Numerical Solution of Partial Differential Equations in Science and Engineering

0

2

0.25

dt

:

0.25

+ k<j>

d^

:

0.75

dt

+ k-,

0.75

^Tfloo

Ά —

^^elo.25 ^Tjn 7 5

•By a spline, p. we refer here to a function on the region D which, in the vicinity of a point within D, can be represented by a polynomial of a specified degree, m say. This function and its first m — 1 derivatives must be continuous and the mth derivatives are square integrable. The knots of the spline ρ are the real numbers / , t„ at which we constrain the behavior of p. 0

Introduction to Finite Element Approximations

59

A n important characteristic of the collocation scheme is illustrated in this expression. Because the collocation point is not located at a node, the algebraic equation is dependent only on information associated with its host element. T h u s the second equation in (2.2.22) does not link Γ to 7", as was the case in the methods above. Evaluation of the coefficient matrix and incorporation of the initial condition yields 3

ACT;

(2.2.23)

kT

e

Equation (2.2.23) provides the following solution for the coefficients: [Γ,,

Γ , 2

Γ ] = [1, 3

0.667,

0.555].

Although it is apparent that the collocation scheme performed relatively poorly in this simple example, the method is nevertheless attractive from several points of view. The coefficient matrix is easily generated, even for very complex operators. The unusual matrix structure (not all collocation matrices are triangular and amenable to immediate forward substitution) suggests the possibility of computationally efficient equation solvers, and the straightforward and relatively simplistic nature of the underlying theory suggests that the approach should be readily understood by the practitioner, who may have a minimal background in numerical methods. It should also be pointed out that collocation is generally used either with basis functions of higher-order continuity (e.g., spline functions in C * where m > 0 ) or with trial functions exhibiting global support (e.g., a single higher-degree polynomial is used over the whole domain of interest). It is particularly interesting to examine the relationship between collocation and Galerkin's method. To do this, let us digress for a moment and reconsider the matrix equation that arises out of (2.2.15), the Galerkin formulation, when a one-point Gaussian quadrature is used to evaluate the integrals appearing in the coefficient matrix. Although the integrals in this case are easily evaluated, in general it is advantageous to operate in a dimensionless ξ coordinate system where - 1 « £ ξ < 1 . Gaussian quadrature information is readily available in standard texts for this coordinate system. T o employ this approach, a simple m

* C ' denotes a class of functions that have continuous derivatives up to the mth order.

60

Basic Concepts in the Finite Difference and Finite Element Methods

coordinate transformation, as illustrated below, is required.

Cii£*< "*<*>)* +

= / ' , ( t f * .

+

d^

di " | = o - • r ,

v

/

i

4

^ ) i " i = / : ^

^ . ) ^

+

. < Η-» πΗ(- !)· 2

+

|+

: 0

where the Gaussian point is located at ξ-O* and the weighting function is 2. T h e use of single-point Gaussian integration for the remaining elements of (2.2.15) yields

-5 - ± i. -H-.

-

1+

(2.2.24)

+4

0

+4

0

τ

2

Ά T

—1 —

2

kT

e

kT 2

e

3

Equation (2.2.24) is similar to (2.2.16), which was obtained through an exact integration of the Galerkin equations. T h e more interesting comparison, however, involves the collocation matrix of (2.2.23). Indeed, the second and third equations, the only ones that must be satisfied in view of the boundary condition, are completely equivalent. This is easily verified by multiplication of (2.2.23) by a factor of { and addition of rows two and three to obtain row two of (2.2.24). Thus the orthogonal collocation method can be interpreted in this example as being analogous to Galerkin's method when the Galerkin integrations are performed using Gaussian quadrature. This interesting observation can, in fact, be readily extended to more complex differential operators and other basis functions (see Prenter, 1975).

2.2.3

The Choice of Basis Functions

The accuracy and efficiency of a M W R scheme is largely dependent upon the choice of basis functions. In the preceding section we introduced the piecewise linear chapeau basis function. This is perhaps the simplest of the many •Gaussian quadrature is most conveniently performed over the interval - Ι < { < 1 , with the basis functions written in terms of {. The derivation of φ,({) will be outlined shortly, whereupon the relationship should become apparent.

DT/DI = \

Introduction to Finite Element Approximations

61

piecewise continuous polynomials that can be used effectively as bases. Figure 2.5 reveals that the chapeau function spans two elements with two nodes located on each element. In this section we introduce higher-degree polynomials as bases, with the expectation of enhanced accuracy. Linear Polynomial Bases. The chapeau functions serve to interpolate nodal information linearly over each element. In other words, one can obtain a solution anywhere within an element, given only the solution at the nodes. Consider, for example, the most accurate solution we were able to obtain using the Galerkin approach in the preceding section. We have at the nodes [Γ„

Γ ,

Γ ] = [1,

2

0.678,

3

0.571].

Suppose that a solution is required at the point r = 0 . 2 5 , which does not correspond to the location of a node. From (2.2.11) we have 3

T~T= Σ 7>,(0. which for the case of / = 0.25 becomes 7ΐθ.25

* ^Ίθ.25 -

Τ\Φ\lo.25

+

^"klnJS

+

ΆΦΐΙ10.25

= ( l ) ( 0 . 5 ) + (0.678)(0.5) + (0.571)(0.0) = 0.839. The analytical solution yields 7/(0.25)=0.803. Quadratic Polynomial Bases. Let us now consider the possibility of introducing a quadratic interpolation over each element. The interpolating function, which is also a basis function, is of the form $ = a + bt + ct

(2.2.25)

2

over each element. Because this is a quadratic function, three point values are required to uniquely define the coefficients a, b, c. We therefore require three nodes in each quadratic element, as illustrated in Figure 2.6. Three conditions must be imposed on (2.2.25) to evaluate the three constants a, b, and c. We require the basis function for node / — 1 to be unity at node / — 1 and zero at the remaining two nodes. In matrix notation, this constraint becomes a (2.2.26)

0

b

0

c

«2

Basic Concepts in the Finite Difference and Finite Element Methods

NOD*

Figure 2.6. Quadratic basis functions φ, appear in (ft) and a function represented by the series f1-ιΦ -ι + 7ΐΦ, + 7',+ ιΦ,+ ι. in (a). 7

(b)

1

This equation can be solved to yield (2.2.27a)

a

= ~

(2.2.27b)

b =

(2.2.27c)

c =

·'*«+!

1

The basis function is now obtained by substituting (2.2.27) into (2.2.25) to give (2.2.28a)

_

Φί

ί ^±ΐ)-{η±1ΐ±ΐ\ {

ι =

Δ,_,/

\

.'

2

ί

Δ,_,

+

/'

Δ,_| '

where Δ , _ , =(/,. - ί,_, X r , , - f,_,). An expression analogous to (2.2.26) may be obtained for each of the remaining two nodes in the element by imposing the same type of conditions as indicated above. The resulting basis functions are +

(2

.. 2

28b)

Λ =

(!^).(^±^), £. +

introduction to Finite Element Approximations

where Δ, =(i, - r,_ , X f

(+

x

«3

and

-t,)

- /,χ/ , - /,_,). where Δ , Whereas the quadratic basis functions defined through (2.2.28) can be used directly in the MWR, the integrations, particularly for the Galerkin method, become somewhat tedious. It is advantageous to develop the higher-degree polynomial basis functions in the dimensionless ( coordinate system (— 1 < ξ < 1) described in the preceding section. This choice of coordinate system facilitates numerical integration by Gaussian quadrature. The quadratic basis functions can be developed in the ξ coordinate system in a manner analogous to that employed above for the t coordinate. The resulting functions are defined in Table 2.4 and will appear as indicated in Figure 2.7a. To illustrate their development, let us derive the dimensionless equivalent of (2.2.28a) using the approach outlined in (2.2.2S), (2.2.26), and (2.2.27). The quadratic polynomial used as a point of departure would be 1 +

ί +

*U) = a + bS +

(2.2.29)

ce.

By introducing the requirement that φ _ | | _ , = 1 and φ _ , | = φ _ , | , = 0 , we obtain the matrix equation 0

Γ

(2.2.30)

0 .0.

=

1 1 .1

Γ

-1 0 1

0 1.

a b

c,

TABLE 2.4. Linear, Quadratic, and Cubic Basis Functions Defined in the Dimensionless ξ Coordinate System Degree Linear

Quadratic

Cubic

Function

Form(-l<$
*-,(*)

Φ,α)

id +

Ί )

Φ^α) φ.(«")

ItO + t)

Φ-ι(*{)

M-n

Φ-Ι/3(«

loV-i ~H+\)otMH-m -\)

Φ./3(Ί)

τέ(-3{ -Ί +3{ + 1)ΟΓ-^(3€ + ΐΧ{ -1)

ΦΜ)

+ 9 { + { - 1 ) or ^ ( 1 - { X 9 € - 1 )

3

2

2

2

2

2

3

2

+ 9 € -1 -1) or A<1 + $X9€ - 1 ) 2

2

64

Basic Concepts in the Finite Difference and Finite Element Methods

(b)

Figure 2.7. Quadratic (a) and cubic (b) basis functions defined in local ξ coordinate system.

which provides the following solution for a, b, and c: (2.2.31)

[a, b, c] = [0,

-l2,

i].

Substitution of (2.2.31) into (2.2.29) yields the required basis function, (2.2.32a)

Φ-,(0=-«(1-0·

The remaining two bases can be formulated in an analogous manner and the resulting equations are (2.2.32b)

Φο(0=ι-£ 2

and (2.2.32c)

Φ.(€) = *€(1 + 0 ·

To demonstrate the use of higher-degree polynomial basis functions defined in the dimensionless | coordinate system, let us reexamine, using quadratic bases, the problem solved earlier using the Galerkin method with chapeau functions. The governing equations are (2.2.9)

^ + k(T-Te) = 0

Introduction to Finite Element Approximations

65

subject to Γ(0) = 1, Γ, = 0 . 5 , k = 2 , and 0 < f < 1. The approximating equations generated using the Galerkin method are similar to those expressed by (2.2.15). However, because a single element constitutes the entire domain 0*s χ «ε 1, the coefficient matrix is full. (2.2.33)

/„'(

λ

*

1

+

/

' ( i h / „ ' (

0

ΤΙ

CkT^dt

o

J

= 73

CkTfodt

o

J

CkTfrdt 0

The integrals appearing in the coefficient matrix may now be evaluated using the bases defined in the dimensionless ξ space. A typical integration proceeds by first transforming from / to ξ.

( 2 2 M )

/ ' ( ^ r ^ . « ) **.(i)*.«>)<* +

0

- / . ' , ( ^ $ • . ( « + *•.(«•.(«) Because d£/dt

(2.2.35)

= 2, (2.2.34) becomes

^(^ίΙφ,ίΟ + Λφ,ίΟΦ,ίΟ)* k

=£J^*,(0+t*.<0*.(OI<«2' One has the option at this point to integrate directly or to employ numerical integration. Whichever route is followed, the evaluated integrals are substituted

66

Basic Concepts in the Finite Difference and Finite Element Methods

into (2.2.33) to give the final matrix equation: k 15 8A: o+ 15 k 15

-1+2* 2

15

3

15 30

k 30 k 15 2k 15

1 6

h

lr k e

= tr k e

iT k e

Solution of (2.2.36) yields [Γ,,

T, 2

Γ ] = [1,

0.696,

3

0.571].

Now using the quadratic interpolation function to find values between nodes, we obtain f(0.25) = 7 > , ( 0 . 2 5 ) + 7- φ (0.25) + 7 > ( 0 . 2 5 ) 2

2

3

= ( l ) ( ! ) + (0.696)(t) + ( 0 . 5 7 1 ) ( - i ) = 0.826. Similarly, 7X0.75)=0.611. In this example the quadratic element formulation provides less accuracy at the nodes than the linear elements, but intermediate values are more accurate. In fact, the linear and quadratic elements in this case are both of accuracy 0(h ) (see Gray and Pinder, 1976). 2

Cubic Polynomial Bases. Although there is no theoretical limit to the degree of polynomial one could generate as a basis function, the cubic is the highest degree found in general use. The reasons are threefold: (1) the higher the degree of the polynomial, the more computational effort required to solve the approximating algebraic equations arising in the M W R , (2) the computational effort required to generate the coefficient matrices using numerical integration increases as the degree of the polynomial increases, and (3) higher-degree polynomials lead to larger element sizes that may be undesirable in problems involving nonhomogeneous media. The cubic basis functions are defined in Table 2.4, illustrated in Figure 2.7Λ, and obtained in the same manner as the quadratic bases. The cubic basis function described above interpolates along the element using a Lagrange third-order polynomial. Although this choice of function is a natural extension of the quadratic case, there is another cubic basis, known as a Hermite polynomial, which can be particularly effective in certain problems. We will turn our attention soon to the development of this basis function.

Introduction to Finite Element Approximations

67

Lagrange Polynomials. T h e basis functions developed thus far can be easily formulated directly using Lagrange polynomials. T h e Lagrange polynomial, /,(£), is given by

,„ _

U-U-··(£-*,-.)(*-*,+.)···»-*,,) α-€ο)···(*,-ί,-,)α-^,)···α-υ'

m

,KK)

Let us consider the case of linear basis functions where η = 1 a n d only one term of the polynomial is involved. Because only two nodes are available, we know that for φ , ( | ) with i — — \ we require «*,· , = 1 a n d ξ, = — 1; similarly for ι = 1, ι = — 1, and <* = 1. Using these values of £ , we obtain +

(

A

A —;M

- _ Azii+i - _Lli

and i-i,-

As a second example, consider the quadratic bases. For the basis function defined for the center node of an element, ξ = 0, the three values of £ to be considered are ξ,_, = — 1, £ , = 0, £ , = 1. Selecting those terms of the Lagrange polynomial containing these values, we obtain k

1 +

The basis function ψ _ , involves the following values of £ : k

From the definition of / " we can write

^

_,2i i =

~

]

U-U.)U-^,->-2)

(ξ,-ί,

+

1

)(Ι,-ξ,

+ 2

)

(*-ο)(*-0 * -1 2

=

(-1K-2)

=

2

The function of φ, can be obtained in an analogous manner. Hermite Polynomial Bases. Hermite polynomials are cubic splines that display, in addition to second-order derivative continuity ( C ) over the element, first-order derivative continuity ( C ) between elements. Thus Hermite interpolation not only interpolates knot values of the function b u t also interpolates knot values of a given number of consecutive derivatives. This property is particularly attractive for a variety of reasons when the solution 2

1

68

Basic Concepts in the Finite Difference and Finite Element Methods

resulting from one simulation step must be differentiated for use in a second step. This situation arises, for example, when a flow-potential function is differentiated to obtain a velocity field, which, in turn, is used as a known parameter in the convective diffusion equation. Because the Hermite is a cubic polynomial, four parameters are required to define it uniquely. Thus four conditions must be imposed to evaluate these parameters. In the case of the Lagrange cubic described earlier, we simply required each basis function to be unity at the node at which it was defined and zero at the remaining three nodes in the element. The conditions imposed on the Hermite, however, reflect the new constraint of first-order continuity between elements. To see how these alternative constraints arise, consider the general expression of a function Τ in terms of Hermite bases, dT

N

(2-2.37)

2

T~T=

j=\

Tj*>j +

-7T*ij,

where Ν is the number of nodes. Examination of ( 2 . 2 . 3 7 ) reveals that the Hermite expansion differs from earlier approximations in two important ways. 1.

Each node or knot has two undetermined coefficients (i.e., 7J and dTj/dt).

2.

Each node is identified with two basis functions

υ

As in the case of C ° polynomials, the basis functions and ψ , , should be defined such that T \ = Tj. In addition, we now require a similar constraint on the derivative, that is, 0j

t = j a t

dT dt

_

dT

J

To establish the relationship between the basis functions and the derivative constraint, let us examine the derivative of ( 2 . 2 . 3 7 ) , dT

df "

dt ~ dt

d^

=

L

,

7=1

> dt

d ^ d ^ dt

dt '

Inasmuch as the basis functions are more conveniently expressed in ζ coordinates, while we require our solution in the global t coordinate system, we now apply the chain rule of differentiation to yield

Introduction to Finite Element Approximations

69

The term *oj dt

dt dt

d T

J

will always vanish at the nodes because d4> /dt at all nodes. To assure that

is defined such that it is zero

0j

dT

dT

j dt '

dt

then d^j

dt

dt

dt

must be unity at node j and zero at all other nodes. This is equivalent to forcing d$ /dt to vanish at all but the yth node, where we impose the constraint that d$ /dt equal dt/dtIt is reasonable to assume that Xj

Xj

Ar _ A/

dt_

dt * Δ{ "

2 ·

is not N o t e that for a variable spacing (i.e., Αχ, ^ Δ χ , . , ) dt/dt^t/At continuous. Given these constraints, we are now prepared to establish the form of φ and φ . Let us assume, as before, that the interpolating function can be written in terms of a cubic polynomial defined in the dimensionless t coordinate system, 0 ;

υ

(2.2.38)

*

Oj

= a

+ bt

0

+ c t

o

+

2

0

d e 0

and (2.2.39)

dt

•b

0

+ 2c t

+

0

Because of the constraints imposed on φ the particular case of φ :

3d t . 2

0

we can establish the following for

0 ;

ο ι

(2.2.40a)

4>o\\-\-' - o~ o

(2.2.40b)

ΦοιΙι =°=a

(2.2.40c)

(2.2.40d)

[ a

0

d

0[

+

b

o~ o>

c

+ b + c + 0

0

dt

d, 0

= 0 = 0 + Z ) - 2 c + 3i/ , 0

dt

d

= 0 = 0+b

0

0

+ 2c

0

0

+

3d . 0

70

Basic Concepts in the Finite Difference and Finite Element Methods

These equations can be written in matrix form as -1

-1

(2.2.41)

1

«0

1

0 0

bo o do

3

c

3

0

The solution of this equation yields the vector of coefficients 2

bo Co

(

' 4

2

-3

3

0

A.

1 - 1 -1

0 - 1

1

-

1

1

' 1"

-1

0 0

1 1

2"

' — J.I

4

-3 0

0

1

The basis function

(2.2.42a)

* b i = i ( * - l ) t t + 2).

A similar argument can be developed to yield an analogous expression for the other node in the element. The resulting equation is (2.2.42b)

Φθ2 = - ί ( ί + 0 ( * - 2 ) · 2

The interpolation function for the first derivative is also derived from a cubic polynomial, (2.2.43)

= a + b i + cxe +

4,

x

n

x

d e. x

Imposition of the appropriate constraints on (2.2.43) yields (2.2.44a)

Φ | _ , = 0 = A , - A, + C | Μ

(2.2.44b)

(2.2.44c)

Φ

d* di {

di

| , = 0 = Α , + Α + ο, + Λ , ,

ι

Ί/Φ,,

(2.2.44d)

-d,,

ι

1

= γ

1

=0+A -2c,+3i/ , l

l

=0=0+A,+2c,+3ii . 1

Introduction to Finite Element

71

F r o m (2.2.44) we obtain the matrix equation 1 1 (2.2.45)

-1 *l

-2

C

l

=

2

" 0 0 At 2 . 0 .

which yields 2 -3

2 3

0

0

1

1

1 1

-1 -1

-1 1

0 0 Δί 2 0

8

The basis function associated with the derivative of Τ is now obtained by substitution into (2.2.43): (2.2.46a)

•„ = χ ( ί + ΐ)(ί-ΐ)

2

An analogous development yields the corresponding function at the other node in the element (2.2.46b)

φ,2 = γ ( ί + 0 ( ί - ΐ ) · 2

It is sometimes advantageous to define the basis function only in terms of ξ. T h u s we can express the new basis function as

"

_


where φ',, is given by (2-2.47)


2

and is illustrated in Figure 2.8c. Similarly, from the function φ (2.2.48)

Ι 2

we obtain

Φ, = Η€ + ΐ Γ ( * - ι ) . 2

T o clarify the concepts involved in the use of Hermite polynomials, let us consider once again our example problem using Galerkin's method. T h e

72

Basic Concepts in the Finite Difference and Finite Element Methods

•Ojlt)

•lift)

0.5 1

'2*·

x

3

(b) Figure 2.8. Hermite cubics in global and local coordinates where Φ

=Φ'

ώ

(modified after Doherty, 1972).

objective is to solve the equation dT + k{T-T ) dt

(2.2.9)

=0

e

subject to T(Q) = l,

T =0.5,

k=2,

e

0
The first step in generating the approximating algebraic equations is to expand the unknown function to give dT

2

(2.2.49) j = \

where, because the problem consists of only o n e element, there are two nodal points. Because we are using Hermite polynomials, the finite element net will be modified slightly from the previous case (Figure 2.5), as indicated in Figure 2.9. Galerkin's method requires that the residual obtained through substitution of (2.2.49) into (2.2.9) be made orthogonal to all bases. This can be expressed algebraically as

(2.2.50)

/ ' { - £ + *(Γ-7;)}φ Α=0. ο

β Ι

ι = 1,2;

α=0,1.

Introduction to Finite Element Approximations

Φ

t /°

73

φ

4

\

t

0.0

i.O

^2 0

Figure 2.9. Hermitian basis functions for example problem.

ζ

t

defined

By substituting (2.2.49) into (2.2.50). we obtain

- CkT 4> ,dt = 0 , e

i = l,2; « = 0,1.

a

Equation (2.2.51) can be written in matrix form as (2.2.52)

άφ dt + k\\ dt + * φ |Φιι dt + d+\ k <*Φπ |Φθ2 dt 0l

*Φ|| )ψ

0

d(

0 1

0

dt

0]

>ot+ k \Φ\2 dt

ι 2

02

+ Α:φ | φ

0 1

02

+ Αφ

Ίφοι+ k<j> 'Φοι dt doi dt + k<(>Ιφιι $02

dt

+ k<j>Φθ2 02

dt + kΙΦΐ2 02

— +

^02

/ίφ,, | φ

0i

d

*Φιι)φι

12

Ι 2

—τ- + ^φ

| 2

fkT^dt

^φ,,

+ Α:φ | φ |2

dt fkT^ dt

0 2

φ

1 2

02

dt

CkT# dt n

Jt\

74

Basic Concepts in the Finite Difference and Finite Element Methods

T o solve (2.2.52) for the undetermined coefficients 7} and dTj/dt, it is necessary to perform the integrations indicated in the coefficient matrix. U p o n integration, the matrix equation becomes (2.2.53a) 0.242

0.205

0.757

-0.162

0.00476

0.0190

0.162

-0.0309

-0.243

- 0.0381

1.24

-0.00476

0.0381

0.00238

0.205

0.0190

0.500 0.0833

dt

0.500 dT,

0.0833

dt

Because 7", is known from the initial condition [i.e., 7ΚΌ)=1], the first row of (2.2.53a) can be eliminated and the final expression is (2.2.53b)

0.0190

0.162

-0.0309

-0.0381

1.24

-0.00476

0.00238

-0.205

0.0190

dT, '

0.0833

dt T dT

2

2

dt

=

0.500

0.00476 — -0.243

-0.0833

0.0381

The second term on the right-hand side of (2.2.53b) contains information derived from the initial condition. By inverting the coefficient matrix* the unknown parameters are obtained. (2.2.54) dT, dt T dT

2

56.3 —

2

1.68 11.1

dt

7.45 0.996 9.80

89.7 2.48 68.2

-0.937

0.0785 0.743 -0.121

=

0.570

.

-0.131

The parameter T = 0 . 5 7 0 is the temperature at t = 1 and compares very well 2

'Although solutions of matrix equations are often expressed formally in terms of matrix inverses, in computations other techniques such as Gaussian elimination are used whenever possible.

Introduction to Finite Element Approximation*

75

with the analytical value 0.568. The temperature at the midpoint (r = 0 . 5 ) is easily obtained from the original expansion (2.2.49). 2

dT

7=1

By evaluating d> and
f ( 0 . 5 ) = (1.00)(0.500) - (0.937)(0.125) + (0.570)(0.50O) + (0.131)(0.125)=0.684. T o three significant figures, this solution is identical to the analytical. It is readily apparent that the Galerkin method employing the Hermite cubic generates a solution to our example problem that is superior to other methods investigated. It should be kept in mind, however, that this approach generated three simultaneous equations, whereas the schemes considered earlier generated only two. It is apparent from the coefficient matrix of (2.2.52) that considerable effort is involved in performing the integrations required in the Galerkin method when more complex bases, such as Hermites, are employed. Consequently, it is worthwhile at this point to investigate the use of Hermites in conjunction with the collocation method. In the collocation approach, the M W R formulation is written

(2.2.55)

f*{^

+ k(f-T )}8{t-t,)a=0, e

1 = 1,2,3,4,

where /, are the locations of the collocation points. When the collocation points are chosen to be specifically the Gauss points, a more accurate numerical scheme results. As mentioned earlier, this is known as orthogonal collocation. We use two Gauss points within the element and two additional points at the nodes to account for the boundary condition. N o t e that if the number of elements in the problem were increased, all additional collocation points would be selected at Gauss points. In contrast to our earlier example (2.2.52), this time let us perform our calculations in the ξ coordinate system. T h u s we write (2.2.55) as

(2.2.56)

fJ^f {f-T )^-i,)f^=Q, t+k

e

,- = 1 , 2 , 3 , 4 .

76

Basic Concepts in the Finite Difference and Finite Element Methods

Equation (2.2.56) yields the following matrix equation: (2.2.57) d

^oi di di

+

,

di

+

+

l

<

+k

t>oi , di

d<

+

d

*m

di

k

+k

£=

-I

£ = 0.577

rfi

+

*<*di

£ = - 0.577

di

dt ^ d i

£ = 0 577

£=

+k

di

^

, . .

+"di

rf€ **»* £ = -

-I

, , .

+k

{ = - 0.577

^ d i £=1

^ d i

k

di

- i

0.577

dt

, ,

+

di

^

^ d i

k

^ οι

''fe

H

+

φ

di

dt " d i t=

,

+

+

£ = 0.577

* " d i £=1

k

k

+"dt

* » d i £= - 1

1

dT,

Λ

^12

dt

*o2 £ d

+"di

+k

. . .

di ^Φΐ2

£=1

di

dt *»di

dt

£ = -0.577

+

k

+

k

dt *»di

T

£ = 0 577

dT

..

+

2

2

dt

£=1

'di

kT^kT^ 'di kT^-

The coefficient matrix of (2.2.57) can now be evaluated using the basis function definitions given in (2.2.42) and (2.2.46): 0 + 0.500*

0.500Δί+0

- 0.500 + 0.442*

0.144Δ/ + 0.0657*

-0.500+0.0576*

-0.144Δ* +0.0176*

0+0

0+0

(2.2.58)

0+0

0+0

0.500 + 0.0576*

-0.144Δί-0.0176*

0.500 + 0.442*

0.144Δ.-0.0657*

0+0.500*

0.500Δί+0

kT dT, dt

e

2 kT

e

2 kT

t

τ

2

2 kT

dt

2

2

dT

t

Introduction to Finite Element Approximations

77

Substitution of the appropriate values of k and T yields e

(2.2.59) 0.500 ~

1.00

0.500

0

0

0.384

0.275

0.615

-0.179

-0.384

-0.109

1.384

0.0126

1.00

0.500

0

0

dT, dt T dT

2

2

d'

0.500 0.500 0.500

Because the value of T, is given via the boundary conditions, we can again eliminate row 1 and solve the resulting matrix equation: (2.2.60)

0.275

0.615

-0.179

-0.109

1.384

0.0126

dT, ' dt T dT

2

2

dt

=

dt T dT

2

dt

2.83

-2.03

0.227

0.573

-0.454

=

2

0.500

1.00

0

dT, '

1.15

0.500-0.384

0 116

0.500 + 0.384

0.884

0.500

0.500

1.06

0.116

-0.936

0.0669

0.884

0.566

1.86

0.500

-0.139

The calculated value of Γ , (0.566), compares favorably with the analytical value, (0.568). When the computed coefficients are substituted into the series approximation to Τ and the result evaluated at / = 0 . 5 , the calculated value of 7 " | is 0.683, compared to the analytical solution of 0.684. We have summarized in Table 2.5 the accuracy of the numerical schemes considered thus far in solving the O D E 2

0 5

(2.2.9)

4£ + k(T-T ) t

= 0,

0
r(o)=i. Although this comparison is interesting qualitatively, one must keep in mind that the sample problem consisted of only one element. A more quantitative analysis is considered in a later section.

78

Basic Concepts in the Finite Difference and Finite Element Methods TABLE 2.5. Comparison of Solutions Obtained with Several Weighted Residual Schemes"

Solution Method

/=0.0

f =0.5

< = 1.0

1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.678 0.667 0.684 0.696 0.684 0.683 0.684

0.571 0.555 0.579 0.571 0.570 0.566 0.568

Galerkin linear bases Collocation linear bases Subdomain linear bases Galerkin quadratic bases Galerkin Hermite cubic bases Collocation Hermite cubic bases Analytical "Problem definition is given in (2.2.9).

Calculation of Derivatives. There are occasions when it is necessary to obtain not only the unknown function but also its derivative. T h e Hermite formulation provided this information directly, but it is also available using earlier formulations. It is only necessary to differentiate the approximating expression

Σ

f=

Wt)

7= >

to obtain -

(2.2.61)

ι

τ;-^-

-

7= 1

1 Τ,—ξ-

—.

7=1

Because the coefficients T are known and the basis function derivatives readily obtained, (2.2.61) is easily evaluated. When linear bases for the derivative at t = 0 (using the Galerkin solution for T ) are used, we have }

}

dt

di

l= 0

+ T ({)(4) 2

dt ,

T

A

2

,

W

= 0

= - 2 . 0 0 + 1.36= - 0 . 6 4 0 .

Similarly, we obtain using quadratic bases

Λ

#

,=o

dt

>

T (= 0

{

l

)

{

2

)

+ r (2)(2)+r,(-i)(2)=-0.787. 2

Introduction to Finite Element Approximations

79

The analytical value of the derivative at r = 0 is - 1.00. In summary, the linear, quadratic, and Hermite bases, when used with Galerkin's formulation, yield derivative values of —0.640, —0.787, and —0.937, respectively, compared to the correct value, — 1.00. Let us now consider calculation of the derivative at the location i = 0 . 5 . This is important because it represents an interelement boundary for linear basis functions. Because the linear bases are C ° continuous, the derivative dfy /dt\ is not formally defined. The most common approximation of the derivative at this point is the average of d$j/dt\ . and d$ /dt\ _ _ , where 0 < ε < Δ ί . Derivatives calculated using the three bases are summarized in Table 2.6. The Hermite bases provide a significantly more accurate solution for the derivative, whereas the quadratic scheme is only marginally better, overall, than the linear case. N o t e that the problem encountered in defining the derivative at the edges of the linear element also arise at the edges of quadratic and C ° continuous cubic elements. l=0

l=(jiJi t

2.2.4

t=0

s

5

t

Two-Dimensional Basis Functions

The extension of the weighted residual method to higher dimensions is relatively straightforward, provided that regular rectangular subspaces are employed. We now focus on this special case and examine the more conceptually difficult irregular subspace problem in Chapter 3. As in the earlier discussion of one-dimensional schemes, we illustrate the two-dimensional case using an elementary example. The discretized domain, which we return to shortly, is illustrated in Figure 2.10. Lagrangian Basis Functions. The simplest way to generate a twodimensional basis function is to form the product of two one-dimensional bases, one written for each coordinate direction. This approach can be applied whether linear, quadratic, cubic, or Hermite cubic bases are used. When C ° continuous Lagrange interpolating polynomials are used, we refer to these as Lagrangian bases and the associated finite elements as Lagrangian elements.

TABLE 2.6. Temperature and Temperature Derivatives Computed Using Linear, Quadratic, and Hermite Cubic-Basis Functions

Function: Location:

/=0.5

Analytic Linear Quadratic Hermite

0.684 0.678 0.696 0.684

f

/ = 1.0

f =0.0

0.569 0.571 0.571 0.570

-1.00 -0.640 -0.787 -0.937

df/dt

f =0.5 -0.368 -0.429 -0.429 -0.378

80

Basic Concepts in the Finite Difference and Finite Element Methods

y 2.0

\i

6

9

Γ5

—Η 0 ·

0.5

m

0,0

0.5

i.O

i.5

2.0

-hFigure 2.10. Discretized domain for two-dimensional heat flow. Finite element nodes are indicated by dots and the elements themselves by Roman numerals (from Pinder and Gray, 1977).

As an example, the two-dimensional Lagrangian basis functions arising out of the linear bases of Table 2.4 are (2.2.62a)

*,U.H) = [ K 1 - O ] [ i 0 - * ) ] .

(2.2.62b)

• (€.η)=[Κΐ + ί)][Κΐ-ιΐ)],

(2.2.62c)

• (€.ι») = [1(1 + €)][Ηΐ + ΐ|)].

(2.2.62d)

2

3

fctt.»)

= U ( l - 0 ] [ i O + i|)].

Figure 2.11a illustrates φ,. The other functions have the same shape. Lagrangian quadratic, cubic, and Hermitian cubic two-dimensional basis functions are presented in Table 2.7a and illustrated in Figures 2Mb, c and 2.13. N o t e that the Lagrangian element requires interior nodes when quadratic or cubic basis functions are generated. Serendipity Basis Functions. Although Lagrangian bases are the logical extension of their one-dimensional counterparts, there is another choice which preserves many of the important properties of the Lagrangian bases while decreasing the degrees of freedom by eliminating interior nodes. Examination of Figure 2.11 b and Table 2.7a reveals that the basis function at the central node spans only the rectangular domain illustrated. In light of this observation, let us now examine the possibility of defining a set of two-dimensional bases that are quadratic along each side of the rectangle of Figure 2.11, yet have no

Introduction to Finite Element Approximations

81

>

Figure 2.11. (a) Two-dimensional basis function that is linear along each side, (b) Two-dimensional Lagrangian basis function that is quadratic along each side. Note the occurrence of a central node where the basis function must be zero.

center node. In a manner analogous to that used to obtain the one-dimensional basis functions, we first write a polynomial expression quadratic in ξ and in η. (2.2.63)

φ = a + bi + CTJ + di

1

+ «?η + / ξ η + gξ η + Λξη + / ξ η . 2

1

2

2

2

There are nine coefficients to be evaluated in this expression, which requires the specification of nine constraints. The Lagrangian scheme involves nine nodes, and consequently the nine conditions are easily formulated. In fact, the bases we derived by taking the product of two one-dimensional bases will arise

82

Basic Concepts in the Finite Difference and Finite Element Methods

(c)

{Continued)

Figure 2.11.

Figure 2.11. (c) Two-dimensional Lagrangian basis function that is cubic along each side. Note the occurrence of four interior nodes where the basis function is defined to be zero.

out of (2.2.63) when each basis function is required to be unity at the node for which it is defined and zero at all other nodes. Because we wish to obtain a basis function with no interior node, there can be only eight constraints imposed. Thus the general polynomial of (2.2.63) requires simplification. Let us achieve this objective by dropping the last term, leaving the expression

= a + bi+cv

(2.2.64)

+ d£ + eT) + / ξ η + g | r j + Λξη . 2

2

2

2

By requiring each basis function Φ,(ξ,τ)) to be unity at node i and zero at all other nodes, we arrive at the matrix equation for the particular case of φ,:

(2.2.65)

1 0 0 0 0 0 0 0

1 1 1 1 1 1 1

.ι

-1 0 1 1 1 0 -1

-ι

-1 -1 -1 0 1 1 1 0

1 0 1 1 1 0 1 1

1 1 1 0 1 1 1 0

1 0 -1 0 1 0 -1 0

-1 0 -1 0 1 0 1 0

-1 0 1 0 1 0 -1 0

a b c d e f g _h

I PΡ-

Ε 8

I

PP" ΆΛΙ·

+

PP-

+

--— ΡΡ-

PP-

w

X

ΡΡ-

+

+

>>

ΙΒ S Υ 5

Ρ-

PΡ"

•A

+ χ

X

—

Ν*

+

c

Ρ· Ρ-

ΡΡ-

+

+

+ Χ

+

(Ν «Α»

Ρ

+

I

i

+ +

Ρ-

I

&

ΡΡ-

f

+

PP-

Ί I

+

Ρ-

«*<·

+

+

I

Ρ&>

χ

Ρ·

***

i

ι

~

χ.

χ

+

+

ΡΡ-

W

V

W

Ρ Ρ· 7

**?

***

+

Ν/

+ + Ρ; Ρ"

Ν ,

«Α*

I

Χ

I

fN Ρ"

V *

I

_

fN *JU>

Ηί5 Η2

-I»

+ Χ Ν*

+

Ρ-

I

Ν*

I

sg

Mi>

2

"8

!3 Υ α

.a

Β

Ν,

sε G Υ

«*?

Ρ

U

U

7

•G

•G

ΙΛ

<75

C Υ Ό

C Υ Ό

+1 +1 II II

83

P-

PΥ

•9

Side no

α

II

Cubic Comer node

II

~- —|fi

Ο

Interioir node

Ο

—

~

AA?|<»

Ρ?|<»

ΡΡΡΡ·

Ρ-

+ χ

Ρ-

+

Ρ-

χ

-

Ρ(Ρ

Ρ-

+

Ρ£

Ρ· Ρ-

Ρ·

+

ΡΡ-

1

I

ι *Α? ΑΛ*

+

~

+ «Α»

*** I

+

+

I

Η£

α. =5

+1 II II P-m

C

+1 II

Interioi• node.

+1 +1 II II

a

+ >—·

Ρ· Ρ-

S ω

+ W

Ρ·

**#

•a

-1_

*A,

is

+

PP-

ON

C

84

Basic Concepts in the Finite Difference and Finite Element Methods TABLE 1.1b.

Basis Functions Formulated Using a Mixed Formulation

Corner node e, = i ( i + « , X i + i i , )

where β and /8 are defined as (

η

Side

fit

A,

1 2

Linear Quadratic Cubic

n

1 2

2

in,

-i

g) 7

Side node Quadratic Φ, = 1(!-ί )(ΐ + ηη,) ί,=ο,

i 8

η, = ±ι

2

Cubic Φ,=Α(1-ί Χ1+9ξί,)(1 + η»),) Φ, = π(1 + ίί,Χ1-η )(Ι+9τ,η,)

_

£, = =*, ", = ±1 {, = ±1, η = ± \

2

2

Equation (2.2.65) yields the vector of coefficients a

I 4

b c d

0 0 I 4

1

e

4

f

1

G

4

4

h

1 1

.

4

J

which, when substituted into (2.2.64) gives (2.2.66)

Φι = ί ( - 1 + ξ + η + ξ η - | η - η | ) · 2

2

2

2

By employing a scheme analogous to the one used for φ , , one can obtain bases for the remaining corner nodes. AH corner node bases are described by (2.2.67)

φ, = I ( 1 + « , ) ( 1 + τ,η,)(«,. + τ,η, - 1),

where £, — ± 1, η, = ± 1, depending on the location of the corner node. Φ, = { ( ΐ - έ ) ( 1 + ητ,,) 2

for

£,=0

Introduction to Finite Element Approximation

85

(-1,1)

(-«.-4) (C)

Figure 2.12. (a) Two-dimensional serendipity basis function that is quadratic along each side, (b) Two-dimensional serendipity basis function that is cubic along each side, ( c ) Two-dimensional mixed serendipity basis functions. Note that the function may be linear, quadratic, or cubic along any given side.

Basic Concepts in the Finite Difference and Finite Element Methods

86

and

Φ, = Η1 +«,)(!-V)

for

η,=0.

Corresponding cubic and Hermite bases are summarized in Table 2,7a. Bases of this class are generally referred to as serendipity basts functions. It is interesting at this point to examine the significance of dropping the ζ η term in formulating the interpolating polynomial (2.2.64). The quadratic and cubic serendipity bases are presented in Figure 2.12a and b and can readily be compared with the Lagrangian bases of Figure 2.11. Note that although an overall similarity exists, the Lagrangian bases are somewhat different in the interior because of the additional constraints imposed on their formulation. It can be shown, using a type of analysis to be presented later, that the addition of the | t j term in general has little influence on the overall accuracy of the approximation. To this point we have tacitly assumed that one type of element, be it linear, quadratic, or cubic, would be used to represent all regions in a particular domain. There are occasions, however, where higher-degree elements may be required in one region but not another. This situation generally arises when the solution is not a smooth surface, such as in the vicinity of singularities or abrupt changes in medium properties. To change from one element type to another requires a transition element. This element must be sufficiently flexible that a different degree of basis function can be used along each side (see Figure 2.12c). The formulation of these bases is presented in Table 2.7a (Pinder and Frind, 1972). 2

2

2

2

Hermitian Basis Functions. Although the two-dimensional Hermite bases are generated through a straightforward multiplication of the one-dimensional functions of Figure 2.9, they are somewhat more difficult to apply than the C ° continuous cubics mentioned above. The two-dimensional expansion of the trial function T(t) now contains, in general, six unknown parameters per node: T oTj/dx. dTJdy, d Tj/dxdy, d T /dx , and 3 7 ; / θ . ν . For any point within an element, we obtain* 2

p

2

2

2

2

}

(2.2.68)

dx dx . dyW

J

'The superscript I is introduced to denote the C' Lagrangian Hermites.

(c)

id)

Figure 2.13. Two-dimensional basis functions formulated using Hermite polynomials.

This expansion requires four bases per node: φ ^ , φ', , φ , and φ ' . Their algebraic form is given in Table 2.7a and they are illustrated in Figure 2.13. Examination of Figure 2.13 reveals that these bases manifest the following properties: 0

3φ'ιο at

3φρι

01

η

9 ΦΊι 2

{ = £,, η = η ,

the node where it is defined

and 9φ\η ^Φοι ^ φ\ι -τ— /= -τ— /- . y ' = 0 σξ ση σξση 2

at all other nodal locations,

While (2.2.68) provides the general formulation, it is possible to rewrite this expansion in a form more clearly related to the one-dimensional expression (2.2.49). This is accomplished by defining new bases in the following way: (2.2.69)

f=ly

97"*

4

0j

+

+

^ φ '

^ +

J

37*

+

Ώ, φ

^

J

8^7"*

+

^

x

v

J

88

Basic Concepts in the Finite Difference and Finite Element Methods

where

0j ~ Φοο, ·

31'

ι _

ι

dy

\

dy

\

2

Φχ.ν>-Φ||>

dx

dx

Comparison of (2.2.69) and (2.2.49) reveals that both are written in fundamentally the same form but with a different number of unknown parameters and bases. In the case of zero-order continuous polynomials, we found that an incomplete polynomial could be used to formulate bases. These were designated serendipity basis functions. This approach may also be used to considerable advantage in developing two-dimensional Hermite bases. The resulting approximating expansion is

(2.2.70)

37. f-

3*

Σ

^Φοο, + ^ Φ ι ο , - ^ ' + di 7= 1

+

J

37:

dr

9Γ,

dx

ι^Φου 3;r™;av

dy d ^

where we have omitted the superscript 1 from the bases to emphasize that they are not first-order continuous. These bases are presented algebraically in Table 2.7a and graphically in Figure 2.14. Equation (2.2.70) may be rewritten in the form of (2.2.69) as

(2.2.71)

Introduction to Finite Element Approximations

89

(c)

Figure 2.14. Two-dimensional serendipity basis functions formulated using Hermite polynomials.

where Φο,

=

Φοο,.

_ dx Φ^-Φ>ο>9Ϊ^ _ dy Φ>·7-Φιο,3^

+

+

dx Φου^^' ^y Φοι,9^·

Examination of (2.2.71) and (2.2.69) reveals a considerable saving in computational effort when serendipity rather than standard Hermitian bases are used. The degrees of freedom per node are reduced from six to three with serendipity bases. Experiments conducted using both types of bases in conjunction with the Galerkin formulation suggest that the two schemes are of comparable accuracy. The advantage of the serendipity scheme is even more pronounced in three-dimensional formulations.

90

Basic Concepts in the Finite Difference and Finite Element Methods

2.2.5

Approximating Equations

Each of the weighted residual schemes introduced to solve the one-dimensiona problem can be extended to two space dimensions. As illustrative examples, we solve the following problem using Galerkin's method with linear basis functions, and a similar problem using collocation and Hermite cubics. Example 1 Consider the problem of two-dimensional, time-independent heat flow in a rectangular plate with a heat source located at the center of the plate. The governing equations for this problem are (2.2.72a)

£(T)

(2.2.72b)

T(x,2)=\,

= T

xx

+T

= Q,

yy

(2.2.72c) (2.2.72d)

T (x,0)=0,

(2.2.72e)

T (2,y)

= 0,

(2.2.72f)

Q(x,y)

=

y

x

QJ(x-\)8(y

where χ and y are Cartesian coordinates, Q is the heat source, and δ is the Dirac delta function. The discretized domain of interest in this problem is illustrated in Figure 2.10. w

Step 1 Define the trial functions 9

(2.2.73)

T(x,y)^f(x,y)=

Σ

T^(x,y).

7=1

For convenience, it is advantageous to define basis functions in local (£, η) coordinates whereupon (2.2.73) becomes, for each element, 4

(2.2.74)

T(x,y)~f(x,y)=

Σ

Τ^(ξ,η)

7=1

and φ (ξ, η) are bilinear chapeau functions such as those illustrated in Figure 2.11a.

Introduction to Finite Element Approximations

91

Step 2 Formulate the integral equations using Galerkin's procedure. This is the special case of the M W R scheme wherein the weighting function has been defined as the basis function. O n e can also interpret Galerkin's scheme as the requirement of orthogonality between the residual R and each basis function, f R(x,y)t,{x,y)dxdy=0,

(2.2.75)

i = l,...,9,

A

J

where R(x,y)=t(f)-Q

(2.2.76)

and £ is defined by (2.2.72a). Substitution of (2.2.72a) and (2.2.76) into (2.2.75) yields [(f

(2.2.77)

A

J

xx

+ T -Q)*,(x,y)dxdy=O yy

/ = 1,...,9.

t

It is advantageous, but not mandatory, to apply Green's theorem to (2.2.77) to incorporate second- and third-type boundary conditions directly into the set of integral equations. Equation (2.2.77) becomes (2.2.78)

/ ( 7 > „ + 7 > „ + Qt,)dxdy-/(f l

+ f ,/, )φ,ds = 0,

x x

J

A

(

J

1

= 1

S

9,

where l and l are direction cosines with respect to the normal to the curve S, the boundary of the domain A. In this particular problem, the second term of (2.2.78) will be used to conveniently define the zero-gradient N e u m a n n boundary conditions of (2.2.72d) and (2.2.72e). Substitution of the trial functions, as defined by (2.2.73), into (2.2.78) yields the following set of algebraic equations: x

(2.2.79)

y

Jf J Σ Γ , ( φ „ φ , + φ,,φ ,) + (2Φ, J dx dy χ

-f (f l s

x x

κ

+ f l )*,ds=0, y y

/ = 1,...,9.

Because φ, is defined such that it is nonzero only over elements adjacent to node ; (see Figure 2.10 and 2.1 l a ) the integrations of (2.2.79) may be performed piecewise over each element and subsequently summed. Thus

•2

Bask Concepts in the Finite Difference and Finite Element Methods

(2.2.79) may be written Σ

(2.2.80)

f ((Σ

-\

WA/

(f I

s

x

x

Φ>ΑΙ)

+

t I )4>, ds^j =

+

y y

+

0,

Q4>1\

dy

i =

9,

where A is the area of element e, S is the curve bounding A , direction cosines. e

e

c

and l

are

a

Step 3 Formulate the matrix equation. In general, it is advantageous to express (2.2.80) in terms of the local (ξ, η) coordinate system to facilitate integration (see Figure 2.IS). This is easily accomplished provided that the relationship between derivatives of φ· in each coordinate system is readily available. T o obtain this relationship, we once again employ the chain rule to obtain

Hj _ Hj dx

3Φ>

8£ ~ dx 9ξ

dy

9η

_ 3Φ, dx

3JC

9η

|

dy dt

HJ dy dy

9η'

This set of equations is conveniently written in matrix form as

3Φ; (2.2.81)

dy'

3Φ, 9η

dx

9η

' 9φ, " dx

Η

3φ/ or

9JC

dy

9η

or

9φ,

' 3φ, " 9* 9Φ =[J) 9η

' 9Φ " 7

dx

;

dy

dy

9Φ>

dx ΘΦ; dy

9φ, 9η

where [J] is the Jacobian matrix. In our example problem [ / ] evaluated as (2.2.82)

1.0 [J\ = 0.0

0.0 0.5

(2.2.83)

1.0 0.0

0.0 2.0

is easily

As we shall see in the section on elements of irregular shape, the Jacobian matrix is not always this easily obtained.

Introduction to Finite Element Approximtioni

(0,0)

(2,0)

Figure 2.15. Rectangular element in (x,y) formed into coordinates.

93

<-t,-t>

(-4,0 coordinates and the same element trans-

Equations (2.2.80)-(2.2.83) may now be combined to yield the following equation defined in the local (ξ, η) coordinate system: 4

±

(2.2.84)

9

/

(/_'/_' ( t

- [

( t j

x

TA44> j4> + 4φ^φ ) (

+

f y l y )

φ,

ds^J

+

νί

fl

=

0,

i

β φ ι

=

^

WJr,

9,

N o t e that in changing the limits of integration to span the local coordinate system we introduced the relationship dxdy=dct[J]didr). Because the elements in our example problem are all of the same size, the four integrals appearing in each element in (2.2.84) are identical for each element. The assembly process, that is, the transformation from element to global integrations, is easily seen with the assistance of Table 2.8. Here we have tabulated the global nodal numbers corresponding to each element nodal number (element nodes are numbered counterclockwise from 1 to 4). For example, to find the global node corresponding to the local node 2 in element I, we move right along the second row to the column marked I, which corresponds to element I. Here a global nodal number 4 is given. Similarly, if

TABLE 2.8. Relationship between Local Node Numbers and Global Node Numbers for Problem of Figure 2.10

Global Nodes (Global Node Numbers)

Element Nodes (Local Node Numbers)

I

II

III

IV

1 2 3 4

1 4 5 2

2 5 6 3

4 7 8 5

5 8 9 6

Basic Concepts in the Finite Difference and Finite Dement Methods

94

TABLE 2.9.

1

\

7

I 1 7

ί •χ J

4

Evaluated Integrals for One Element in Local Coordinates for Equation (2.2.84)

1

2

3

4

ι3

1 6

\

1

1 6

1 6

2 3

1

I 3

1 3

1 6

6

_ 3X


_ 1 6

_X

a

3

6

we wish to establish the integral values associated with global matrix coefficient 7 = 4, 7 = 5, the first step is to find the equivalent local numbers. Examination of Table 2.8 reveals that these two global nodal identifiers appear together in columns marked I and III. Thus these two elements (I and III) contribute information to the global integral. The assembly procedure now calls for extracting information concerning integration at the element level from Table 2.8. From column I we see that global nodes 4 and 5 correspond to element nodes 2 and 3 in element I. Table 2.9 tells us that the contribution to the global integral attributable to these nodes is found in row 2, column 3; it is — This value is now placed in matrix location (4,5) of the global matrix. We are not through, however, because there is also information to be retrieved from element III. Table 2.8 locates this information in position (1,4) of the element matrix of Table 2.9, it is also — £· This value summed with the previous integral value is now placed in the same

TABLE 2.10. Evaluated Integrals for the Four Elements of Figure 2.10 in Global Coordinates for Equation (2.2.84)"

X 1 2 3 4 5 6 7 8 9

1

2

3

13

_ 1 6

0

_ i

6

0

_ 1 6 _ 1 3

0 0 0 0

i1

3 6 _ 1 3 _ 1 3 1 3

0 0 0

_ 1 6

a 3 0

_ 31 1

(.

0 0 0

4

5

6

7

8

9

-1 i31

1 3 1 3

0 _ i

0 0 0

0 0 0

3 8 3 1 3 1

0 0 0 0

- 1 3 4 J

0

4 3

_ 11

—1 03

1 6 _ 1 3

0

_ ΐ1 3 _

_ 3 1 6

0

_ 1 6 1 3"

0

1 3X — i _ —1 0

3 6

06

1 3

" i _ 1 3 1 6 4 3 1 6

_ 1 3 _ 1 6

0

_ 6X I 3

"Note that these values are obtained using the information from Tables 2.8 and 2.9 only.

Introduction to Finite Element Approximations

95

global matrix location (4,5). Combination of the two integrals yields the final value of - \ , which appears in Table 2.10 in row 4, column 5. It is important to understand the two-step integration process outlined above, because this is the procedure used in nearly all finite element programs. In general, the element coefficient matrix (Table 2.9) is different for each element because of changes in either element geometry or parameter values. Moreover, it is often necessary to perform the integrations of (2.2.84) numerically. Because of the local coordinate system we have selected, this is easily accomplished using Gaussian quadrature (as will b e outlined later). Superficially, it appears that an unnecessarily complicated procedure has been introduced to evaluate rather simple integrals. Introduction of these basic ideas at this point, however, simplifies the extension to more challenging concepts in later chapters. Step 4 Solve the matrix equation. It is now possible, with the information of Table 2.10, to solve our example problem. The matrix equation arising out of (2.2.84) is _

_ 1 _ 1

2 3 1

—1

0

_

3 1 6

_

1

4

6 6

3

0 0 0 0

0

6

I

3

~~ 1 0 0 0

τ,

-X

o

x

x

Ά

j{f l

x x

τ< T

6

3

0

()

3

«

- 1

3

3

-i

+

T l )^ds

+

t l )^ds

+

f l )^ds

y y

y v

y y

0

T

7

f{f l

x x

9

+

t l )^ds y v

0 0

Ά f{f l

x x

+

I 6

3

3

5

T

τ

x

_

_ 13

-\

f {TJ

j{t l

6 Γ) υ

- i

- i

3

s

3

3 i 3

-I

_ 1 6

0 0 0

υ

_ i 3

—1

v

0 0 0

3

3

3 _ i 3

3 .1 6

3

3

i

.1

—1

- i

υ

0

3

6

- l

1 6

f l ) ds y v

9

X

3 _ 1 6

0

0 0 0 -

j

—

i

-

-{ —

0 0 0 0

_ 1

3

J-

_

f

_

6

1 6

0

1 6 2 3

Basic Concepts in the Finite Difference and Finite Element Methods

96

The line integrals appearing o n the right-hand side of (2.2.85) represent flux boundary conditions along the perimeter of the problem area. Through summation of element integrals, these line integrals vanish along interelement boundaries. The point source becomes a value specified at node 5 through the evaluation of the Dirac delta function. In general, when Dirichlet boundaries are specified, it is necessary t o expand a n d evaluate the line integrals. However, when C ° bases are used, one can condense rows containing the known temperature values out of the matrix equation. Rows 1, 2, 3, 6, and 9 may thus b e eliminated. The corresponding columns may also be eliminated by placing information pertaining to these nodes on the right-hand side of equations 4, 5, 7, and 8. The reduced matrix equation is _

_ 1 3 _ 1 3 _ 1

Ι

3 £ 3 _ 1 3

(2.2.86)

_

1 2

Ά τ

6

Ί

-β„ + Ι 0

1 3

I

3

1 2

or [A]{b)

(2.2.87)

=

{f).

Equation (2.2.87) may be solved directly to yield 0.783'

Ά

0.525

Τι

0.655

Ά

0.783

The coefficients T , Γ , Γ , and Γ represent the values of the temperature at nodes 4, 5, 7, and 8 because the basis functions are defined such that φ, is unity at node ι and zero elsewhere. The analytical solution for this problem is (Tang, 1979) 4

5

7

8

T =

U+V+W,

where u

=

2

S sin[(/i + j-)ir(a - j>)/a]cosh[(ii + j >u-x)M π

y

=

2

„ίο

(n + ^)cosh(,t +

S cosh[(n + ±)vy/a]nn[(n + * nio (« + i)cosh(n + i)7r

])7r i)wx/a]

sin[(« -r4)wx/a]sin[(n + i)w£/a] VV

_2

S

~*Λ>

sin\i[{n + ^)ιτ(α-y)/a]cx)sh[(n

(« + i)cosh[(« + i)ir]

+

\)ιτη/α]

Introduction to Finite Element Approximations

97

E L E M E N T NUMBER

(0,0)

(4,0)

(2,0)

(a) COLLOCATION ^ POINT

(1,-4)

M.-O (b)

Figure 2.16. (a) Collocation net in ( x , y) coordinates, (b) One element of collocation net in (£, n) coordinates.

where a is the length of the side of the square (a = 2), and ζ and η are the χ and y locations of the heat source, respectively. The analytical solution for the nodal locations 4, 5, 7, and 8 is 0.756, 0.127, 0.719, and 0.756, respectively. It is evident that the numerical solution at the singular point (1,1) is not very accurate. This is not unusual inasmuch as we are attempting to represent a rapidly varying function with four bilinear elements. We will see in Chapter 4 that special trial functions can be used to circumvent this difficulty.

Example 2* In this example we consider the solution of Laplace's equation on a square region using the collocation method. T h e problem is illustrated in Figure 2.16 •The calculations for this example were made by Michael Celia, a research assistant in the Department of Civil Engineering, Princeton University.

98

Basic Concepts in the Finite Difference and Finite Element Methods

and the governing equations follow: (2.2.88a)

£o(«) = « « + «

v v

=0,

(2.2.88b)

u (0,y)=0.

0<>><2,

(2.2.88c)

«,(2,^)=0,

0
x

(2.2.88d)

« ( x , 0 ) = 3 - x,

0<x<2,

(2.2.88e)

I I ( J C , 2 ) = 3 - x,

0<x<2,

where u may have a number of physical interpretations such as electrical potential, temperature, or hydraulic head. Step 1 Define the trial functions. In this case we use cubic Hermite polynomials as basis functions. Thus the trial function for each element is written [see (2.2.69)]* 4

(2.2.89)

u(x,

y)<*u(x,

a t/ ^-ώ 3JC dy 2

1

+

.

Σ

y)=

d u.

d U,

dx

dy

,

+

317. , ^

y

j

2

2

+

W.

Ujtij + J^tij

2

φ'

,

where Φο/

Φοο, -

-

dx Φ>ο,3^

Φν7-Φ.ο,9|>

dx dx Φο.,9ην+Φ, £

.

2

+

υ θ

+

dy Φ ο υ ^ + Φ.υ

η 7.

3 y dx .

3x 3 ^ 3η 3 ?

9 £ 9

θ η

y

3η 3 £

J

3x 3 X

,

_

,

dy dy .

•Note that the .x and ν subscripts do not denote partial derivatives in this equation.

Introduction to Finite Element Approximations

99

Because we are using square elements in this example, dx/dr\ the following simplifications are possible: (2.2.90)

φ!,, = φ'

Φ ί „ = φ' „ = 0, ν

ι

_

ι

dy dx .

ι _

ι

— dy/di

= 0 , and

|07

dy .

Combination of (2.2.90) with (2.2.89) yields the appropriate trial function for a single element: dU

4

(2.2.91)

u(x,y)~u(x,y)=

2

dxdyV^dn

U^

+^ \

Wj

dU

Ά

χ

0

j

^ J

+

a

v

jftin^J

di J

It can be seen through a comparison of Figure 2.16a and b that dx/d£ dy/df\ = {- in this particular example.

=

Step 2 Formulate the approximating algorithm using the collocation procedure. In this case of the M W R , the weighting function is chosen to be the Dirac delta function specified at the 16 Gauss points ( x , , >>,). Thus we obtain (2.2.92)

f (R(x,y)8(x-x,,y-y,)dydx=0,

i = l,2

16,

where R(x, y) is the residual defined as R(x,y)=t (u)

(2.2.93)

0

and £ ( •) is defined in (2.2.88a). Substitution of (2.2.93) into (2.2.92) yields 0

(2.2.94)

jj£ (u)8(x-x„y-y,)dydx=0, 0

, = 1,2,..., 16.

We now transform (2.2.94) from ( x , y) to (ζ,η) space, not because it is particularly valuable in this example, but rather to provide a foundation for more complex formulations to follow in later sections. In (ξ, η) coordinates

100

Basic Concepts in the Finite Difference and Finite Element Methods

(2.2.94) becomes, for each

element,

f(Q (u)S(i-i , - )dei[J]d dt

(2.2.95)

o

k V Vk

= 0,

V

* = 1,2,...,4,

where [J] is the Jacobian matrix of (2.2.81). Let us now substitute for the operator £ ( « ) in (2.2.95): 0

(2.2.96)

(((u

+ u )Stt-i , - )det[J]d di

xx

vv

k v Vk

k=

= 0,

v

\,2,...,4.

From the definition of the Dirac delta function, we can rewrite (2.2.96) as (u

(2.2.97)

xx

+ u )det[J]\ =0, yy

* = 1,2,...,4,

(tiii

where (£*, T J ) is the coordinate of the &th Gauss point. Introduction of the definition of the trial function (2.2.91) into the collocation expression (2.2.97) yields, in a given element, a

(2.2.98)

Η I

+

l u'dx^

+ ^

2

3 x 3 ν 3r, H

J

3t/. dy

9 >

^

2 -·>) φ

+

dx

+

\

U

dv dy

2V0>J

dxdy

^ j j - ^

^ 2 V

VUj dy

, J

j

d

V

dx d£

J

+

9

2

"ioy

^

, dy *" 2

J

det[7]

= 0,

* = 1,2,...,4.

Step 3 Formulate the matrix equation. To evaluate (2.2.97) we must calculate the determinant of the Jacobian matrix and establish a transformation for the second-order derivatives from the ( x , y) to the (ξ, η) coordinate system. dy/dη, T h e Jacobian is obtained directly from the definitions of 9 x / 9 £ and which are observed to be \ from Figure 2.16, as mentioned earlier. Thus we

Introduction to Finite Element Approximations

101

obtain dx Κ det[/] = dx

(2.2.99)

dy 3? dy dη

4"

i

ο

T o transform the derivatives, we once again use the chain rule expansion (2.2.100) 3 Φ_ 2

3*2

d ΙΒφ\ dx { dx ) Α/^Μ dx \ 3£ dx

=

_/3£\ I dx)

2

3Φ 3£ 2

2

3 £3φ 2

+

3x

2

3£

+

+

^ 9 η \ dη dxj

/^^^_3_\/3φ3|3φ37ΐ\ \ dx 3£ dx dy j \ 3£ dx 3ij dx j

=

3{ 3η 3 φ dx dx 3£3τ,

/3η\ 3φ \ 3JC/ 3 ^

2

2

2

3^,.3φ 3JC 3 ^ ' 2

Because of the geometry, this particular problem (2.2.100) reduces to (2.2.101.)

'

and similarly (2.2.101b)

i^^(M) dx

1

3y

=

de\dx)

^ = 3 W M 2

2

( 4

de

2 =

3T,M W 9

^

3 J *

3η

(

), }

4

)

.

2

The final equation to be evaluated at each collocation point is obtained by substitution of (2.2.101) and (2.2.99) into (2.2.98): (2.2.102)

* = 1,2

4.

102

Basic Concepts in the Finite Difference and Finite Element Methods

It is important to recognize that (2.2.102) is written for one element only. It contains 16 unknowns while generating only four equations. When this equation is evaluated for all four elements, we will have 16 equations, but there will now be 36 unknowns. However, the number of unknowns will reduce to 16 when the appropriate boundary conditions are imposed. Step 4 Solve the algebraic equations. T o solve the algebraic equations associated with (2.2.102), we first evaluate this equation for each element. The resulting element equations are Η,,

Η,-,

//•·>

H

Η 22

Η 23

24

#3.

#32

Η-33

34

#4,

Η,42

#43

#4

2]

(2.2.103)

Η 14 =0

or [/7]{χ}=0, where the elements of [H] and {x} are row and column vectors, respectively, that is, 1 / Θ

ι 2

a \ 3 |

2

1

2

3

2

3

2

2

3η

W. 3(7, 3 t7, U ——3x ' dy 3 x 3 y 2

It should be noted that [H] is not a square matrix inasmuch as there are only four collocation points (k = 1,2,3,4) and 16 undetermined coefficients in {x}. Once the element coefficient matrix [H] is generated for each element, we must assemble this information into a global matrix. We recognize that many of the undetermined parameters in {x} are common to two or more finite elements. This assembly process generates a (16 X 36) global coefficient matrix. The associated global vector of unknowns is (2.2.104)

{G} = [ { g } „ { g M g } 3

{«ΙοΓ,

Introduction to Finite Element Approximations

103

where Γ

dU

W

m

dU \ 2

m

T

m

T o obtain a tractable set of equations, we impose the four boundary conditions specified in (2.2.88). This reduces the number of unknowns to 32. It is possible, however, to extract additional information from the boundary conditions. For example, because u = 0 along the side defined by x—0, the cross derivative d/dy (d/dx) must also vanish along this side. Thus we have generated an additional constraint on the global system of equations. Similar arguments lead to the following additional constraints: x

3(7 (2.2.105a)

= - 1 ,

y=4,6,

3 t7, 2

' < > ' . (2.2.105b) dx~dy~=°< y We now have a total of 20 constraints (see Figure 2.17), 16 degrees of freedom, and a tractable system of equations. The solution to this set of equations and a comparison with the analytical solution is given in Table 2.11. Because the corner nodes are singular in terms of 3M/3JC, an alternative set of constraints, in which Wj/dx — — l,j = 1,3,7,9, is also possible. In this case, however, it is more difficult to argue the disappearance of the cross derivative. Before leaving this introduction to the fundamental concepts underlying the finite element method, let us consider briefly the relationship between finite element and finite difference methods. =

1

2

3

7

8

U * 2 6

U = 3 3

U,»3«0

U» * 0 2

3

6

9

9

u9< t U , «0 U, -0 9

> 9

2

5

8

U„ «0

U< * 3 4 U„ » 0 φ Uxyl* 0

U«8*0 U,, «0

> 2

8

4 U4 • 2

7

u · t 7

U,

· 0

7

U,, »0 7

Figure 2.17. Dependent variable constraints for the collocation formulation using Hermite polynomial bases.

Basic Concepts in the Finite Difference and Finite Element Methods

104

TABLE 2.11. Collocation and Analytical Solutions for Problem Described by Equation (2.2.88) and Illustrated in Figure 2.17

Collocation Solution

Dependent Variable

—1.553 2.384 8.325 X H T ' 1.553 3.322X10" 5.427X10" 1.999 -5.715X10"' -8.240X10" -1.256XIO" 1.474X10" -5.427X10" 1.553 1.616 7.372 X 1 0 " -1.553

Uy, U

Analytical Solution

2

Uy2 Uyi

Percentage Error

2.325 6

16

0.0000

2.5 0.0

0.0000

0.0

2.000 0.0000

0.0 14 0.0

0.0000

0.0

1.675 0.0000

3.5 0.0

2

u U u

x

t

x

S

A

5

-0.5000

1 7

Uyi

1 5

u

x

v

S

16

Uyt

2

u

x

r

t

Uyi u,

1 7

Vyl Uy,

2.3

RELATIONSHIP BETWEEN FINITE ELEMENT A N D FINITE DIFFERENCE M E T H O D S

In Sections 2.2.1 a n d 2.2.2 we considered the relationships between the various weighted residual approximations. Let us n o w compare the finite element a n d finite difference methods to establish whether there are any features common to both. Consider once again the two-dimensional heat flow problem of the preceding section, but with the nodes renumbered as illustrated in Figure 2.18. Because node (0,0) is interior to the domain, the equation associated with that node can b e used t o demonstrate the general form of the finite element approximation. In other words, row 5 of the coefficient matrix of (2.2.85) does not reflect the influence of boundary conditions imposed on the system. When this row of the matrix equation is expanded, the resulting equation is (2.3.1)

i ( 7 - i . - i + 7 - . + 7'_ . + 7 . _ - 8 7 Ό . Ο - τ - ^ , + Γ , . . , ,

,

l

0

1

l

0

l

+ T

K0

+ T) lA

= Q. w

Although this expression cannot b e readily interpreted in terms of standard finite difference notation, it can b e rearranged to make the relationship more

Relationship Between Finite Element and Finite Difference Methods

105

y -1,1

2.0

n

1.5 -i.o

I

0.5

•

0.5

0.0

-hFigure 2.18. Discretized domain for two-dimensional heat flow. Finite element nodes are indicated by dots and elements by Roman numerals (numbering as used in the finite difference scheme) (from Pinder and Gray, 1977).

apparent. The rearranged form (which is obtained primarily through ingenuity and insight) is (2.3.2)

i

{(ϊ"ι.ι -2J"o.i +

4(r,.

)+

^-1.1

-2Γ

0

0 - 0

+

Γ_ ,. ) 0

+ ( Γ . _ - 2 Γ , _ + Γ _ . _ ) } + * { ( Γ . - 2 Γ , + 7', _ ) 1

1

0

ι

1

1

ι

+ 4 ( Γ . - 2 Γ , ο + Γ ._ ) + ( Γ _ , - 2 Γ _ 0

1

0

0

1

1

1

1

Ι

1 < 0

0

ι

1

+ Γ. ,_ )} 1

1

It can be shown (Gray and Pinder, 1976) that (2.3.2) is a special case (h = k — 1) of the more general expression (2

α χ\

h

k

(

T

\,\~

2 T

o,\

+

T

~\,\

,

7 4

*l,O~

Γ,,.,-ΣΓο,., + Γ , , , , , ] A I ^ ο,\ τ

2 7

ο.ο

7

ο.-ι

+

O,0

+

7

'-l.0

ftfcf 1

J

2

+

2 7

Τ'-ι,ι

Γ,,,-ΣΓ,,ο + Γ,,.,

6

27'-ι,ο + '

k

2

1,-1

106

Bask Concepts in the Finite Difference and Finite Element Methods

Finally, if we assume node (0,0) to be the more general point of reference ('«j\ obtain an expression that is clearly related to a finite difference approximation: w

(2

34)

e

Mcf^+i^+i 6 [

27]

J +

h

^T

! + 7]_, j ! +

l + i

h

2

2

+• hk j

27]+, + T y

6 1 ^Ί-i.j+t

7j -

[ + 1

t7

*2

27]_,

2r, + r, _ J

Λ

+ 7j_,

y

+l

y

1

2

^_

Equation (2.3.4) can be written more compactly using the symbolic notation of Table 2.1: (2.3.5)

^ { f t , +48X + i X } + l

r l

It is apparent from (2.3.5) that the finite element equations can be interpreted in terms of weighted-average finite difference approximations. But can the weighted averages themselves be interpreted in terms of standard finite difference theory? T o examine this relationship further, let us consider the well-known Simpson's rule of numerical integration. Simpson's rule for one-dimensional integration in a two-dimensional space is

k

dT

5

b

90dx dy 2

4

'"'

References

107

which provides a fourth-order-accurate integration using three discrete values. Replacement of the continuous derivatives of (2.3.6) by their difference counterparts yields the following: (2.3.7)

± Γ η

y

J

k

=

Vj+>

= y,-,

8X dy k

=

k

{«Χ,-,

+

in

k

aT

5

6

|

When (2.3.7) is substituted into (2.3.5), we obtain

U

~ }j - \

x

k

~

t~

x

I

Division by hk gives the alternative form

1 /• v* = v,+1 δΖ* L 2

1 r k - •*!+1 δ» 7Λ , x

π

It is now apparent that the finite element equations, defined over a rectangular subspace and formulated using chapeau basis functions, can be interpreted as integrated averaged finite difference approximations. Because this expression is 0(h ) in approximating the derivative and 0 ( Λ ) in performing the integration, the overall approximation is of 0(h ), the same as that achieved by the finite difference method on the same grid. However, the two approaches do not always generate schemes of the same order of accuracy, as demonstrated in later chapters. Moreover, the order of accuracy is only one indicator of computational performance and we see that, in fact, the solutions obtained using the finite element and finite difference methods on the same grid can be quite different. Although the analysis presented above involved only secondorder derivatives, an analogous development is possible for higher dimensions and for derivatives of other orders. 2

4

2

REFERENCES Biezcno, C. B. and J. J. Koch, "Over een Nieuwe Methode ter Berekening van Vlokke Platen met Toepassing op Enkele voor de Techniek Belangrijke Belastingsgevallen," Ing. Grav., 38, p. 25-36, 1923. Bubnov, I. G., "Report on the Works of Professor Timoshenko Which Were Awarded the Zhuranskii Prize," Symp, Inst. Commun. Eng., 81, All Union Special Planning Office (SPB), 1913. DeBoor, C. and B. Swartz, "Collocation at Gaussian Points," SIAM J. Numer. Anal., 10, 4, p. 582-606, 1973.

108

Basic Concepts in the Finite Difference and Finite Element Methods

Doherty, P. C , "Unsaturated Darcian Flow by the Galerkin Method," U.S. Geological Survey Computer Contribution, Program C938, 1972. Finlayson, B., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972. Galerkin, B. G., "Rods and Plates. Series in Some Problems of Elastic Equilibrium of Rods and Plates," Vestn. Inzh. Tekh. (USSR), 19, p. 897-908, 1915. Gartner, S., "Tragermethode—ein Neuartiger Defektabgleich zur Berechnung von Anfangswert— Problem mit der Methode der Finiten Elemente," Dissertation, Tech. Univ. Hannover, 1976. Gray, W. G., and G. F. Pinder, "On the Relationship between Finite Element and Finite Difference Methods," Int. J. Numer. Methods Eng., 10, p. 893-923. 1976. Mikhlin, S. G„ Variational Methods in Mathematical Physics. Macmillan, New York, 1964. Pinder, G. F. and E. O. Frind, "Application of Galerkin's Procedure to Aquifer Analysis," Water Resour. Res., 8. I. p. 108-120, 1972. Pinder, G. F. and W. G. Gray, Finite Element Simulation in Surface and Subsurface Hydrology, Academic Press, New York, 1977. Prenter, P. M., Splines and Variational Methods, J. Wiley, New York, 1975. Salvadori, M. G. and M. L. Baron, Numerical Methods in Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1961. Slater, J. C , "Electronic Energy Bands in Metals." Phys. Rev., 45. p. 794-801, 1934. Tang, D. H., personal communication, 1979.

Finite Elements on Irregular Subspaces

3.0

INTRODUCTION

The basic concepts of finite element analysis were introduced in Chapter 2. In that discussion we tacitly assumed that the domain of interest was subdivided into elements of rectangular geometry similar to the standard finite difference approach. One of the advertised advantages of finite element schemes, however, is the ability to consider a wide range of element geometries. Although the most commonly encountered element is the triangle, there are many other shapes, including some higher-order forms with curved sides. This chapter is devoted to the description of commonly encountered finite element forms and their utilization. We borrow many of the transformation concepts introduced in Chapter 2 to facilitate integration over irregular elements. For the triangular element with straight sides, we develop relatively simple integration formulas. The more exotic elements, however, require the use of numerical integration. The triangular element is considered first, although in some ways its development is more conceptually difficult than the more general formulations presented later.

3.1 3.1.1

TRIANGULAR ELEMENTS The Linear Triangular Element

The triangular element with straight sides is universally associated with the finite element method. It is the simplest element that permits an accurate representation of an irregular domain. Because of the difficulty encountered in integrating over irregularly shaped triangles, it is necessary to introduce, at the outset, a convenient natural coordinate system. 109

1-3»*

Figure 3.1. Triangular finite element in Cartesian and natural coordinates.

(t>) 3.1.2

Area Coordinates

The natural coordinates we use are called area coordinates and are analogous to the (ξ, ij) system introduced in Chapter 2 to facilitate integration over rectangles of varying geometry. As the name implies, area coordinates are related to the area of a triangle, as indicated in Figure 3.1. Each coordinate L, is defined in terms of the triangular subarea whose base is the line L, = 0 , and vertex is the point £., = !, that is, (3.1.1)

A '

1* =

A '

-+,

where Λ = Σ ? , Λ , . Examination of the properties of L, reveals that they fulfill the requirements of bases defined on a triangular subspace. That is, L, is unity at node / and zero at any other node, and L, + L + L = 1. These functions can be readily expressed in terms of global ( x , v) Cartesian coordinates using the geometrical relationship of Figure 3.1. The area of a triangle is written in terms of the Cartesian coordinates of its vertices as =

2

1 (3.1.2)

A = ±det

Λ y-i

3

111

Triangular Elements

Consequently, the L , area coordinate and basis function is defined by

det (3.1.3) det

1

x

2

1

x

3

1

χ

y .

y

2

1

x.

y\

1

x

yi

1

*

2

3

or

(3.1.4)

x y3

1 2A

2

2

\yi

x

-xjy

_

2

*2>Ί

Λ "Λ

*3 *2

Λ

χ, — χ

>Ί " Λ

Χ

_

3

*|

— 2

Let us now examine how a typical integral arising in Galerkin's method can be evaluated using the L, basis functions. Consider the integral (3.1.5)

dx

dx

dxdy,

where A is the area of a triangle of arbitrary geometry such as illustrated in Figure 3.1a. Our first task is to rewrite the derivatives in this integral in terms of area coordinates. From the chain rule, we obtain (see also Section 2.2.5) dL,

(3.1.6)

dx

dL,

dL,

3L, ,

ν

'•I

dx ,

3L

+ ^

dL . 2

2

dx

In (3.1.6) it is important to note that the third coordinate is a function of the other two, L = L ( L , , L ) , because only when this is clearly understood are the partial derivatives with respect to L , and L meaningful. Notice that the derivative for 7_ is expressed only in terms of L , and L , 3

3

2

2

3

2

dL 3 _ dx

dL,

dL 2

dx

dx '

This is readily verified by substitution of L for L, in (3.1.6). Similar expressions are generated for the derivatives with respect to y . By differentiating L , , L and L as defined in (3.1.4) with respect to x, we can evaluate the right-hand side of (3.1.6) directly: 3

2

3

112

Finite Elements on Irregular Subspaces

Substitution of (3.1.7) into the integral (3.1.5) yields (3.1.8)

(yz-y\) 9L,

{y^-yM^xdy

9L,

Before attempting to evaluate this expression, it is worthwhile to introduce the following relationship for integrating polynomials in L,\ (LT>L^L^dxdy J

(3.1.9)

= 2A-

A

m[!m !m !

. , (m, + m + m + 2 ) ! 2

3

3

2

where m , , m , and w are positive integers. Because, in this example, the integrand of (3.1.8) is a constant, m, = m = m = 0 , and we obtain 2

3

2

.

(3.1.10a)

4A

3

9L,

(Λ >Ί) -

\ dL

2

t

9L,

3L,

(ν ->Ί)·Μ

3L,

3

or (3.1.!0b)

/· 9L 9L, ι l ^ j ^ s ^ d x d y ^ ^ y ^ - y ^ K y ^ - y ^ ) ,

where the i and 7 indices are cyclic 1 < i, j < 3. For example, when 1'· — 3, / + 1 refers to vertex 1. The formula for integration of the y derivative product is obtained in an analogous manner to that presented above and is

(

3

1

n

)

1 4l"(*<+2

C dL dL J ~dyJy dxdy=

A

V 2 V > ) -

-

-

Another commonly encountered integral is (3.1.12)

I =

JL,Ljdxdy.

When ϊφ j , wi, = m = 1, m = 0 , and (3.1.9) yields 2

(3...13,

/

3

/

Λ

Λ

φ

=

2

, ( β Ο

=

^ .

Triangular Elements

113

When the subscripts are the same (i.e., i = j), m , = 2 , m = m ~0, obtain 2

fL,L,dxdy

(3.1.14)

= 2A^

3

and we

= lA.

Whenever the integrand is a polynomial, (3.1.9) leads to an efficient integration formula. In the event that axisymmetric problems are encountered, however, the radius appears in the denominator of the integrand and it is n o longer a polynomial. Under these circumstances, either an approximate average radius over the triangle may be assumed with the resulting constant radius taken from under the integral, or numerical integration may be used. Several schemes have been developed for numerical integration over triangles. They have the general form / = ff(x,y)dxdy

(3.1.15)

Σ /(*>.^>Η

=

J =

Λ

1

in global coordinates, and in local coordinates /= C

(3.1.16)

( ~ g(L,,L )2AdL,dL l

L2

2

= 2A 2

g{L ,L )w u

2j

2

jy

7=1

where we have used the relationship (Connor and Will, 1969) dxdy =

(3.1.17)

2AdL,dL . 2

Integration such as indicated in (3.1.16) is performed over the element indicated in Figure 3.1b and (3.1.17) is the area transformation. Note that the integrand has also changed in going from (3.1.15) to (3.1.16) because the derivatives with respect to the global coordinates must now be expressed in the L, coordinate system [see (3.1.6)]. Appropriate values of n, L, and w are tabulated in Table 3.1 for polynomial forms of g up to fifth degree. Although the derivatives 3 L , / 9 x and dL,/dy are easily evaluated, this may not be true for more complicated basis functions. In general, it is necessary to formulate the Jacobian matrix and invert it to obtain the χ and y derivative expressions. Thus, applying the chain rule to the general basis function φ, = <j> {L , L , L (L , L )}, one obtains r

l

l

2

3

t

2

f

114

Finite Elements on Irregular Subspaces T A B L E 3.1.

η

Polynomial Degree

Quadrature Formulas for Triangular Finite Elements

1

^2 f

r

1 Linear

1 0.333 333 0.333 333

0.333 333

0.500 000

3 Quadratic

1 0.500 000 0.500 000 2 0.000 000 0.500 000 3 0.500 000 0.000 000

0.000 000 0.500 000 0.500 000

0.166 667 0.166 667 0.166 667

4

1 2 3 4

0.333 0.600 0.200 0.200

333 000 000 000

0.333 0.200 0.600 0.200

333 000 000 000

0.333 0.200 0.200 0.600

333 000 000 000

-0.281 0.260 0.260 0.260

250 417 417 417

1 2 3 4 5 6 7

0.333 0.797 0.101 0.101 0.059 0.470 0.470

333 427 287 287 716 142 142

0.333 0.101 0.797 0.101 0.470 0.059 0.470

333 287 427 287 142 716 142

0.333 0.101 0.101 0.797 0.470 0.470 0.059

333 287 287 427 142 142 716

0.112 0.062 0.062 0.062 0.066 0.066 0.066

500 970 970 970 197 197 197

Cubic

7 Quintic

From Brebbia and Connor. 1973.

Equation (3.1.18) can be written in matrix form as 3φ,

(3.1.19)

dL,

3L

2

2

L,

2

or " 3φ, " (3.1.20)

dL, 3φ,

3L

3φ,

Li dx d, dy dL '-' . dy

dL, dx

3φ,

dL

dy

3*

= [J\

' d, ' dx 3φ,

dy

2

where \J] is the Jacobian matrix. The χ and y derivatives of the basis functions, defined in the local coordinate system, can now be obtained from ' 3φ, " (3.1.21)

dx

3φ,

dy

=uv

l

dL, 3φ,

3L

2

Triangular Elements

115

which is analogous to the expression obtained for rectangular elements [i.e., (2.2.81)]. Although the evaluation of the elements of the Jacobian matrix is relatively straightforward in the case of regularly shaped rectangles, it is somewhat more cumbersome for triangles. The transformation of the general triangular element to the equilateral of Figure 3.\b is achieved using the same functions as those adopted as bases for the unknown solution t5(x, y). The appropriate relationship is (3.1.22a) i= l

(3.1.22b) ;=l

where x, and y are Cartesian coordinates of the nodes of an irregular triangle. It is easily shown that when φ, is specified as L there is a one-to-one correspondence between the nodes in the local and global system, and the straight-line geometrical relationship between nodes in both systems is preserved. To evaluate the Jacobian matrix equations, (3.1.22) are differentiated to yield (

r

dx dL, dy_ dL,

'2

1

= 1

dx dL,

'dL,

= 1

i

=l

_4

dy_ dL,

3*1

= 2y, dL, ι

-ι

'dL, 9φ,

As an illustrative example, let us consider the case of linear triangles wherein φ, = L These expressions become r

dx dL, dy_ dL,

3x dL,

x, - x ; 3

dy_ dL,

'•y\-yi>

/-I

and the Jacobian matrix is (3.1.23a)

c

c

i - *

3

i~ 3 x

yi-yj y-L~y-i

116

Finite Elements on Irregular Subspaces

The determinant of the Jacobian matrix, det [J], is generally referred to as the Jacobian. (3.1.23b)

de\[J] = (x,-x,){y

- y,)-(x -x,)(

2

(3.1.23c)

2

Λ

[/]

Λ

_

Λ

-

-y,)\

y]

J'I

det[7)

Examination of (3.1.23b) and the determinant of (3.1.2) reveals that they are identical, which indicates that

aet[J] =

(3.1.24)

2A.

Thus the incremental area relationship of (3.1.17) can be rewritten dxdy=dei[J]dL dL ,

(3.1.25)

l

2

which is a well-known relationship and was employed earlier in (2.2.84). Equation (3.1.23c) may now be used in conjunction with (3.1.21) to obtain the global derivatives. For example, the case of φ, = L , yields ' 3L, ' (3.1.26)

dx 3L,

1

yi ~ r

3L,

Λ - y\

3

~ 2A

x,

2/1

3L,

31,

-x.

3L,

which is easily verified by comparison with (3.1.4). 3.1.3

The Quadratic Triangular Element

The preceding section introduced the basic principles behind triangular finite elements using the linear triangle as an example. As in the earlier case of rectangular elements (Section 2.2.4), we now extend our linear analysis to consider the case of quadratic basis functions. T h e quadratic element is illustrated in local coordinates in Figure 3.2a. In contrast to the linear triangle, three nodes must now appear on each side in order to allow a quadratic approximation. The quadratic basis functions are generated by first assuming a general second-order approximation of the unknown function tJ over the element. The polynomial is then constrained such that u ( x , , y )—V at the ith nodal location (/' = 1,2,... ,6). The starting point is a quadratic polynomial written in the local coordinates L, and L that is, i

l

2

(3.1.27)

φ,• = a + bL + cL + dL] + eL\ + }

2

fL L . 1

2

Triangular Elements

117

L M 3

Figure 3.2. Quadratic (a) and cubic (b, c) triangular elements.

We now impose six constraints for each basis function. Each function must be unity at the node for which it is defined and zero at all other nodes. For the basis function at node 1, we obtain 1 1 0

" 1' 0

1 0

I 2

ι

0 0

0 0

0

0 0

0 i

0

4

ι

2

2

0

0 0

0 i 4

i 4

0 0

Equation (3.1.28) may be solved for the vector of unknown coefficients to yield (3.1.29)

[a,

b,

c,

d,

e,

/ ] = [0, r

- 1 ,

0,

2,

0,

0) . T

When an equation analogous to (3.1.28) is generated and solved for each basis

118

Finite Elements on Irregular Subspaces

T A B L E 3.2.

Linear, Quadratic, and Cubic Basis Functions for Triangular Elements

Functions Quadratic

Linear

Basis

Cubic |Z. (3L -l)(3L,-2) 1L,L (3L,-1) |L,Z. (3L -I) K (3L -l)(3L -2) |L L,(3i. -l) K L (3L -I) iL (3L -l)(3L -2) iL L,(3L,-l)

2L]-L,

Φι Φι Φ

^3

Φα

—

4L L

Φ Φ* Φι

—

2L]-L,

—

—

4L L, — —

—

—

—

—

4L L t

3

5

φ Φιο 9

1

2

2L\-L

2

2

2

3

2

2

2

2

2

3

—

φ»

l

2

2

2

3

3

3

3

3

3

27L,i. Z. 2

3

function, the formulas recorded in Table 3.2 are obtained. Because these bases are polynomials in L,, the integrals generated in the Galerkin formulation can generally be evaluated using (3.1.9). When numerical integration is required, as in the case of elements with curved sides (discussed later), (3.1.16) can be used. 3.1.4

The Cubic Triangular Element

The extension of the concepts introduced for quadratic elements to consider cubic polynomial bases is relatively straightforward. The point of departure is the cubic polynomial expansion (3.1.30)

$=a

+ bL + cL + dL\+eL\ x

+ hL L\ l

1

+ iL, +

+ fL L ]

2

+

gL]L

2

jL\,

which involves 10 coefficients. Nine of the nodes required to define a cubic basis can be conveniently located along the edges of the triangle as indicated in Figure 3.2b. The tenth node is located, for convenience, at the centroid of the triangle. The vector of unknown coefficients can be readily evaluated using an expression analogous to (3.1.28). The resulting basis functions, once again defined in local L, coordinates, are tabulated in Table 3.2. In our discussion of cubic elements in Chapter 2 we suggested the possibility of using Hermite polynomials as basis functions. Recall that this approach replaced function values at interior and side nodes with function derivatives at corner nodes. In the two-dimensional case, rather than solve for the function ύ, at / = 1,2,..., 16, we sought

Triangular Elements

119

TABLE 3 3 . Cubic Basis Functions for Elements with Continuous 3«/dx and 9A/9y at Nodal Locations Basis Function

Functional Form

0ooi

/. (L,+3L +3L3)-7L,L L3

0ιοι

L ( a Z , - a L ) + (a -

0011

L (A L -A3L )+(/> -* )L L,L

0002

L^(L +3Z. +3L )-7L L L

0102

L|(a,L -

0012

L (b L,

0003

L](Lj + 3 L , + 3 L ) + 7 L , L L

0103

Z.|(a L -e /- ) + (e,-e )£, L Z-

2

2

2

a )L L L

2

3

1

2

i

2

3

t

2

3

2

2

2

3

2

3

3

3

I

1

Β ί.,)+(Β 3

3

- b.L^+i^

2

3

2

2

2

1

l

2

-

b )L L L, 3

2

2

3

- a,)L,L i-3

2

2

3

2

t

2

3

1

2

3

0013 27L,L L

0004

2

a,

= *A

~ *,

3

= >> ~ Λ

After Felippa, 1%6.

A similar approach can b e developed for triangular elements. In lieu of obtaining 10 values of the function we seek the function and its χ and y derivatives at the corner nodes of the triangle and the function alone at the centroid. N o t e that unlike the complete Hermite element, here d c/dxdy is not continuous at the nodes. This yields the 10 degrees of freedom we require to correctly define cubic interpolation. T h e element corresponding to this type of formulation is illustrated in Figure 3.2c and the corresponding bases, which are obtained in the same manner as the Hermites of Chapter 2, are given in Table 3.3. These bases represent the unknown function ύ within a given element as 2

(3.1.31)

ύ=

A «i 2 Uj**j + Σ

in a manner analogous to (2.2.71).

d

u

,

^

dx

a

i

da ; ^

dJt

^

120

Finite Elements on Irregular Subspaces

Because the basis function at the centroid interacts only with the nodes within the element for which it is defined, it can be eliminated from the global system of equations. In other words, one can express the coefficient at the centroid as a linear combination of the other nine coefficients. This is true for either type of cubic formulation. Thus one decreases the number of unknowns in the final matrix equation. The value at the centroid can, of course, be calculated later from the corner values. This procedure is commonly called "static condensation" in the structural engineering literature. 3.1.5

Higher-Order Triangular Elements

In the preceding section we examined the use of elements that exhibited continuity of the first derivative at nodal locations. It is also possible to require additional continuity constraints at the expense of calculating additional degrees of freedom at each node. Several elements of this kind have been formulated for structural engineering applications where an accurate representation of the strain is required along with displacement. This situation arises, for example, in the bending of plates. In general, a more accurate representation of the first derivative is achieved with elements of higher-order continuity. This may not always be the case, however, in the neighborhood of a discontinuity, where a large number of lower-order elements may be superior. It must also be kept in mind that, given a specified number of degrees of freedom, the larger bandwidth of the higher-order element results in increased computer storage and computational effort as compared to lower-order elements.

3.2

ISOPARAMETRIC FINITE ELEMENTS

In Chapter 2 regular rectangular elements were considered, and earlier in this chapter we considered triangular elements. It would appear, from the evidence presented to this point, that triangular elements would be preferred to rectangles because of the flexibility of the triangle in generating a net. In this section we show that irregular quadrilateral elements can be formulated that extend the utility of the regular rectangular element developed earlier. 3.2.1

Transformation Functions

In the isoparametric finite element approach, elements of irregular shape are transformed or mapped into elements of regular geometry to facilitate integration. Most of the concepts involved in this procedure have been introduced previously in a disguised form. They are repeated, in part, in this section for the sake of clarity and continuity. Consider an irregularly shaped quadrilateral element with straight sides and four nodes, one located at each corner of the element (see Figure 3.3a).

Μ,-Ο

(ο,-η

( b)

χ

(d)

Figure 3.3. Isoparametric elements: ( a ) linear quadrilateral; {b) quadratic quadrilateral with curved sides; (c) mixed quadrilateral; (d) quadratic triangle with curved sides.

122

Finite Elements on Irregular Subspaces

Further, let us assume that we wish to evaluate an integral arising out of the Galerkin formulation which has the form

7-/-ί < **> + · M A A 3φ

8ψ

9

Formally, the integration of (3.2.1) is to be performed over the entire x — y domain. In practice, however, the integral is evaluated elementwise where the elements may be of irregular shape such as those illustrated in Figure 3.3: (3.2.2)

/_

2 j ^ - -

+

J

- - j

d

x

d

y

,

where Ε is the number of elements. An obvious computational problem arises when we attempt to evaluate the elementwise integrations over this irregular geometry. The solution to this difficulty was introduced, in part, in Chapter 2. We transform (3.2.2) into an integral in the local coordinates (ξ, η) and integrate numerically. T o transform from (x, y) to (£, η) space requires the formulation of a transformation or mapping function. We develop such a function using a finite expansion analogous to that developed for the trial function ύ. The expansions for the χ and y coordinates are 4

(3.2.3a)

*=

(3.2.3b)

y=

Σ */*,(£.l)

( = 1

Σ ΛΨ,(Μ).

where, as earlier, x and y, are nodal coordinates in (x, v) space. These expressions are of the same form as (3.1.22) introduced earlier for triangular elements. Notice that the functions ψ, are formulated in the (ζ, η) system. T o establish the form of ψ,(£,η), we impose two constraints: t

1.

There must be a one-to-one mapping of the nodes between the two coordinate systems.

2.

There must be a linear variation in χ and y along the sides of the square element in (ξ, η) space.

The second constraint leads us to choose a transformation expression of the form (3.2.4a) (3.2.4b)

y = a + b i + c T\ + 2

1

2

d tri. 2

Isoparametric Finite Elements

123

It is readily apparent that χ and y vary linearly along ξ(η = ± 1) and along i»(* = ± D .

When (3.2.4) are evaluated at the nodal locations in each coordinate system, we obtain an equation of the form (using χ as an example)

"V (3.2.5)

1

1

-1

1

1

-1 1 1

-1 1 -1

*3

1

-1

"a."

ft,

Equation (3.2.5) is an expression of the first constraint (a). Solution of (3.2.5) yields

ft,

(3.2.6)

_ 1 ~ 4

1

1

1

-1 -1 1

1

1

-1

1

-1

1

1

Substitution of (3.2.6) into (3.2.4a) gives the required series expansion for x: (3.2.7)

* = 4 [( * l + * 2

+ (x

t

* 3 + *4 ) + (

+

~X

+ Χ

2

3

_

* I + *2 + * 3 ~ * 4 ) £ + ( ~ * 1 ~ * 2 + * 3 + * 4 ) η

~χ )ξη]. 4

T o establish the form of ψ,, we must rearrange (3.2.7) in a form more directly comparable to (3.2.3a). Thus we rewrite (3.2.7) as (3.2.8)

χ =

ί ( ι - ί - η + ίη)Λ, + Ηΐ + Ι - η - ί η ) * 2

or, equivalently, (3.2.9)

x

= [i(l- )][i(i- ,)] + [i(l + |)][i i- ,)] i

T

+ [i(l + i)][i(l +

X)

™)]^3

(

T

+ U(l-0][Hl

+

X2

")]*4.

124

Finite Elements on Irregular Subspaces

It is now apparent through an examination of (3.2.9), (3.2.3a), and (2.2.63) that the functions ψ, are, in fact, the basis functions φ, formulated for the rectangular element of Chapter 2. When the transformation or mapping functions and basis functions are the same, such as in this case, the formulation is called isoparametric. It should be evident that we would encounter no difficulty in using a higher-degree polynomial for the bases than for the transformation functions. In Figure 3.4 there is no advantage in using quadratic functions for the transformation because the element sides are straight with midside nodes at the midpoints and, therefore, exactly described by linear functions. On the other hand, the unknown function ύ may behave such that quadratic interpolation is warranted. When the transformation is accomplished using functions of lower degree than those used as bases, the formulation is called subparametric; when the transformation function is of higher degree than the basis function, the element is called superparametric. There are occasions when it is advantageous to use basis functions of different degrees in different locations within the domain of interest. This can be achieved through the use of a mixed element such as illustrated in Figure 3.3c. This element may employ different basis functions along each side and thus may act as a transition element. The two-dimensional bases for this element are given in Table 2.76. To evaluate the integral of (3.2.2), it is necessary to express the derivatives in terms of local coordinates. The integrands of (3.2.2) contain χ and y derivatives of the basis functions, while the bases are defined in terms of | and η. Using the chain rule, one can relate the required χ and y derivatives to the readily computed £ and η derivatives by (3.2.10a)

—ay"-—g^af+ —97-θξ

x

t

Figure 3.4. Subparametric element in global (x,y) and local (ξ,η) coordinates. Boxed nodes only are used for geometry transformation [i.e., only corner nodes have prescribed (x, y) coordinates]. Linear mapping gives straight sides with midside nodes at the midpoint of the side.

Isoparametric Finite Elements

9φ{ξ,η) 3η

(3.2.10b)

=

9φ{ξ,η) dx

dx 3η

125

3 φ ( £ , η ) dy 3ν 3η

t

Here χ and y derivatives are defined implicitly because (3.2.3) lends itself to computation of partial derivatives of χ or y with respect to ξ and η. On the other hand, χ or ν derivatives of ξ or η are not always readily calculated. Equation (3.2.10) can be rewritten in matrix form as

(3.2.11)

3JC

dy'

di

3?

Η

9φ(£,η) 3η

dx

dy 3η

3η

' 3φ({.ΐ) 3χ 3Φ(€,«) 1

^

J

or dφ (3.2.12)

di

dφ dx

3φ

dφ

where [ / ] is the Jacobian matrix. The derivative relationship we require is readily established as 3φ dx

(3.2.13)

3φ dy

3φ

= [·/]

3ξ 3φ 3η

The efficacy of this approach depends upon our ability to efficiently evaluate the Jacobian matrix. By differentiation of (3.2.3), it is apparent that [J] is easily obtained from the relationship " dx (3.2.14)

dx 3η

dy' dt

' 3φ,

3φ

dt

dy 3η

3φ,

3ί 3φ

3η

3η

2

3φ

3

di 2

dφ

'*\ Xl

y\ >2

3

3η

The two matrices on the right-hand side of (3.2.14) are known: the first a function of i and η is obtained through differentiation of φ, and the second contains the global coordinates of the nodes. Although the integrand of (3.2.2) is now expressed in local coordinates, the limits of integration and the differential area must also be changed. The

126

Finite Elements on Irregular Subspaces

appropriate relationship is [see, e.g., (3.1.25)] j^)dxdy

(3.2.15)

= f

J'

(-)det[J]did . V

Combining (3.2.2), (3.2.13), and (3.2.15) yields an integral expressed entirely in (ξ, η) coordinates: (3.2.16)

t\ r\ ' . +

1

/ dy 3Φ,

dy 3φ, \

^ - , J _ , d e t [ y J l 3η Η 1

/

/ dy 3φ,

dy 3φ, \

31 3η / d e t [ / ] \ 3η 3ξ

3£ 3η /

3x 8ψ, , dx 3φ, \

d l t T 7 l l ~ ^ ^

+

1

1

/

3χ3φ,

^ " 3 T / o ^ i ~ ^ W

3χ3φ,\

"3T^)

+

r

i

l

. , .

d e t [ y ] i /

^ '' i

where d e t [ / ] , the Jacobian, is given by ,3.2,7)

d

e

,

U

1

= | | | Z _ | £ | .

Whenever an integral involving a polynomial expression in φ(£, η) is encountered, it is readily evaluated because only (3.2.15) needs to be applied. As an example, consider the following commonly encountered integral: (3.2.18)

^ φ , ( £ , η ) φ , ( ξ , η)

= f

J*

φ,{ξ,ν)φ,(ζ,7,)ά«ν]α·ξάη.

There is n o need in this case to modify the original integrand. However, the right-hand integral has an extra factor which, in some cases, is dependent on ξ and η. 3.2.2

Numerical Integration

Integrals of the form (3.2.16) and (3.2.18) can be evaluated using several numerical procedures. The most commonly used approach is based on Gaussian quadrature. In this scheme the integral, which is defined over the range — 1 to 1, is approximated as the weighted sum of the integrand evaluated at specified points in the interval —1 to 1. Consider, for example, the function /= f(£). The appropriate Gaussian quadrature expression is

(3.2.19)

Σ *,/(?,)+Λ,„ /=ι

Isoparametric Finite Elements

TABLE 3.4.

where

127

Gaussian Quadrature Formulas: Weighting Functions H>, and Coordinates of Integration Points £,·

η

Polynomial Degree

I

Linear

0.000 000

2.000 000

2

Cubic

0.577 350

1.000 000

3

Quintic

0.774 600 0.000 000

0.555 556 0.888 889

4

Septimal

0.861 136 0.339 981

0.347 855 0.652 145

±£

H>,

(

w, = weighting factor £,=coordinate of the i t h integration point (Gauss point) η — total number of integration points Λ = 2 " '(«!) (2« + 1 ) ~ ' [ ( 2 « ) ! ] - ^ { 2

+

4

3

Λ

-1<0<1.

«ζ

A polynomial of degree In - 1 can be exactly integrated using η integration points. T h e weighting coefficients a n d integration points are computed using Legendre polynomials and the scheme is frequently called Gauss-Legendre quadrature. Tabulated values of w, and £, for up to four points are found in Table 3.4. The two-dimensional equivalent of (3.2.19) for a polynomial of degree up to 2n — 1 in ξ or η is obtained in a straightforward manner, (3.2.20)

f _ f _ f ^ )

d

^ =

Σ

Σ^,/α,ΐι,),

7 = 1,=i

where the locations TJ and corresponding weighting factors w are obtained from Table 3.4. T h e three-dimensional extension of (3.2.20) is 7

(3-2.21)

y

f_f_f_M^i) i ^s= d d

ΣΣ

i ^ . A € , . i » , . f a ) .

k=17=11=1

Although exact integration of terms of the form of (3.2.16) may be necessary in some instances, in selected cases fewer Gauss points than formally required may be used with satisfactory results.

Finite Elements on Irregular Subspaces

128

Examination of Table 3.4 reveals that the Gauss points are always located on the interior of the domain of integration ( — ! < £ , · < 1). When this domain consists of an element, the Gauss points then fall within the element. Because of the simple form of the bases at the nodes, it is sometimes desirable to use the nodal points themselves as integration points. Although this is not permitted using Gaussian quadrature, it can be accomplished using a highly accurate scheme based on Lobatto formulas. Lobatto formulas require one more integration point than Gauss formulas to achieve a particular order of accuracy. T h e appropriate Lobatto integration formula is

/ ' ftt)dt

(3.2.22)

'

= wj(-l)-r i

*,/(£,)+*„/(])+

n

R. n

1=2

For Lobatto

Λ

- = (^)( ) '" [("-0^2η-1)- [(2ι.-2)!]- ^ί 2

α

,

, >

3

ί=β

\<θ<\. Selected weighting functions and coordinates of integration for Lobatto formulas are given in Table 3.5. It is apparent from this table and the integration formula (3.2.22) that some integration points lie on the perimeter of the

TABLE 3.5. Lobatto Integration Formulas; Weighting Functions κ>, and Coordinates of Integration Points £,

η

Polynomial Degree

3

Cubic

1.000 000 0.000 000

0.333 333 1.333 333

4

Quintic

1.000 000 0.447 214

0.166 667 0.833 333

5

Septimal

1.000 000 0.654 654 0.000 000

0.100 000 0.544 444 0.711 111

Fron Abramowitz and Stegun. 1964.

= €,

w

,

Isoparametric Finite Elements

129

domain of integration. Moreover, when this domain is a standard element, some of these boundary points correspond to nodal locations. 3.23

finite

Isoparametric Serendipity Hermitian Elements

The formulation of finite elements employing Hermitian polynomial bases was introduced in Section 2.2.4. Recall that we developed two approaches in that section, one that preserved the C ' continuity of the bases and a second which guaranteed C continuity only at the nodal points and C ° continuity along the element edges. Because the first approach generates two additional unknowns when a coordinate transformation is required, (i.e., d Tj/dx and d Tj/dy ), let us consider the C ° continuous case. This approach results in three degrees of freedom at each node (i.e., 7", 9 7 J / 3 x , 8 7 J / 3 y ) . ]

2

2

2

2

Consider the isoparametric Hermitian finite element illustrated in Figure 3.5. The principal feature that distinguishes this element from those illustrated earlier is the lack of side nodes on a curved side. Information required to describe this curvature is now provided at the corners. As in earlier isoparametric formulations, we use the basis functions as coordinate transformation or mapping functions. Thus a point {x, y) in the global coordinate system is expressed in terms of the nodal locations (x,,y ) and the basis functions as (see van Genuchten et al., 1977) t

(3.2.23a)

χ(ξ,·η)=

Σ ΦοοΛ+Ψιο^ + Ψοι^

(3.2.23b)

,(ξ,η)=

2 ΦηΛ + ΦηΜ

-Μ)

(-4,-0

+

Φο^,

H.t)

«,-η

Figure 3.5. Isoparametric Hermitian finite element with side lengths £,. The angles defined by the element sides £, and £ and the χ axis are designated a and β, respectively. 4

y

Finite Elements on Irregular Subspace

130

where (3.2.24a)

Φοο>

(3.2.24b)

Φιο

(3.2.24c)

Φοι>

=

*(ΐ + ^ ) ( ΐ + η η , ) ( 2 + « , + η η , - £

- Κ , ( ΐ - £ ) 0 + «,)( 2

= 7

2

™,)'

1 +

=

and η = ± 1 depending upon the local coordinates of they th node. Because of the properties of these bases, the coefficients of the Jacobian matrix can be readily evaluated provided that the coefficients 3 x / 9 £ , 9.x . / 9 η , 3 y , / 3 i , and 3 > , / 3 η can be determined. These coefficients can be evaluated geometrically when the element side lengths are known; consider, for example, the approach 7

y

(3.2.25a)

ycosa,,

οξ _

(3.2.25b)

3? dx

(3.2.25c)

1

9η

J

y

=

sin a , , -fcosfi , J

dyj _ £

(3.2.25d)

3η

=

y s i n ^ .

where side k defines the angle β with the χ axis at the node j as illustrated in Figure 3.5. Unfortunately, the side lengths £ are not generally known when curved elements are employed. However, along an element boundary, (3.2.23) and (3.2.24) reduce to a one-dimensional form, for example, y

d

Xi

(3.2.25e )

x(0 =

(3.2.25f )

y{£Y-

(3.2.25g )

x(ij) =

Σ j=\

ν

Σ j=\

(φοο,Λ + Φ \

Φ ο ο Λ + Φ\0]~$£

for s i d e η = ± 1,

2

Σ

7=1

(3.2.25h )

y{i))-

Σ

0j j + Φ< X

9*, 3η

(φοο^+Φο,,-^)

\

for side

ξ=±\.

Isoparametric Finite Elements

131

tj can be calculated from

(3.2.26)

with the integration performed numerically along side j. The first estimate of £ is usually based on the straight-line distance between the two nodes. Examination of (3.2.23), (3.2.25), and (3.2.26) reveals that an iterative procedure is in fact necessary to evaluate the Jacobian. Given an estimate of £ , (3.2.25) is evaluated; this result is introduced into (3.2.23) to obtain 9 v / 9 £ , and so on, which are finally used in (3.2.26) to establish a new estimate of £ . The overall procedure is repeated until a suitable value of £ is obtained. A final estimate of the Jacobian matrix coefficients is then made. Because these Jacobian elements are only used for the side-length transformation, they are no longer needed after the £. are calculated. The Jacobian matrix coefficients for the interior Gauss points are different. Fortunately, this procedure need not be repeated unless the mesh geometry is modified. There are two main advantages to employing Hermitian isoparametrics. Whenever a first derivative is required, it is easily obtained, particularly at the nodes. In addition, there is a saving in computational effort over C ° cubic elements. Numerical experiments conducted using the two-dimensional transport equation reveal that 25 to 40% fewer degrees of freedom are required to obtain the same accuracy achieved by C ° cubic elements (van Genuchten et al., 1977). T h e principal disadvantage involves the introduction of N e u m a n n - t y p e (prescribed gradient) boundary conditions. We now overcome this difficulty by using a normal-tangential coordinate system more suited to the Hermitian isoparametric element. 3.2.4

Isoparametric Hermitian Elements in Normal and Tangential Coordinates

We have presented one approach to the solution of field problems using isoparametric Hermitian finite elements, and it is probably a worthwhile investment to present an interesting alternative approach developed by Frind (1977). The particular problem he considered to demonstrate his technique involves the calculation of the fluid potential in a wedge-shaped region of an axisymmetric radial flow field in a porous medium. Flow occurs from the perimeter toward the center of the wedge (see Figure 3.6). The governing equations are (3.2.27a)

v ( A : v w ) = 0,

(3.2.27b)

u=u

c

on

S , 2

132

Finite Elements on Irregular Subspace

Figure 3.6. Wedge-shaped grid for radial flow (after Frind, 1977). ^9"

(3.2.27c) du

(3.2.27d)

5 and5 ,

on

3

4

where Κ is the hydraulic conductivity of the porous medium. Let us employ Galerkin's method of weighted residuals to obtain a set of approximate integral equations (3.2.28)

f{V'{K

Vu)) dxdy=Q,

i = 1,2,..., 14,

ai

A

J

α = 0,σ,τ,στ,

where the symbol φ represents four new types of Hermitian bases: αι

Φο,'

Φ.,'

Φτ/'

3

n

Φοτ,'

d

which will be described shortly and « is defined as

(3.2.29)

ϋ

ύ= 2

i , d U, + ^ Φ „ + ^ Φ ,+ ^ Φ W

7= '

"Α,

2

W

τ

τ

β

ν

.

Although one can readily solve (3.2.28) directly using Hermitian cubic bases, it is sometimes advantageous to apply Green's theorem so that the natural boundary conditions appear directly in the integral operator and only first-order derivatives require transformation:

(3.2.30)

f {Kvu)'V dxdyai

i = 1,2,..., 14,

JKvu'ηφ„,ds α=0,σ,τ,στ.

=0,

Isoparametric Finite Elements

133

To establish the functional form of the bases