthly, Vol. 110, pp. 353-355, Apr. 2003. [8] D. W. SCHWENDEMAN, Dynamics of Automotive Catalytic Converters in Mathematical Modeling: Case Studies from Industry, E. Cumberbatch and A. Fitt (Eds.), Cambridge, 2002. [9] R. BIRD, W. STEWART, E. LIGHTFOOT, Transport Phenomena, Wiley, 1960. [10] A. ISERLES, Numerical Analysis of Differential
Equations, Cambridge, 1996.
[II] H. RHEE, R. ARIS, AND N. R. AMUNDSON, First Order Partial Differential Equations, Vol. I: Theory and Application of Single Equations, Dover, 2001. [12] B. J. McCARTiN, Exponential Fitting of the Delayed Recruitment/Renewal Equation, J. Comp. Appl. Math., Vol. 136, pp. 343-356, 2001. [13] G. BLAIR, Design and Simulation of Four-Stroke Engines, SAE Press, 1999.
[14] S. BENJAMIN, C.ROBERTS, AND J. WOLLIN, A Study of Pulsating Flow in Automotive Catalyst Systems, Exp. in Fluids, Vol. 33, pp. 629-639, 2002.
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A Framework Linking Military Missions and Means P.J. Tanenbaum* and W.P. Yeaket 1
Introduction
The Missions and Means Framework (MMF) , developed mainly by Deitz and Sheehan [6, 7], is a conceptual structure to support reasoning about the jobs that military units at any echelon must perform and the units' wherewithal to accomplish those jobs. It supplies, for the first time, rigorously specified linkages between two mature disciplines: military operations and the development and management of the resources these operations require. In this paper, we describe the problems (in both disciplines) that MMF was intended to address and we summarize the framework itself by explaining its origins. We then introduce an order-theoretic approach to reasoning about readiness of tactical vehicles and finally describe a recent application of MMF to illustrate its effectiveness.
2
Background
The U.S. military today is undertaking a massive transformation to modernize— some would say revolutionize—itself and all its activities. This effort has been likened to changing the carburetor on a motorcycle one is riding down a bumpy motocross track at 70 miles per hour. Almost everything the Department of Defense does is being scrutinized for enhanced efficiency, effectiveness, and relevance. Recognizing that the Cold War is past and faced with enemies who are not conventional states and who practice asymmetric warfare through such means as terrorism, assassination, and targeting of civilians, U.S. defense leaders appreciate that they must seek to improve each of the primary tools at their disposal: doctrine, *U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005 U.S.A. tQRSA Corporation, Aberdeen, Maryland 21001 U.S.A.
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organization, training, materiel1 leadership, personnel, and facilities (known collectively in military jargon as DOTMLPF, but referred to henceforth as means). The Army, in particular, is committed to becoming lighter and more agile while retaining the overwhelming lethality and survivability it has long enjoyed. The events of September 11 and the wars in Afghanistan and Iraq have strengthened the case for such a transformation and have increased the sense of urgency in reshaping the force and how we equip it. Central to the Army's transformation is the Future Combat Systems program, which is developing a family of 18 "network-centric" platforms— manned or unmanned vehicles—designed to operate as a system of systems. But changing everything at once does pose significant problems. For one thing, the seven components of the means are tightly interdependent, so changes always have second- and higher-order effects that must be managed. Added to this is the ever-mounting complexity of the problem of even communicating requirements between those who use and those who develop solutions in terms of the means. Consider a decades-old example. U.S. soldiers fighting in Vietnam requested better sandbags because burlap, the materiel of choice at the time, was known to disintegrate quickly in Vietnam's tropical environment. The Army's Weapons Systems Laboratory arranged to have several million nylon sandbags shipped out; rotting bags were no longer a problem. But nylon was far too slippery: the new sandbags could not be stacked without sliding off one another, and thus were nearly useless. The soldiers fighting the war had not specified how their sandbags had to function, and the engineers back home had not thought to ask. When they are asked, soldiers in the field—the operators—can elaborate on the capabilities they require of their materiel. But a thorough, systematic, and unambiguous tracing of these materiel requirements back to the underlying mission requirements is quite difficult. And everything said here about materiel applies just as well to the other six elements of the means.
3
What is MMF and where did it come from?
MMF is probably best understood as an extension of the logical structure that has evolved in the domain of the analysis of ballistic survivability (and dually, lethality) of materiel. Developed in the mid-1980s by Deitz and others [2, 3, 5], the socalled Vulnerability/Lethality (V/L) Taxonomy addressed a previous generation of longstanding methodological shortcomings. The war gaming community, one of the primary consumers of V/L data, had come to expect such outputs as probabilities of "killing" a vehicle or expected loss of function. These metrics are very highly aggregated and understood to be, in some vague sense, averaged over all possible combat scenarios. A more serious problem with such metrics is that since they are so abstract, they are not even empirically observable, so the models that produced them could not, even in principle, be validated. The V/L Taxonomy solves these problems by addressing explicitly several intermediate steps that the earlier approaches glossed over. It is based on four spaces and operators between them. The first space corresponds to the relevant 1
The word materiel is defined as "equipment, apparatus, and supplies of a military force".
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physical interactions, as when a bullet or the blast or fragments from an artillery burst hit a target. The second space corresponds to the components of the targets, the state of which, in general, is changed by the interactions. The I-to-II operators capture the effects that the interactions have on the components, either damaging or repairing them. The third space corresponds to the capabilities of the vehicle's subsystems and systems. The II-to-III operators express the dependence of those capabilities on the components that supply them. The first three spaces of the V/L Taxonomy and the operators between them constitute the core of the ballistic V/L domain, but they do not address larger so-what questions. While a tanker would likely be interested to learn whether his vehicle can maintain its maximum rated speed, for instance, what he really wants to know is whether the capabilities his vehicle provides are adequate to his immediate task of winning a battle. That question is in the realm of the fourth space of the taxonomy, which corresponds to mission utility. Here, too, are the ill-defined metrics so typically demanded from the V/L analyst, like expected loss of function. Once the problem had been decomposed according to the V/L Taxonomy, it became clear that space IV did not "belong" to the V/L analysts, but to the combat soldiers, those known in the vernacular of the military as the operational forces. V/L analysts' earlier efforts to account for the mission context by capturing the dependence of battlefield utility on vehicle subsystems had depended on experienced soldiers, who were asked to express that dependence as a sum of probabilities of damage weighted subjectively to account for such factors as the significance and likelihood of a mission type. But, whatever this weighted sum might have meant, the combat simulations actually used it as though it captured the specific condition of a particular vehicle in a particular context. As it happens, the war fighters had a similar problem of their own. Those who plan military operations and those who train the fighting units have a very highly developed and rigorous approach. Reminiscent of the lengthy checklists that astronauts use to manage every minute detail of a space mission, this thoroughgoing approach was developed to ensure that all participants speak a common language and to minimize the risk that something important might be overlooked. Such careful coordination is obviously crucial, since so many lives are at stake. But there was no way of tying the plan (or training) to the means at its foundation with anything like the rigor that characterizes the rest of the process. This Military Decision Making Process and the V/L Taxonomy turn out to be dual to one another. Each is, in a way, the solution to the problem experienced with the other. And the key to fitting everything together is the Missions and Means Framework [6, 7]. To illuminate this linkage, we must explain the two central constructs of the process. The first is a hierarchy of levels of focus, from the strategic national, through the theater and operational, and down to the tactical level. Starting with a strategic national goal, planners derive a military mission, which they decompose into operations and tasks. This decomposition process is recursive, with tasks at each echelon becoming missions for subordinate commands and agencies. The leaves of this recursion are all tasks, and some (e.g., deliver fire) are atoms. The other central construct for this process is a rigorous dictionary of tasks. Each task is annotated with appropriate conditions and standards, which
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describe respectively the environment in which the task is executed and criteria for key aspects of its performance and its effectiveness. This dictionary is embodied in such documents as the Army Universal Task List [1] and, across all the services, the Universal Joint Task List [10]. MMF links the mission-to-task decomposition of the war fighters to the taxonomy of the engineers and scientists and supports the posing and answering of the central question: how does a collection of capabilities stack up against the tasks that must be performed to accomplish a mission? The tasks generated by the Military Decision Making Process, together with their conditions and standards, provide neither more nor less than the substance of the III-to-IV operators that complete the V/L Taxonomy. When a planner needs to assess whether his tasks can be accomplished, MMF provides the link into the V/L Taxonomy to chase the answer all the way down to the hardware level. And, going the other direction, when a materiel developer needs to evaluate a system, MMF supplies the bridge into the mission space, thus providing the context essential for ensuring relevance.
4
Platform-level readiness and the demands of the mission
Among the formal requirements for the Future Combat Systems are diagnostic and prognostic systems embedded in each platform. The rigorous approach that MMF provides for reasoning about system capabilities may also be put to use to define this kind of onboard, automated evaluation of readiness, as one of us (Tanenbaum) has demonstrated. Consider a recovery vehicle like the M88A2 Hercules, which is essentially a battlefield tow truck. The possible net degradations to its recovery capability may be built up from elements such as r\ = cannot tow and r2 = cannot hoist. Let 7£ = {ri,..., r-fc} be the full set of such elements. Other capability categories such as mobility and communication would also have degradation elements similar to those described here for recovery. A Hercules might suffer either of r\ and r<2 independently, and neither can be generally called more important than the other. One could imagine contexts in which either of towing and hoisting was a critical function for the mission and the other was unnecessary. Without specifying the particulars of a mission, one cannot make many universal assertions about the relative criticality of the individual capability elements or collections of them. The strongest statement that holds indisputably is that suffering additional degradation never improves a platform's readiness. Thus, the containment relation D on the 1k possible states of the recovery capability—namely, the subsets of 7?.—is a reasonable first pass at ordering the capability states by severity to obtain the is-a-lesser-capability-than relation. In practice, though, there are two kinds of constraints that allow us to prune this Boolean lattice because they make some capability states unattainable. First, there are semantic constraints among the elements. With regard to the mobility capability, for instance, the meaning of the element m^ = complete immobilization incorporates both mi = reduced max. speed and m^ — reduced maneuverability.
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The other class, the system-design constraints, result from the way the vehicle is engineered. If, for instance, a functioning local area network is essential for communicating data beyond the platform, then, while it may be possible to suffer Xi = no external data communication without suffering £4 = lost LAN, the vehicle cannot suffer £4 without x\. Because of these constraints, the severity order is, in general, a subposet S of the Boolean lattice (2 , D). Establishing a particular mission for the platform to perform—decomposed into the basic tasks with the associated conditions and standards—fully determines whether each of the possible states implies sufficient recovery capability to allow the Hercules to complete its mission successfully. If we color each s G S green or red, according as it suffices or not for the current mission, we can obtain an overall sense of how robust the platform is to the demands of that mission. Putting the question somewhat more rigorously, from the entire space of possible degradation states of the vehicle's capability to perform its recovery functions, in which portions can the vehicle reside and fulfill whatever recovery role is necessary for successful completion of the mission? From the complexity of all these posets (one for each capability category) and the varying demands on a platform's capabilities over the course of a mission, how can one boil out some kind of useful summary, as for instance, for development into an on-board display? This application is not well suited for the order theorist's first instinct: Hasse diagrams are just not optimum for effective communication of critical information to your typical combat soldier under fire. We propose a partition of the set of states into parts, or bins, that are reasonably homogeneous and lend themselves to arraying along a linear axis. Let -< be the weak order on the states such that si -< 82 if and only if hts(si) < hts(s2) 5 where hts(s) is the height of s e S, the greatest cardinality of any chain in S with maximum element s. Then the levels, or maximal antichains, of (S, -<) may be thought of as bins that themselves form a chain B\ -<' B<2 -<' • • • -<' Bh of the same height as S. This total order -<' on the partition II = {-Bi, • • • , Bh} of S is a defensible l-to-h ranking of capability-state bins by overall level of severity. Note that the ranking -<' is founded only on the assumption that readiness is a decreasing function of degradation and on the semantic and system-design constraints, so it is independent of the mission context. We produce the boiled-down summary we sought of the robustness of the platform to the demands of the mission by transferring to the bins the green-red coloring of the states that encodes adequacy for a particular mission. From the state coloring x'-S —> {green,red}, we define the bin coloring x':II —> {green,red} bv
Thus, any yellow bins may be seen to contain states lying on either side of the frontier, a curve through the Hasse diagram of S that separates the states adequate to the immediate job from those that are inadequate. This colored binning provides an overall view of the capability states that the system could conceivably enter and
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the extent to which the current tasks stress them. An option to improve the quality of this summary is based on the notion of the weak discrepancy of a poset [9, 4, 8]. The idea is to replace the chain (II, -<') of bins based on the height in (5, -<) of the states by another chain of similar construction but based instead on an optimal weak extension of (5, D), which Trenk shows can be found in polynomial time [9].
5
System-of-systems readiness and mission success
The crux of MMF in the materiel domain is the process of comparing the platformlevel tasks at the leaves of the mission-to-task decomposition to the capabilities the platform can currently provide, where the context in which to decide the adequacy of these capabilities is found in the conditions and standards associated with each task. The benefits of MMF, however, are fully realized only by unwinding the recursive decomposition and determining the success or failure of each higher-level task based on whether its subtasks were performed successfully. This unwinding allows us to move beyond the platform level and assess the performance of the system of systems in terms of the success of the so-called collective tasks. Having already bridged the gap between the V/L Taxonomy and the mission decomposition by comparing tasks to capabilities, we can now apply the established logic of the Military Decision Making Process to work our way back up through the collective tasks and finally evaluate the success or failure of the entire mission. This evaluation considers the status of the subtasks and whether any is critical to the collective task. If the system failed at any critical subtask, we consider whether the risk incurred by that failure is acceptable in the current mission context and, if not, whether there is a possible adjustment of resources or an alternate course of action that will address the problem. If no such options exist, we conclude that the mission cannot succeed. This process has recently been demonstrated by a team including the authors and others from the Army Research Laboratory, the Defense Modeling and Simulation Office, the Dynamics Research Corporation, and the ORSA Corporation. Asked by Mr. Walter Hollis, the Deputy Undersecretary of the Army for Operations Research, to develop a demonstration to show how MMF could be applied to the evaluation of a system of systems, we prepared a realistic battle scenario, performed the mission-to-task decomposition, and then simulated executing the battle plan while generating a plausible stream of component-level state changes. This demonstration allowed us to explore ways in which the course of combat and the success of the mission could be influenced by the continually changing status of the platforms' components. As a poet might say, we have shown that, by explicitly treating every link in the chain of causality, it is possible to trace the loss of a kingdom all the way back to the want of a nail.
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6
Conclusion
The Missions and Means Framework breaks new ground in the accurate modeling of the domain of military planning and operations. This breakthrough should have significant impact on efforts in modeling and simulation to support fields as varied as materiel acquisition, concept analysis, testing and evaluation, war gaming, training, readiness reporting, and trade-off analysis. In truth though, while we have achieved tremendous increases in expressive power and in clarity of reasoning in this military application domain, we have done it without dipping particularly deep in the mathematical well. It seems likely that still greater improvements are to be had by applying more sophisticated mathematical tools. The time-varying and largely emergent nature of the system-of-systems approach to organizing a force and executing a mission creates a qualitatively new level of challenges for those who hope to model and understand it.
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Bibliography [1] The Army Universal Task List, U.S. Army Field Manual 7-15, 2003. [2] P. H. DEITZ AND A. OZOLINS, Computer Simulations of the Abrams Live-Fire Field Testing, U.S. Army Ballistic Research Laboratory Memorandum Report BRL-MR-3755, 1989. [3] P. H. DEITZ AND M. W. STARKS, The generation, use, and misuse of "PKs" in vulnerability/lethality analyses, The Journal of Military Operations Research, 4 (1999), pp. 19-33. [4] J. GIMBEL AND A. N. TRENK, On the weakness of an ordered set, SIAM J. Discrete Math., 11 (1998), pp. 655-663. [5] J. KLOPCIC, M. W. STARKS, AND J. N. WALBERT, A Taxonomy for the Vulnerability/Lethality Analysis Process, U.S. Army Ballistic Research Laboratory Memorandum Report BRL-MR-3972, 1992. [6] J. H. SHEEHAN, P. H. DEITZ, B. E. BRAY, B. A. HARRIS, AND A. B. H. WONG, The military missions and means framework, in Proc., 2003 Interservice/Industry Training, Simulation &: Education Converence, (2003), pp. 655663. [7] J. H. SHEEHAN, P. H. DEITZ, B. A. HARRIS, A. B. H. WONG, B. E. BRAY, AND E. W. EDWARDS, The Nexus of Military Missions and Means, U.S. Army Materiel Systems Analysis Activity Technical Report TR-737, 2004. [8] P. J. TANENBAUM, A. N. TRENK, AND P. C. FISHBURN, Linear discrepancy and weak discrepancy of partially ordered sets, Order, 18 (2001), pp. 201-225. [9] A. N. TRENK, On k-weak orders: Recognition and a tolerance result, Discrete Math., 181 (1998), pp. 223-237. [10] Universal JOINT Task List (UJTL), Chairman of the Joint Chiefs of Staff Manual CJCSM 3500.04C, 2002.
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Computational Topology for Geometric Design and Molecular Design Edward L. F. Moore*, Thomas J. Peters* Abstract The nascent field of computational topology holds great promise for resolving several long-standing industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, aerospace, shipbuilding and consumer product industries. Commercial CAGD packages depend upon complementary geometric and topological algorithms. The emergence of geometric modeling for molecular simulation and pharmaceutical design presents new challenges for supportive topological software within Computer Aided Molecular Design (CAMD) systems. For both CAGD and CAMD systems, splines provide relatively mature geometric technology. However, there remain pernicious issues regarding the 'topology' of these models, particularly for support of robust simulations which rely upon the topological characteristics of adjacency, connectivity and non-self-intersection. This paper presents current challenges and frontiers for reliable simulation and approximation of topology for geometric models. The simultaneous consideration of CAGD and CAMD is important to provide unifying abstractions to benefit both domains. In engineering applications it is a common requirement that topological equivalence be preserved during geometric modifications, but in molecular simulations attention is focused upon where topological changes have occurred as indications of important chemical changes. The methods presented here are supportive of both these disciplinary approaches. ""University of Connecticut, Department of Computer Science and Engineering ^ University of Connecticut, Department of Computer Science and Engineering, Department of Mathematics. Partial funding for both authors was from National Science Foundation grants DMS0138098, CCR 0226504 and CCF 0429477. All statements in this publication are the responsibility of the authors, not of the National Science Foundation.
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1
Introduction
Topology is the branch of mathematics that studies the properties of geometric objects which are preserved under continuous deformation [26]. Whereas geometry is concerned with rigid form, size and location of objects, topology is concerned with deformation, connectivity, and associativity of objects. The field of computational geometry as its own discipline has been pervasive in CAGD, computer graphics, and robotics for more than twenty years [29]. The term 'computational topology' first appeared in 1983 to emphasize the role of the topological adjacency relationships within CAGD [23]. More recently, efforts have arisen to formalize computational topology into its own discipline [28], so as to improve reliability in geometric computing, varying in scales from the atomic to the astronomical. This paper simultaneously considers topological issues in CAGD and CAMD, offering new perspectives and research directions for advancing these fields. It does so by first exploring the motivation and use of topology for CAGD applications, and by formalizing communication about topology for the field of CAGD in Section 2. Next, contemporary uses of topological methods for CAMD are discussed in Section 3 within the context of the formalisms introduced for CAGD. Possibilities and motivation for improving CAGD and CAMD are discussed throughout with an emphasis placed on topological equivalence. Section 4 provides conclusions and future directions for applications for CAGD and CAMD.
2
Challenges in Computational Topology for CAGD
Surprisingly, all contemporary CAGD systems lack the reliability and robustness required for automation of engineering analysis tasks such as computational fluid dynamics (CFD), finite element analysis (FEA) and optimization of design processes. These engineering applications rely heavily on the accuracy of the model provided by the original CAGD system. An intermediate step between CAGD and analysis is the generation of a piecewise linear (PL) mesh which introduces added errors from the mesh approximation of a free-form surface. While meshing is typical in CAGD, it is expected to become more prominent in CAMD as more sophisticated surface models of molecules evolve. During model approximations for engineering analysis, topological anomalies that now arise include extraneous self-intersections, unwarranted gaps, and incorrect connectivity - all of which can cause problems in a subsequent analysis. According to Farouki [12], a core problem with modern CAGD systems is in the underlying mathematics used for representing models, and the source of many of these problems arise from the algorithmic issues inherent in computing approximations. The mismatch between approximate geometry and faithful topology has historically caused reliability problems not only in CFD and FEA, but also for scientific visualization and engineering applications. As CAMD matures, with more sophisticated geometric models, the accompanying visualizations and simulations are expected to experience similar approximation problems. Hence, it is useful to consider this approximation problem from the viewpoints of CAGD and CAMD simultaneously, while preserving topology during these approx-
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imations by meshed representations. An important challenge for the computational geometry and topology communities is the development of rigorous, robust and tractable methods for guaranteeing that computational approximations of curves and surfaces preserve critical topological characteristics. The emphasis here is upon geometric models having at least C2 continuity. However, the approach presented is consistent with a broader perspective where important model discontinuities should also be preserved to maintain topological equivalence during approximation. For example, too much algorithmic smoothing on a geometric model with curvature discontinuities could lead to topological changes, just as well as too coarse an approximation could result in topological changes. Both types of topological changes could result in inaccuracies in the associated analysis. The 'topology' of a CAGD model is frequently expressed as a critical property for reliable engineering simulation. However, the use of the term 'topology' is often informal, sometimes even to the point of being misleading. There is little use in speaking universally of the topology of a model. Rather, it is crucial to decide which topological characteristics of a model should be preserved or changed to support effective engineering simulations. Hence, the use of precise topological terminology is important. The following subsections propose a foundation for communication about topological equivalence for both CAGD and CADM. An article of this length cannot presume to be comprehensive about modeling topology. Hence, selective attention will be given to those specific topological properties that the authors judge to be most central to modern industrial modeling. 2.1
Combinatorial and point-set topology
In CAGD, combinatorial topology1 is used to represent the adjacency relations between constituent geometric elements of a model [43]. The topological instantiations are typically known as vertices, edges, and faces, which correspond to the geometric entities of points, curves, and surfaces, respectively. The combinatorial topology provides no information about how an object is embedded in 3-dimensional space. For example, Figure 1 shows that the same combinatorial data can describe a completely different embedding. Note that each object would have exactly the same adjacency graph, even though one object is topologically a circle and the other object describes a trefoil knot, as described further in the next subsection. A central definition in point-set topology [26] is that any two subsets A and B of M3 are topologically equivalent if there exists a function / : A —>• B, such that / is continuous, 1 — 1, has a continuous inverse and maps onto all of B. Such a function is known as a homeomorphism. The two objects in Figure 1 are also homeomorphic. The important summarizing observation is that neither combinatorial topology nor point-set topology provides sufficient capability for more subtle topological characteristics that are crucial for today's increasingly sophisticated geometric models. The new frontiers in topological modeling are presented in the next subsection. 1
Combinatorial topology is often referred to as symbolic topology or adjacency topology in the CAGD literature
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Figure 1. Same combination represents different 2.2
objects.
Topological Equivalence
While the relations of combinatorial topology and the equivalence relations defined by homeomorphisms are quite powerful and useful, they fail to distinguish how an object is embedded in three-dimensional space. To expand upon the discussion of Subsection 2.1, consider a very simple closed curve, the circle. It is the special planar case of an unknot, which is pictured in Figure 2(a). The important intuitive generalization from the planar circle to the unknot is that the unknot is any closed curve in M3 formed from certain permissible deformations of the circle, where the types of permissible deformations will be formally described in the following definition of ambient isotopy. A more complex knot, the trefoil is depicted in Figure 2(b). Its essential distinguishing characteristic is the presence of 3 crossings, as shown by the hidden surface rendering in Figure 2(b). The knot with four crossings is shown in Figure 2(c), and is called the figure-8 knot. It is easy to see that all these knots are homeomorphic to each other, yet it is often important to distinguish among these curves 2. Indeed, this can be accomplished with additional topological techniques. Such techniques are based on a stronger notion of topological equivalence than homeomorphism, known as an ambient isotopy. The formal definition of an ambient isotopy follows. Definition. If X andY are subsets ofM3, then X and Y are ambient isotopic if there exists a continuous function H : M3 x [0,1] —> M3 such that for each t € [0,1], H(-,t) is a homeomorphism such that • H(-,ty is the identity and
• H(X, 1) = Y. Consider the unknot, the trefoil knot and the figure-8 knot in Figures 2(a), 2(b) and 2(c), respectively. None of these knots can be continuously deformed into the other without breaking a strand. Hence, none of these three knots are ambient isotopic, but they are homeomorphic. It is worth noting that the existence of an ambient isotopy between two sets can be very hard to detect. Consider the tangled mess in Figure 3, and note that 2
For a listing of standard knot types, the reader is referred to [1, 32].
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Figure 2. (a) Unknot. (b) Trefoil knot, (c) Figure-8 knot. it is ambient isotopic to the circle, meaning it can be 'untangled' without breaking a strand to form the circle 3.
Figure 3. This tangled mess is ambient isotopic to the unknot. Recently, considering generation of self-intersections during perturbations of geometric objects has been fundamental for effective algorithms to preserve isotopy class during these perturbations [3, 4, 5]. This recent work on computational ambient isotopy builds upon the considerable computational geometry literature for detection of self-intersections in curves and surfaces [19, 20, 44].
3
Frontiers in Computational Topology for CAMD
This section describes integration of theory for useful molecular modeling. It is composed of four sub-sections. Subsection 3.1 describes how the abstract mathematics of knots has been useful in forming conceptual models of molecules. Subsection 3.2 then describes an example of how some of this abstract theory could be used as the theoretical basis for molecular simulation algorithms. Subsection 3.3 then points out the difficulties in implementing efficient algorithms for even some relatively simple problems in knot theory, leading to reasonable pessimism about the effectiveness of knot theory for practical implementations. The section is concluded in Subsection 3.4 with an alternative approach to overcome that expressed pessimism. The proposed focus then becomes on approximations to create practical models, such that these approximations maintain topological equivalence under ambient isotopy. 3
An animation of the knot in Figure 3 can be seen using Robert Sharein's KnotPlot tool [36]. Knot images in Figures 2, 3, and 5 are partially created from KnotPlot.
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The authors' specific mathematical contributions are then articulated, suggesting how these approximations can be useful in molecular simulations. 3.1
The Conceptual Role for Knots
For CAGD and computer graphics, static models are often sufficient to represent complex shapes for product design. However, form and function are integrally related in the life sciences, and form is rarely static. Biochemical processes are dominated by dynamic changes which modify function. Existing geometric methods for simulating these dynamics are computationally intensive. While geometry is the correct mathematics for capturing static form and rigid motion changes, topology is more focused upon how an object deforms in time. Hence, the faithful integration of computational topology and geometry is emerging to model dynamic changes in molecular function within the domain of CAMD, and such simulations have the opportunity to leverage and improve more than two decades of experience with geometric models within CAGD. The same formalisms about topology discussed in Section 2 may be equally well suited for advancing CAMD. Note that contemporary use of computational topology for CAMD by the bio-molecular community is mostly combinatorial, particularly when simulating molecules by traditional 'ball-and-stick' models. Such combinatorial models are composed topologically of vertices and edges, where the vertices correspond to individual atoms and the edges represent the bonds between atoms4. Current 'textbook' methods classify protein structure as a hierarchy of four subset structures (See [8, 34]), where the secondary structure contains critical combinatorial topology information. A model of the protein, carbonic anhydrase I, is shown in Figure 4(a), and its corresponding secondary structure topology diagram of its peptide chain is shown in Figure 4(b). The Nl and (72 represent the ends of a fragment of the chain. The circles represent alpha and 3io helices, and the triangles represent beta strands. (The interested reader can obtain more information on these diagrams from [24], and can easily produce such diagrams on the internet [40] directly from protein structural information contained in the Protein Data Bank(PDB) [31]. The PDB ID for this particular protein is Ihcb. For the reader who wishes to see a more detailed color image of the carbonic anhydrase I protein in Figure 4(a), an online image can be viewed from the PDB.) In addition to the combinatorial approach to macro-molecular modeling, the mathematics and computer science communities have also begun to formulate a topological perspective of bio-molecules. For example, Edelsbrunner and his associates create new computational topology algorithms to model macro-molecules [9, 10, 21]. The role of knots in understanding molecular dynamics in DNA was discovered nearly two decades ago by Wasserman et. al. [41, 42] and Sumners [38]. These researchers applied topology to predict knots in DNA molecules, and confirmed their results with experimentation. They also showed that specific enzymes can 4
Geometrically, these are well known as simplicial 1-complexes [6].
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Figure 4. (a) Carbonic anhydrase I (Ihcb). (b) Topology diagram. cause a change of knot type, and advocated the use of topology to characterize the role of enzymatic action on DNA structure, since no matter how much a DNA molecule is twisted or distorted, without 'breakage' caused by reaction with specific enzymes, the DNA topology remains invariant. Although DNA is commonly thought of as a long, thin double helix, in reality, DNA mostly exists in a supercoiled form - meaning it is twisted and tangled in order to be in a state of minimum energy [37]. Furthermore, two or more strands of supercoiled DNA can also be linked together to form what biologists call catenanes. Note that links are simply a generalization of knots in the mathematical knot theory literature [1, 32]. Such linking and supercoiling of DNA is studied in the life sciences, and presents challenges to the modeling of DNA structure and function [35]. It is interesting to note that knots have been found to naturally occur in the primary structures of proteins. Taylor [39] describes an algorithm which he used to scan 3,440 protein structures in the Protein Data Bank, and found that eight of these contained knots. The protein in Figure 4 is one of these, and contains the trefoil knot. Since knot theory is emerging as useful mathematics for research in the disciplines of molecular modeling and geometric design, some related questions about efficiency for knot recognition algorithms are presented in the next section. 3.2
Algorithmic Efficiencies for Knots
A knot can be visualized as a closed loop of string. The way in which the strands of a knot are entwined is far more important than the size or shape of the knot, and helps distinguish different types of knots. Examples of knots are shown in previous sections of this paper. The important point to note is that topological equivalence of knots is determined by the notion of ambient isotopy. The interested reader is referred to the excellent text by Adams [1] for the introductory details of knot theory that are not included in this article.
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A fundamental question in knot theory is how to determine when two knots are equivalent. An active area of research in knot theory today is finding mathematical methods that determine topological invariants of knots. Such invariants are useful for classifying and distinguishing different types of knots, and are typically given by a polynomial of one or two variables. For example the well-known Jones polynomial [18] of the simple trefoil knot depicted in Figure 2(b) is given by V(i) = —t~4 + t~3 + t~l. However knot invariants are typically very hard to compute, and may become intractable as the knot complexity increases. Computation of the Jones polynomial is known to be ^tP-hard [17]. In addition, it is known that even the problem of determining whether an object is knotted or not is in NP [14], which presents challenges for developing practical applications. Hence, these authors have shifted their attention to determining when an object and its approximation are ambient isotopic, even while the knot type of either may not be known. Both effective theory and algorithms for determination of knot equivalence have been developed [3, 4, 5, 33] and should gain an increasing prominence in CAGD and CAMD. 3.3
Knot representations
Computations for knots must be performed on some representation of the knot. Typically, knots are represented combinatorially by identifying crossings with their adjacent crossings on an oriented knot projection. An example of an oriented projection of the trefoil knot is shown in Figure 5. Note that each crossing is labeled numerically, and the sign on the crossing indicates whether it is an over crossing or under crossing.
Figure 5. An oriented projection of the trefoil knot with labeled crossings. These knot representations are important for algorithms to distinguish between geometric objects in differing isotopy equivalence classes. For example, the intersection between two surfaces could result in a knotted intersection curve, yet current methods for surface intersections do not specify the isotopy class of the intersection set. This is shown hi Figure 6, where the intersection curve of the two surfaces in Figure 6 (a) would result in the unknot, and the two surfaces in Figure 6(b) would result in the trefoil knot. In these surface intersection examples it is important to produce a knot, but in other instances, a knot may be an unwanted artifact of a poor approximation. The next subsection discusses how an approxi-
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mation algorithm can change the isotopy class of a geometric object, a predicament that has only recently been considered in the literature [2, 3, 5, 22, 33].
Figure 6. Surface-to-surface trefoil knot.
3.4
intersection resulting in the (a) unknot. (b)
Algorithms for isotopically equivalent approximations
Approximations are central to geometric modeling and molecular design for computational efficiency, and problems can arise in preserving the isotopic equivalence during an approximation [2, 22, 33]. The relevant summary of the previous material of this section is to advise the reader that fundamental knot computations are not likely to be efficient for practical computations. In particular, performing explicit computation of the isotopy class of a knot and its approximant in order to determine isotopy equivalence is unlikely to be tractable. Hence, a more subtle approach is required, which includes additional specific information about the type of approximation process being undertaken. The goal then becomes to demonstrate that specific approximation processes will produce isotopic approximations. Note that this is similar, in spirit, to an approach throughout topology. Namely, topology has many theorems that discuss which topological characteristics (e.g., compactness, connectedness, closure, Euler number) are preserved under specific types of functions (e.g., continuous, homotopic, homeomorphic, diffeomorphic). To motivate that approach, the difficulties that can arise if approximation is done without incorporating specific topological constraints are shown now. Then, recent work by these authors, as well as others will be summarized to indicate progress on this problem. Consider the smooth cubic-spline curve in 3-dimensions shown in Figure 7(a). Note that spline curves are often approximated by piecewise linear or piecewise polynomial curves for display and analysis. A good piecewise linear approximation that maintains equivalent topology is shown in Figure 7(b). However, it is easy to see that a poor approximation, shown in Figure 7(c), can result in a different knot. This particular example results in the figure-8 knot. Hence, a poor approximation as shown in Figure 7 could be detrimental to
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Figure 7. (a) Smooth spline, (b) Good approximation, (c) Knotted approximation.
many analyses. Therefore it is important to consider ambient isotopy as the relevant topological equivalence relation. Recent work [2, 3, 5, 22, 33] develops sufficient conditions for preserving ambient isotopy of manifolds, with quantitative bounds on piecewise linear approximations, which may be useful for practical computational applications. Developing algorithms to preserve the isotopy class on resultant approximants is expected to have an important role in both CAGD and CAMD. Recent results in that regard are now stated. Approximation of spline curves is fundamental for visualization and simulations, where these approximations are often obtained by subdivision methods. Hence, it becomes important to understand when the control points obtained under repeated subdivision form a PL curve that is ambient isotopic to the original spline. Sufficient conditions are given in the following theorem by the present co-authors. The details of the proof are contained in a pre-print by the present authors [25]. Theorem 1: Let B be a non-self-intersecting C2 Bezier curve with regular Bezier parametrization in M3. Then subdivision will produce a control polygon of B that is ambient isotopic to B, provided that nontrivial knots are not introduced during this approximation process. (Note that B may be open or closed.) The criterion about nontrivial knots is, admittedly, informally expressed, above. The interested reader is referred to the full paper [25] for a rigorous formulation of this condition about knots. However, the role of this non-knotting hypothesis was unexpected. After all, the intuition is that the control polygon can be made arbitrarily close to the curve, so after sufficiently many subdivisions one would naively expect them to have the same topological characteristics. However, the authors do not see how to eliminate the non-knotting hypothesis. This will continue to be investigated, but it has implications for the broad use of subdivision to approximate curves and surfaces. Namely, there is some risk, that if this nonknotting condition is violated, then that specific approximation will not have the same topological characteristics as the original geometric object. The authors know of no geometric design or molecular design system that now explicitly checks this
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criterion. Hence, this observation has broad software design implications, since the problem of recognizing when a PL knot is the unknot continues to attract considerable theoretical interest [7, 13, 14], but the development of practical algorithms remains elusive. Note that subdivision is a recursive algorithm. Theorem 1, as stated, provides the valuable insight that subdivision can be used to create an ambient isotopic approximation. However, the stopping criterion can now only be detected by specific geometric checks for this containment. Since these geometric tests are computationally expensive, ongoing work is exploring if the number of iterations can be analyzed in advance to change the algorithm from its current recursive form into a more efficient iterative style. For surfaces, the ambient isotopy problem is considerably more difficult. However, there are some initial results, to which co-author Peters has contributed [30, 33]. These are specialized to the circumstances of compact connected C2 manifolds without boundary, which are embedded in M3. Such manifolds are broadly assumed within CAGD as the bounding surfaces for design objects containing compact volumes. Similar utilities are expected within CAMD. Again, the proof relies upon a recursive approach to determine if the approximation error is within a specific upper bound. However, the resulting surface is no longer from subdivision, but is a PL surface created with rectangular patches that are parallel to the standard three co-ordinate planes associated with a right-hand co-ordinate system within R3. In the re-statement of that theorem, below, this will be referred to as a piecewise box approximation. Theorem 2: Let M be a compact connected C2 manifold without boundary, which is embedded in IR3. Then for an appropriately chosen value of e > 0, there exists a piecewise box approximation of M such that the approximation error is less than e and is ambient isotopic to M. While it is easy to see the applicability of these theorems to CAGD, their usefulness to CAMD remains speculative, but promising. The proposed path to that transition is to consider that ball-and-stick models have been common in molecular modeling. Recently, such models have even been integrated with fast methods for computing their orbitals [11], where the coupling with a static model of a benzene molecule is used as an example. However, simulation algorithms must extend beyond these restrictions on two fronts 1. by replacing the ball-and-stick models with surface and solid models that more fully capture volumetric and physical properties of macro-molecules, and 2. by replacing static models with dynamic ones. Arguably, the fulfillment of objective #1, above, could be achieved for static molecular models with current spline modeling capabilities of existing CAGD systems. However, the creation of accurate molecular models would either entail the use of high degree splines or multiple intersecting lower degree splines. Hence, the efficiency demands for objective #2, above, would lead naturally to approximation of these models with PL models, including the approximation of knotted intersection
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curves that were discussed in Subsection 3.3. The coiled nature of macro-molecules (proteins, DNA, RNA), some of which are known to contain knots, will provide challenging test cases for algorithms based upon the above theorems to deliver ambient isotopic approximations merely for the static cases. Moreover, during simulations, it is expected that life scientists will either require that the isotopy class of a molecule be preserved, or that they will be informed when the isotopy class changes as a result of a chemical or biological process. There remain significant challenges to support such simulations, but the preceding two theorems are presented as fundamental foundations towards that goal.
4
Conclusion
This paper was motivated by the authors' observation that communication about topology is often ambiguous in the literature concerning geometric and molecular modeling. The importance of guaranteeing topological equivalence during approximation is stressed. The criterion proposed for topological equivalence is stronger than the traditional usage based upon homeomorphism. This stronger criterion of ambient isotopy additionally characterizes how an object is embedded in R3. This permits distinguishing between different types of knots, which are all homeomorphic. The isotopy equivalence class of a design model is an important topological aspect of a model. However, computations to determine knot type are known to be intractable, thereby presenting major algorithmic challenges to determining the isotopy equivalence class of a geometric model. Because of this problem, these authors present sufficient conditions for approximations to be isotopically equivalent to the original model, instead of trying to explicitly determine the isotopy equivalence class of each approximation. This comparative approach holds great promise for both CAGD and CAMD. While CAGD models are typically static, it is noted that in the life sciences form and function are inextricably related, and form is rarely static. Topology is more suited than geometry to capture the increased emphasis upon dynamic change in macro-molecular simulations. Hence, it is important to be precise about which topological specializations should be modeled for simulation and preserved during approximation for both CAGD and CAMD. Opportunities exist to leverage topological issues in CAGD and CAMD to benefit both domains, where the transition to support CAMD is discussed in detail in the immediately preceding section. The development of sophisticated algorithms capable of correctly capturing all topological characteristics for simulation is a challenge, and remains on the frontiers of modern computational and mathematical sciences.
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Bibliography [1] C. C. ADAMS, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W.H. Freeman and Company, 2001 [2] N. AMENTA, T. J. PETERS AND A. RUSSELL, Computational topology: ambient isotopic approximation of 2-manifolds, Theoretical Computer Science, 305, 2003, 3-15. [3] L.-E. ANDERSSON, S. M. DORNEY, T. J. PETERS, N. F. STEWART, Polyhedral perturbations that preserve topological form, Computer Aided Geometric Design, (12)8, 1995, 785-799. [4] L.-E. ANDERSSON, T. J. PETERS, N. F. STEWART, Self-intersection of composite curves and surfaces, Computer Aided Geometric Design, (15)5, 1998, 507-527. [5] L.-E. ANDERSSON, T. J. PETERS, N. F. STEWART, Equivalence of topological form for curvilinear geometric objects, International Journal of Computational Geometry and Applications, (10)6, 2000, 609-622. [6] R. H. BlNG, The Geometric Topology of 3-Manifolds, American Mathematical Society, Providence, RI, 1983. [7] J. S. BIRMAN AND M. D. HiRSCH, A new algorithm for recognizing the unknot, Geometry and Topology, 2, 1998, 175 - 220. [8] C. BRANDEN AND J. TOOZE, Introduction to Protein Structure, 2nd edition, Garland, 1999. [9] T. K. DEY, H. EDELSBRUNNER AND S. GUHA, Computational topology, Invited paper in Advances in Discrete and Computational Geometry, B. Chazelle, J. E. Goodman and R. Pollack eds. Contemporary Mathematics, AMS, Providence, 1998. [10] H. EDELSBRUNNER, Biological applications of computational topology, Chapter 63 of Handbook of Discrete and Computational Geometry, eds. J. E. Goodman and J. O'Rourke, CRC Press, Boca Raton, Florida, to appear. [11] G. FANN, G. BEYLKIN, R. J. HARRISON, K. E. JORDAN, Singular operators in multiwavelet bases, IBM J. Res. Dev. 48, 2004, 161-171.
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[12] R. T. FAROUKI, Closing the gap between CAD model and downstream application, SLAM News, 32(5), June 1999. [13] W. HAKEN, Theorie der Normalflachen, Acta. math, 105, 1961, 245 - 375. [14] J. HASS, J. C. LAGARIAS AND N. PIPPENGER, The computational complexity of knot and link problems, Jounal of the ACM, 46, 1999, 185-211. [15] M. W. HlRSCH Differential Topology, Springer, New York, 1976 [16] C. M. HOFFMAN, Geometric and Solid Modeling, Morgan Kaufman, 1989. [17] F. JAEGER, D. L. VERTIGAN AND D. J. A. WELSH, On the computational complexity of the Jones and Tutte polynomials, Proc. Cambridge Philos. Soc., 108, 1990, 35-53. [18] V. F. R. JONES, A polynomial invariant for knots and links via Von Neumann algebras, Bulletins of the American Mathematical Society, 12, 1985, 103-111. [19] S. KRISHNAN AND D. MANOCHA, An efficient surface intersection algorithm based on lower-dimensional formulation, ACM Transactions on Graphics, (16)1, 1997, 74-106. [20] D. LASSER, Calculating the self-intersections of Bezier curves, Computers in Industry, (12)3, 1989, 259-268. [21] J. LIANG, H. EDELSBRUNNER, P. Fu, P. V. SUDHAKAR AND S. SUBRAMANIAM, Analytical shape computation of macromolecules: II. inaccessible cavities in proteins, PROTEINS: Structure, Function, and Genetics, 33, 1998, 18-29. [22] T. MAEKAWA, N. M. PATRIKALAKIS, T. SAKKALIS AND G. Yu, Analysis and applications of pipe surfaces, Computer-Aided Geometric Design, 15(5), 1998, 437-458. [23] M. MANTYLA Computational topology: a study on topological manipulations and interrogations in computer graphics and geometric modeling, Acta Polytechnica Scandinavica, Mathematics and Computer Science Series 37, Finnish Academy of Technical Sciences, Helsinki, 1983. [24] I. MlCHALOPOULOS, G. M. TORRANCE, D. R. GILBERT AND D. R. WESTHEAD, TOPS: an enhanced database of protein structural topology, Nucleic Acids Research, 32, 2004, 251-254.
[25] E. L. F. MOORE AND T. J. PETERS, Integration of computational topology and curve sudivision, pre-print, www.cse.uconn.edu/~tpeters. [26] J. R. MUNKRES, Topology, 2nd Edition, Prentice Hall, 1999. [27] M. NEAGU, E. CALCOEN, AND B. LACOLLE, Bezier curves: topological convergence of the control polygon, in Mathematical Methods in CAGD: Oslo 2000, T. Lyche and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, 2001, 347-354.
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[28] Emerging challenges in computational topology, Report from the NSF-funded Workshop on Computational Topology, Miami Beach, FL, June 11-12, 1999. [29] J. O'ROURKE, Computational Geometry in C, Cambridge, 1998. [30] T. J. PETERS, J. BISCEGLIO, R. R. FERGUSON, C. M. HOFFMANN, T. MAEKAWA, N. M. PATRIKALAKIS, T. SAKKALIS AND N. F. STEWART, Computational topology for regular closed sets (within the I-TANGO project), invited article, Topology Atlas, vol. 9, no. 1 (2004) 12 pp., http: //at. yorku. ca/t/a/i/c/50. htm. [31] Protein Data Bank, http://www.pdb.org/ [32] D. ROLFSEN, Knots and Links, AMS Chelsea Publishing, Providence, 2004. [33] T. SAKKALIS, T. J. PETERS AND J. BISCEGLIO, Application of ambient isotopy to surface approximation and interval solids, CAD, Solid Modeling Theory and Applications, G. Elber & V. Shapiro (ed.), 36 (11), 1089 - 1100. [34] T. SCHLICK, Molecular Modeling and Simulation: An Interdisciplinary Guide, Springer, 2002, 61-89. [35] T. SCHLICK, Modeling superhelical DNA: recent analytical and dynamic approaches, Current Opinion in Structural Biology, (5), 1995, 245-262. [36] R. SHAREIN, http://www.pims.math.ca/knotplot/ [37] R. R. SINDEN, DNA Structure and Function, Academic Press, 1994. [38] D. W. SUMNERS, Lifting the curtain: Using topology to probe the hidden action of enzymes, Notices of the AMS, 42(5), May 1995, 528-537. [39] W. R. TAYLOR, A deeply knotted protein structure and how it might fold, Nature, 406, 2000, 916-919. [40] Toplogy of Protein Structure, http://www.tops.leeds.ac.uk/ [41] S. A. WASSERMAN, J. M. DUNCAN AND N. R. COZZARELLI, Discovery of a predicted DNA knot substantiates a model for site-specific recombination, Science, 229, 1985, 171-174. [42] S. A. WASSERMAN AND N. R. COZZARELLI, Biochenmical topology: applications to DNA recombination and replication, Science, 232, 1986, 951-960. [43] K. J. WEILER, Topological Structures for Geometric Modeling, Ph.D. Thesis, Comput. Syst. Engin., Renselaer Polytechnic Inst., 1986. [44] C. WONJOON, T. MAEKAWA AND N. M. PATRIKALAKIS, Topologically reliable approximation of composite Bezier curves, Computer Aided Geometric Design, (13)6, 1996, 497-582.
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Discretize then Optimize John T. Betts*, Stephen L. Campbell 1 Introduction Computational techniques for solving optimal control problems typically require combining a discretization technique with an optimization method. One possibility is to Discretize Then Optimize, that is first discretize the differential equations and then apply an optimization algorithm to solve the resulting finite dimensional problem. Conversely one can Optimize Then Discretize, that is write the continuous optimality conditions first and then either discretize them or discretize a functional analytic method for solving the necessary conditions. The goal of this paper is to compare the two alternatives and assess the relative merits. There are numerous variations on how to solve optimal control problems. We are interested in problems which may be high dimensional and which can have several inequality constraints with complicated switching strategies between them. We also are interested in situations where functions are not given by "simple" formulas but may involve complicated computer implementations. These considerations are typical of many industrial applications and guide some of the discussion that follows. Our intention is not to review the many software packages that are available and in production. Additional references are in the bibliography of [2]. Here we focus on the more fundamental issue of the order of the discretization and optimization processes. The genesis of this paper was the earlier technical report [4]. Some of the ideas, including some discussions of the example in Section 2 appear in [5, 6]. In addition to presenting an improved understanding of the problem described in [4], this paper contains results, discussion, and the consideration of issues not found in [5, 6].
* Mathematics and Engineering Analysis, The Boeing Company, P.O. Box 3707, MS 7L-21, Seattle, Washington 98124-2207 tNorth Carolina State University, Department of Mathematics, Raleigh, NC 27695-8205. Research in part by the National Science Foundation under DMS-0101802, ECS-0114095, DMS0209695, and DMS-0404842.
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2
High Index Partial Differential-Algebraic Equations
Our ultimate goal is to optimize systems described by nonlinear (e.g. Navier-Stokes) partial differential equations subject to inequality constraints. In these problems one often wants the control to be applied on the boundary. As a model problem we will focus on a particular example which can be used to illustrate several points. Heat transfer can be described by the partial differential equation
where the spatial domain is 0 < x < TT and the time domain is 0 < t < 5. Conditions are imposed on three boundaries of this domain,
The input temperatures Uo(t) and u^(t] at the ends of the domain are viewed as control variables. In addition, the temperature over the domain is bounded below according to
where
is a prescribed function with a = .5, b = .2, and c = 1. Finally, we would like to choose the controls uo(t] and uv(t} to minimize
For our example we set the constants Qi = Q2 = 10~3. This is typical of the situation in practice where the primary interest is minimizing some function of the state but small weights are added to the controls to help regularize the problem numerically. One way to solve this problem is to introduce a discretization in the spatial direction. We take a uniform spatial grid, Xk = k6 = k^ for k = 0 , . . . , n. If we denote y(xk,t} = yk(t), then we can approximate the partial differential equation (1) by the following system of ordinary differential equations:
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This (method of lines) approximation is obtained by using a central difference approximation to |^| and incorporating the boundary conditions (3) and (4) to replace ?/o = UQ and yn = u^. As a consequence of the discretization, the single constraint (5) is replaced by the set of constraints
The boundary conditions (2) are imposed by setting 7/fc(0) = 0. Furthermore, if we use a trapezoidal approximation to the integral in the or-direction, then the objective function (7) becomes
2.1
State Vector Formulation
Since we will consider several aspects of this problem, it is helpful to introduce a more compact notation. We define a normalized time t = 62r and use "/" to denote differentiation with respect to r in which case
Using this notation we can rewrite the differential equations (8) as
where the state vector is yT = (7/1,7/2, ••• ,2/n-i), and the control vector is UT = (UO^UTT). The symmetric, tridiagonal, (n — 1) x (n — 1) matrix F has —2 on the main diagonal and 1 on the two adjoining diagonals. The only nonzero elements in the rectangular (n — 1) x 2 matrix G are G\^ — 1 and <7 n _i ; 2 = 1- We can also write the objective function (10) as
where ^4 is an n — 1 x n — 1 diagonal matrix with entries 283 and B is a 2 x 2 diagonal matrix matrix with entries 62(6 + 2#i), S2(8 + 2*72). It is important to note that the direct transcription approach can be applied to nonlinear problems, even though this example involves the linear differential equations (12) and quadratic objective (13).
3
Direct Transcription Results
The direct transcription method utilizes a "Discretize Then Optimize" philosophy. Specifically, the differential equations (12) and objective function (13) are replaced by discrete approximations subject to the inequality constraints (9), leading to a
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large sparse nonlinear programming problem. After solving the NLP the discretization (in time) is refined until a sufficiently accurate solution is obtained. This approach is implemented in the S0CS software and is described in [2, 7]. The results presented here utilize the SQP option. If we set n = 20, then the solution obtained using the direct method is illustrated in Figure 1.
Figure 1. y(x,t) using a direct method and n = 20. Checking the output from SOCS we see that the only place that the constraints are ever active is along the middle of the spatial interval. In Figure 2 we plot the constraint surface and the solution y(xiQ,t). We see that the solution appears to ride the constraint surface. This is called a constraint arc.
Figure 2. Constraint surface g(x,t)
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andy(x\o,i).
This impression is reinforced in Figure 3 which shows the x = X\Q slice of both the constraint surface and the solution graphed in Figure 1.
Figure 3. Slice of solution and constraint surface along x = XIQ.
4
The Indirect Approach
The "Optimize Then Discretize" philosophy requires explicit calculation of the optimality conditions. In order to simplify further analysis let us assume that n is even, and define ra = ra/2. In order to solve the necessary conditions it will be necessary to determine the constraint arcs (intervals), if any, where the constraints are active. It will also be important to determine the isolated points (touch points) where the constraints are active. There are results in the literature stating when touch points [15] and constraint arcs [11] cannot exist. But examples like the tool path planning problem [1] show that it is sometimes impossible to tell ahead of time what this switching structure looks like and that it can be very complicated. Looking at the computational output of the direct transcription approach and checking where the inequality constraints are active we find they are only active at xm. Looking then at Figure 2 and Figure 3 it appears that this one inequality constraint from (9) is active on a constraint arc. We proceed to apply the indirect approach under this assumption. The derivation will be useful not only in discussing the numerical results but also in showing what is involved with an indirect solution. We have one inequality:
where e is the n — 1 vector with:
Denote the k-th derivative of
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since eTGu = 0, where the definition of y' from (12) is used to simplify the expression. The process of differentiation followed by substitution can be repeated leading to the expression
when k < m and
The k-th derivative of the constraint function is given by
which can be computed from the defining expression for g(x,t). Let us define
for k = 0,1,... , m — 1. Thus for 0 < k < m, equation (17) becomes
and equation (18) is just
5 5.1
Optimality Conditions Unconstrained Arcs (s < 0)
During the time when no constraint is active we derive necessary conditions by considering the Hamiltonian
Recalling that FT = F for our problem, it then follows that the adjoint equations are
The Maximum principle yields 0 = H^ = £?u+GT A which can be solved analytically to give an expression for the optimal control u = — B~1GT\.
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5.2
Constrained Arcs (5 = 0)
When the constraint is active we have the Hamiltonian
In this case the adjoint equations are
The Maximum principle yields
which can be solved analytically to give an expression for the optimal control
On the constrained arc we require s^ = 0 so if we substitute (28) into (22) and rearrange we obtain
Solving (29) for /^ we obtain the following expression
5.3
Boundary Conditions
Inspection of the direct solution (Figure 3) suggests that there is one constrained arc so we assume that the optimal solution is defined on three distinct regions
where the constrained arc occurs for r\ < r < r^ and the other regions are unconstrained. Obviously from (2) we must have
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If we define the vector
then at the beginning of the constrained arc (T = r\) we must satisfy the tangency conditions
Note that it is equally valid to impose these conditions at the other boundary (r = T2), however, we retain the convention established in [9] and [13]. The adjoint variables at the boundary must satisfy:
where TT is an ra-vector and from (21) and (33) we have
If we define the vector JVj" = [g^\g^\ • • • ,9^] then we must also have
The following simple continuity conditions must be imposed
Finally, we must impose the terminal condition
5.4
Optimality Conditions: Summary
Summarizing the results of the previous sections, we must solve a multi-point boundary value problem. When the solution is unconstrained we must satisfy the differential equations
and the constrained arcs satisfy the differential equations
with boundary conditions (32), (34), (35), (37), (38), (39), (40), and (41).
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6 6.1
Computational Comparison-Direct vs Indirect Direct Method
The SOCS software was used to obtain solutions for a number of different spatial discretizations n. Figure 1 illustrates the optimal temperature profile corresponding to the n = 20 case. Table 1 presents a summary of the computational results. The first column of the table gives the size of the spatial discretization. Column two gives the number of mesh refinement iterations required to achieve a discretization error of 10~7 or approximately eight significant figures, and column three shows the number of grid points required to achieve this accuracy. The "discretize then optimize" approach implemented in SOCS requires solving a sequence of large sparse nonlinear programming problems, and the next three columns summarize information about the NLP performance. Column four gives the total number of gradient/Jacobian evaluations required to solve all of the NLP problems. The number of Hessian evaluations is given in the next column, followed by the total number of function evaluations including those needed for finite difference perturbations. For this example, the NLP subproblem on each mesh is just a quadratic program (QP), however, small inaccuracies in the gradient information preclude "perfect" computational behavior. The final column gives the total CPU time to compute the solution on a SUN Blade 150 workstation. Default tolerances were used for all of the NLP subproblems which guarantee that all discrete constraints were satisfied to a tolerance of e — ^fe^ K> 10~8 where £m is the machine precision. Since the direct method does not require special treatment for constrained arcs these results were obtained using a single phase formulation.
1]able n 4 6 20 40
6.2
Refn 8 8 13 16
1. Direct Method (One Phase) Grid 172 317 1063 2105
Grad 32 32 50 61
Hesn 8 9 14 18
Func 959 908 1617 2057
CPU 7.84 26.4 474 3685
Indirect Method
In contrast to the direct method, the indirect method must be modeled using more than one phase because the optimality conditions are different on constrained and unconstrained arcs. If one assumes there are three phases as suggested by Figure 3, then the SOCS software can also be applied to the boundary value problem summarized in Section 5.4. That is, it can be used as the boundary value problem solver of a indirect method. The numerical results for this approach are presented in Table 2. The first row of the table corresponding to the case n = 4 demonstrates the expected behavior for the method. The mesh was refined 5 times, leading to a final grid with 183 points. The solution of the boundary value problem required 29
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Table 2. Indirect Method (Three Phases) n
Refn
Grid
Grad
Hesn
Func
CPU
4 4f 6t 20 40
5 5 6
183 183 229
29 51 150
0 21 52
419
2.46 8.61 62.8
2777
Fails to Converge Fails to Converge f Slack Variable Relaxation
908
gradient evaluations, and 419 total function evaluations and was computed in 2.46 seconds. Since there were no degrees of freedom (and no objective function), no Hessian evaluations were required. For all other cases tested it was either difficult or impossible to compute a converged solution. In an attempt to improve robustness, the boundary value problem was formulated as a constrained optimization problem by introducing slack variables. Specifically, equality constraint boundary conditions such as (37) of the form ^(x) = 0 were formulated as tj)(x) + s+ — s~ = 0 with the slack variables s+ > 0 and s~ > 0 followed by a minimization of an objective F(x) — XXS+ + s~)- The second and third row in Table 2 present results obtained using the slack variable formulation (denoted by f). In both cases solutions were obtained with converged values for the slacks s < e, however, the rate of convergence was very poor, as indicated by the number of Hessian evaluations. It was not possible to obtain a converged solution using either technique when n > 6.
7 7.1
Analysis of Results The Quandary
The numerical results presented in Sections 6.1 and 6.2 pose a number of controversial issues. It appears that the direct method solves the problem in a rather straightforward manner for all values of the spatial discretization n. Yet when the inequality constraint is active s(y,r) = 0, the combined differential-algebraic equation has index m = n/2 + 1. Thus, referring to Table 1, it appears that the direct method has solved a DAE system of very high index (11 or 21). However, usually the only way to solve a high index DAE is to use index reduction [8] since no known discretization converges for index 21 DAE's. In fact, the default discretization in SOCS, like most discretizations, fails to converge if the index is greater than three [10]. In short, it appears that the direct method works, even though it shouldn't! In contrast, the index reduction procedure has been performed for the indirect method. Specifically, the necessary conditions given by (29) explicitly involve the rath time derivative of the constraint s(y,r) = 0. In fact, consistent initial conditions for the high index DAE are imposed via the tangency conditions (34). In short, it appears that the indirect method does not work, even though it should! In order to explain this quandary, we first suspected a bug in the code. To address this issue we compare the direct and indirect solutions for the cases n — 4
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and n = 6. Figures 4 and 5 plot the difference between the direct and indirect solutions normalized by the largest absolute error. At least visually the solutions for the state variable (temperature) appear to agree quite closely. There is a slight discrepancy between the control variable (temperature at rr = 0, TT) solutions, however, this possibly can be attributed to differences in the formulation. Specifically, the direct method has a single phase, and the control approximation is continuous by construction. In contrast, the indirect method has three phases, and consequently can introduce a discontinuity in the control at the phase boundary. This still does not explain why the direct method works and the indirect method fails.
Figure 4. Normalized Difference Between Direct and Indirect Solutions n — 4, (Max. Error — .000171597,).
7.2
The Explanation
In order to explain what is happening we take a microscopic look at the results obtained by the direct method. Figure 6 plots the quantity s(y, r)/e where e = y/£^ ~ 10~8 and Em is the machine precision. Since e is the nonlinear programming constraint tolerance, we can view s(y, r)/e as a normalized path constraint error. More precisely, |S(T/,T)| < e when — 1 < s(y,r)/e < 1. Thus in Figure 6 we are plotting the regions when the path constraint is within convergence tolerance. Conversely when s(y,r)/e < —I the mathematical constraint (14) is strictly satisfied. Regions corresponding to strict satisfaction of the path constraint are shaded gray. From Figure 6 it would appear that when n — 4 there is exactly one region where the path constraint is within tolerance. However, forn = 6 it appears there are three distinct regions where the path constraint is within tolerance. And for n = 20 and n = 40 there may be as many as five regions where the path constraint is within tolerance. At least graphically, this suggests that the number of constrained arcs is not always one! It may be three, or five, or perhaps some other value. It also suggests that the number of constrained arcs may change with the spatial discretization n.
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Figure 5. Normalized Difference Between Direct and Indirect Solutions n = 6, (Max. Error = . 00621083 j.
How many constrained arcs (or touch points) are there? Even though a graphical analysis cannot answer the question we immediately have an explanation for the apparent failure of the indirect method. In order to explicitly state the necessary conditions (42)-(43) with boundary conditions (32), (34), (35), (37), (38), (39), (40), and (41), we assumed there was only one constrained arc. Clearly, if the number of constrained arcs is wrong, the optimality conditions are wrong! At this point, it is important to note the result from [11] which says that for problems of the type we consider here, there cannot be any constraint arcs at all if n/2 is an odd integer greater 2. There can be only touch points. The computational complexity of the problem in this paper as n increases prevents a more careful examination of this behavior. In [6] we develop a parameterized fixed dimensional problem with a similar structure for which we are able to compute the solutions much more accurately. What we see there is an increasing number of touch points except that the deviation from the constraint was several orders less than 10~7. For this problem the theory held but was numerically meaningless since the constraint deviation was below normal error bounds. Academic problems are often "simple." In our experience real world applications often contain very complicated constraints. For these applications it is seldom possible to predict when the constraints will be active or compute the requisite necessary conditions. Also, as illustrated above and in [6], the theoretically predicted behavior may occur at or below the computational tolerances used to solve the problem, leading to an extremely ill conditioned indirect solution. Thus the difficulties exhibited by this example can occur in real world problems where it is even harder to predict what will happen. To fully understand how the direct method successfully obtained a practical solution without having to solve the necessary conditions and without having to determine ill conditioned small scale behavior, it is helpful to review how the sparse non-
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Figure 6. Inequality Constraint Error linear programming algorithm treats the discrete problem. During the final mesh refinement iterations an implicit Runge-Kutta (Hermite-Simpson) discretization is used and for this method the NLP variables are: x = [yl, ui, U2, y2, U2, us,... , yM, UM] T - Specifically, the NLP variable set consists of the state and control variables at M grid points y^ = y(rj), Uj = U(TJ), and the control variables at the midpoints fj = (TJ + Tj-i)/1 of the M — 1 discretization intervals Uj = U(TJ). The differential equations (12) are approximated by the defect constraints
and equations estimating the midpoint values yk, fk [2] • Of particular relevance to this discussion is the treatment of the (continuous) state inequality constraint (14), which is approximated by enforcing it at the gridpoints and midpoints
The (2M — 1) constraints (45) are treated as inequality constraints by the nonlinear programming algorithm.
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Table 3 summarizes this information for the four spatial discretization cases. The first row gives the total number of discrete inequality constraints (2M— 1) that are enforced by the nonlinear programming algorithm. The second row of the table gives the number of inequality constraints that are within tolerance, which corresponds to the information displayed in Figure 6. Row three tabulates the number of constraints treated as "active" by the nonlinear programming algorithm. Row four gives the percentage of constraints within tolerance that are also considered active by the NLP. Table 3. The NLP Active Set Spatial Discretization, n Inequalities s Inequalities within tolerance, \s\ < e Active Inequalities Percent Active
4
6
20
40
343 102 59 58
633 149 52 35
2125 475 38 8
4209 1083 40 4
Clearly this data suggests that only a small fraction of the discrete constraints within convergence tolerance are actually treated as "active" inequalities by the nonlinear programming algorithm. Now, the active inequality constraints at the NLP solution satisfy a constraint qualification test. More precisely the underlying quadratic programming algorithm constructs the set of active inequality constraints such that the gradients are linearly independent. In essence the active set is constructed to satisfy so-called linear independent constraint qualification (LIQC). Conversely, we suspect that if all of the inequalities within tolerance had been included in the active set, the resulting Jacobian matrix would be rank deficient. However the solution does satisfy the more more general Mangasarian-Fromovitz constraint qualification (MFQC) as demonstrated recently by, Kameswaran and Biegler [12]. Can the direct approach be used to solve a high index (greater than three) DAE? Suppose the high index path constraint had been treated as an equality constraint. This would require treating every grid point as an equality constraint, thereby leading to a singular Jacobian matrix. In addition, as noted earlier, the discretization does not converge for high index DAE's. The direct approach will fail if the path constraint is treated as an equality. The solutions presented exploit the fact that the path constraints are inequalities not equalities. In other words the direct approach solves the high index DAE conditions by directly enforcing a small subset of the discrete conditions, and trivially satisfying the others. This explains why the direct approach works when it shouldn't! Since the NLP active set is determined using the Lagrange multipliers for the discrete problem, the behavior of the direct method also demonstrates a theoretical issue of some interest. In [3] we prove for a class of higher index state constrained control problems that the discrete solution converges to the "continuous" optimal control using the theory of consistent approximations [14]. Although the proof is applicable only for a trapezoidal discretization, it does not rely on the convergence of the discrete NLP Lagrange multipliers to the continuous adjoint variables. In
153
fact, it is shown in [3] that the multipliers fail to converge to the adjoint variables. In [3], it was also shown that for problems where there was an active constraint arc, that the direct method could stabilize an unstable discretization by using small perturbations off the constraint arc. We suspect that the consistent approximation ideas will apply here, but as yet have not extended the results in [3] to this situation.
8
Comparison with Shooting Methods
The transcription approach described here does not distinguish between state and control variables. Discrete values for both the differential (state) and algebraic (control) variables are computed simultaneously. In contrast direct shooting methods introduce a finite dimensional approximation to the control (e.g. by discretization). The approach exploits the fact that given a control, one can evaluate the dynamics and evaluate the cost. Thus the problem can be considered as minimizing F(u}. This approach appears attractive because the optimizer applied to F sees a much smaller problem. This is particularly true when, as is often the case, the dimension of the control is much smaller than the dimension of the state. The shooting method relies on a decomposition of the space into differential and algebraic variables. By "solving" the differential equations, one can determine the functional dependence of the state on the control y(u). In essence, a shooting method implements an elimination of variables technique. However, there is a reduced flexibility that comes about because the evaluation of F requires a choice of u and then the integration of the dynamics by a numerical integrator. For the inequality constrained problems we have discussed earlier this would be complicated and not practical because of the ill conditioned switching. It would also not be practical, without problem reformulation on some better behaved inequality constrained problems because when the constraints are active a high index DAE may result which the integrator would not be able to integrate. This raises the possibility that a direct transcription method may be able to solve a problem with high index dynamics, arising from inequality or equality constraints, that a shooting method cannot because there is an alternative control choice, perhaps unknown to the user, which results in index one dynamics. Preliminary computational studies suggest that this is the case but space limitations do not allow us to pursue this point further here.
9
Conclusions
This paper presents a comparison of two techniques for solving optimal control problems, namely the direct versus the indirect method. For the purpose of demonstration we consider a model problem that requires solving a partial differential equation subject to an inequality constraint on the state. After discretization using a method of lines, the problem can be stated as an optimal control problem with linear ordinary differential equations, subject to a high index path inequality constraint. A preliminary examination suggests that the model problem has a single constrained arc, and the appropriate optimality conditions are derived for this case. However we also demonstrate that the assumed arc structure is incorrect.
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The example illustrates that to successfully use the indirect method it is necessary to guess the number of constrained arcs, the number of touch points, the location of the constrained arcs, and the correct index of each constrained arc. In general it is extremely difficult to determine this information a priori, and failure to do so will cause the indirect method to fail. Even if the correct necessary conditions can be found they may be too ill conditioned for the indirect approach to solve them. In contrast the direct method works because the underlying nonlinear programming algorithm determines the number, and location of the constrained arcs. The NLP algorithm regularizes potential ill-conditioning by ignoring numerically insignificant behavior. In short, the direct method is a robust technique for solving optimal control problems especially when state inequality constraints are present.
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Bibliography [1] J. T. BETTS, Parametric Tool Path Trajectory Optimization, Technical Document Series SSGTECH-98-006, Mathematics and Engineering Analysis, The Boeing Company, PO Box 3707, Seattle, WA 98124-2207, Mar. 1998. [2]
, Practical Methods for Optimal Control using Nonlinear Programming, Society for Industrial and Applied Mathematics, Philadelphia, PA., 2001.
[3] J. T. BETTS, N. BIEHN, AND S. L. CAMPBELL, Convergence of Nonconvergent IRK Discretizations of Optimal Control Problems with State Inequality Constraints, SIAM Journal on Scientific Computing, 23 (2002), pp. 1982-2008. [4] J. T. BETTS AND S. L. CAMPBELL, Discretize then Optimize, Mathematics and Computing Technology Report M&CT-TECH-03-01, Mathematics and Computing Technology, The Boeing Company, PO Box 3707, Seattle, WA 98124-2207, Feb. 2003. [5] J. T. BETTS, S. L. CAMPBELL, AND A. ENGELSONE, Direct transcription solution of inequality constrained optimal control problems, Boston, MA, 2004, Proc. 2004 American Control Conference, pp. 1622-1626. [6]
, Direct transcription solution of optimal control problems with higher order state constraints: Theory vs practice. Submitted, 2004.
[7] J. T. BETTS AND W. P. HUFFMAN, Sparse Optimal Control Software SOCS, Mathematics and Engineering Analysis Technical Document MEA-LR-085, Boeing Information and Support Services, The Boeing Company, PO Box 3707, Seattle, WA 98124-2207, July 1997. [8] K. E. BRENAN, S. L. CAMPBELL, AND L. R. PETZOLD, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Vol. 14 of Classics in Applied Mathematics, SIAM, Philadelphia, PA., 1996. [9] A. E. BRYSON, JR. AND Y.-C. Ho, Applied Optimal Control, John Wiley & Sons, New York, NY, 1975. [10] S. L. CAMPBELL, N. BIEHN, L. JAY, AND T. WESTBROOK, Some Comments on DAE Theory for IRK Methods and Trajectory Optimization, Journal of Computational and Applied Mathematics, 120 (2000), pp. 109-131.
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[11] D. H. JACOBSEN, M. M. LELE, AND J. L. SPEYER, New necessary conditions of optimality for control problems with state variable inequality constraints, J. Math. Anal. Appls, 35 (1971), pp. 255-284. [12] S. K. KAMESWARAN AND L. T. BIEGLER, A Further Analysis of the Betts and Campbell Heat Problem, tech. rep., Chemical Engineering Department, Carnegie Mellon University, 2004. [13] H. J. PESCH, A Practical Guide to the Solution of Real-life Optimal Control Problems, Control and Cybernetics, 23 (1994), pp. 7-60. [14] E. POLAK, Optimization: Algorithms and Consistent Approximations, Springer-Verlag, New York, 1997. [15] H. SEYWALD AND E. M. CLIFF, On the existence of touch points for firstorder state inequality constraints, Optimal Control Applications &; Methods, 17 (1996), pp. 357-366.
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Transferring Analyses onto Different Meshes D.A. Field" Abstract Different applications of a CAD-model often require very different finite element decompositions for their respective mathematical simulations. Moreover one analysis can require mesh dependent results from a prior analysis. In many of these situations constant functions define the initial conditions on each element of the unanalyzed mesh. This paper investigates the transfer of mesh dependent information from an analyzed mesh to an unanalyzed mesh. Instead of presenting the implementation of a two dimensional algorithm, this paper distills the mathematics involved into one dimensional problems. Solutions to and characteristics of these problems extend to higher dimensions. The initialization of a structural analysis with mechanical properties predicted by the of stamping sheet metal serves as a prototype example.
1
Introduction
Mathematical models that simulate physical behaviors of a solid object, often rely on different finite element meshes for different types of simulations. Moreover one analysis can require mesh dependent results from a prior analysis. Initializing a structural analyses with properties predicted by simulating stamping sheet metal motivates this paper on the transfer of mesh dependent results from one finite element analysis to another. In this case, among transferable quantities such as yield strength and plastic strain, transferring the predicted thickness of each element in the stamping mesh to elements in the structural mesh provided the prototype example. Although stamping and structural meshes typically depend on the same CADmodel, major geometric differences between the stamping and structural meshes make transference of numerical data between the meshes difficult. Stamping meshes "General Motors Research, Development &; Planning Center, Warren, MI
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usually consist of triangular shell elements but meshes consisting of only quadrilateral shell elements or mixtures of triangular and quadrilateral elements also occur. On the other hand, structural meshes favor quadrilateral shell elements but triangular elements regularly intermingle quadrilateral shell elements. Discretizations of CAD-models produce structural meshes and meshes that provide the geometry of dies used in simulating the stamping process. Stamping meshes approximate the final shape of sheet metal formed by stamping operations. Adaptive finite element techniques used to simulate stamping sheet metal create stamping meshes with elements varying in size by several orders of magnitude. Sometimes final meshes contain elements which do not conform to their neighbors. On the other hand, initial structural meshes tend to have significantly fewer elements of roughly uniform size. The process of mapping the output of a finite element analysis onto the mesh used for a subsequent finite element analysis has a natural division into two essential tasks, registration and assignment. Both tasks use approximations. Registration reorients one mesh so that it optimally coincides with the other mesh. Although registration depends on myriad approximations, computational geometry dominates this initial task. In this paper all meshes have been registered. Much research has been published on registration which encompasses aligning clouds of points to a variety of surface; references [4, 5] offer reviews of recent literature and include the standard iterative closest point algorithm (ICP) of Besl and McKay [1]. Since this literature documents many general methods and many very fast implementations, the task of registering two finite element meshes should be relegated to using available software. The second task first establishes correspondences among the elements in the mesh of the original analysis and the elements in the mesh of the subsequent analysis. In its simplest form, a mathematical model of stamping sheet metal produces a piecewise constant function / defined on a finite element decomposition of a domain F where / approximates a continuous but unknown function h; i.e. the thickness over F. On a different decomposition or partition of F another approximation of h, say g, must be defined. The determination of g makes no assumptions on h nor on how well / approximates h. g essentially approximates /. Thus the remaining step in the second task, and the main topic of this paper, involves the mathematics of determining constants used by g to approximate the piecewise constant function /. The investigation into approximating / by g begins by considering two distinct meshes or partitions of the simplest and most easily registered domains, two arbitrarily placed copies of a generic interval T = [a, b]. Extensions to one dimensional piecewise linear curves in higher dimensional situations will clarify the contributions of geometry and dimensionality to this approximation problem. In these cases / is a simple curve. This paper will focus on one dimensional piecewise linear meshes defined on planar and in three dimensional spaces. Since these domains comprise the boundaries of surface meshes or lie on solid finite element meshes, meshes of these one dimensional domains provide a natural initial class of finite element meshes to study. Approximation methods with characteristics that extend to higher dimensions will receive major emphasis. The results of applying an extension to the surface meshes
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associated with stamped sheet metal will be reported elsewhere. The second section of this paper concentrates on partitions of F. It presents methods for local and weighted local averaging and then gives them a parametric formulation. This section presents expressions for errors in approximating / by g. Section 2 also discusses implications from constraints such as maintaining extremal values. The third section of this paper focuses on extending the initial domain F to planar piecewise linear curves. It also identifies several alternate methods. The fourth section outlines the applicability of the methods for planar piecewise linear curves to spatial piecewise linear curves.
2 2.1
Averages Two Methods
Let {xi}^, a = x\ < x% < ... < XN = b partition the interval F. Denote by {ci}± the values of the piecewise constant function /(#) defined on F.
On a different partition {£7}^, a = £1 < £2 < ••• < £M = b, of T define the piecewise constant function g(x),
where the values 7, will depend of the values Cj. Begin the approximation of f(x) with g(x] by overlaying the partitions {xi}± and {£?}i^ and define the sets Since Xj ^ 0, an easy, natural and routinely used approximation when post processing finite element meshes computes the value j j by simply averaging the set of Q'S contained in Xj]
Call this assignment Local Averaging or LA. Although extremely easy to calculate, LA does not weight the average in Equation (4) proportionally to the amounts intervals overlap in the definition of Xj. Define Weighted Local Averaging or WLA by
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In 2- or 3- dimensions different meshes of the same object generally do not register exactly. Relative to F an imprecise registration corresponds to a domain rg, where g partitions an interval whose endpoints do not coincide with XQ = a and XN = b. Since Equation (5) implies a partition of unity on each subinterval in the partition of Tg when Tg = F, maintaining a partition of unity on each subinterval [£j, £j+i], when Tg 7^ F would require changing the denominators in Equation (5) for at least the first or last subinterval, [£1,^2] or [£M-IS£M]> when F5 extends beyond the interval [a,b].
2.2
Properties of LA and WLA
Expressing errors of approximation provides a good start for comparing the properties of LA and WLA.
expresses the error of approximation by LA in the Max or ^oo-norm. The error of approximation for WLA in the ^oo-norm has the similar expression
The error of approximation of f ( x ) by g(x) in the Z^-norrn,
uses the constants that appear in Equations (6) and (7). Evaluating the integral in Equation (8) over the subintervals of g partitioned by the subintervals of / produces the square root of a weighted sum of squares of the constants, namely,
Consider the case where M = 2; i.e. {£j} = {^1,^2}- LA produces the singleton value,
and WLA yields a corresponding singleton value,
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If M = 2 and the vertices in the partitions {xi} are uniformly distributed, then 7LA = 'JWLA • For partitions of g with more than two subintervals, uniformity in the distributions of x^ in the subintervals of g can make WLA and LA equivalent. Sometimes g must preserve certain qualitative characteristics of /. For example, in three dimensions each shell finite element in a surface mesh can have a different uniform thickness, g can be constrained to preserve the volume of material. In one dimension the comparable conserved quantity would produce
for a > 0, 1 < i < N — 1, and for 7,> 0, 1 < j < M — I . Without positivity of Q and 7j,
has the geometric interpretation of conserving the sum of rectangular areas given by heights |Q| and widths Xi+i — Xi. When Ci > 0, Equations (12) and (13) can also refer to preserving cumulative distributions. For a > 0, 1 < i < N — 1, let Si — Y^k=ick- For 7j > 0, 1 < j < M — 1 and Sj = Z)L=i7fc choosing 7, so that the 8j 's approximate the Si's corresponds to cumulative distributions functions addressed by the Kolmogorov-Smirnov goodness-of-fit test [10]. Among conserved quantities, the values in {7^} occasionally must also attain the extreme values in the set {cj}. Achievement of this constraint depends on the partition. Clearly, the singleton partition {^1,^2}? a = £1 and 6 = £2, cannot attain a maximum and a minimum if the set {Q} contains more than one distinct value. To enforce this constraint when {ci} contains at least two distinct values, the partition {£j} must have two or more intervals. Suppose Ci > 0, 1 < i < A7", and the two intervals in the partition {^1,^2,^3} have lengths o;s and UL where us < UL. If Cimin and Cimax represent distinct minimum and maximum values in {Q} respectively, then including the attainment of extremal values Cimin and Cimax along with Equations (12) and (13) as conserved quantities restricts the values of 71 and 72 of g and limits g to two values; namely v\ or V2 where
Consequently, a necessary condition for g with partitions {£/}, 1 < j' < M, M > 4, to attain these three positive conserved quantities (the area V in Equations (12) and (13) ), the minimum Cimin and the maximum Cimax] is
2.3
Parametric formulation
The intrinsic value for parameterizing the piecewise linear mappings on F lies in its direct application to piecewise linear planar and spatial curves. This subsection
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introduces parameterization by using B-splines. The advantage of using B-splines becomes obvious from the notation introduced by Carl DeBoor [2]. Given a partition {U}, ti < t^ < ... < tN of an interval, the lowest order Bsplines, Ni^(t), represent characteristic functions on the partition; for 1 < i < N—1,
The next order B-splines, the hat functions AT ii2 (t), 0 < i < N, defined by
requires prepending to> ^o < ^i and appending tN+1, tN < tN+1 to {ti}. Parameterizations in software for Computer Aided Design typically use Ni^(t) and distribute ti e [0,1] uniformly. Use Equations (16), (17) and the partitions {ti}, ti = iri~p Q
Similarly, for {?;}, Tj = ^T, 0 < j < M + 1 and 0 < r < 1,
The use of B-splines in Equations (18) and (19) simplifies the formulation of a minimization in the I/2-norm to establish values for 7j. Minimizing C,(x), with respect to 7.,, 1 < j; < M — 1, i.e.
produces M — 1 linear equations, dC/d^j = 0, and the solutions
WLA produces the same value of 7^.
3
Planar Piecewise Linear Curves
The elemental case of piecewise constant approximations over an interval [a, 6] not only obviates the registration but, more importantly, omits the geometric problems intrinsic to piecewise linear domains of two piecewise constant functions / and
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Figure 1. Line n lies orthogonal the common domain and the ranges of f and g.
g. The portions of / and g plotted on the same graph in Figure 1 initiates the following brief discussion of some important geometric aspects in approximations with piecewise constant functions. In Figure 1 consider the coordinate (x, g(x}) that lies on a line segment parallel to the £-axis and bounded in the ^-direction by consecutive points £/ and £J+i in the partition {^jf^. As in typical approximations with functions, the domains of the functions / and g coincide. More significantly, in Figure 1 the coincidence of domains implies that the direction of lines normal, N for example, to the domains coincide and are also normal to the piecewise linear segments in the range of / and g. The coincidence shows that essential differences in the geometry of the domains have been eliminated and not amplified by normal projections. In sharp contrast to approximating piecewise constant functions over an interval [a, 6], the errors from approximating piecewise constant functions over piecewise linear curves and finite element meshes have distinct geometric components. One source of error comes from the registration which defines a rigid transformation that typically optimizes the coincidence of / and g in a least squares sense. In particular, registration usually minimizes the squared distances of each vertex in the moved mesh with the closest point on the surface of the fixed mesh. In some cases registration uses prescribed vertices on each mesh. Differences in the discretizations of finite element meshes can be measured with respect to the two meshes or relative to the CAD model upon which each mesh depends. Therefore the functional values of g, the 7j's, depend on registration errors, on the geometric differences of the meshes
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and on the errors in the functional values of /. Without loss in generality, in the remainder of this paper meshes have been registered in some optimal geometric sense. When approximating a piecewise constant function by another, it is extremely important to recognize that the two piecewise linear curves have intrinsically different domains. Even though the parameterizations discussed in Section 3.2 provide a surrogate common domain, the most influential step in assigning constants to a piecewise constant function defined over a piecewise linear approximating curve is the assignment of correspondences among points and line segments in the two domains. In the remainder of Section 3 {(xi,^)}^ and {(£i,?7i)}i^ denote the vertices in the piecewise linear curves corresponding to / and g, respectively. Li denotes the line segment between vertices (xi,yi) and (xi+i,yi+i) and Aj identifies the line segment between ( £ j , r j j ) and (£j+i,77j + i). On the interior of Li and at (xi,yi), f ( x , T/) = Ci and ,on the interior of Aj and at (£j, rjj), g(£, 77) = 7j. At the endpoints ( x N , y N ) , and (£ M ,77 M ), f ( x N , y N ) = CN and#(£ M ,77 M ) = 7M. 3.1
Parallel Projections
With regard to their respective coordinate axes, the linear segments of / or g can have arbitrary orientations. Nevertheless, when projecting {Li}-^ ~l and {Aj}i ~ onto the coincident x-axis and £-axis respectively produces y as a function of x and 77 as a function of £, the projections also produce the essential constituents required to apply the contents of Section 2. Minor modifications such as adjusting Equation (5) for mismatched ends of the partitions or for the lengths of projected edges as weights in WLA may be necessary. If projecting {Li}^~1 and {Aj}^"1 onto the coincident y- and rj-ax.es respectively produces y and 77 as one-to-one functions of x and £ respectively, the contents of Section 2 apply as well to partitions {yi}^ and {T?;}^. In both cases the projections take their direction from normals to the coordinate axes. Using the Ty-axis rather than the £-axis generally produces different values of 7j. For example, use the £-axis in Figure 2. Since x± < £3 < £4 < #5, either LA or WLA imply 73 = 04. Using the 77-axis, LA and 773 < 7/4 < ye < y$ < 774 imp 7s = (cs + 04 + GS + Ce)/4 but WLA would apply Equation (5) with 774 — 773 in th denominators and segments 774 — 7/5, 7.75 — y6, yQ — 7/4 and 7/4 — 773 in the numerator Furthermore in Figure 2, since the endpoint £5 lies to the right of #7, to preserve a partition of unity on [£4, £5] for WLA, the denominator in the weights for 74 should be xj — £4 instead of £5 — £4. Combinations of rotations and translations cannot always put {Aj}l ~ in a position to consider #(£,77) a function of £. Certain ameliorations can still make Equation (5) usable. For instance, use absolute values in the denominator in Equation (5) when £ J+ i < £,. Even when {Aj}^"1 can be treated as a function of d^iV), a value of 7j can be inappropriately influenced by a line segments in {Li}. In Figure 3, consider {Aj}^"1 positioned to treat g(£,rj) as a function of £. The vertical projections of the line segments L^ and Aj+i overlap, but the proximity of the pair (£j+2, ^+2) and (^+2,2/1+2) and the pair (£j+s,?7.j+3) and (0^+3,^+3),
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Figure 2. Projections onto the coordinate axes produce different values for 73. demark Li+z as exclusively corresponding to Aj+2 and implies it correspondence to Aj+2 and not A^+I. Clearly, local convexities in both {Aj}i ~ and {Li}l ~ exert a dominant influence on which line segments in {Li}l -1 should contribute to individual values of 7^. Parallel projections onto coordinate axes offer ease of computations by eliminating rigid transformations. Unfortunately, entire planar piecewise linear curves cannot always be repositioned for an appropriate parallel projection onto to a single line. To avoid inappropriate parallel projections such as the example in Figure 3, local convexities of {Aj}^"1 and {Li}^~1 allow localized and appropriate parallel projections onto lines other than coordinate axes. Grouping {Aj}^"1 into convex subsets of contiguous line segments where the segments in a subset enjoy convexity with respect to each other offers computational geometers a well defined problem to explore. Approximation properties of continuous functions with respect to these groupings have an old history dating from at least 70 years ago; variation diminishing methods investigated by Schoenberg [11] refer to Fejer's paper from 1933
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Figure 3. The projection of Li+2 onto the £-axis should not influence the calculation o/7j. [6].
In the more general case, normal projections, use projective directions that coincide with the directions normal to the segments Aj. The value of 7j depends only on the segments Li which intersect the perpendicular projectors of Aj. The normal projection approach has its own deficiencies. For example, in Figure 4 strips orthogonal to and as wide as the segments of {Aj}^ fail to account for the line segment Li. Switching the roles of {I/iJi -1 and {Ajjj -1 generally creates a worse situation because every 7j must be given a value and therefore no Aj can be left out. 3.2
Parameterizations
Although the projections in Section 3.1 retained nearly all the features of piecewise constant functions over the interval [a, 6], the geometry of piecewise linear planar curves introduced complications. Parameterizations of the planar curves can remove these geometric complications. Parameterizations have another advantage over projections; most of the formulas in Section 2 remain applicable with no essential changes. The domains of / and g however do force a notable exception in
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Figure 4. Using projections normal to the edges of g omits the contribution from Li. Equations (18) and (19). The parametric representations in Equations (18) and (19) become
In Computer Aided Design standard practices often approximate the arc length of a curve by accumulated chord length. As the term indicates, the accumulated chord length denotes an incremental summing the lengths of chords that approximate the curve. In the context of this paper the the line segments of a piecewise linear curve define chords . Starting from a vertex v on a piecewise linear curve C, the accumulated chordal distance from v to an arbitrary point p on a line segment of C equals the length of contiguous line segments of C that join the vertices p and v. Dividing any distance from v along C by the entire length of C creates a continuous linear parameterization of C on interval [0,1]. By starting at
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Figure 5. Parameterizations based on arclength can produce unexpected values for {7j}. the appropriate ends on the two piecewise linear curves upon which / and g depend and using their accumulated chord lengths, all the formulas in Section 2 apply with a = 0 and 6 = 1. Parameterizations with the accumulated chord length only consider the lengths and traversed direction of piecewise linear curves. The relative positions of / and g do not matter. In the top of Figure 5 / has 6 segments, Z/j, 1 < i < 6, of unit length, g has three segments, Aj, 1 < j < 3, and shares vertices x\ and Xj with f; #1 = £1 and x7 — £4. Let c\ = c<2 — —03 = —c4 = 1 and c5 = CQ = 0. Since A3 lies inside the union of L5 and L&, 7s should equal zero because 05 = CQ = 0. Yet with the accumulated chord parameterizations displayed in the bottom of Figure 5, neither LA nor WLA gives 73 the value of zero on AS .
3.3
Minimal Distances
The following two statements motivate the use of minimal distances. When a segment in {Li} contains a point q located at the minimal distance from p on g the value / at q should have the most influence on the value g assigns to p. Of equal merit but different in outcome, the value of / at a point q should influence the values of the points on g closest to q. To initiate the exploitation of minimal distances among points on / and points on g the following example emphasizes the
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Figure 6. Pairs of points vary according to the roles of f and g in calculating minimal distance.
important observations stated in the previous sentences. As illustrated in Figure 6, of all points on / the point q on / minimizes the distance from p on g to /. Of all points on g the same point p minimizes the distance from q, q 7^ q on / to g. Figure 6 thus illustrates an asymmetrical characteristic of minimal distances; namely, the point from which one calculates a minimal distance is crucial, q and q could easily have belonged to two different segments of /. Minimal distances possess the very important feature that tangency of circles rather than intersections of normal lines determine locations of points and a fortiori the values 7^. Circles produce the key consequence that a point, or points, on / of minimal distance to a point on
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Figure 7. Voronoi diagram of f and g. 3.4
Medial Projection
The medial projection method proposed in this section relies on Voronoi diagrams. The dual to Voronoi tessellations, Delaunay triangulations, provides the basis for one of the earliest algorithms that computes medial projections; see [12]. Although this projection method has weaknesses similar to those possessed by the parallel projections and minimizations presented in the preceding two subsections, the medial method has the very desirable feature that it treats the proximity of points with equal priority to / and g. The Voronoi diagram of linear segments defines a set of of points equidistant from at least two segments; that is, a point on the Voronoi diagram lies at the center of a circle tangent to at least two segments. Tangency at an endpoint shared by two segments counts twice; i.e. tangency occurs at each segment. The tangencies of a circle also identify corresponding segments. Projections along radii and normal to the points of tangency meet (perpendicular radial intersection) at the center of the tangent circle and thereby treat / and g equally. In Figure 7 gray segments denote the Voronoi diagram of two piecewise linear planar curves /, with solid vertices, and p, with hollow vertices. The thinner gray segments denote angle bisectors bisecting the angles formed by intersections from adjacent segments in / and in g. To determine values 7^ the medial projection uses the Voronoi diagram except for the angle bisectors bisecting the angles formed by two contiguous segments. The thicker gray segments enclosed by / and g in the Voronoi diagram define the medial projection of / and g. In general the Voronoi diagram of / and g contains parabolic segments and straight line segments. The medial projection captures a one dimensional subset in the intersections of normal lines projecting from the line segments in / and g. In this sense the medial projection represents a variant of the projection methods discussed in Section 3.2. The tangency of circles characterizing Voronoi diagrams implies that from
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Figure 8. The incenter of a triangle can be used to construct the medial axis. each point on the medial projection at least one line radiates and intersects perpendicularly with /, a perpendicular radial intersection, and at least one line radiates and intersects perpendicularly with g. The intersection of the radius of a circle tangent at the common endpoint of consecutive segments of / or of g counts as a perpendicular radial intersection. A single or many points on the medial projection can correspond to a single point on / or on g. Parabolic segments of the medial projection indicate many-to-one correspondences with either / or g. Some portions of the medial projection have well known geometric properties that simplify their calculation. For instance, in Figure 8, the two angle bisectors in the medial axis intersect at the incenter of the triangle. The geometry of / and g implies that the medial projection contains only segments of parabolas and segments of angle bisectors. These segments, as subsets of medial axes and Voronoi diagrams, have been extensively studied; see [3, 7, 8, 9, 13] for computational algorithms. Even though the medial projection has no preference for / or
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contrast to this well behaved segment Aj, a radially deficient subsegment O^' in & segment Aj/, i.e. a subsegment of p's where rg(p) cannot be defined, can inherit a constant value in many ways. For example, since the medial axis associates a constant value of h with each of the two nonradially deficient subsegments adjacent to fife', assign Aj/ the average the two values. Alternatively, each p on a radially deficient subsegment of Aj/ has at least one point q on / which minimizes the distance of p to /. The set of p's on this radially deficient subsegment generates a finite number of subsubsegments upon which for all p in a subsegment the value of /(q), q minimally distant from p, remains constant. Apply LA or WLA to calculate 7j andij/. A third type of segment Aj. contains one or more points p* where many points p on the medial axis correspond to p*; i.e. p £ r~ 1 (p*). The points p* can be omitted when applying LA or WLA on nonradially deficient subsegments of Aj* because Aj* can have only a finite number of points p*. The null length of a point p* legitamizes the omission of the p*'s in evaluating LA or WLA on Aj*. Contrariwise due to the line segments that define the domains of / and g, r~ 1 (p*) can be partitioned into a finite number of subsections where all the points in a subsection corresponds to the same value of /. The lengths of the subsections in r~l (p*) then support the claim that the p*'s should be factored into an evaluation of LA or WLA on Aj.. Use LA or WLA on r~l(p*) to assign a value to r~1(p*). This value, or a value weighted by the length of r~ 1 (p*) appropriately normalized with respect to the lengths of constant valued r^ (p'), p' contained in the nonradially deficient subsegments of Aj., with a second application of LA or WLA calculates 7 j*. 3.5
Remarks On Methods For Planar Piecewise Linear Curves
By parameterizing planar piecewise linear curves on the same interval, say [0,1], with partitions {ti} and {TJ} replacing {zj} and {£j}, the formulas in Section 2 can be evaluated mutatis mutandis. On the other hand comparable formulas for other extensions of the elemental case for piecewise constant approximations on an interval [a, b] require prudent adjustments. In Equations (6) and (7) the expressions for errors in approximation contain only the values Ci and 7j and depend on projections onto intervals {[£j, £j+i]}i ~ 1 In this section / and g have different planar domains; namely {Lj}^1 and {Aj}^~l respectively. Therefore, to extend Equation (3), the definition of Xj used in Equations (6) and (7), let IIj denote an arbitrary parallel projection from Aj into Lj.
defines the planar equivalent to Xj. Substituting projections from Li into Aj instead does not change Hj. In general the union of the Hj's will not contain every c^. Whereas in Section 2 maximizing over the jt/l-indexed intervals [£j,£j + i), 1 < j < M — 1, or interchanging the roles of the intervals [xi,Xi+i) and [£j,£j+i) and maximizing over the it/l-indexed intervals [o^iCi+i], 1 < i < N — 1, made sense, the indices 1 < j < M — 1 and projections IIj(Aj) must be used with Ej.
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To use Equations (6) and (7) for the minimization or medial projection methods requires a substantially different definition in Equation (24). With the notation || • || declaring the Euclidean norm, for a segment Aj define tyj, an analog to Hj, by
With the critical assumption that g has no radially deficient points, define a corresponding analog for medial projections by
Beyond the possibility of missing Q'S in tyj and
j may be empty. The last paragraph in Section 3.4 addressed this issue. Whereas Equation (8) relies on integration over the common domain of / and g, Equation (9) remains applicable with substitutions Ej or fyj for Xj. Having to use the Euclidean norm in Equation (9) produces
where Vj = (£j,ijj)- To calculate the weights in Equation (27) the denominators in Equation (5) become the lengths of Aj and the numerators become the length of Aj's subintervals whose points p and h(r~l(p)) yield the same constant contained in the set {Q}. Suppose / and g approximate an unknown curve. Substitute the medial projection to obtain an unbiased surrogate curve. A geometric component can be distilled from the approximations by using this surrogate curve in an expression based on the first integral in Equation (20). Assume that g does not contain any radially deficient points and consider
In Equation (28) the first term inside the integral emphasizes the geometries and locations of / and g and the second term, though influenced by the relative locations of / and g, emphasizes the constants values of / and g. The interpretation of Equations (12) and (13) ) as conservation of area motivates restating Equations (12) and (13) ) as
where
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4
Spatial Piecewise Linear Curves
Extending parameterizations from planar to spatial piecewise linear curves retains the benefits, liabilities and formulations from Sections 2 and 3.2 with minor adjustments for three dimensional coordinates. The parallel projection methods must now use parallel planes instead of parallel lines. The formulation of minimizations in Section 3.3 remains intact by replacing circles with spheres. In general the computational burden of three dimensional variants of two dimensional algorithms increases with the increase in dimension. However, extending medial projections requires sophisticated alterations. For example, for planar piecewise linear curves / and g each point on the medial projection is the center of a unique circle tangent to / and to g. Among the points equidistant from a pair of points, q on / and p on g, at most one point, if one exists, lies on the medial projection if and only if the point centers a unique circle tangent to / and g at q and p respectively intersecting / and g nowhere else. In three dimensions for a pair of points, q on / and p on 5, there may exist infinitely many spheres tangent to / and g at q and p and intersecting / and g nowhere else. In three dimensions uniqueness corresponds to the tangent sphere of minimal radius. Whereas determining the medial projection in the plane consists of calculating linear and parabolic segments, the corresponding segments in three dimensional require significantly more arduous calculations.
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Bibliography [1] P. J. BESL AND N. D. McKAY, A method for registration of 3D shapes, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), pp. 239-256. [2] C. DE BOOR, On calculating with B-splines, Journal of Approximation Theory, Vol. 6, (1972), pp. 50-62. [3] F. CHIN, J. SNOEYINK AND C. A. WANG, Finding the medial axis of a simple polygon in linear time, Vol. 1004 Lecture Notes on Computer Science, Springer Verlag, (1995), pp. 382-391. [4] D. W. EGGERT, A. W. FITZGIBBON AND R. B. FISHER, Simultaneous registration of multiple range views for use in reverse engineering for CAD models, Computer Vision and Image Understanding, 69 (1998), pp. 253-272. [5] D. W. EGGERT, A. LARUSSO AND R. B. FISHER, Estimating 3-D rigid body transformations: a comparison for four major algorithms, Machine Vision and Applications, 9 (1997), pp. 272-290. [6] L. FEJER, Gestaltliches iiber die Partialsummen und ihre Mittelwerte bei der Fourierreihe und der Potenzreihe, Zeitschrift fur Angewandte Mathematik und Mechanik, 13 (1933), pp. 80-88. [7] M. HELD, Voronoi diagrams and offset curves in curvilinear polygons, Computer-Aided Design, 30 (1998), pp. 287-300. [8] M. HELD, VRONI: an engineering approach to the reliable and efficient computation of Voronoi diagrams of points and line segments, Computational Geometry: Theory and Applications, 18 (2001), pp. 95-123. [9] D. L. LEE, Medial axis transformation of a planar shape, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-4 (1982), pp. 363-369. [10] A. M. MOOD, F. A. GRAYBILL AND D. C. BOES, Introduction to the Theory of Statistics, McGraw-Hill, 1974. [11] I. J. SCHOENBERG, On variation diminishing approximation methods, On Numerical Approximation, R. Langer, Ed., Univ. of Wisconsin Press, (1959), pp. 249-274.
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[12] T. K. H. TAM AND C. G. ARMSTRONG, 2d-finite element mesh generation by medial axis subdivision. Advances in Engineering Software and Workstations, 13 (1991), pp. 313-324. [13] C. YAO AND J. ROKNE, A straightforward algorithm for computing the medial axis of a simple polygon International Journal for Computer Mathematics, 39 (1991), pp. 51-60.
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Bivariate Quadratic B-splines Used as Basis Functions for Collocation Benjamin Dembart* 9 Daniel Gonsor*, and Marian Neamtu^ Abstract Recently a new approach to multivariate (non-tensor product) splines has been introduced which defines a spline space with several attractive properties, including optimal smoothness and high order of approximation. In this paper, we investigate several important components of the application of this theory to the solution of practical problems. In particular, we look at the construction of spline spaces, discuss boundary conditions over a finite domain, and examine the B-spline collocation matrix (giving numerical evidence that it exhibits excellent spectral properties).
1 Introduction Classical univariate spline functions have a well-established and deserved place in the area of data fitting, geometric modeling of curves, and numerical solution of ordinary differential equations. Bivariate splines (i.e., piecewise polynomials over partitions of planar domains) have also found many applications. In particular, the finite element method and geometric modeling of surfaces in computer aided geometric design have traditionally made use of bivariate splines (see [14] and [2]). In most of these applications, piecewise polynomials of low degree (piecewise constant *The Boeing Math Group, The Boeing Company, P. O. Box 3707, M/S 7L-21, Seattle, WA 98124-2207, ben.dembartQboeing.com. tThe Boeing Math Group, The Boeing Company, P. O. Box 3707, M/S 7L-21, Seattle, WA 98124-2207, daniel. e. gonsorOboeing. com. * Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, neamtuQmath. vanderbilt. edu. Supported by NSF under grant CCF0204174 and The Boeing Company.
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and piecewise linear) are preferred due to their simplicity and robustness. However, for certain applications higher order polynomials are necessary or advantageous, in general because smoothness is required (e.g., conforming finite elements or geometric design, where smoothness can be critical) or because higher order polynomials enable better approximation and, as a result, make it possible to dramatically reduce the number of elements in a partition (such as the number of triangles in a triangulation). Degree two is a natural starting point since it gives Cl smoothness, which is sufficient for many applications, while keeping the increased complexity to a minimum. Still, the step from degree one to two is highly nontrivial and has been much studied, resulting in numerous strategies for constructing smooth piecewise polynomials. Beside the classical tensor product splines, w^hich are usually ill-suited for working on non-rectangular domains and for adaptive meshes, most current methods are based on piecewise polynomials on triangulations of the domain. Unfortunately, for low degrees (2-4), requiring at least Cl continuity is quite prohibitive in that the resulting splines are too rigid and additionally may suffer from low approximation quality. For example, despite much effort, precise results on approximation properties of Cl quadratic, cubic, and quartic piecewise polynomials on arbitrary triangulations are lacking, making it unlikely that they can become useful in applications. To circumvent these difficulties, one usually resorts to either higher degree polynomials (such as the well-known quintic element [17]) or to piecewise polynomials on special triangulations, obtained by refining a given triangulation by dividing each triangle into subtriangles. The well-known quadratic Powell-Sabin [13] and cubic Clough-Tocher [1] elements are classical examples of such spline spaces. An alternative to the above so-called macro-element schemes that has been largely ignored by the applied mathematics and engineering community, is based on the concept of a bivariate B-spline. In this approach, one defines a set of compactly supported smooth functions, called B-splines, and uses them to construct a spline space. One construction in particular [10] closely parallels traditional spline theory in one variable. It leads to a space of spline functions which are, in general, optimally smooth (i.e.. splines of degree n are of class Cn~1} and have optimal approximation properties. These attractive theoretical properties invite the question of whether the Bspline approach can be a useful in practical applications. In this paper we present some first steps in addressing this query. In particular, we look at the construction of spline spaces, discuss boundary conditions over a finite domain, and examine the B-spline collocation matrix (giving numerical evidence that it exhibits excellent spectral properties). The paper is organized as follows. In the next section we describe the motivation for this work, and give some background material. In Section 3, the bivariate B-spline is defined in terms of Delaunay configurations, and in Section 4 we discuss efficient computation of Delaunay configurations. Boundary conditions and collocation sites are covered in the subsequent two sections. In the final section, we present some experimental results.
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2
Background
The application that initiated this investigation was the solution of integral equations of the second kind arising from electromagnetic scattering problems. Such equations can be expressed in general form as
where fi is a subset of the plane and / is unknown. Typically, / is a two- or three-component vector, and k is a matrix. If the solution is expressed as a linear combination of spline basis functions, then ideally the basis would possess the following properties: (1) (2) (3) (4) (5)
Low polynomial degree Local support Stability Good approximation order C1 (or higher) smoothness
Properties (l)-(3) are useful in order to keep the representation of the solution and subsequent calculations as simple and as numerically stable as possible. Property (4) means that for sufficiently smooth solutions, we expect O(hn+l] convergence of the approximant as the partition of the domain is appropriately refined. Here, n denotes the degree of the spline and h is the mesh size of the partition, which in the B-spline case roughly means the size of the largest B-spline support. This convergence property allows for a relatively coarser partition of the domain 17. As for (5), a smooth solution is often necessitated by certain applications. In particular, a smooth solution allows one to handle the not uncommon case where the kernel K is highly singular. In these situations, integration by parts is used to deal with the kernel's singularity, and continuity conditions are used to eliminate boundary terms. A precursor to this investigation is the technical report [18], where the authors considered the Powell-Sabin basis — that is the basis defined on the Powell-Sabin split [13] of a given triangulation by interpolating positional and first partial derivative values at the vertices. The resulting basis functions are Cl piecewise quadratic, with support consisting of collections of triangles surrounding a vertex. The report [18] is mainly concerned with the spectral properties of the collocation matrix formed by the Powell-Sabin basis, since this essentially determines the reliability and effectiveness of the basis in the solution of equation (1) with an iterative solver. The authors of that report discovered that the conditioning of the basis collocation matrix is much more dependent on the choice of derivative values at the vertices than on the choice of collocation points. In fact, the authors give experimental evidence that for appropriate scaling of derivative values, the conditioning of the basis collocation matrix is very good. However, there are several outstanding issues. First, although the authors supply a method for scaling the derivatives based on the local shape of the triangulation, there is no guarantee that this will be successful
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for arbitrary triangulations. For example, it is not clear how sensitive the conditioning is to local smallest angle in the triangulation. Another issue is that even for the successful scaling values, three or more collocation points are required for each macro triangle, thus increasing the complexity of the problem, especially for triangulations with large numbers of triangles. Finally, an issue related to conditioning is the number of basis functions that overlap at an arbitrary point in the domain. It is known that an excessive number of overlaps usually results in poor conditioning of the basis collocation matrix. In the case of the Powell-Sabin basis, this turns out not to be a problem since the number of overlaps for a given so-called micro triangle is 9 — just a bit more than the ideal value of 6 (the dimension of the bivariate quadratic polynomials). Our numerical experiments suggest that this issue is also not a concern for the quadratic B-spline. Thus, when using these functions for collocation, the resulting collocation matrices are nearly optimally sparse.
3
Quadratic B-splines and Delaunay Configurations
The two main ingredients in the definition of bivariate B-splines are simplex splines and Delaunay configurations, which we now briefly review. We will restrict ourselves to the quadratic case and refer the reader to [7] for a detailed account of the theory of simplex splines. A bivariate simplex spline is a piecewise polynomial function defined by a set of points in R 2 , referred to as knots (correspondingly, knotset refers to a collection of knots). A simplex spline of degree zero is defined by a collection of three noncollinear knots, xl.x2.x3, say, and is simply a normalized characteristic function of the triangle whose vertices are the knots. More specifically, if we denote by M ( - I x 1 . x2.x3) the constant (i.e., degree 0) simplex spline with knots {x1.^2.^3}, then
where [...] denotes the convex hull of a set of points and int [...] denotes the interior. For the sake of accuracy and numerical stability, it is important to appropriately define the constant simplex spline on the boundary of [x1, x'2.x3}. See [16] for one such definition. Simplex splines of degree greater than zero can now be defined recursively by means of the Micchelli recurrence relation [6]. This recursion expresses a simplex spline in terms of simplex splines of one degree lower. Thus, to evaluate a quadratic simplex spline, one first expresses it as a combination of linear simplex splines, which in turn are evaluated as combinations of constant simplex splines. If a set of knots is denoted by X, and its cardinality by #-X", then the recurrence relation for a simplex spline of degree n = #X — 3 has the form
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Figure 1. Quadratic simplex splines of varying orders of continuity.
where the numbers A;, which depend on x, satisfy the system of equations
This recurrence relation can be implemented in a numerically satisfactory manner, see [4] and [9] for more information. A function M(-\X) defined in the above manner is clearly zero outside the convex hull of the knots X. Moreover, if the knots are in general position (i.e., no three knots are collinear) then the simplex spline is optimally smooth. In particular, a linear simplex spline (defined by four knots) is continuous (C°) and a quadratic simplex spline (defined by five knots) is continuously differentlable (C1). If the knots are not in general position, the order of continuity is decreased, which resembles the univariate situation of coalescent knots. The possibility of modeling functions with discontinuities in positional and/or derivative values is important, as we shall see in Section 5, for imposing boundary conditions. Figure 1 depicts several quadratic simplex splines with varying orders of continuity along with corresponding knotsets. The second ingredient in the construction of a bivariate B-spline is the Delaunay configuration [10]. A Delaunay configuration is a collection of knotsets defined as follows. Given a positive integer n and a discrete set K of knots in R2 in generic position (i.e., no four knots are co-circular and no three are collinear), the set An of Delaunay configurations of degree n is the family of all pairs (B.I),B.I C K,#B — 3.#/ = n, such that the circle circumscribed to the three knots in B contains the knots / and no other knots from K. The knots / will be referred to as the interior knots of the configuration (B, I) and the knots B will be called boundary knots. We will denote by J the set of all interior knotsets, i.e., the set of all I C K, #1 — n, such that (B. I) € A n , for some B. In the case n = 2, each configuration (B, I) e A2 has #(B(JI) = 5 and hence the knots B U / determine a quadratic simplex spline. We now define a quadratic
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B-spline as follows [10]. Let / € J and let B(I) := {B\(B,I} e A2}, that is, the set of all boundary knotsets whose circumcircles contain the same fixed set / of interior knots of size #/ = 2. The quadratic B-spline, BI, associated with / is the function
Note that the normalization in (4) is chosen so that the collection of B-splines forms a partition of unity when [K] — R 2 (i.e., the convex hull of K is all of R 2 ), that is
If #K is finite, which will be the case in most applications, it can be shown that the B-splines still sum to one in a well-defined subset of [K]. The collection of B-splines, {Bi,I G X}, turns out to be a basis for a spline space that satisfies the requirements listed in Section 2. In particular, the B-splines have compact support since each is the sum of a finite number of simplex splines. As we shall see in Section 7, numerical experiments suggest that the B-splines are also relatively well localized, i.e., relatively few B-splines are nonzero at a particular point in the domain. In addition, under mild conditions on the geometry of the knots, similar to those in classical finite element analysis, the B-splines form a stable basis of the spline space
in the sense that for 1 < p < oc,
for some constants B > A > 0 [12]. Moreover, £2 has optimal approximation order [12], which is a direct consequence of the fact that £2 contains all bivariate quadratic polynomials [11]. As mentioned previously, for knots in general position, quadratic simplex splines are Cl and hence so are the associated quadratic B-splines. We will address the case of arbitrary knot positions in Section 5. When working with spline functions it is customary to associate the spline coefficients in a B-spline expansion with physical control points that help capture the geometry of a spline curve or surface. To understand how this can be done for the quadratic splines considered here, we first need to recall the notion of a Greville site. In univariate spline function theory, certain knot-averages, called Greville sites, play an important role. Here we define a bivariate analog of Greville sites. Such sites also generalize the familiar Bezier sites arising in the theory of Bezier polynomials defined on triangles. A Greville site £/, corresponding to a quadratic B-spline BI with interior knots / E J, is simply the average of those interior knots, that is, for / — [u. v} c K,
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Thus, in the quadratic case, the Greville site is the midpoint of the edge connecting the two interior knots, which is a direct generalization of the univariate situation. Greville sites have several geometric properties related to the spline space <$2In particular, there is a one-to-one correspondence between the Greville sites and the B-splines (and hence also the associated B-spline coefficients i.e., the coefficients in the representation of a spline as a linear combination of the B-splines). A plot of these points (see Section 7) gives a rough idea of the location of the B-splines in the given domain since each site £/ is typically strictly inside the support of BI, the only exception being special situations arising from imposing boundary conditions (to be discussed in Section 5). Second, associating each B-spline coefficient c/ e R in the expansion
with the Greville site £/, one obtains the spline control points Cj :— (£/,c/) e R3. / 6 J, with analogous properties as the classical control points in one variable and also the bivariate tensor product setting. For example, if these control points are coplanar then the spline surface itself is planar. This follows from the result [10] that if [K] = R 2 , then
for all linear functions / and all x £ R 2 . The above equation also holds true in the case where K is finite, i.e., for bounded domains [K], provided that the knots on the boundary of [K] have been appropriately placed.
4
Computation of Delaunay Configurations
From the previous section, we see that the construction of B-splines requires calculating degree-n Delaunay configurations of the knotset K. Thus, an efficient algorithm for computing Delaunay configurations is one requirement for the practical application of multivariate B-splines. The brute force approach is to search through all triples from K, then choose those containing precisely n knots inside their circumcircle. Clearly, this approach would be prohibitively expensive for even moderately sized sets K, since the complexity of such an algorithm is O(N3), where N = #K. A more efficient algorithm for computing Delaunay configurations has been proposed in [8], which achieves the (optimal) complexity O(NlogN). Here, we give a sketch of the algorithm for the quadratic case n = 2. The reader is referred to [8] for the general case and for proofs. The algorithm is recursive in nature and builds the configurations of a given degree using information about configurations of one degree lower, beginning with degree zero (i.e., the Delaunay triangulation of K) which is well understood from an algorithmic point of view. The algorithm has three main steps. Step 1. Compute the Delaunay triangulation AQ of K. There are several efficient algorithms for performing this task, see e.g., the survey [15]. It is well known that
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Figure 2. A cell and the corresponding constrained Delaunay triangulation.
the optimal complexity of this calculation is O(NlogN). Step 2. Compute the Delaunay triangulation of each cell of AQ. A cell of AQ is the planar region consisting of the union of triangles sharing a vertex (i.e., a knot) of AQ. Thus, a cell is a polygonal domain, not necessarily convex. By triangulating a cell, we mean computing the Delaunay triangulation of the knots defining the boundary of the cell. In fact, this is a constrained Delaunay triangulation if the cell is nonconvex since the triangulation is required to preserve the boundary of the cell. Figure 2 gives an example of a cell and its corresponding Delaunay triangulation. On the left is the cell denned by knot 1, and on the right is the Delaunay triangulation, consisting of triangles AC := {{2,6. 7}. {2. 5,6}. {2.3. 5}, {3. 4. 5}}. From our example, one can see that the Delaunay triangulation of the cell gives the set of all boundary knotsets B for the interior knot / = {!}. In particular, we have Ac1 = &(!)• This implies that the set of pairs ({2,6, 7}, {!}), ({2,5,6}, {!}), ({2,3,5}, {!}), ({3,4,5}, {!}) belongs to AI, and in fact that these are all possible configurations whose interior knot is 1. We can proceed vertex by vertex to complete the construction of AI. Therefore, we have reduced the computation of AI to calculating a series of local, constrained Delaunay triangulations. It can be shown that the complexity of this entire computation is O(N) i.e., linear in N, and hence even better than the cost of computing AQ. Step 3. Compute the Delaunay triangulation of each cell o/Ai. This last step is virtually identical with Step 2, except that a cell is now a combination of degree-1 configurations, see [8]. Hence, the complexity of this step is still linear in N. We end this section by pointing out that the above algorithm is also advantageous in that it allows us to relax some of the assumptions we made concerning Delaunay configurations. First, it is possible to handle effectively cases where the knots are not in generic position. In particular, if the knots are all distinct but such that there are more than three cocircular knots, then it is known that the Delaunay
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triangulation of such a knotset is not uniquely defined. This is because there may be several triangulations that minimize the smallest interior angle of the triangulation, which, in the generic case, is equivalent to saying that the triangulation is Delaunay. Thus, in the nongeneric case we can simply use any one of these different triangulations. The same applies to the Delaunay triangulations of the cells in Steps 2 and 3 of the algorithm. Moreover, a similar approach can be used if there are multiple-fold knots. Second, it is sometimes too restrictive to require that all collections of knots of size five, used to define quadratic simplex splines and consequently B-splines, be associated to Delaunay configurations as we defined them in Section 3. This is often the case even for triangulations i.e.. for Delaunay configurations of the lowest degree. For example, when we desire to triangulate a set of points in a nonconvex region, a Delaunay triangulation of such a set of points will not necessarily match the triangulated region since the edges of the boundary of this region will in general not coincide with the edges of the Delaunay triangulation. Hence, for this reason it may be preferred to consider a constrained Delaunay triangulation, which will preserve the boundary edges. A similar situation occurs for Delaunay configurations of higher degrees, for example when we wish to construct a quadratic spline space over a nonconvex region fi. The algorithm for obtaining Delaunay configurations can be used in the same manner as before, except that we must start it off with a constrained Delaunay triangulation of the nonconvex 0, and then continue with constrained local Delaunay triangulations in Steps 2 and 3 of the algorithm.
5
Boundary Conditions
Let O be a closed bounded domain in R 2 , which is the domain of definition of some approximation problem (e.g., the problem of approximating the solution of an integral equation or fitting a function from discrete data associated with points in 17). To define an approximation space <S-2, we first need to select a set of knots K. Obviously, the case where K is infinite is of theoretical interest only. In practice, we want to select a finite set K of knots. However, we cannot limit ourselves to knotsets K C 0, which are in general position, because in that case each spline s e S-2 and its first partial derivatives will generally vanish on <9Q, the boundary of fi. This is a consequence of the fact that for knots in generic position the quadratic B-splines are globally C1. Hence, if their support is a subset of 0, they cannot be used to approximate functions that are nonzero on <9Q. The most straightforward approach to handling this difficulty is to include points exterior to 17 in the knotset, thus ensuring the existence of B-splines not identically zero on d£l. Note that the spline space generated by such a knotset, restricted to fi, will still contain the quadratic polynomials, and also have the partition of unity property on 0. Beside the obvious inelegance of such an approach, there are also a number of practical difficulties. For example, how to choose an appropriate number and location of knots outside 0 is not obvious. Also, we need to place the knots so that the support of each B-spline overlaps 0 adequately, otherwise the B-spline basis could be poorly conditioned, see [5] for a similar phenomenon in
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the case of tensor-product B-splines. Another consideration is that knots exterior to 0 can lead to Greville sites that are also exterior to £1, which eliminates them as potential collocation points, see next section. A better alternative is to require that K C 0 but allow multiple and collinear knots along the boundary, similar to the imivariate case. First observe that a perturbation argument can be used to define a spline space corresponding to coalescent or collinear knots. Namely, by perturbing the knots such that they are in generic position, constructing the corresponding B-splines, and letting the size of the perturbation go to zero, will, in the limit, give rise to a collection of B-splines that inherit the properties of B-splines defined for generic knots. Namely, such limiting B-splines will still sum to one and their span will contain all quadratic polynomials. This follows by the fact that the limit of a B-spline is still a B-spline. More precisely, if {Xk} is a sequence of knotsets of a given fixed size, converging to X, then the limit of the simplex splines M ( - \ X k ) , in the distributional sense, is the simplex spline Af(-\X). If the support of A-1(-\X] is nondegenerate, then the convergence is in fact also pointwise at all points of continuity of M(-\X). Since a B-spline is a sum of simplex splines, it follows that the perturbed B-splines also have a well-defined limit. It should be noted that the space obtained using the above argument will depend on the perturbation in that two different perturbations, no matter how small, will in general yield different Delaunay configurations and hence also different sets of B-splines, spanning different spline spaces. However, this should not come as a surprise since the same phenomenon occurs also in the case of constant and linear splines. For example, if the knots form a regular lattice, e.g., if they are a subset of the integer lattice Z2, there are many different triangulations, all obtained as a limit of the Delaunay triangulation of a perturbed knotset. Each such triangulation gives rise to a different spline space of piecewise constant splines. Example 1. Let 0 be the unit square in R 2 . Figure 3 (left) shows a simple layout of knots in Q. Note that the corner knots are triple knots, hence we use three numbers to label them. For simplicity, we only selected one simple knot inside of fJ. Although none of the knots lie outside of Q, they still give rise to an appropriate spline space. In particular, this space contains functions that are nonzero along the boundary. To obtain a spline space corresponding to these knots, consider the perturbation depicted in Figure 3 (right). Thus, the three-fold knot 1-3 is "pulled apart" into three separate knots numbered 1.2.3 in the figure. The edges in the figure contain information about the Delaunay configuration of the perturbed knotset. In particular, if two knots are connected by an edge, this means that there is a Delaunay configuration of degree two, whose set of interior knots are these two knots. The black dots in the figure depict the Greville sites corresponding to the B-splines associated with these configurations. For example the knots 3 and 13 are connected by an edge, since they are the interior knots of the configurations (which cannot be read off Figure 3): ({1,2,7}, {3,13}), ({2,4,5}, {3,13}), ({2, 5,10}, {3,13}), ({2,7,9}, {3,13}), ({2,9.10}, {3,13}). Thus, the B-spline corresponding to the set / = {3.13} is the sum of five normalized simplex splines corresponding to the above five configurations. The Greville site associated with this B-spline (i.e., the average of knots 3
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Figure 3. Multiple knots and their perturbation.
Figure 4. Multiple knots and their perturbation.
and 13) is numbered 5 (in smaller font) in the figure. The support of the B-spline is the union of the five triangles {1, 2.7}, {2,4, 5}, {2,5,10}, {2. 7,9}, {2,9,10}. Thus, in the limiting case, i.e., when the size of the perturbation is zero, the support of this B-spline coincides with the domain 0. A more serious issue than the issue of the dependence of the limiting spline space on the perturbation is the possibility that the limiting spline space will contain functions that are discontinuous in the interior of O. Example 2. Consider the case of linear splines corresponding to the knots in Figure 4 (left). For simplicity, we chose a single double knot in the lower left corner of 0. For the perturbation shown in Figure 4 (right), there are seven Delaunay configurations of degree one: ({2,4,6}, {!}), ({1,3,6}, {2}), ({1,5,6}, {4}), ({3,4,6}, {5}), ({1,2,4}, {6}), ({2,3,5}, {6}), ({2,4,5}, {6}). Thus, there are five different linear
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Figure 5. The B-splines B\, B^, and their sum B[.
B-splines spanning the corresponding spline space, whose Greville sites are identical to the knots 1.2.4. 5.6, shown as black dots in Figure 4 (right). It can be seen that as the perturbed knots 1 and 2 merge, the B-splines corresponding to the Greville sites 1 and 2 (i.e., in this case, just the simplex splines with knots (2.4.6.1} and {1.3.6.2}) become discontinuous along the edge connecting the double knot 1-2 with knot 6 (since three collinear knots in a linear simplex spline produces a discontinuity). A consequence of the knot-multiplicity is that the (desired) discontinuity of the spline across the boundary 50 at its lower-left corner propagates inside 0. Such a discontinuity will be present in the spline as long as the B-spline coefficients of the above two B-splines are different. Thus, to guarantee that our spline space contains continuous functions only, in the presence of knot-multiplicities, we must identify the B-splines whose Greville sites are identical. That is, instead of considering these B-splines as separate functions, we should consider their sum and view this sum as a single B-spline, whose Greville site is the double knot 1-2. The resulting (adjusted) spline space corresponding to the given knots will then be, in this example, the span of four B-splines, all of which are continuous in 0, namely the B-splines whose Greville sites are 1-2, 4, 5, and 6. The idea explained in the previous example translates directly to the quadratic case. Example 3. (Continuation of Example 1) Consider again the case of the triple knot 1-3. The value of a quadratic spline in the considered spline space at that knot depends only on the B-spline coefficients, say GI and 03, associated with the Greville sites 1 and 3 (see Figure 3), since all other B-splines vanish there. However, both B-splines associated with these coefficients, shown in Figure 5 (left), say B\ and £3, are discontinuous along the edge connecting the considered triple knot with knot 13 (note that in Figure 5 the peaks correspond to the values of the functions at knots 1-2). However, one can show that if c\ = 03, then the discontinuity of the spline c\B\ + C3J3s along the mentioned edge cancels out. Thus, as in Example 2, we must replace the B-splines B\. B% with a single new B-spline, B{ say, which is the sum of
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Figure 6. New Greville sites and C1 continuity conditions.
the two discontinuous B-splines, B( — Bi+By,. The Greville site for B( will be the same as those of B\ and £3 since, in the limiting case, they are identical. We can do a similar ''consolidation" of B-splines for all other cases associated with multiple knots. For example, from Figure 3 we see that we must identify the Greville sites 13 and 17; 12 and 14; 15 and 16; 7 and 10; 6 and 8; 18, 19, and 16; 9 and 11. Figure 6 (left) shows the resulting new Greville sites. Note that the described procedure guarantees that the spline space S-2 i.e., the space spanned by the newly defined B-splines, gives rise to the usual univariate quadratic splines when restricted to the boundary edges. In our particular example, since there are no knots inside any of the four boundary edges, the functions along these edges are just ordinary quadratic polynomials. In fact, the restriction of each of those B-splines that do not vanish identically along an edge to that edge is just a univariate Bernstein polynomial, and hence the boundary Greville sites can also be viewed as the well-known Bezier sites. The above-described procedure can be implemented very efficiently. The algorithm for determining Delaunay configurations, discussed in the previous section, can be easily modified to handle multiple (and collinear) knots. As a result, the actual algorithm does not use knot-perturbations, it merely treats double or triple knots as separate when it comes to their labels, but as identical as physical points in space. As said, the process gives rise to a spline space of functions continuous inside the domain fi. However, knot-multiplicities can, and in general will, also lead to discontinuities in the first derivatives in 0. This is again a consequence of the fact that some of the boundary B-splines, which are designed to be discontinuous across the boundary of O, are C1-discontinuous inside 0. To cope with this phenomenon, the key is not to allow arbitrary combinations of such defective B-splines, but only special combinations that will not compromise the smoothness of the spline. We illustrate the idea with an example.
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Figure 7. B-splines associated with Greville sites B. C. D.
Example 4. Consider again the setup in Example 3. Figure 5 (right) and Figure 7 shows plots of the B-splines corresponding to the Greville sites in Figure 6 (right), labeled A, B, C, D (as is the case in Figure 5, the top corner in the figure corresponds to the point A). The plots illustrate that the B-splines are not C1, but only C°continuous along the edge AC. This means that an arbitrary linear combination of such B-splines will also suffer from this defect. Thus, we need to restrict our choice of linear combinations to those that preserve the C1-continuity of the spline. To explain this, we will take advantage of a geometric interpretation of the spline coefficients as control points, mentioned in Section 3. This interpretation is similar to the well-known Bezier points of a polynomial in that the control points can be used to formulate continuity conditions of the spline. Consider the Greville sites A, B. C, D, denote the corresponding B-splines as BA, BB-, BC, BD, and the associated B-spline coefficients by CA-CB-CC-CD, respectively. As mentioned, the linear combination
is in general not C1 along the edge AC. It turns out that the condition that guarantees that a spline is C1 along this edge is that the control points, i.e., the points (A.CA);(B.CB};(C.CC};(D.CD} e M3, are coplanar. This is schematically depicted in Figure 6: the shaded region adjacent to site A expresses the condition that the four corresponding control points should be coplanar. In this example, there are four such coplanarity conditions needed for the resulting spline (i.e., a linear combination of all twelve B-splines) to be C1 in the interior of Q. For a general polygonal domain Q, the number of coplanarity conditions depends on the number of multiple knots, which typically depends on the number of corners of Q.
6
Collocation at Greville Sites
In previous sections we were concerned with the construction of a spline space over a bounded domain with a number of attractive properties. Functions in this space are generally smooth piecewise quadratics and can be expressed as linear combinations
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of compactly supported basis functions. These and other useful theoretical properties make these splines a prime candidate as tools for solving various approximation problems, ranging from data fitting to solving partial differential and integral equations. In this section we will consider a particular type of approximation problem, interpolation of data, with an eye on potential application in the area of solving integral equations of the second kind by collocation. In particular, when solving the integral equation (1) by collocation, one attempts to solve the discretized problem
where E C 0 is a finite set of the so-called collocation points. In this setting we seek a solution which is a quadratic spline s E tS2, corresponding to a given set of knots K. This leads to a linear system of equations for the spline coefficients cj.I E 1-. Thus, we need a method for choosing the collocation points E that guarantees the existence of a unique solution to (6). In this paper we propose to select the set E to be the set of the Greville sites, namely The motivation for this choice of knots is a special case of (1)—namely the case where the kernel is identically zero, which results in pure interpolation. This special case is useful since it is known that for a large subclass of problems of type (1), the spectral properties of the discrete problem (6) are governed by the spectral properties of the interpolation problem. Moreover, there are good reasons to expect that the interpolation problem is solvable with the proposed choice of E. In particular, it follows from the well-known Schoenberg-Whitney Theorem that the interpolation problem is uniquely solvable in the univariate case when the interpolation points are chosen to be the Greville sites. In the bivariate case, it is not unreasonable to expect that this might also be the case. First, the number of Greville sites exactly matches the number of spline coefficients. Second, if follows from equation (5) that the square linear system
is solvable when / is linear. Although thus far a rigorous proof of the solvability of the interpolation problem for arbitrary functions / is missing, in Section 7 we present numerical evidence that Greville sites are a good choice as collocation points. We remark that the general problem of choosing appropriate collocation points is highly nontrivial. For example, in [18] Powell-Sabin splines were used, which are also smooth quadratic splines, with collocation points either edge midpoints or triangle centers. For both choices, however, the number of collocation points exceeds the spline dimension (i.e., the number of unknown coefficients), resulting in an overdetermined system. Also, the authors of that report discovered that the conditioning of the collocation matrix depends dramatically on the normalization of the cardinal basis functions that are naturally associated with the Powell-Sabin splines.
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We end this section by discussing the continuity of the spline resulting from the proposed method. As we know from Section 5, the spline function that is a solution of the collocation problem is, strictly speaking, not C1. This is due to the fact that in the presence of multiple knots, typically at the corners of the domain fJ, a spline in the corresponding spline space ^2 is C1 only if certain coplanarity conditions of the spline coefficients are satisfied (see Example 4). Thus, if a true C1 continuity is required, then the need to impose these conditions is at odds with the strategy proposed above, in which the Greville sites are used as interpolation/collocation points. For in general one cannot both interpolate at all Greville sites and also satisfy the mentioned conditions. This means that in the presence of knot-multiplicities one has to solve a constrained problem as opposed to unconstrained. For example, one could attempt to minimize the discrete least squares error at the Greville sites, subject to the coplanarity constraints. However, in our initial experiments, in which we merely interpolated a smooth function at the Greville sites (as opposed to solved the integral equation (1) by collocation), we refrained from imposing the smoothness conditions and solved the unconstrained problem only. Thus, the resulting interpolating surfaces, such as those shown in the next section, are not exactly C1 along edges emanating from the corners of 0. Still, those surfaces are approximately C1, at least for relatively dense sets of knots. This follows from the observation that when interpolating a smooth function /, the values of this function at those Greville sites that are involved in the coplanarity conditions are approximately coplanar. Thus, in Example 4, the points (A, f ( A ) ) , (B, f ( B } } , (C, f ( C } } , (D, f(D}} e R3 are approximately coplanar, where the approximating plane is the tangent plane of / at A. It remains to be seen whether the described simplified strategy is acceptable for the full collocation problem, especially in the presence of a singular integral kernel.
7
Experimental Results
In this section we present the results from several experiments. First, we experimented with two collections of data sets. The domain in both cases is taken to be the unit square [0,1] x [0.1], where the first collection consists of uniformly distributed sets of n x n knots, and the second consists of semi-randomly distributed knotsets. A semi-random knotset is formed from a uniform knotset by applying a random perturbation to the individual knots. A representative example is depicted in Figure 8, where squares denote knots and circles denote Greville points. Each of the experimental knotsets determines a spline space from which we interpolated the function f ( x ) = sm(xy)ex+y at the corresponding Greville points. Tables 1 (uniform knots) and 2 (semi-random knots) list the resulting condition number of the B-spline collocation matrix, the maximum error at a uniform sampling of points in the domain, and the rate of convergence as the number of interpolation points is approximately doubled. It is evident from these results that the conditioning of the collocation matrix is very good, and the rate of convergence is on the (optimal) order of h3.
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Figure 8. Two knotsets with corresponding Greville points - 6 x 6 uniform knotset on left, a semi-random knotset on right.
Mesh 3x3 6x6 12 x 12 24 x 24
No. distinct knots 9 36 144 576
No. B- splines 16 64 268 1108
Condition no. 1.70 3.47 3.42 3.41
Error 2.35e-2 1.43e-3 1.27e-4 1.40e-5
Convergence rate 16.39 11.27 9.10
Table 1.
No. distinct knots 9 36 144 576
No. B-splines 17 74 309 1292
Condition no. 4.41 20.60 36.34 62.21
Error 6.39e-2 5.25e-3 6.37e-4 9.46e-5
Convergence rate 12.18 8.24 6.73
Table 2. In Figure 9 we depict the results of interpolating (at the Greville points) the well-known Franke function [3] on a nonconvex domain. In particular, the domain is the unit square with a region removed from the interior. Illustrated are the Franke function over this domain and two interpolants (one for 39 randomly generated knots and one for 148 uniformly placed knots).
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Figure 9. Original function (left), spline fit for 39 random knots (right), spline fit for 148 uniform knots (center).
Our final experiment was to calculate the condition number of the collocation matrix for 300 randomly generated knotsets (each with fifty knots). The dimension of these matrices was approximately 70 x 70, depending on the number of B-splines (note that the number of B-splines depends not only on the number of knots but also on their precise location). The results are shown in Table 3. The last two columns list the maximum number of nonzero entries in each row and column of the system matrix, respectively. In particular, there are at most seven nonzero Bsplines at any given collocation point since the maximum number of nonzero entries in each row is 7. Note also that the median number of nonzero row entries was 6 while the median number of nonzero column entries was 10. These values are a good indication of the sparsity of the matrices and hence of the localization of the B-spline basis. Max cond. no. 52.62
Mean cond. no. 12.79
Median cond. no. 10.69 Table 3.
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No. row 7
No. column 12
Acknowledgments We would like to express our appreciation to the anonymous referees for a thorough and helpful review of this paper. Also, we thank Dr. Jifeng Xu of the Boeing Company for several insightful discussions regarding integral equations and the application of the Powell-Sabin basis to such equations. The third author thanks the Boeing Company for supporting his visits.
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Bibliography [1] Clough, R. W., and J. L. Tocher, Finite element stiffness matrices for analysis of plates in bending, in Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright-Patterson A.F.B., 1965. [2] Farin, G., Curves and Surfaces for CAGD, 4th ed., Academic Press, 1997. [3] Franke, R., Scattered data interpolation: Comp. 38 (1982), 181 200.
Tests of some methods, Math.
[4] Grandine, T., The stable evaluation of multivariate simplex splines, Math. Comp. 181 (1988), 197 205. [5] Hollig, K., Finite Element Methods with B-splines, Frontiers in Applied Mathematics, 26, SI AM, 2003. [6] Micchelli, C. A., A constructive approach to Kergin interpolation in R fc : Multivariate B-splines and Lagrange interpolation, Rocky Mountain J. Math. 10 (1980), 485-497. [7] Micchelli, C. A., Mathematical Aspects of Geometric Modeling, CBMS-NSF Regional Conference Series in Applied Mathematics, 65, SIAM, Philadelphia, 1995. [8] Mulansky, B., and M. Neamtu, On Delaunay configurations, manuscript. [9] Neamtu, M., Multivariate B-splines and their evaluation, Memorandum no. 598, University of Twente, 1986. [10] Neamtu, M., What is the natural generalization of univariate splines to higher dimensions?, in Mathematical Methods in CAGD: Oslo 2000, T. Lyche and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, 2001, 355 392. [11] Neamtu, M., Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay, Trans. Amer. Math. Soc., to appear. [12] Neamtu, M., Delaunay configurations and multivariate splines: Approximation order and B-splines, in preparation.
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[13] Powell, M. J. D., and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3(4) (1977), 316-325. [14] Schwarz, H. R., Finite Element Methods, Academic Press, 1988. [15] Schumaker, L., Triangulation methods, in Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker, and F. Utreras (eds.). Academic Press, New York, 1987, 219-232. [16] Seidel, H.-P., Polar forms and triangular B-spline surfaces, in Computing in Euclidean Geometry, D. Z. Du and F. Hwang (eds.), Lecture Notes Ser. Cornput., 1, World Sci. Publishing, River Edge, 1992, 235 286. [17] Zlamal, M., On the finite element method, Numer. Math. 12 (1968), 394-409. [18] Xu, J., and B. Dembart, Using Powell-Sabin splines as basis functions for scattering problems: An experimental study, Technical Report M&CT-TECH03-03, The Boeing Co., 2003.
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Part III: The Process of Mathematical Modeling, Practice & Education Developing mathematical responses to challenging industrial problems requires mathematical models and modeling expertise appropriate for the problem at hand, often involving close collaboration of academic and industrial researchers. Three questions arise: How does one develop a mathematical model to respond to an industrial need where none has existed before? How does one develop the kind of close academic and industrial collaboration needed? Finally, how does one develop the expertise needed to do all this? The articles in this section address, to a limited degree, these questions. In the first, A Complex Systems Approach to Understanding the HIV/AIDS Epidemic, Carl Simon and James Koopman of the University of Michigan, follow the development of a mathematical model for HIV/AIDS. Such predictive tools for disease are important tools for corporations in protecting the health of workers, as this article discusses. Furthermore, the mathematical modeling techniques developed can be adapated and extended to other industrial problems. In that regard the specific methodology here is an exemplar for other applications. The next article A Combined Industrial/Academic Perspective on Fiber and Film Process Modeling by David Carlson, Institute for Defense Analysis, and Christopher Cox, Clemson University, explores academic and industrial collaboration. The authors show that there is value to industry in providing an experimental test bed with leading edge equipment for experimentation, and there is value to faculty and students in being able to discover the subtleties and intricacies of mathematics within practical industrial problems. The final article, Helpful Hints for Establishing Professional Science Master's Programs by Charles MacCluer of Michigan State University, and Leon H. Seitelman, is a broad expression of the role of education in training mathematicians for industry, with an emphasis upon developing the mathematical maturity needed to understand what industry is doing and why it is doing it. Such education far transcends mere course content in its goal of providing the perspective that is necessary for synthesizing disparate mathematics into useful models.
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A Complex Systems Approach to Understanding the HIV/AIDS Epidemic Carl P. Simon*^ and James S. Koopman^ 1
Abstract
A complex system is one that includes elements normally omitted in classical models in biology, economics, epidemiology, ecology, etc. Such elements include: heterogeneity and diversity, nonlinear dynamics away from equilibrium, organizational structure and nonrandom mixing, feedback and learning, and emergence of macro phenomena from micro level assumptions. Focusing on the classical dynamic model of the spread of HIV, this paper illustrates how one is naturally drawn in the study of industry and health issues to including elements of the complex systems approach in order to understand the phenomenon thoroughly and to draw effective policy interventions. In the case of HIV, such public health interventions can have major impacts upon the medical, transportation and insurance industries.
2
Introduction: Complex System Modeling
A major goal of science is to find useful and valid descriptions and explanations of phenomena in the natural world. The search for such descriptions and explanations can take many paths: incrementalist or systems-based, simple or complex, static or dynamic. One natural way to study a social, biological or physical phenomenon is to break the underlying system down into its components, understand how each com* Professor of Mathematics, Economics, and Public Policy, University of Michigan tDirector, UM Center for the Study of Complex Systems * Professor of Epidemiology, University of Michigan
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ponent works, and finally understand how the components fit together to form the whole system. One focuses on a single component, ignoring the rest of the system, at great peril. Many of the failures of science, from DDT to epidemics of drug resistant viruses, have resulted from focusing incrementally on the performance of individual components and ignoring the "big picture." To study how a system and its components work, one usually builds a model - mathematical or verbal - of how the various components of the system operate and interconnect. In this approach, one usually starts with the simplest model that can shed light on the process under discussion. Then, one adds more and more detail to the model until it captures the important characteristics of the underlying phenomenon. 2.1
Complex Adaptive Systems
Most of the basic models we teach and study in, say, biology, ecology, economics, and epidemiology, assume that: (i) (ii) (iii) (iv)
there are relatively few components - all fairly identical, there is little or no structure to their connections, the underlying system is a static one in some kind of equilibrium, and the whole is a simple, linear sum of its parts.
For example, every microeconomics text assumes that there are just a few (kinds of) consumers and two or three goods, that there is no structure to who trades with whom, that the economy is sitting in competitive equilibrium, and that consumers do not change their preferences - neither from education nor advertising. Furthermore, there is a large gulf between the microeconomic study of individual and firm decision-making and the macroeconomic study of the behavior of global markers and markets. On the other hand, the complex systems approach begins with the assumptions that
. the whole is more than the sum of its parts, . . . . . .
3
the components that make up the system are many and varied, this heterogeneity (or diversity) is important for the behavior of the system, the individuals get feedback from their actions and adapt in response, contacts between agents are not random, the network of those connections plays a key role in the outcomes, and finally the interest is in studying how macro-system behaviors emerge from assumptions about the underlying micro-components.
Disease Spread as a Complex Adaptive System
To gain some insights and how these various approaches compare and contrast, we will take an important, but difficult, problem and see how much progress can
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be made in understanding it through various approaches, beginning with a linear biostatistics approach, moving to a simple dynamic "compartmental" model and finally including the key aspects of a complex systems approach - one aspect at a time. We will use the spread of HIV/AIDS through a population of gay men. The basic equations are fairly simple, but one quickly finds that to gain insights that make a difference, one must go beyond the homogeneity, non-dynamic, random mixing, no-behavior-change approach that one uses to begin the study. One at a time, one is led naturally to the introduction of the core complex systems concepts: nonlinear dynamics and feedback, heterogeneity and non-random mixing of agents, changing behaviors, and ultimately the use of agent-based modeling. We believe that a thorough consideration of most real world problems entails this progression from simple models to more complicated models and eventually to the inclusion of complex systems concepts and techniques. We have chosen the spread of HIV as an example not only because we have worked intensely in this area but also because this progression from simple to complex occurs so naturally and inevitably in this context. Furthermore, as illustrated by Ford Motor Company's HIV project in South Africa, illustrative and especially accurately predictive models are important tools for promoting the health and safety of an industry's work force or of a government's population. 3.1
Static Individual Risk Approach
Actually, the most common approach in epidemiology is not systems oriented, but one based on hypothesis testing about risk behaviors [23]. In a simplified version of this "individual risk approach," one writes out a large table with "study subjects and their infection status" labeling the rows and possible risk factors labeling the columns. One uses correlation or some sort of statistical analysis to see what combinations of risky behaviors are likely to lead to the result "subject infected" or "subject not infected." Despite its prevalence in the epidemiological literature, there are some important drawbacks to this approach. In particular, the underlying statistical analysis requires the independence assumption - outcomes in one individual are independent of outcomes in others, a curious assumption for a communicable disease like HIV. Furthermore, this approach relies heavily on linearity, and it gives insights on correlations not causality. Finally, it turns out that the basic bottom line outcome measures, like the odds ratio (the odds of disease given a risk factor relative to the odds of disease without that risk factor) and the incidence rate ratio (percent growth of infection in a population with a risk factor relative to the percent growth of infection in a population without that risk factor), can vary dramatically over time when the system is far from equilibrium and are thus difficult to interpret. 3.2
Adding Dynamics
The antithesis of the individual risk approach is the dynamic model approach, in which one constructs a mathematical or computer model that comes close to capturing the essence of the disease transmission system under study. In the process,
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one must stay concerned about questions such as: does the model capture what is happening, and how are its results dependent on the modeling choices made. Simple models may be studied analytically, but more complex models may require computer simulations to gain insights. At some point, stochastic effects may need to be considered - especially for such an inherently stochastic phenomenon like disease spread.
3.3
Advantages of Mathematical Modeling
Models are invaluable tools for our understanding of the spread of a contagious disease like HIV. They can guide us to which data to collect and how to use the collected data effectively. They can help estimate important factors in the spread of the disease. They can form the basis of simulation experiments in a realm where field experiments are often impractical or unethical. They can lead to more careful formulation of the assumptions used to study disease mechanisms. And, of course, they can be used to identify trends and make forecasts about the future path of the disease. Models have led to major breakthroughs in our understanding of disease spread. Sir Richard Ross's 1911 malaria model introduced the key concept of a threshold for epidemic spread and used it to show that malaria could be controlled without killing every mosquito in the infected area. In 1952 MacDonald used a similar model to show that control of adult mosquitoes is more effective than control of larvae. Models have been used to forecast the onset and spread of worldwide influenza epidemics and to design vaccination programs for childhood diseases like measles and rubella.
4
Introduction to Dynamic Epidemiological Models
To understand more clearly the roles of simplicity and complexity in modeling disease spread, we give a short introduction to the basic epidemiological models. See [1, 2], for example, for details. In general, we divide the population N under study into four mutually exclusive classes: susceptibles 5, infectives /, removed R (those neither infective nor susceptible, often those with permanent or temporary immunity), and exposed E (those currently infected but not yet infective). One uses these four initials to classify diseases and disease models. For example, in an SIS model, infected individuals are immediately susceptible once they recover (as in gonorrhea). In an SIRS (respectively SIR) disease, recovered infecteds have temporary (respectively, permanent) immunity. HIV/AIDS is usually considered an SI disease since infected individuals rarely recover. To construct a simple SIS model, let X denote the number of susceptibles, Y denote the number of infectives, and TV = X + Y denote the size of the population under study. Keeping things simple at first, let a denote the rate of recovery, c the average number of possibly disease transmitting contacts per individual per period (sex acts for an STD), and j3 the probability of transmission per contact. Assuming
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random contacts, the number of new infections per period is:
where X • c is the number of contacts by susceptibles, (Y/N) is the probability that such a contact is with an infective (in a world of random mixing) and (3 is the fraction of those contacts that actually transmit the infection. Including the rate of recovery a leads to the simple dynamic:
Finally, bring in some vital dynamics by letting /j, denote the birth rate and (background) death rate. Assuming no disease-related death and that all individuals are born susceptible leads to the dynamic:
Let At —> 0 to turn these difference equations to differential equations,
Because the underlying flow balances, the population size TV remains constant. Substitute X = N — Y into the bottom equation of system (1) to get the single differential equation
in the single variable Y. Equation (2) is the quadratic "logistic" equation, the simplest nonlinear differential equation and the equation used to study population growth. One effective way to study an autonomous one-variable equation like (2) is to graph dY/dt against Y by graphing the right hand side of (2) as a function of Y. Since Y factors out of (2), this graph goes through the origin; its other intercept is
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Figure 1. dY/dt vs Y
Figure 2. Phase diagrams for Equation (1) 4.1
Basic Reproduction Number as a Threshold
There are two cases to consider, depending on whether in (3) r * < 0 (i.e., c/3-(a + fjL) < 0),
or Y* > 0 (i.e., c(3 - (a + /z) > 0),
(4)
These two cases are illustrated in Figure 1. In the left side of Figure 1, dY/dt < 0 whenever Y > 0; and therefore, Y(t) decreases monotonically to 0, i.e., the disease always dies out. In the right side of Figure 1, dY/dt is positive and so Y(t) increases when 0 < Y < Y*; and dY/dt is negative and so Y(t] decreases when Y > Y*. In other words, Y(t) tends to Y* for any initial 1^(0), as summarized in Figure 2. Rewrite the threshold condition c(3 — (a + /j,) < 0 in (4) as [c/3/(a + ^)] < 1We can summarize the above discussion as: c(3 < 1 implies (a + /j,)
Y(t) —» 0 for any initial 1^(0); (5)
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The ratio RQ = cf3/(a + p) gives a simple threshold for distinguishing disease die-out from endemic equilibrium. The numerator of RQ is roughly the rate of new infections per person in a population of susceptibles; the denominator is the rate that individuals leave the infected state through cure or death. It is natural to call RQ the basic reproduction number of the infection. It is roughly the number of infection transmissions per infective, in a population of susceptibles, during the period of the infection. In fact, RQ plays many roles. For example, as (5) suggests, the endemic prevalence (Y*/N) is 1 — (l/Ro] when RQ > I. So far, we have moved from a static individual risk model to a dynamic population level model. But our model is far too simple yet to use it for estimating parameters, evaluating intervention strategies, or making forecasts, especially in its assumptions of homogeneous population and random mixing. 4.2
Adding Heterogeneity
The simplest way to add heterogeneity to this model is to divide the population into subgroups by some criterion, for example, race, age, sex, frequency of sexual activity, type of sexual activity, or some other behavioral characteristic. Let Xi denote the number of susceptibles in subpopulation i and Yi the number of infectives in i. Let NJ = Xj + Yj, the size of subpopulation j. To capture between-group mixing, let Pij denote the proportion of contacts by a typical person in group i that are with members of group j and let f3j denote the transmission probability from a contact with an infective in group j. (More generally, we could have (3 = (3ij depend also the subgroup i of the susceptible.) Now, the analogue of system (2) describing the transmission dynamics is:
It turns out that this more complicated system is still amenable to mathematical analysis. In fact, in a classic paper of epidemiologic modeling, Lajmanovich and Yorke [28] proved that system (6) behaves pretty much like system (2) in that either all solutions tend to the disease-free equilibrium or all solutions tend to a positive endemic level of infection. One can easily add a removed (e.g., immune) class Zi to system (6). The resulting system of 3n equation turns out to be much more difficult to analyze. Twenty years after Lajmanovich and Yorke's SIS result, Hethcote and Thieme [11] proved the corresponding result for SIR and SIRS systems: either every solution tends to the disease-free equilibrium, or every solution tends to a unique endemic equilibrium. Their proof is one of the more difficult and technical formal proofs in mathematical epidemiology.
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5
Modeling the Spread of HIV
To model any specific infection, one must make some adjustments to system (6) to take into considerations special characteristics of that infection. In the case of HIV, the most obvious characteristics are that HIV passes through a number of fairly well-defined stages, that HIV is a lethal disease, and that susceptibility does not begin at birth for the homosexual transmission system under study. To handle the first, we divide the category Y of infecteds into a number ra of classes corresponding to the various stages of HIV: Yi for the period of primary infection during which the immune system first becomes activated, > 2 , . . . , Ym for the long asymptomatic period with each Yi characterized by some decreasing interval of CD4 count in the blood system, and finally Ym+i full blown AIDS. To handle the second characteristic, we will no longer be able to treat the population total N as a constant, so that JV's in the denominator of (6) will now change over time and the resulting system will no longer be a quadratic one. Furthermore, there is no longer a recovery rate G^, but now a rate ki of moving from stage i to stage i + \. To handle the third characteristic, we assume that persons become susceptible at some rate C7, instead of at the birth rate /j,N. We will assume that the susceptible population is divided into groups by level of sexual activity Cj. Figure 3 gives the resulting compartmental diagram with the definitions of the parameters summarized in Table 1. System (7) is the underlying system of compartmental differential equations. (See, for example, [19].)
System (7) is a fairly "complex" system to analyze. Because the system is no longer a quadratic system like (6), it is fairly difficult to derive general mathematical results for (7). Simon and Jacquez [32] compute, at least for the case of homogeneous mixing (no subgroups), a formula for the threshold RQ and show that for RQ < I every solution of (7) tends to the no-disease equilibrium, while for RQ > 1, every solution of (7) tends to a unique positive endemic equilibrium. System (7) includes some elements of "complex adaptive systems" especially the nonlinear dynamics and a modicum of heterogeneity and nonrandom mixing. On the other hand, it still lacks some key elements of the HIV transmission system, including effects of drug treatments, behavior change, and differences in human system response.
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Figure 3. HIV Compartmental Diagram Table 1. Meanings of the Variables and Paramters in System (7) susceptibles in subpopulation i individuals in population i who are in stage h of HIV individuals in population i within full blown AIDS for each group j mortality rate nsmission probability, probability that a contact between a susceptible in group i and an HIV-infected in group Yjh transmits HIV fractional transfer rate from proportion of contacts of a person in group i that are with persons in group j contact rate that characterizes group ?'.
6
Mathematical and Epidemiological Goals
Now that we've set out the basic equations (7) of HIV spread, let's stop to list what we would like to use this model to accomplish. We began with the following six goals: 1. Construct and study more realistic mixing mechanisms ((/>ij)), especially looking for mechanisms that would relate well to collected data. Study the effects of these mixing regimes on the dynamics of system (7). 2. Compute RQ for these more realistic and more complex models. 3. Construct and study stochastic versions of system (7) to understand the benefits and costs of working with the deterministic system (7), even though the real world transmission dynamics is much more stochastic. 4. Compare the output of system (7) (with realistic choices of parameters in Table 1 to real world data to estimate important parameters of the underlying system, especially the transmission probability /3. 5. Use these comparisons of model and data to explore public health implications
of the resultant findings. 6. Include vaccination strategies and drug treatment strategies.
7
Structure and Non-Random Mixing
Quite a few mixing formulations pij have been used in the literature in system (6) and its variants to study disease spread. The simplest is, of course, that of system (1): no subgroups and random mixing in a homogeneous population. The next simplest is a population with multiple subgroups but no mixing between groups, so that p^ = 1 for i = j and 0 otherwise, a fairly uninteresting situation we'll call "restrictive mixing." The next simplest is the multi-group version of random mixing, often called "proportional mixing" or "proportionate mixing." Here, the probability p^ that a person in group i interacts with a person in group j is simply the fraction of group j's contacts relative to the contacts of the total population:
In this case, no group has its own identity for the dynamics sake. Until the 1980s, nearly all multi-group models of disease spread used proportional mixing, an especially strange hypothesis for sexually transmitted diseases, among others. In the mid-1980s, a number of authors [12], [13], [14], [19], [25] worked with a somewhat a more general form of mixing: what is now called "preferred mixing," a convex combination of restrictive mixing and proportional mixing. In preferred mixing, each group retains some self-identity by having each group member reserve some mixing for within-group mixing with the rest of the mixing occurring randomly with the rest of the population. In this case, if n is the fraction of group i's mixing reserved for other members of group i,
Preferred mixing is probably the most complex form of mixing that is still mathematically tractable. For example, as we will indicate below, Simon and Jacquez [32] showed how to calculate RQ for preferred mixing. Unfortunately, the assumption of preferred mixing is still a mathematical convenience that is a bit too rigid for most data sets. As a general but still manipulable mixing in this dynamic framework, we [17] introduced "structured mixing," for ease in matching the dynamics of system (6) with data from real population epidemics. Divide the population two different ways: first, as above into "population subgroups" by characteristics that make a difference for transmission patterns (e.g., contact rates with different groups) but may be difficult to estimate by individual subjects, and second, into "activity groups," settings in which sexual partnerships are formed - information more obvious to each individual (e.g., place, age and sexual activity preference of partners). Using surveys and estimates, assign population subgroups to the various activity groups by formulating how each of the population
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subgroups divides its contacts among the various activity groups. Within activity group k, let Ni(k) denote the number of population group i members and let Ti(k) denote the effective rate at which members of population group i seek new sexual partners in setting k, both of which could be discernable from interviews. Then, C(k)ij, the number of new sexual partnerships formed per unit time between members of population groups i and j in activity group k is (dropping the &'s):
For more flexibility (at the cost of more complexity), one can include a "mutual acceptability coefficient" Q(k)ij into the numerator of the above expression. We [26] call this variant of structured mixing "selective mixing." It is especially valuable when multiple races or sexes are part of the susceptible population. Structured (and selective) mixing are conceptually simple but mathematically complex. They do include most mixing formulations used in the literature and have the capability of relating better to data sets, provided the appropriate questions are asked in interviews. Castillo-Chavez et al. [4] use a different approach to present a general classification of all mixing mechanisms that could be used in systems (6) or (7).
8
Calculation of RQ for Complex Contact Structures
For the mixing models we have studied, there exists a threshold parameter RQ, which is roughly:
roughly the number of new cases generated by an infective person in a population of susceptibles during the length of the infection, in other words, the basic reproduction number of the infection. For RQ < 1, the disease-free equilibrium is globally asymptotically stable; for RQ > 1, the disease-free equilibrium in unstable and intuitively a unique endemic equilibrium is stable. For restricted mixing,
where
For proportional mixing,
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Figure 4. Markov Diagram for a Stochastic SIS Model For preferred mixing,
Notice that each of these .Ros are of the form (8). See [18], [32] for details. Given the complexity of (9), it's hard to imagine a formula for RQ for mixing formulations beyond preferred mixing; but, see also Diekmann and Heesterbeek [8].
9
Stochastic Models
Disease spread is such an inherently stochastic process that it is reasonable to ask whether stochastic variants of (1), (6), or (7) have dynamics similar to their deterministic counterparts. For example, we have seen that the simple SIS model (1) has either a globally stable disease-free equilibrium or a globally stable endemic equilibrium depending on a threshold RQ. What about the stochastic analogue of (1)? Figure 4 shows the states and transitions for the stochastic version of (2). We continue to use N for the fixed size of the population. Now the number of infectives is a random variable Y, with realizations y, which are restricted to integer values between 0 and N. In Figure 4, /zy is the transition rate for Y decreasing from y to y — 1 (recovery) and 7^ is the transition rate for Y increasing from y to y + 1 (new infection). More precisely,
Here, o(h) represents the usual "error term" or remainder in the class of functions / such that /(*)/* -»• 0 as t -»• 0. If we write (2) as
ignoring vital dynamics for simplicity sake, then it is natural to set
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in the stochastic analogue of (1) for discrete populations. Note that (11) implies that 70 = 7jv — 0 and ^o = 0- Actually, the fraction in equation (11) should be y/(N — 1), but nearly all the work in the literature uses y/N, and we follow that convention here. Model (lOa, lOb, lOc) is the classical stochastic birth-death process with birth rates 7y, death rates f i y , and independence of births and deaths. As usual in such models, we let Then,
Write these equations as:
Letting At
0, we find
for y = 0,1,..., N, where 7-1 = 70 = 7JV = 0 and ^o = /A/v+i = 0. Equations (12) are the Kolmogorov forward equations for the stochastic SIS model (lOa, lOb, lOc) under consideration. In matrix form, these become:
Figure 4 presents the simple underlying Markov process for this linear system. Just a glance at Figure 4 unambiguously reveals the long run behavior of this system since the system has one and only one absorbing state: the no-disease equilibrium y = 0. By standard theory of Markov processes, with probability one the infection dies out!! This is a surprising difference from the deterministic system (1) in which we found two long run stable behaviors: no disease or endemic disease. It appears that the stochastic system does not allow for endemic disease. Why do the deterministic and stochastic formulations yield such different results? Let
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Figure 5. Solution of System (12) for N — 100, R0 = 2. denote the mean of y in distribution p(t). One can easily show using system (12) that if RQ = Sfj- < 1, then my(t} —> 0 monotonically, just as in system (1). Suppose on the other hand, that this RQ > 1. Figure 5 presents a computer solution of system (12) for N = 100 and RQ = 2. Notice the convergence to a bimodal distribution with modes at y = 0 and at the deterministic endemic equilibrium - at least for the first 200 time units. In fact, this bimodal distribution persists for fairly large t. Eventually, for some unrealistically large £, of order 106, the endemic mode weakens and the system eventually converges to the absorbing no-disease state. The endemic state is called in this situation a "quasi-stationary" solution, it is a stable solution for a long, but not infinitely long period of time. The bottom line is that, for all intents and purposes, the stochastic SIS system (12) does behave just like its deterministic analogue (1) for large enough population sizes. Depending on whether or not RQ > 1, the system tends to a no-disease or to an endemic equilibrium. See [16], [27] for details. One direct way to find and analyze quasi-stable distributions is to work with stochastic dynamics of system (12) conditioned on non-absorption. For such an analysis, one replaces Pi(t] in (12) by
Wheree the denominator is the probability that infection is present at time t, the dynamics for q is given by the system:
a quadratic system of differential equations!! A distribution q* is stationary solution of this system if and only if ( 0 q* ) is a quasi-stationary solution of (12).
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The analysis in this section works for SI and SIR systems and systems with multiple stages of infection, like (7), provided the population under study is homogeneous and mixes randomly. However, realistic, complex mixing formulations add another layer of difficulty to the analysis. For example, Koopman et al. [21] present a simple SIS model in which individuals mix either with their neighbors or in a central location. In this case, they show that, depending on the mixing, the stochastic model can have much different behavior than the corresponding deterministic one, even calling for different intervention techniques. For further connections between deterministic and stochastic approaches, see [5], [20],
10
Estimating the Transmission Probability 3
After spending a few years developing our HIV transmission model and appropriate mixing formulations, we turned to our ultimate goal of using the model to estimate key parameters of the transmission process. Our estimation procedures focused on the transmission probability (3j in (7), where j denoted stage of infection. Studies [6], [20] of monogamous pairs had established that (3 was in the 0.0001 to 0.001 range during HIV's long asymptomatic period, and that it rose dramatically after HIV turned to AIDS. However, estimations of (3\ during the one- to three- month long period of "primary infection" were especially difficult, mostly because so few infections had been identified that early. For similar reasons, most HIV epidemics had not been identified until they were well established in the community. The open squares in Figure 6 show what we know of the Chicago HIV epidemic: no good cumulative data until the epidemic had reached 80% of its endemic level. Two notable exceptions were that very first San Francisco epidemic for which there were blood samples collected because of a simultaneous hepatitis b outbreak in the same community [3], and the much later Bangkok epidemic, which had occurred late enough that its onset was somewhat anticipated. The dark squares in Figure 6 indicate the progression of the San Francisco epidemic. Note in particular the rapid early rise; the epidemic had reached 20% of endemic level in two to three years, and 50% in four years. The Thailand data showed a similar rate of increase. Our path was a fairly straightforward one. Roughly speaking, working with the San Francisco data, we estimated the background infection and mixing parameters directly from the data collected. Then, we compared the output of our model for various values of (3\ with the epidemic curve in Figure 6. Figure 7 shows the model's outputs for (3\ — .04, .06, .08, .10 respectively, assuming that the next stage /?s are in the 0.001 to 0.005 range. Note that the larger /?i, the quicker the initial rise in the number of infecteds. Even with /?i — 0.10, it takes nearly 50 years for the epidemic to reach 20% of its endemic level. After checking that factors such as behavioral change or core group saturation did not affect our results significantly, we [15] concluded that (3\ is substantially larger than the /% for later stages and that individuals in their primary infection stage are responsible for significantly large amount of HIV transmission. Later, we [22] showed that partner selection criteria and changes in sexual activity because of aging would point to large amount of transmission during primary infection, even if fli were not that much larger than
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Figure 6. HIV Spread in San Francisco and Chicago
Figure 7. Y(t) for various values of f3\. /?2- When infection with a high (3\ is introduced into a population whose members have a high rate of sexual partner change, a large group is infected in a rather short period. This is followed by a period of low infectivity, as the group moves from high to low infectivity almost like a cohort; hence, the rather distinctive shape of the epidemic curve in Figures 6 and 8. Figure 8 presents that "bathtub-shaped" (3 curve that our analysis suggests. There is a similarly shaped curve for the amount of HIV virus in the blood stream over time [1], [29], [31]. Such a curve makes intuitive sense. The amount of virus in the system initially rises until the immune system becomes thoroughly activated. Once the immune system kicks in, it gains the advantage and the amount of virus in the blood system remains at a low level. However, the HIV virus slowly destroys the immune system and eventually wins the war; after a long asymptomatic period,
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Figure 8. Bathtub-shaped (3 Curve the amount of virus in the blood (and the infectivity) rises again. Recent research (see [7] for a summary) showing that high infectivity correlates with high viral load in the blood stream completes the argument. 10.1
Implications of a high 3}
Our analysis suggests that the narrow two-month period of primary infection at the beginning of infection plays a major role in the transmission process. The next step is to understand some of the implications of this observation. Who is especially at risk for infection during this small window? One such group includes those who have a regular partner whose partner gets infected outside the partnership. In this case, one is likely to share sex during the partner's window of primary infection and to be much more vulnerable to infection oneself. So, multiple partnerships play a large role in the transmission system of HIV. A high 0i also points to the dangers of relying on HIV tests on one's partner. If one's partner is in primary infection when risk of transmission is especially high, he may not come up as positive yet on many HIV tests, especially the simple athome tests marketed in the 1990s. In fact, given the very high transmissibility during primary infection when HIV tests are likely to be negative and the very low transmissibility during later stages, it may be safer to have sexual contact with a person who has tested positive than with a sexually active person with many partners who still tests negative. Finally, these HIV transmission models and their analyses yield some suggestions for developing HIV vaccines. One usually thinks of a vaccine working by keeping an uninfected person uninfected, mostly by activating the immune system through the introduction of a little bit of infection into the system. However, the thought of getting a little HIV or HIV-like antigen in one's system is unsettling, especially since HIV works by eventually destroying the immune system itself. On the other hand, vaccines can work just as well by lowering the contagiousness of an infected person as they can by preserving the uninfected status of an uninfected
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person. The Salk polio vaccine worked very successfully by lowering contagiousness of the infecteds without preserving anyone's susceptibility. Could an HIV vaccine work via herd immunity this way? We [24], [33] argued that it could; furthermore, we argued that current vaccine tests that focus on susceptibility effects and ignore contagiousness effects may very well reject a successful vaccine. These arguments were given coverage in the April 6, 1998 US News and World Report description of our work. The challenge would be to develop vaccine trials for this contagiousness effect - tests that would have to include individuals and their partners.
11
The Necessity of a Complex Adaptive Systems Approach
At the outset of this presentation, we characterized a complex adaptive systems (CAS) approach as one that includes heterogeneity of agents and actions, dynamics away from equilibrium, non-random contact structures, agents' adapting to feedback they receive from their actions, and the emergence of macro-level behavior from micro-level assumptions that include these aspects. We started by describing the dominant linear risk assessment approach of epidemiology - an approach that includes few or none of these aspects, one that focuses on correlation rather than causation, and one whose independence assumptions may be particularly inappropriate for studying contagious diseases. We then introduced the dynamic model and the thresholds that separate convergence to a disease-free world from an endemic disease. In the dynamic compartmental context, we introduced heterogeneity of agents and of activities and spent some time discussing how to model non-random mixing. We pointed out that when one includes realistic assumptions important for the spread of HIV, such as, multiple levels of contact, disease-related death, multiple stages, and heterogeneous immune systems, the resulting dynamic models leave the domain of models that are tractable by pencil-and-paper mathematical analysis and move into the realm of study by numerical analysis and computer simulations. One of the main conclusions of our analysis is the important role of the primary infection period in the transmission process and the resulting necessity to include multiple partnerships, with flexible starting and ending times, into the construction and analysis of the model. However, it is very difficult, if not impossible, to include flexible multiple partnerships in a differential equation model. For such models intrinsically have no individuals and no concrete starting or ending times of partnerships. They keep track of aggregates and averages. Dietz's dynamic models [9], [10] of pairs offer plenty of insights, but they do not have the flexibility needed to capture the critical aspects of the HIV epidemic. One is naturally led to the consideration of models [5] that focus on individuals, models such as discrete event models (like GPSSH, AnyLogic) or agent-based models (like Swarm, Ascape or Repast) that are key techniques in the complex adaptive systems approach. In both cases, the focus is on individuals who have specific contacts in specific moments of time with other specific individuals. These individuals have rather well-defined choices of partners or partner-types, and of sexual activities. Agents can change these preferences by what they experience or what
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they see others experience. Though difficult to analyze mathematically, these models can be much more realistic and can capture the key details of the transmission system. In particular, they can include the stochastic components that permeate the transmission process. This stochastic element implies that the modeler must work with many, many runs of the simulated model with different random seeds, with different initial conditions and with different parameter values before drawing conclusions. But by carefully constructing these agent-based simulation models and by docking them convincingly to the more elementary compartmental models we presented earlier, the modeler has a powerful tool for understanding the spread of a contagious infection like HIV and more importantly for suggesting and evaluating interventions that may be able to halt the epidemic.
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Bibliography [1] R.M. ANDERSON, R.M. MAY, Epidemiological parameters of HIV transmission, Nature, 333 (1988), pp. 519-522. [2] R.M. ANDERSON, R.M. MAY, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, England, 1992. [3] P. BACCHETTI, A.R. Moss, Incubation period of AIDS in San Francisco, Nature, 338 (1989), pp. 251-253. [4] S. BLYTHE, C. CASTILLO-CHAVEZ, J. PALMER, M. CHENG, Toward a unified theory of mixing and pair formation, Math. Biosci., 107 (1991), pp. 379-405. [5] S.E. CHICK, A.L. ADAMS, J. KOOPMAN, Analysis and simulation of a stochastic, discrete-individual model of STD transmission with partnership concurrency, Math Biosci., 166 (2000), pp. 45. [6] R.A. COATES, L.M. CALZAVARA, S.E. READ, ET AL., Risk factor for HIV infection in male sexual contacts of men with AIDS or an AIDS-related condition, Am. J. Epidemiol., (1988), pp. 4729-4739. [7] E. DAAR, Treatment of Primary HIV Infection, Medscape General Medicine, 4(4) 2002. [8] O. DIEKMANN, J.A.P. HEESTERBEEK, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley, Chichester, England, 2000. [9] K. DlETZ, The role of pair formation in the transmission dynamics of HIV, in Sesquicentennial Invited Paper Sessions, M.H. Gail, N.L. Johnson, eds., American Statistical Association, Alexandria, VA, 1989, pp. 609-621. [10] K. DIETZ, K. HADELER, Epidemiological models for sexually transmitted diseases, J. Math. Biology, 26 (1988), pp. 1-25. [11] H.W. HETHCOTE, H.R. THIEME, Stability of the endemic equilibrium in epidemic models with subpopulations, Math Biosci., 75 (1985), pp. 205-227.
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[12] H.W. HETHCOTE, J.W. VAN ARK, Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs, Math. Biosci., 84 (1987), pp. 85-118. [13] H.W. HETHCOTE, J.A. YORKE, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomath. 56, Springer-Verlag, Berlin, 1984. [14] H.W. HETHCOTE, J.A. YORKE, A. NOLD, Gonorrhea modeling: A comparison of control methods, Math. Biosci., 58 (1982), pp. 93-109. [15] J.A. JACQUEZ, J. KOOPMAN, C.P. SIMON, I. LONGINI, Role of primary infection in epidemics of HIV infection in gay cohorts, Journal of A.I.D.S., 7 (1994), pp. 1169-1184. [16] J.A. JACQUEZ, C.P. SIMON, The stochastic SI model with recruitment and death: comparison with the closed SIS model, Math Biosci., 117 (1993), pp. 77125. [17] J.A. JACQUEZ, C.P. SIMON, J. KOOPMAN, Structured mixing: Heterogeneous mixing by the definition of activity groups, in Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez, ed., Lecture Notes in Biomathematics 83, Springer, Berlin, 1989, pp. 301-315. [18] J.A. JACQUEZ, C.P. SIMON, J. KOOPMAN, The reproduction number in deterministic models of contagious diseases, Comments Theor. Biol., 2 (1991), pp. 159-209. [19] J.A. JACQUEZ, C.P. SIMON, J. KOOPMAN, L. SATTENSPIEL, AND T. PERRY, Modeling and Analyzing HIV Transmission: The Effect of Contact Patterns, Mathematical Biosciences, 92 (1988), pp. 119-199. [20] J. KOOPMAN, S.E. CHICK, C.S. RIOLO, A.L. ADAMS, M.L. WILSON, M.P. BECKER, Modeling contact networks and infection transmission in geographic and social space using GERMS, Sex. Transm. Dis., 27 (2000), pp. 617626. [21] J. KOOPMAN, S. CHICK, C.P. SIMON, C. RIOLO AND G. JACQUEZ, Stochastic effects on endemic infection levels of disseminating versus local contacts, Mathematical Biosciences, 180 (2002), pp. 49-71. [22] J. KOOPMAN, J.A. JACQUEZ, G. WELCH, C.P. SIMON, B. FOXMAN, S. POLLOCK, D. BARTH-JONES, A. ADAMS, K. LANGE, The role of primary HIV infection in the spread of HIV through populations, Journal of A.I.D.S., 14 (1997), pp. 249-258. [23] J. KOOPMAN, I. LONGINI, J.A. JACQUEZ, C.P. SIMON, ET AL., Assessing risk factors for transmission, Am. J. Epidemiol., 133 (1991), pp. 1199-1209. [24] J. KOOPMAN, C.P. SIMON, J.A. JACQUEZ, Assessing contagiousness effects of vaccines and risk factors for transmission, in Modeling the AIDS Epidemic, E. Kaplan and M. Brandeau, eds., Raven Press, 1994, pp. 439-460.
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[25] J. KOOPMAN, C.P. SIMON, J.A. JACQUEZ, J. JOSEPH, L. SATTENSPIEL, AND T. PARK, Sexual partner selectiveness effects on homosexual HIV transmission dynamics, Journal of Acquired Immune Deficiency Syndromes, 1 (1988), pp. 486-504. [26] J. KOOPMAN, C.P. SIMON, J.A. JACQUEZ, AND T. PARK, Selective Contact Within Structured Mixing; With an Application to the Analysis of HIV Transmission Risk from Oral and Anal Sex, in Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez, ed., Springer-Verlag Lecture Notes in Biomathematics 83, 1989, pp. 316-348. [27] R.J. KRYSCIO, C. LEFEVRE, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Probab., 27 (1989), pp. 685-694. [28] A. LAJMANOVICH, J. YORKE, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), pp. 221-236. [29] J.A. LEVY, Mysteries of HIV: challenges for therapy and prevention, Nature 333 (1988), pp. 519-522. [30] D. OSMOND, P. BACCHETTI, R.E. READ, ET AL., Time of exposure and risk of HIV infection in homosexual partners of men with AIDS, Am. J. Public Health 78 (1988), pp. 944-948. [31] C. PETERSON, C.M. NIELSEN, B.F. VESTERGAARD, ET AL., Temporal relation of antigenaemia and loss of antibodies to core antigens to development of clinical diseases in HIV infection, Br. Med. J., 295 (1987), pp. 567-569. [32] C.P. SIMON, J.A. JACQUEZ, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations, SIAM J. Appl. Math., 52 (1992), pp. 541-576. [33] C.P. SIMON, J. KOOPMAN, Infection transmission dynamics and vaccination program effectiveness as a function of vaccine effects in individuals, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases, Part II: Models, Methods and Theory, C. Castillo-Chavez, S. Blower, P. van den Driessche and Abdul-Aziz Yakubu, eds., Springer-, New York, 2001, pp. 143155.
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A Combined Industrial/Academic Perspective on Fiber and Film Process Modeling * C. David Carlson, Jr.^ and Christopher L. Coo$^ 1
Abstract
A combined industrial/academic perspective on modeling of fiber and film processes will be presented. The development of realistic simulations necessitates mtiltidisciplinary teams comprised of researchers from both academia and industry. The primary fiber and film processes will be described, along with a representative mathematical model. Challenges in developing realistic simulations will be highlighted. Factors affecting collaboration between industry and academia will be considered, along with examples of successful interactions. In particular, a simulation package that is being designed with industry guidance will be discussed, along with opportunities for further work in modeling of fiber and film processes.
2
Introduction
The Navier-Stokes equations are considered by many to be the foundation of fluid mechanics (see [11], for example). Yet there are many interesting fluids (or perhaps we should say fluid-like materials) whose behavior is not represented by the Navier-Stokes equations. These other fluids fall under the general classification of non-Newtonian fluids, with polymeric fluids included in this group. The treatise by "This work was supported in part by the ERG program of the National Science Foundation under Award Number EEC-9731680. ^Institute for Defense Analyses, 4850 Mark Center Drive, Alexandria, VA 22311 ^Formerly at Mitsubishi Polyester Films, Inc., Greer, SC § Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975 ^ Corresponding author, e-mail: [email protected]
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Coleman, Noll, and Markovitz, [10] presents a development of the governing equations for a particular class of non-Newtonian flows. Petroleum-based fibers and films are polymeric materials, which are known to exhibit properties that are quite difficult to represent in a mathematical model. This is especially true in the context of fiber or film processing, where the material can undergo phase changes as velocity, stress, and temperature change. Many books (e.g. [1], [3], [6], and [12]) and papers (e.g. in the journals [32] and [33]) have been published on modeling polymeric material processes. There are still many open problems to address (see, for example, [2] and [21]). The complexity of these processes calls for an interdisciplinary approach in which modeling is intertwined with experimental characterization and verification, and motivates the formation of collaborative efforts between academic and industry scientists. We present a joint perspective, from academia and industry, on modeling fiber and film processes. A brief overview of typical processes is provided in Section 3. Examples of governing equations and a discussion of the challenges in numerical modeling of fiber and film processes are presented in Section 4. Section 5 contains a discussion of challenges and opportunities associated with collaboration between industry and academia, along with aspects of one ongoing collaborative effort between industry and academia. As the title suggests, this paper is primarily expository in nature. The testbed described in Section 5, which resulted from and supports university-industry collaborative efforts, is distinctive in the following ways. In a single location, there is a complete, advanced, three-step system for improving current fiber and film processes, and more importantly, for developing novel materials and production methods. The second step, modeling, is the focal point of the system. Step 1 is experimental characterization, e.g. rheology which provides material parameters for the model. Experimental validation, which comprises Step 3, includes state-of-the-art structure measurement. Each one of these steps is being supported (with infrastructure and personnel) as a research area, ensuring that the testbed stays at the developmental forefront. The testbed is enabling a multidisciplinary team of faculty and students, whose home departments include mathematical sciences, engineering (chemical, mechanical, and electrical), materials science, and computer science, to be relevant to the current and future needs of the fiber and film industry.
3
Typical Fiber and Film Processes
Typical processes for manufacturing polymeric fibers and films are listed in order of increasing complexity in Figure la. Wet spinning and dry spinning are classified as solution spinning processes. These processes are used for intractable materials, i.e. those that would degrade in the more common melt processes which involve high thermal and pressure gradients, have extremely high melt temperatures, or are of such high viscosity that melt processing is not possible even if the material is thermally stable. Unlike the melt processes, diffusion plays a strong role in solution spinning. A thorough description of wet and dry spinning can be found in [29]. The three remaining processes (fiber melt spinning and the two film processes) are composed of five stages, shown in Figure Ib.
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Figure 1. Aspects of common fiber and film processes: (a) Processes listed in order of complexity, and (b) Melt-process stages In the melting & mixing stage, polymer is propelled continuously in a screw barrel through regions of high temperature and pressure where it is melted and compacted, [24]. The filtration stage, where debris particles are removed from the molten polymer, presents an interesting porous media problem, especially because of the non-Newtonian nature of the fluid. At the forming stage and beyond, the processes differ greatly from one another. In the forming stage the fluid is pumped through a die which, for a fiber process, resembles a shower head, often with a thousand or more holes which are each less than 1 mm in diameter. The blown film process uses a ring with air blowing through the middle to create a bubble, and the cast film process uses a coathanger die, which ends in a very high aspect ratio slit. Fiber melt-spinning, film blowing, and film casting are illustrated in Figures 2a, 2b, and 3, respectively. Materials produced by these processes (in the order just given), include Nylon and polyester fibers, plastic bags, and packaging wrap. For each process, the molten polymer cools rapidly and solidifies after it passes through the die. For melt-spun fiber and blown film, the region between the die and the first take-up roll can be several meters in length. The distance between the cast film die and the first chill roll is often only 1 to 3 inches. Modeling of this post-die region is especially challenging because of the presence of free surfaces, shown in Figure 2b (the surface of the bubble) for blown film and Figure 3 (labeled Neck-in and Die-swell) for film casting. Fiber die-swell occurs similarly to the cast film situation. Fiber (film) die-swell occurs in the vicinity of the spinneret (slit opening), yet it can have a significant effect on process conditions and final properties of the material, [20]. The stretching or post-draw stage consists of a series
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Figure 2. (a) Fiber Melt Spinning, and (b) Blown Film Processes
Figure 3. Film casting initial stages of rolls and, in some cases, applied heat which keeps the temperature of the material above the glass transition temperature, and in the case of semi-crystalline materials such as PET, below the material's cold crystallization temperature. During this
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stage final structural properties of the fiber or film are locked-in. This stage of the oriented film process is especially complex, involving several heating and stretching phases. The typical arrangement is to have stretching in the machine direction followed by stretching in the transverse direction, as shown in Figure 4. There are many variations on each of these processes. Our intent has been to provide a brief description of their main characteristics.
Figure 4. Cast film process draw phases
4 4.1
Modeling Fiber and Film Processes Governing Equations
A mathematical model for the flow of a polymeric fluid consists of the standard momentum, mass, and energy balance equations, plus a constitutive equation representing the manner in which the stress depends on the velocity gradient. The form of the latter equation distinguishes polymeric fluids from Newtonian or Navier-Stokes flows. The general equations for flow of an incompressible continuous medium are
where p is the fluid density, v is velocity, — is the material derivative of v, a is at the total or Cauchy stress tensor, and f is a forcing term, [23]. Heat transfer effects play an important role, especially past the die region, so the conservation of energy equation is normally included also, e.g.
where T is temperature, Cp is the heat capacity, k is heat conductivity, the rate of strain tensor, 'V , is defined as
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and o" : 'j — 2, Vij'lji- The extra stress tensor can be written as a combination of i,j
the pressure p, and the extra-stress tensor, r, i.e.
where 8 is the identity tensor. Though the standard Newtonian constitutive model,
does not suffice for most polymeric fluids, in some cases the Generalized Newtonian model, where 77 is the fluid viscosity, can be used. In this equation, the shear rate, 7, is defined as
Two common examples of generalized Newtonian models are the power law model,
and the Carreau model, These models are implemented as fits to rheological data, where n, the power law exponent, is between 0 and 1, and the constants m, 770, (the zero-shear-rate viscosity) and A are fitting parameters. The Carreau model is a variation on the power law model, allowing for Newtonian-like behavior when the shear rate is low, and power law behavior for higher shear rates. A typical Carreau model plot of shear rate vs. viscosity is displayed in Figure 5. Many fluid flows associated with polymer processing are classified as viscoelastic, because they exhibit properties of both a viscous fluid and an elastic solid. The Carreau model captures shear-thinning behavior observed in polymeric fluids, but can not predict elastic effects. Viscoelastic constitutive models are normally differential or integral equations. One of the more popular differential viscoelastic models is the Giesekus model, [6]. It can be expressed in terms of a polymeric part, TP, of the extra stress tensor, where T has been split according to
and the Newtonian part, r n , is a constant multiple of 7. This would correspond, for example, to a polymer dissolved in a Newtonian solvent. In this case the Giesekus model is
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Figure 5. Typical Carre.au model curve In (4), A is a relaxation time, a is called the mobility factor and is between 0 and 1; rjs and rjp are the solvent and polymer viscosity, respectively, with Tn = 77,5-7. The upper convected derivative, TP(I) is defined as
In the non-isothermal case, viscosity, 77, and other terms in the constitutive models will vary with temperature. A non-isothermal form of the Giesekus model can be found in [17]. An important subject which is beyond the scope of this manuscript is the development, during processing, of microstructure including crystallization and molecular orientation. An understanding of structure-process conditions-final property relationships will enable producers to optimize their processes. A discussion of the equations for modeling crystallization, along with references to other sources, is presented in [17]. A thorough development of a variety of constitutive models can be found in [6]. Worth mentioning also is the pom-pom model, a recently-developed constitutive model that has the potential to be effective for a wide range of polymer flow conditions, [25].
4.2
Numerical Solution
A typical computational domain for an axisymmetric fiber, during the forming stage, is shown in Figure 6. This domain is similar to that for a center longitudinal slice of a cast film, in the vicinity of the film die. The task is to solve equations (l)-(4), on this domain, for v, p, T, and T. From [17], we have the boundary conditions
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Figure 6. Computational domain Boundary inflow wall outflow symmetry free surface
Conditions to specify specify v, T, and rp v = 0, T = J-wall vz = Vtake-up, o- : nn = 0, VT • n = 0 vr — 0, er : nt = 0, VT • n = 0 v - n = 0, a : nn + ^H = 0, cr :• I'll —
T~drag
Table 1. Boundary conditions
for the axisymmetric case given in Table 1. Here n and t are the unit normal and tangent vectors, respectively, H is the mean curvature, Ca is the capillary number, and Tdrag is drag-induced shear stress. Finding an accurate solution of the equations governing viscoelastic flow, for example on the domain just described, is not an easy task. We describe some of the challenges here. Nonlinearities. The nonlinear term v-Vv in (1) is often dropped, at least in regions where the flow is considered to be a creeping flow. However, the constitutive models, whether generalized Newtonian or viscoelastic, are strongly nonlinear. Another source of nonlinearity is in the mapping equations used for the free surface, [9]. Flow Complexities. The region displayed in Figure 6 is a simplified version of reality, yet even for this domain there are complicating factors. In the region upstream of the die outlet (where die-swell begins), referred to as a 4~t°~ 1 contraction, there will often be vortices in the corner. There is a corner singularity at the die outlet, and there are complex interfaces, e.g. air-polymer and, in the case, of coextrusion, a polymer-polymer interface. Instabilities. There are well-known flow instabilities that can occur in practice (see, for example, sharkskin in the literature). Numerical instabilities are also easily induced, [15]. Issues of Scale. Because the complex behavior observed in polymer flows is directly
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linked to molecular properties, the ultimate goal in modeling is to develop a multi-scale approach which incorporates molecular dynamics, macroscopic behavior, and intermediate levels. Currently, molecular dynamics simulations are used to predict crystallization rates as a function of stress and orientation, [18]. A multi-scale approach with respect to time is also desirable, in the sense that multi-mode constitutive models, using a series of relaxation times, are able to predict viscoelastic effects more accurately than single-mode models. Numerical Complexity. The size of the algebraic system generated by a discretization method for equations (l)-(4) grows dramatically as the mesh spacing decreases, even for 2D problems on simple domains, [7]. This motivates the implementation of advanced iterative techniques which can be ported to a parallel processing environment, especially for 3D problems. The primary method used to solve the equations governing viscoelastic flow is the mixed finite element method, ([2], [12], [15], and [17]). The approaches vary with respect to finite element basis functions, the method used to accommodate the hyperbolic nature of the constitutive model, and the iterative approach used to linearize the problem.
5 5.1
On Collaborative Efforts between Industry and Academia Challenges and Opportunities
Some of the differences between industry and academia which need to be considered in a collaborative setting are summarized in Table 2. There are exceptions to these Industry Short term deliverables Team effort Revenue-oriented Proprietary work
Academia Long term studies Individual effort Knowledge-oriented Free exchange of ideas
Table 2. Differences between industry and academia
characteristics in each case, but the comparison is generally true. Except with respect to coursework, academic researchers work under few deadlines. The indepth nature of university discovery-based research precludes the following of a strict schedule. Businesses often train their employees to work as teams, bridging and in fact building upon differences. The evaluation system in academia rewards those who are able to work independently and publish papers. (This stereotype is actually not as true as it once was, especially as funding agencies are encouraging the submission of large proposals by interdisciplinary groups). The results of industry research are often kept in-house, in order to maintain an edge over the competitor.
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This is in contrast to the drive in academia to contribute to the archival literature in ones field. None of these differences are insurmountable, and in fact in many fields, including those related to fiber and film processing, there is a great deal to be gained from collaboration between the two realms. The opportunities which motivate collaboration between industry and academia easily outnumber the challenges. We list several of these opportunities here. Reduction of industrial R&D facilities. In the highly competitive economic times which many industries are currently experiencing, on-site research and development efforts are being scaled back. With a significantly smaller investment, companies can still have research questions answered through contracts with university researchers. Universities are becoming more business-like. As state funding for public universities shrinks, faculty are being encouraged to help increase the revenue stream. Though industry contracts will not supplant large grants from government and private funding agencies, they still provide a means to support faculty, and students in particular. Industry provides interesting problems. Real-world applications are often difficult and don't fit a textbook form. However the challenge of tackling an industrially relevant problem can be invigorating when university researchers are given the chance to find innovative solutions. Potential hires for industry (and academia). Industry representatives can get to know some of the best and brightest students, since these are the ones who are assigned outside funding by faculty. Industry researchers who are considering making the leap to academia can get to know students, faculty, and administrators before taking that step. Complementary perspectives can lead to new solutions. Industry/academic collaboration provides a balance of practical and abstract thinking. Participants from each side are challenged to think outside of the box. Resources. The obvious resources provided by an institution of higher learning are bright students and faculty expertise. Universities also have well-stocked libraries and an international network of colleagues who can answer questions. Industry scientists have a wealth of experience which can help guide academic research. Another valuable set of tools which industry can offer is in personnel development, including time-management, team-building, leadership training, and safety. Testbeds for innovative approaches. The reduction of industrial R&D facilities provides an opportunity for academic institutions, if support (space and funding) is available to develop these laboratories. Industry pilot-scale facilities, where still available, can be used to validate models developed by students and faculty. A description of a current testbed in an academic setting is given in Subsection 5.4.
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5.2
The Center for Advanced Engineering Fibers and Films
Numerous successful collaborative projects between industry and academia have been enabled by the Center for Advanced Engineering Fibers and Films (CAEFF). CAEFF is an NSF Engineering Research Center, established in its current form in 1998 with funding from NSF, the State of South Carolina, and approximately 20 industry partners. A recently updated list of CAEFF industry partners is available on the CAEFF website, [30]. While the majority of CAEFF facilities and researchers are at Clemson University, MIT and Clark Atlanta University are core partners with Clemson. The CAEFF vision statement summarizes its purpose: The Center for Advanced Engineering Fibers and Films (CAEFF) provides an integrated research and education environment for the systems-oriented study of fibers and films. CAEFF promotes the transformation from trial-and-error development to computer-based design of fibers and films. This new paradigm for materials design - using predictive numerical and visual models that comprise both molecular and continuum detail - will revolutionize fiber and film development. There are several ways in which CAEFF industry partners interact with CAEFF students and faculty. These include: Directed projects. Industry partners may support directed projects which address specific research needs. These projects vary in scope and technique required. A common approach consists of three stages: material characterization (e.g. rheology measurements), modeling, and experimental validation. The latter often involves on-line property measurement. Intellectual Property issues are addressed in an Industry Partners Membership Agreement, developed with the input of CAEFF industry partners in the early days of the Center. From a practical standpoint, an arrangement can often be made for a proprietary project to include a non-proprietary subproblem for which results can be made public. Summer undergraduate research projects. During each summer, CAEFF hosts an NSF REU program. Ideas for the REU projects often come directly from industry, and an industry scientist participates in the project as a mentor. Plant trips. Visits to manufacturing facilities are always enlightening for students and faculty members. Visitors gain a sense of scale with regard to speed, size, and complexity. A healthy exchange of questions and answers always occurs during plant trips. Exchange visits and internships. One CAEFF member company allowed one of their research scientists to spend a semester on campus to help guide experimental characterization and verification efforts. His salary was considered as an in-kind payment towards the membership fee. Graduate students may participate in an industry residence program, spending a semester or more at an industry facility.
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Thesis committee member. Industry scientists are serving on thesis committees, having been granted adjunct faculty status. 5.3
The CAEFF integrated model software package
CAEFF industry partners are providing guidance in the development of an integrated model software package called FISIM (Fiber and film SIMulation). The design of this code is an interdisciplinary effort, bringing together a variety of contributors including applied mathematicians, chemical engineers, and software engineers. Two primary design criteria for FISIM are flexibility and extensibility. The code must be flexible with respect to the user's expertise, the complexity of the problem (e.g. dimensionality), and the process stages which are incorporated in the model. Extensibility is needed for adaptation to new materials and processes. FISIM is written primarily in C++ using generative programming techniques [8]. The goal is to produce a highly modular, maintainable, and efficient code which addresses the numerical difficulties listed in Subsection 4.2. The leading commercial software package for polymer process modeling is POLYFLOW™, [35]. The design criteria for the CAEFF modeling package warrant a fresh approach, with developers having full access to source code. The ID flow-induced crystallization model of Doufas and McHugh for fiber spinning is included in FISIM for the case when quick turnaround is needed and the assumption of variation of dependent variables only in the axial direction is acceptable, [14]. This module in the FISIM package is an executable generated from Matlab source code by the Matlab compiler, [34]. The finite element formulation for the higher dimensional viscoelastic model is the DEVSS-G/SUPG approach, [17], and the nonlinearity in the constitutive model is handled through fourth order Runga-Kutta time iteration, [7]. Newton's method is used to linearize the mapping equations associated with the free surface. Visual renderings are being developed by graduate students in the Clemson Digital Production Arts program. CAEFF industry partners have provided advice throughout the development of this software, including critiques of pre-release versions. A screen shot of the Welcome page for FISIM is displayed in Figure 7. The prototype application for the code is composed of three stages of a fiber melt spinning process: filtration, forming, and post-draw (stretching). The interface page which allows the user to select the stage for modeling is shown in Figure 8. Figure 9 illustrates the means by which the geometry for the spinneret capillary is specified. A similar interface is used to specify other process and material parameters. The code directly accesses a polymer database from which material properties (e.g. density and viscosity) can be downloaded and used as input values, [30]. A plot of the calculated fiber shape near the spinneret is displayed in Figure 10. Simulation results are rendered using calls to the VTK graphics package, [36]. 5.4
The CAEFF melt-spun fiber testbed
CAEFF testbeds for advanced polymer process studies have generated many theses and journal articles, and addressed research problems posed by Center industry
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Figure 7. Introductory page for FISIM modeling package
Figure 8. User interface for process selection partners. The Engineering Research Centers Association defines testbed as experimental proof of concept, technology demonstration, pre-prototype, [31]. Based on this definition, the CAEFF vision statement presented in Subsection 5.2 could be interpreted as stating that a major goal of CAEFF is to replace experimental (lab-scale) testbeds with virtual or computational testbeds. Several current projects in CAEFF combine experiment and computation. Here we present a brief discussion of the CAEFF fiber melt-spinning testbed, which consists of experimental characterization, the integrated model, and experimental validation. An exciting new direction for the experimental effort is in the determination of structure development during
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Figure 9. User interface for setting die shape
Figure 10. Die-swell simulation fiber or film processing. Characterization of on-line molecular-level development is being accomplished using X-ray diffraction and Raman spectroscopy [19]. Accurate determination of material rheological parameters is essential to successful model implementation. An example of viscosity vs. shear rate data for polypropylene is displayed in Figure 11. This data was generated using two rheometers. Steady and dynamic data (low to mid shear rate) are generated using a cone-and-plate
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rheometer, and high shear data (labeled ACER on the figure) is generated using a capillary rheometer. Rheology data is also needed to determine parameters for the constitutive equation. In Figures 12 and 13, results of the integrated model (1-D) for isotactic polypropylene (specifically Basell Pro-fax™ polypropylene homopolymer) fiber melt-spinning is displayed against on-line experimental measurements. The take-up roll speed is 1000 m/min, the spinneret temperature is 220° C, and the mass flow rate is 0.43 g/min. The non-physical slight rise in the computed temperature at the end of the spinline, an area for further study, may be related to the stiffness of the system of ODE's in the mathematical model. Polypropylene is a highly viscoelastic and shear-thinning polymer which is difficult to model as it is being processed [13]. To the best of our knowledge, there are very few papers which correlate modeling results with experimental data for polypropylene. A paper by Bhuvanesh and Gupta compares crystallinity results only, [5].
Figure 11. Rheological data (viscosity vs. shear rate) for isotactic polypropylene
6
Conclusions and Suggestions for Further Work
Accurate modeling of polymeric fiber and film processes is a daunting task which can be accomplished through interdisciplinary teamwork. This is one of many fields
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Figure 12. Model results vs. experimental measurements for diameter of melt-spun polypropylene fibers
offering opportunities for collaboration between industrial and academic scientists. Research centers, such as CAEFF, provide an environment for fruitful interaction between academia and industry. Clearly, it is not necessary to have a large-scale research center to pursue rewarding collaborative efforts. Discussions can be initiated by inviting industry scientists to give colloquium lectures, or visiting manufacturing facilities. Conferences such as the 2003 SI AM Conference on Mathematics for Industry: Challenges and Frontiers provide a venue for building partnerships between industry and academic researchers. Opportunities still abound in modeling of fiber and film processes. Hot topics include multi-scale modeling and bio-based materials. Current research involving multi-mode constitutive models is leading to more accurate models of complex flows, [26]. Though ID models have been shown successful in matching experimental data, there is a clear need for continuing development of higher dimensional simulations. Henson et al. showed that significant differences occur between ID and 2D models when comparing calculated temperature and stress distributions, [16]. One-dimensional models cannot predict die-swell, a phenomenon which may effectively change the initial fiber radius by 30 to 50 percent, [28]. Rigorous mathematical analysis is still lacking for many fiber and film models, especially due to their highly nonlinear nature. Significant advances in this regard include the work of Greg Forest and colleagues, e.g. [4, 27], and Michael Renardy, e.g. [21, 22]. New developments in discretization methods, linearization techniques and parallel solvers will lead to the design of new processes and materials that will impact many industries, including medicine, agriculture, transportation, and apparel.
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Figure 13. Model results vs. experimental measurements for temperature of melt-spun polypropylene fibers
7
Acknowledgements
The authors thank Edward Duffy, Gary Lickfield, Sandeep Sant, Stanley Sims, Barr von Oehsen, Kate Stevens, Srinivasa Varkol, and Xiaoling Wei in the Center for Advanced Engineering Fibers and Films for experimental and simulation results used in Section 5. Gratitude is also extended to the reviewers of this manuscript for their helpful comments.
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Bibliography [1] J.-F. AGASSANT, P. AVENAS, J.-Pn. SERGENT AND P. J. CARREAU, Polymer Processing: Principles and Modeling, Carl Hanser Verlag, Munich, Vienna, New York, 1991. [2] F. P. T. BAAIJENS, Mixed finite element methods for viscoelastic flow analysis: a review, J. Non-Newt. Fluid Mech., 79, 361-385, 1998. [3] D. G. BAIRD AND D . I . COLLIAS, Polymer Processing Principles and Design, Butterworth-Heinemann, 1995. [4] S. E. BECHTEL, J. G. CAO, AND M. G. FOREST, Viscoelastic free surface jets and filaments, invited contribution for Proceedings for the Symposium on Rheology and Fluid Mechanics of Nonlinear Materials, ASME International Mechanical Engineering Congress and Exposition, Dallas, TX (1998). [5] Y. C. BHUVANESH AND V. G. GUPTA, Computer simulation of melt spinning of polypropylene fibers using a steady-state model, Journal of Applied Polymer Science, Vol. 58, 663-674, 1995. [6] R. BIRD, R. ARMSTRONG, AND O. HASSAGER, Dynamics of Polymeric Liquids, Vol. 1, Wiley, Second edition, 1987. [7] A. CAOLA, Y. L. Joo, R. C. ARMSTRONG AND R. A. BROWN, Highly Parallel Time Integration of Viscoelastic Flows, J. Non-Newt. Fluid Mech., 100, 191-216, 2001. [8] K. CZARNECKI AND U. W. ElSENECKER, Generative Programming: Methods, Tools, and Applications, Addison-Wesley, 2000. [9] K. N. CHRISTODOULOU, L. E. SCRIVEN, Discretization of free surface flows and other moving boundary problems, J. Comp. Phys. 99, 39-55, 1992. [10] B. D. COLEMAN, H. MARKOVITZ, AND W. NOLL, Viscometric Flows of NonNewtonian Fluids. Theory and Experiment, Springer Tracts in Natural Philosophy, Volume 5, NY: Springer 1966. [11] M. S. CRAMER, Foundations of Fluid Mechanics Website, http://www.navier-stokes.net/nsintro.htm, 2004.
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[12] M. CROCHET, A. DAVIES, AND K. WALTERS, Numerical Simulation of NonNewtonian Flows, Elsevier, 1984. [13] B. M. DEVEREUX AND M. M. DENN, Frequency response analysis of polymer melt spinning, Ind. Eng. Chem. Res., 33, 2384-2390, 1994. [14] A. K. DOUFAS, A. J. McHuGH, AND C. MILLER, Simulation of melt spinning including flow-induced crystallization Part I. Model development and predictions, J. Non-Newt. Fluid Mech. 92, 2766, 2000. [15] V. J. ERVIN AND W. W. MILES, Approximation of time-dependent viscoelastic fluid flow: SUPG approximation, SIAM J. Num. Ana., 41, (2), 457-486, 2003. [16] G. M. HENSON, D. CAO, AND S. E. BECHTEL, AND M. G. FOREST, A thinfilament melt spinning model with radial resolution of temperature and stress, J. Rheology 42(2), 329-360 1999. [17] Y. L. Joo, J. SUN, M. D. SMITH, R. C. ARMSTRONG, R .A .BROWN, AND R. A. ROSS, Two-dimensional numerical analysis of non-isothermal melt spinning with and without phase transition, J. Non-Newt. Fluid Mech. 102, 37-70, 2002. [18] M. S. LAVINE, N. W. WAHEED, AND G. C. RUTLEDGE, Molecular dynamics simulation of orientation and crystallization of polyethylene during uniaxial extension, Polymer, 44(5), 1771-1779, 2003. [19] R. P. PARADKAR, S. S. SAKHALKAR, X. HE AND M. S. ELLISON, On-Line Estimation of Molecular Orientation in Polypropylene Fibers Using Polarized Raman Spectroscopy, Applied Spectroscopy 55(5), 534, 2001. [20] M. PASQUALI, Swell properties and swift processing, Nature Materials, Vol 3, August, 509-510, 2004. [21] M. RENARDY, Mathematical Analysis of Viscoelastic Flows, SIAM, 2000. [22] M. RENARDY, Current issues in non-Newtonian flows: a mathematical perspective, J. Non-Newt. Fluid Mech. 90, 243-259, 2000. [23] J. SERRIN, Mathematical Principles of Classical Fluid Mechanics, Handbch der Physik, Springer Verlag, Vol. VIII, 125-263, 1959. [24] Z. TADMOR AND C. G. GOGOS, Principles of Polymer Processing, Wiley, 1979. [25] W. M. H. VERBEETEN, G. W. M. PETERS, F. P. T. BAAIJENS, Differential constitutive equations for polymer melts: The extended Pom-pom model, J. Rheol, 45, pp. 823-843 2001. [26] W. M. H. VERBEETEN, G. W. M. PETERS, F. P. T. BAAIJENS, Viscoelastic analysis of complex polymer melt flows using the extended Pom-Pom model, J. Non-Newt. Fluid Mech. 108 (1-3), 301-326, 2002.
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[27] Q. WANG, M. G. FOREST, AND S. E. BECHTEL, Modeling and computation of the onset of failure in polymeric liquid filaments, , J. Non-Newt. Fluid Mech. 58 97-129, 1995. [28] J. L. WHITE AND J. F. ROMAN, Extrudate swell during the melt spinning of fibers - influence of Theological properties and take-up force, Journal of Applied Polymer Science, Vol. 20, 1005-1023, 1976. [29] A. ZlABlCKl, Fundamentals of Fiber Formation, John Wiley and Sons, 1976. [30] Center for Advanced Engineering Fibers and Films, www.clemson.edu/caeff [31] Glossary of the Engineering Research Centers Association, www.erc-assoc.org/ manual/bp_ch6 _attach6_4.htm [32] Journal of Applied Polymer Science, www3.interscience.wiley.com/cgi-bin/jhome/30035 [33] Journal of Non-Newtonian Fluid Mechanics, www.elsevier.com/ wps/find/journaldescription.cws_home/502693/descrip [34] The Matlab Compiler, www.mathworks.com [35] POLYFLOW Software Package, www.fluent.com/software/polyflow/ [36] The Visualization Toolkit, www.vtk.org
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Helpful Hints for Establishing Professional Science Master's Programs Charles R. MacCluer* and Leon H. Seitelman^ Abstract In recent years, there has been an increased interest in developing business- and industry-focused programs of study in mathematics and science at the master's degree level as an alternative to the traditional programs. The newest programs in mathematics, many of which were begun with seed money from the Alfred P. Sloan Foundation, are now beginning to reach maturity. Many other institutions are considering starting programs of their own. This paper offers suggestions about best practices for setting up such programs.
1 Introduction We offer two perspectives on these new professional science masters (PSM) programs: the focused view of a principal developer of one professional program, and the broader view of an industrial mathematician who conducted a Sloan-sponsored survey of master's degree programs in applied mathematics, with a focus on PSM programs. Our goal is to inform those with an interest in setting up such programs about the "do's and don'ts" of program development. We will also highlight some of the habits that prospective students in such programs need to develop: persistence; working in teams; interacting with nonacademics and accommodating to their needs; developing the important non-technical skills of writing and presentation; and dealing with deadlines and non-technical professionals. * Professor of Mathematics and Coordinator of the proMSc program in Industrial Mathematics, Michigan State University, East Lansing, Michigan. ^Consultant, Glastonbury, Connecticut.
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2
The changing world of work
Motivating this metamorphosis of technical education is the need to respond to changes in the workplace. The global marketplace has increased pressure on all businesses (especially technology-based ones) to produce more for "the bottom line." Greater strides in operational efficiency have certainly resulted, but it is also true that the "skunk works" operations (e.g., Bell Labs, Watson Research Center, Lincoln Labs, etc.) that spawned many past technological revolutions no longer enjoy blue sky funding, and have instead become corporate profit centers. Open-ended research projects have become increasingly unpopular with the many technically unsophisticated decision makers who are driven by increasingly shorterterm, profit-focused business models. We do not endorse this mind-set. It is simply there; knowledge workers must adapt to, and function within it. As mathematicians might say, "It is a boundary condition."
3
Implications for program development
Many traditional technical programs currently allow their business-naive graduates to leave for industrial employment totally unprepared for the environment and working conditions in which they will work, and in which they hope to contribute and advance. A program that couples a strong technical background with a knowledge of business practices will equip students for substantial roles in both product development and organizational management. PSM programs should provide real workplace experience. Only with this kind of real-world exposure can students prepare to contribute within an environment in which deadlines, interdisciplinarity, teamwork and communication are central.
4
Sloan steps up
The Alfred P. Sloan Foundation has supported the development of new PSM programs in a number of scientific fields. The goal of this "seed money" was to discover which program models seemed to be most successful, so that they could ultimately be replicated. Although most of the mathematics-related programs (see http://www.siam.org/students/sloansurvey.htm) are just now reaching maturity, we can make some reasonably general (and widely applicable) statements about what does, and does not, work. What now follows is a realistic (but partial) list of steps necessary for establishing a PSM program. We draw freely from the experiences in constructing the Michigan State University program.
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5
Reaching out to industry
At MSU we received solid support for the PSM programs from the chairs, dean, and provost—the cachet provided by a significant Sloan Foundation PSM start-up award certainly helped. But you must determine if your administration will support industrial outreach efforts on behalf of terminal Master's degrees. They may instead believe that industrial outreach should lead to significant faculty research contracts. PSM programs may not fit with your institution's goals. Let us assume—a major assumption!— that your administration is persuaded. Now the work begins: Developing contacts who can provide student access to realworld problems requires enormous effort. But absent this effort, the chance of program success is minimal.
6
Form an industrial advisory board and take their advice!
An industrial program must be grounded in the realities of the workplace. Faculty have little experience with, or first-hand knowledge of, what actually takes place in the workplace. Advice from people in situ is essential. A board's advice on curriculum, program structure, and professional development is invaluable. It might seem advantageous to pack the board with high-level executives, but the best advice (and cooperation) will come from project-level managers; these managers have fresh experience with new hires and their deficiencies. Try to select alumni for the board; they are likely to have greater patience and a sincere interest in a successful outcome. Choose representatives from companies large and small, local and national. A board of about a dozen has worked well for the MSU program; its current makeup consists of representatives from Ford (2 members), General Motors, DaimlerChrysler, National Security Agency, Veridian, Delta Dental, AAA, S.E. Michigan Council of Governments, McCleer Power, Neogen Corp., and WatsonWyatt. Anticipate that one-third of the board members will be unable to attend any given meeting because of last-minute emergencies at work. The exigencies of business trump longer term priorities. In any meeting of faculty with the industrial board, the dynamics of the meeting will be unproductive unless the industrial people far outnumber faculty. The industry people must feel relaxed and free to brainstorm. Faculty need to listen rather than lecture.
7
Incorporate industrial projects into the curriculum
A portfolio of completed industrial projects is an enormous advantage to your graduates when they interview for industrial positions. Obtaining such projects is relatively simple once you know the "Maki method." D. P. Maki, former Chairman of the Indiana University Mathematics Department, has been soliciting industrial projects for students for decades. In MSU's version of the Maki method, a paid PSM student is given an office with a telephone and an answering machine. The student
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places cold calls to local companies, explaining to the receptionist that the department is seeking projects, and asking for contacts within the company. The student explains our problem-solving service to the contact, and appeals for "help in better educating our students." The first project is often offered on a complimentary basis. The student calls back repeatedly; industry people are busy and depend on persistence. The rule is simple: Never give up on a contact until they have supplied two more contacts. Jot down company names seen in travels about town. Ask faculty for their contacts within local industry. Talk with people on airplanes, at gatherings, and at social settings. Never miss an opportunity to network. Ask prospective project proposers for backburner projects that they would attack, given the availability of time and resources. Offer them a team of three very smart graduate students to work on their project. Their only commitment is the assignment of a company liaison to propose the problem, to occasionally field clarifying questions from the team, and to arrange for a presentation on site at project end. Any interesting question should be welcomed. At MSU no time is spent clarifying and presolving the problem to anticipate its possible mathematical content and tools required. A better student project results when a company's ill-posed problems are left for the students to untangle. This experience should be a problemsolving, rather than a mathematical problem-solving experience. Occasionally team members will continue on with the project as paid consultants after term's end. MSU has far more projects proposed than it has teams to work on them. Other PSM programs substitute internships or co-ops for this industrial experience with mixed successes (since the university loses control over the experience). Some internships are carefully structured and skillfully managed. Such internships rob the student of the inestimable but uncomfortable experience of an ill-defined problem statement and the real possibility of team failure. Other internships are little more than office help. The industrial project, as described above, promotes more effective problem solving and team-focused efforts.
8
Cultivate faculty facilitators
A significant hurdle is the need to enlist faculty volunteers to facilitate the development of the student teams. All faculty members believe that they are completely overloaded and can take on no further work. Moreover, they are reluctant to take on industrial problems, worrying, "What if we can't solve the problem?" They must be reassured that one can always bite off and solve a portion of any industrial problem, almost always to the delight of the company. Most of the work will be in clarifying the problem, at which mathematicians are truly gifted. Choose faculty with little or no experience in the particular problem area. It has been the MSU experience that, the more the faculty knows, the more they meddle; and the more they meddle, the worse the final project outcome. Insist that faculty members serve only as facilitators, not true managers. (This insistence will also ease their fears.) Moreover, because of intellectual property issues, it is best for the faculty not to be involved intimately with the project development.
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Once a program has a project track record, it can begin to charge small amounts for projects to be used as stipends (i.e., bribes) for faculty to facilitate projects. MSU now has more faculty volunteers than it can employ.
9
Provide professional development
The goal of PSM programs (to put it indelicately) is not to train "cubical rats" to serve as cannon fodder for the management MBAs. Instead, we shape our graduates to eventually take on these management roles. PSM graduates must understand the fundamentals of business and must be skilled in writing and oral presentation skills. They will be judged in industry not only by their work but by how well they present their work. So it is imperative that the students be exposed to the fundamentals of business communication as well as project, marketing, and financial management; managerial accounting; intellectual property law; consensus building; etc. At MSU, this is accomplished through 10 intensive weekend modules developed and presented by the MBA faculty. We also invite "headhunters" to visit to instruct students on resume writing, grooming, and preparing for interviews. This continuous exposure to the language and culture of business is an enormous advantage for students when they begin to interview. All of our graduates have easily found excellent positions with an average starting salary of over $ 54,000, a bump of more than $10,000 over an academic Master's. Their present employment can be examined at the Resume link from our program's web site http://www.math.msu.edu/msim. Students at MSU receive a transcriptable Certificate in Business and Communication for their professional training. Information about this Certificate is available at the web site http: //grad. msu. edu/all/bus_mgt. htm
10
Mount a branding campaign
The title "industrial mathematician" is at present relatively unknown. Program developers should take any opportunity to reach out to the business community in order to advertise their product—their "universal problem solvers." A track record of diverse past projects is quite persuasive. Faculty can then plausibly argue that they are training versatile, self-starting, quick-study employees with a deep understanding of computational issues. At the same time, recognize that, while mathematicians are extremely capable of performing well in problem-solving jobs, very few jobs in industry explicitly advertise for mathematicians. For this reason, it is imperative to imbue the PSM students with the idea that they should seek out opportunities by function, rather than by profession.
11
Choose a committed coordinator for the program
This cannot be an add-on duty for a faculty member. The coordinator must be essentially full time. There is a heavy load of day-to-day management: recruiting, teaching, advising, industrial outreach, project development, advertising, resource
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generation, etc. As one looks around the country, the single most effective characteristic of successful programs has been the existence of a committed and energetic coordinator. Programs without committed coordinators are failing.
12
Recruit, recruit, recruit
Establish a splendid web site describing the program—it is your most effective recruiting tool. MSU has tried many recruiting techniques such as writing personallyaddressed, personally-signed letters to all students who took the GRE and expressed an interest in graduate study in mathematics. We also seize on any opportunity to speak to undergraduates at student colloquia or mathematics conferences. But these approaches are too often too late—seniors who would be attracted to industrial work have already transferred to other majors because they were unaware of the rewarding PSM career path. Recruitment must start earlier. To this end Sloan has advertised in the newsletter of the National Career Development Association and similar organizations. Recruiting is the single most troublesome issue, requiring an enormous amount of one-on-one communication with potential recruits.
13
Resources
The A. P. Sloan Foundation has been the single most effective proponent of PSM programs. The Foundation recognizes that national competitiveness in modern, increasingly science-based industries depends on having science-trained professionals at policy-making levels of industry and government. The Foundation has provided crucial seed money for program start ups and continues to support existing programs in many ways. For example, the foundation maintains an extensive clearing house for information about all PSM programs at http://www.sciencemasters.com Detailed information about the 7 PSM programs at Michigan State is found at http://www.ns.msu.edu/prospective/grad/profmasters.htm and information about the industrial mathematics program in particular at http://www.math.msu.edu/msim
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Author Index
Ames, R.,45 Betts, J. I, 140 Campbell, S. L, 140 Carlson, C. D.,222 Cavendish, J. C.,87 Cosner, R, R.,24 Cox, C. L.,222 Dembart, B., 178 Ferguson, D. R.,24 Field, D, A., 158 Gonsor, D., 178 Koopman, J. S.,200 MacCluer, C. R.,242 McCartin, B. J., 100 Moore, E. L F., 125 Neamtu, M., 178 Parmentola, J. A.,5 Peters, T. J., 125 Pratt, M. J.,58 Seitelman, L. H.,242 Simon, C. P., 200 Tanenbaum, P. J., 117 Yeakel, W. P., 117 Young, P. D,, 100
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