A

Integration Formulas

Some useful formulas and results on integration are given below. Proofs can be found in any standard calculus textbook. Let R" denote the n-dimensional Euclidean space and Z+ the set of nonneg­ ative integers. A useful integration formula is

(A.l)

00 n 1 t where r(n) is the gamma function, defined by f(n) = 1 t - e- dt, n > 0, such that n r(n) if n is real, I'()n+ 1 = { n! ifn E Z+.

A special case of formula (A. I) for m , n E Z+ is

1 ,I m n m.n. x (1 - x) dx = ( )1 ' (A.2) 1o m+ n+l .

INTEGRATION BY PARTS. Let f(x) and g(x) be piecewise continuous func­ tions in an interval (a,b) with their first derivatives continuous in (a,b). Then using integration by parts we get

(A.3)

Further, if f(x) and g(x) are piecewise continuous functions in an interval (a, b) with their first and second derivatives piecewise continuous in (a, b), and h(x) is 394 A. INTEGRATION FORMULAS a continuous function in (a, b), then using integration by parts we get

The integration by parts in R", n 2: 2, is defined as a consequence of the gradient and divergence theorems . Let a finite domain n c R2 be bounded by a smooth closed curve I', and let wand F be scalar functions and G a vector function continuous on n. Then

Gradient Theorem: fk"ilF dx dy = i n F ds, Divergence Theorem: fk"il . G dx dy = in. G ds, where n is the outward normal to the curve I', ds denotes the line element that is defined by ds = J dx2 + dy2 . These two theorems lead to the following two 2 useful identities in R :

fln("ilF)Wdxdy=- fin("ilw)Fdxdy + inwFds, (A.6)

- fin ("il 2F)wdxdy = fin v«. "ilFdxdy - i ~: wds, (A.7)

OM . h ld .. U· wereh on = n· v = nx oxa+ ny oy a IS t e norma envanve operator. smg the gradient theorem, the component forms of (A.6), with appropriate variables, are as follows:

fin w ~: dx dy = - fin ~: F dx dy + i nxw F ds, (A.8) fin w~: dxdy= - fin ~; Fdxdy+ i nywFds.

The gradient and the divergence theorems are valid in R3 if the surface integral on the left side is replaced by a volume integral and the line integral on the right is 395 replacedbyasurfaceintegral. Formula(A.8)canbeextended to higherdimensions; for example, in R3 w~: ~: l~n 11k dxdydz = - 11k Fdxdydz+ nxwFdB, w~: ~; l~n 11k dxdydz = - 11k Fdxdydz+ nywFdB, (A.9) ~~ ~: JJLw dx dy dz = - JJL F dx dy dz +Jfan n z w F dB, wherean denotesthe boundary surfaceof n,and dB denotesthe surfaceelement. B

Special Cases

Some special triangular and rectangularelements lead to different stiffness matrices and force vectors. The following three cases are valid for the Laplacian - \72 on a right-angled triangular element with sides a and b, a 2 b, and the location of the local nodes 1, 2, and 3, such that the local node 1 is at the origin.

1. For a right-angled linear triangular element n(e) with base a and altitude b, if the local nodes 1,2, and 3 are at (0,0) , (a, 0), and (a,b), respectively, and the local node 2 is at the right angle (see Fig. B.I (a» , then

(B.1)

and

(B.2)

2. For a right-angled linear triangular element n (e) with base a and altitude b, if the local node 1 is at the right angle and the nodes 2 and 3 are numbered counterclockwise (see Fig. B.l(b», then

(B.3)

and the force vector f (e) is the same as in (B.2). 398 B. SPECIAL CASES

3. For a right-angled linear triangular element nee) with base a and altitude b, if the local node 3 is at the right angle and the nodes 1 and 2 are numbered counterclockwise (see Fig. B.l(c», then

K(e) = _1_ (B.4) 2ab

and the force vector r(e) is the same as in (B.2).

(a,b) (O,b) (O,b) (a,b) 3 2 3 3

2 2 (0,0) (a,O) (0,0) (a, 0) (0,0) (a) (b) (c)

Fig. B.1. Three Cases of a Right Triangle.

4 . For a 4 node bilinear square element nee) of side a (Fig. 5.2 with a = b), the stiffness matrix and the force vector for the Laplacian - '\72 are given by

!1 ~1 -2 j 2 =i -1] r(e) = __(e)a {i} (B.5) [-2 -1 4 -1 ' 4 r t ' -1 -2 -1 4 1

5. For heat transfer problems with convective conductance (3, the following ele­ ments are mostly used, with the respective stiffness matrices. 399

(5a) For a 3-node linear triangular element (Fig. 5.1):

K (e) - -!L!1...(3(e)Z(e) [ 2~ ~1 o~] b - 6

(3(e) l (e) [0 0 0] (3(e) l (e) [2~ 0 + ~ 021 +~ 0 (B.6) 6 0 1 2 6 0

12 (3 f (e) _ 12(e)T 00 Z(e)12 {OIl} b - 2

(B.7)

(5b) For a 4-node bilinear rectangular element (Fig. 5.2):

(B.8)

(B.9)

(5c) For a 6-node quadratic triangular element (Fig. 5.7) :

4 2 -1 0 0 0 1 16 2 0 0 0 (3(e) l (e) K ( e)=~ -1 2 4 0 0 0 b 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 400 B.SPECIAL CASES

0 0 0 0 0 0 0 0 0 0 0 0 /3(e) l(e) +~ 0 0 4 2 -1 0 30 0 0 2 16 2 0 0 0 - 1 2 4 0 0 0 0 0 0 0 4 0 0 0 -1 2 0 0 0 0 0 0 /3(e) l(e) +~ 0 0 0 0 0 0 30 0 0 0 0 0 0 (B.10) -1 0 0 0 4 2 2 0 0 0 2 16 1 0 4 0 /3(e) T I 3l(e) 1 /3 (e)T35l(e) 1 f (e) _ 13 00 13 + 35 00 35 b - 6 0 6 4 0 1 0 0 1 0 /3(e) T 51l(e) 0 + 51 00 51 (B. ll) 6 0 1 4

6. Consider Eq (3.1) with a1 2 = a21 = c = 0, that is,

-- a ( a ll -au )-- a ( a22 -au ) + cu - f = 0, (B.12) ax ax ay ay

whereall,a22 ,and careprescribedfunctionsofxandy.lfall = k1 anda22 = ka. then Eq (B.12) defines heat transfer problems, where k 1 and k2 denote thermal conductivities. We consider the case when all and a22 are bilinear functions ofthe global variables x and y, i.e., all = a ~~) + ag) x + ag) y, and (0) (1) (2) a 22 = a 22 + a 22 x + a 22 y. Then (6a) In the case of a linear triangular element n (e) ,

(e) _ /1 ( a4>~e) a4>je) a 4>~ e) a4> je») Kij - all a a + a 22 a a dx dy O l e ) x x y Y

= 1 2 fl /3i/3j (a (0)ll + a (1)u x + all(2» Y) 4 (A(e») O (e ) (0) (1) (2» ) +'Yi'Yj ( a22 + a 22 x + a 22 Y dx dy 401

1 [ (1) (2) ) = 2 {3i{3j (a ll 0)100 + all lID + all 120 4 (A(e)) (0) (1) (2) )] +In j ( a 22 100 + a 22 lID+ a 22 120 , (B.13)

where 100 , 110• and 120 are defined in (A.25). (6b) In the case of a bilinear rectangular element n (e),

(B.14)

where H ll and H 22 are defined in (6.15), and

8 A. (e) 8¢(e) H llx = X _ '+'_i J_ dx dy = H ll1 + xie) n " , /1Ole ) 8x 8x 8A.( e) 8¢(e) Hlly = y _ '+'_i J_ dx dy = H 1l2 + yie) H 11, O l e) 8x 8x /1 (B.I5) 8A.(e) 8¢\e) H 22x = x _ '+'_i J_ dxdy = H 221 + xie) H 22 , /JO (c) 8y 8y 8A.(e) 8¢(e) H 22y = y _ '+'_i J_ dx dy :::: H 222 + y ~ e ) H 22, /JO ( e ) 8y 8y where

- 2 -1 - 1 -1 ~ 2 1 1 1 -1 H '" s. [..', !,] H'" ~ ~ [!, ,] 12 -1 1 2 - 2 ' 12a - 1 1 3 ~3 ' 1 -1 - 2 2 1 - 1 - 3 1 - 1 1 -1 3 -3 - 2 - 1 221 ~ -~f -']- 1 H 222 = ~ [1 2 2 H 12b - 1 -3 3 l ' 12 -1 - 2 2 -'1] . -1 - 1 1 1 - 2 -1 1 2 (B.16)

7. Bound ary value problems with axial symmetry are governed by

--1- 8 all r -8U) --8 a22-8U) + aA DO U - f ()r, z = 0, (B.17) r 8rC 8r 8z C8z 402 B. SPECIAL CASES

where aoo, all, and a22 are functions of rand z. The stiffness matrix and the force vector are given by

(7a) In the case of a linear triangular element n(e):

where A(e) is the area of the element.

(7b) In the case of a bilinear rectangular element n(e),

K~) (r~e) ll 1l1 (r~e) 22 221 = all H + H ) +a22 H +H ), (B.20)

2 (e) = a b j(e ) {~} f] 12 2' (B.21) 1

where H ll and H 22 are defined in (6.15), and H 1ll and H 221 are defined above in (B.16). 8. In heat transfer problems when qn = /3 (u - uoo ), we have FOR A LINEAR TRIANGULAR ELEMENT:

K~e) = f3lr;) [~ ; ~], or = /3lj~) [~ ~ 0], 6 000 6 01 21

/3l(e) a or = -.!£.. [2a a a1] ; (B.22) 6 1 a 2 1 oo l /3uoolj~ /3uoolk~) f (e) _ /3u r;){ 11 }_ {01 }_ {a } (B.23) b - 2 0'or - 2 l'or - 2 l' 403

FOR A BILINEAR RECTANGULAR ELEMENT: f3l~;) ~ ~~] (3l;~) [~ ~ ~ ~] K(e) = [i = b 6 0 0 0 0 ,or 6 0 1 2 0 ' 000 0000

_ (3li% [~ ~ ~~] _ (3l~~ [~ ~ ~ ~] or - 6 0 0 2 1 ' or - 6 0 0 0 0 ' (B.24) 0012 1002 f(e) = (3u(x,l~;) {~} = (3u oolW {~} = (3u ooli% {~} b 2 0' or 2 l' or 2 l ' o 0 1

or ~ ~u~l~l {~ }. (B25) c Temporal Approximations

Finite difference schemes to compute the first- and second-order time derivatives are considered.

C.l. First-Order Derivative

To compute the vector x E R" in the matrix equation

M x + K x = F, 0::; t ::; to, (C.1) where x = dx]dt, and M and K are known square matrices and F a known vector in H"; we use the B-scheme, which approximates the mean value of x at two consecutive time steps tn and tn + l by the weighted average of x at these two time steps. This scheme is defined by

Xn+! - Xn B . (1 B) ' A = X n+l + - x n , O::;B::;l, (C.2) utn+1 where D.tn+! = tn+! - tn, and the suffix n represents the value of the quantity at time tn. The weight Brefers to some well-known schemes, which are

0, Forward difference scheme, 1, Backward difference scheme, B= (C.3) 1/2, Crank-Nicolson scheme, { 2/3, Galerkin scheme. 406 C. TEMPORAL APPROXIMATIONS

Of these schemes, the forward and backward difference schemes are conditionally stable, while the other two are unconditionally stable. Substituting the B-approximation (C.2) for times tn and tn+! into Eq (C.1), we get

which simplifies to (C.4) where M: = M + B~tn+! K, K = M - (1 - B) ~tn+! K, (C.5) F = ~tn+! (BFn+! + (1- B) Fn) .

Formula (CA) determines the unknown solution of Eq (C.1) at time t = tn +! in terms ofthe known solution at t = tn. Since the solution is known at t = 0, we take t n +! = n ~t, where ~t is an equally spaced time step. Then we start at tn = ° and compute the solution for tn +! = ~t . This process is continued successively, forward in time, moving with the time step ~t each time. This is known as the forward time-marching process. Also, both F nand F n+! are known since the vector F is known at all times. For better results we use smaller step size ~t. The stability analysis shows that if A is the minimum eigenvalue of

det(K - >. M) = 0, then the solution is (i) stable without oscillations if °< A < 1, (ii) stable with oscillations if -1 < >. < 0, and (iii) unstable if A < -1.

C.2. Second-Order Derivative

We compute the vector x in the matrix equation

Mx+Kx=F, (C.6) where x = d2 x/de, M and B are known square matrices, and F is a known vector. C.2. SECOND-ORDER DERIVATIVE 407

Using the Newmark direct integration scheme, we get

.. (1) ., xn+l - x n 0: Xn+l + - 0: Xn = A , (C.7) utn+1

.. + (~ - (3) " - X n + l - X n _ ~ (3 X n +1 2 x., - A2 At' (C.S) U n + l U n+l where the parameters 0: and (3 control the stability and accuracy of the solution. Thus, for example, 0: = 1/2, (3 = 1/4 give an unconditionally stable solution in linear problems, while 0: = 1/2, (3 = 1/6 give the linear acceleration scheme. From (C.8) we get

(C.g) and (C.?) gives (C.10) where

(C.lI)

Substituting (C.?) and (e.8) into (C.6), we obtain for t = tn+l

or

(C.12)

The procedure is as follows. First, we use formula (C.12) to compute X n + l ; then we use formulas (e.9) and (e.IO), in that order, to compute the first and second derivatives (velocity and acceleration) at time t = tn+l' For preassigned values of xo, *0, and Xo at t = 0, we solve (e.9) and (C.10) successively, forward in time, for a prescribed time step tlt, where tn+1 = n tlt, and compute x, X, and xfor time t > O. The increment tlt can be optimized by the condition that the smallest eigenvalue of det((1 - >') ao M- K) = 0 is less than 1. The step size tlt can also beobtained from formula tlt = Tminlrr, where Tmin is the minimum period of natural vibrations associated with the problem. D

Isoparametric Elements

Isoparametric elements are used for regions with curved boundaries using elements for curved sides . n.i. Introduction

The isoparametric concept is based on the principle of mapping a 'reference' element in the e,1J-planeonto the curvilinear element in the xy-plane, the sides of which pass through the selected nodes. Care must be taken to ensure that no gaps occur between the adjacent distorted elements, which means that the interpolation functions must satisfy the required continuity conditions. D.l.l. Local e-System. The transformation from the global coordinate system x to a local system ~, when the origin is at the center of an element and scaled such that ~ = -1 is at the left-end node and ~ = 1 at the right-end node, is given by (see Fig . D.I)

2x - ( x~e ) + x ~e » ) (D .1) e= l(e)

The e-coordinates are called normal (or natural) coordinates since these coordinates are normalized (nondimensionalized) with values between - 1 and 1. The formula (D. I) establishes the transformation between points x ( x~e ) ~ X ~ x~e » ) and the points e(-1 ~ ~ ~ 1). The (Lagrange) interpolation functions for a linear element in normal coordi­ nate system are 410 D. ISOPARAMETRIC ELEMENTS

s b • (D.2a) •I 2 2 - node element (linear)

s • c • (D.2b) I 2 3 3 - node element (quadratic)

9 2 ¢l = --(1 - e)(1/9 - 16 e) ¢2 = ~~ (1 - eH1/3 - e) (D.2c) ¢3 = ~~ (1 - eH1/3 + e)

9 2 ¢4 = -16 (1 + eH1/9 - e ).

Fig. D.l. Linear Elements in the Normal Coordinate System.

Note that the interpolation functions ¢i (ej) are chosen such that

I, ifi = j ¢i(ej) = Dij = { a ·f ·...t· ,It T J, whereej denotes the e-coordinateofthej-th node of the element, ¢i (i = 1, ... , n) are polynomials of degree n - 1 (n being the number ofnodes in the element), and Dij is the Kronecker delta. Another method of constructing the functions ¢i (ej) is as follows. Form the product of n - 1 linear functions e - ej 0 = 1, . . . , i-I,i + 1, .. ., n; j :f- i), i.e.,

Here ¢i = aat all nodes except the node i. Now, determine Ci such that ¢i = 1 at e= ei, i.e.,

This gives the required interpolation functions ¢i associated with the node i as D.l. INTRODUCTION 411

Formula (D.3) can be used to derive the interpolation functions (D.2a), (D.2b), and (D.2c). For example, to derive (D.2c), note that 6 = -1,6 = -1/3,6 = 1/3, and e4 = 1. Then, substituting these values, we obtain

D .1.2. Triangular Elements. The following cases are mostly used.

1 2 (a) (c)

Fig. D.2. Three 2-D Triangular Elements.

Fig. D.2(a) : At the corner nodes:

k~ 1, 2, 2, 3, 3, 1 . { I - 2, 1,3,2, 1,3

At the center node i = 10 and

D.1.3. Rectangular Elements. We present only those cases that are most frequently used.

FIRST-ORDER CONTINUITY FUNCTIONS. Refer to Fig. D.3. Here e, TJ are dimensionless coordinates, defined by

1 = -(x - xc), (D.4) e a with limits ±1. Then a 1 a a 1 a = = (D.5) ax ~a( ay baTJ' and

r rf(x,y) dA = 4ab f(e, TJ) de dTJ. (D.6) JJR rr-1 -1

U2 u. (- 1,1) .-"------+----_ (1,1)

oL.------+--+

(~:-1 )------e(~,~1) Yj o x Fig. D.3. Rectangular Element.

LINEAR FUNCTIONS. The simplest model for a rectangle has only corner nodal unknowns (thus only 4 parameters), which means that the function varies linearly on the boundaries. Appropriate 2-D interpolation functions for u can be generated by evaluating

(D.7) at the nodes and solving for Cl, C2 , C3, C4, which gives

Ul = Cl + C2 + C3 + C4 at (1, I) , U2 = Cl - C2 + C3 - C4 at (-1, I), U3 = Cl - C2 - C3 + C4 at (-1, -I), U4 = Cl + C2 - C3 - C4 at (1, -I), D.l. INTRODUCTION 413 which, on solvingfor CI, C2, C3, C4, gives

UI + U2 + U3 + U4 UI - U2 - U3 + U4 C2 =- CI = 4 4 UI + U2 - U3 - U4 UI - U2 + U3 - U4 C3 =- C4 = 4 4 thus yielding

U = UI + U2 + U3 + U4 _ UI - U2 - U3 + U4 ~ 4 4 UI + U2 - U3 - U4 UI - U2 + U3 - U4 c - 4 TJ + 4 <"TJ

=

QUADRATIC FUNCTIONS. A refined rectangular elementis obtainedby tak­ ing a quadraticvariation of the function u. Thismodelimproves uponthe simplest (linear)rectangleof the previous section. This requiresan additional interiornode on each side, thus having a total of 8 nodes (see Fig. D.4).

4 7 3

8 6

5 2

Fig. D.4. An 8-ElementRectangle.

Wetake 414 D. ISOPARAMETRIC ELEMENTS

which , when solved for the Ci , i = 1, ... , 9, at the 8 nodes, gives

At node I: Ul = Cl - C2 - C3 + C4 + Cs + C6 - C7 - CS, At node 2: U2 = Cl + C2- C3 + C4 - Cs + C6- C7 + cs, At node 3: U3 = Cl + C2 + C3 + C4 + Cs + C6 + C7 + CS , At node 4: U4 = Cl - C2 + C3 + C4- Cs + C6 + C7- CS, At node 5: Us = Cl - C3 + C6, At node 6: U6 = Cl + C2 + C4, At node 7: U7 = ci + C3 + C6, At node 8: Us = Cl - C2 + C4·

On solving for Ci, we get s U = L¢iUi' i = l where

1 1 2 ¢ l = -4(1 - €) (1 - 7])(1 + € + 7] ), ¢s = '2(1 - € )(1 - 7]), 1 1 ¢2 = -4(1 + 0(1 - 7] )(1 + € + 7]) , ¢6 = '2(1 + € )(1 - 7]2), (D.10) 1 1 2 ¢3 = - 4 (1 + €)(1 + 7])(1 - € - 7]) , ¢7 = '2(1 - € )(1 + 7]) , 1 ¢4 = -4(1 - €)(1 + 7])(1 + € + 7]) , ¢s = ~(1 - €)(1 - 7] 2). 2

D.2. Curvilinear Coordinates

To find the transformation from straight to curved sides, we express x , y in terms of curvilinear coordinates as

The choice of € , 7] depends on the element geometry, i.e., linear, quadratic, cubic, or a higher dimension (see Fig. D.5). Generate the interpolation functions in terms of € , 7] and evaluate curvilinear derivatives as follows: D.2. CURVILINEAR COORDINATES 415

The general transformation for a function ¢ is given by 8¢ 8¢ 8~ 8¢ 8T! -=--+--, 8x 8~ 8x 8T! 8x (D.ll) 8¢ 8¢ 8~ 8¢ 8T! -=--+--. 8y 8~ 8y 8T! 8y

3 4 3 4

1111 2 ~aster Element o ~ 4 7 3

8 6

(x2' Y2) 11 1 5 2 Y ()xI'YI (Xs'Ys) ~aster Element ~urved-sideElement o ~ o x

Fig. D.5. An 8-Element Curvilinear Rectangle .

However, explicit expressions for~, T] in terms of x , yare not easily available . So we consider ¢ = ¢(x,y). Then, using formulas for 8¢/8~ and 8¢/8T], similar to those in (D.lI), we get {f!}=J{U}' where J is the Jacobian matrix given by

8x J = 8(x ,y) = 8{ 8(~ ,T!) [ 8x 8t] On inversion we get

~} 1 { f!.!E. - J- {*}f!.!E. ' 8y 8t] 416 D. ISOPARAMETRIC ELEMENTS which gives

(D .12)

where IJI denotes the determinant of J. Note that PI must be nonzero and finite for the transformation to be unique. Also,

(D.13)

If PI #- 0, we have By BfJ (D.14) [ By B~ where ~ = ~(x, y) and fJ = fJ(x, y). It is obvious from (D.14) that the functions ~(x, y) and fJ(x, y) are continuous and differentiable. It is also required that the transformation x = x(~, fJ). y = y(~ , fJ) must be algebraically simple. This would allow us to evaluate the matrix J easily.

D.2.1. One-Dimensional Integrals. For one-dimensional integrals of the type

XN 1 f(x) dx = g(~) d (x(~)) d~, (D.15) 1 d~ lXl -1 where d(x(~)) =J let d~ ,

(D.16) where Xl, ... ,XN are the global nodal coordinates. For example, for a 2-node element n(e), let

where the shape functions

(D.17) D.2. CURVILINEAR COORDINATES 417

Then the new integral is

X 2 l(e) f(x) dx = - 11 g(~) d~ . (D.18) lXl 2 -1 D.2.2. Two-Dimensional Integrals. Consider the double integral

(D.19) where the new variables (~ , TJ) are the natural coordinates, and the integral

(D.20) where the new variables are the triangular coordinates 6,6 (see §5.3). For each coordinate system the Jacobian of the transformation is J, i.e.,

Y] 8x 8y ][ 8x 8 8~ 8~ = J = 86 86 . 8x 8y 8x 8y (D.21) [ - - -- 8TJ 8TJ 86 86

Now, in the case of a linear triangular element the transformation equations are

x (6,6) = d6 ,6) Xl + 2 (6,6) X2 + 3(6,6) X3, (D.22) Y(~1 ,6) = l (~1,6) Yl + 2 (~1,6) Y2 + 3(6,6) Y3, where (Xl, Yl), (X2, Y2), (X3, Y3) are the global nodal coordinates. If we replace the shape functions by their triangular coordinate equivalents, we get

x(6 ,6 ) = 6 X l +6 X 2 +6 X 3 , (D.23) Y (6,6) = 6 Yl + 6 Y2 + (1 - 6 - 6) Y3, because 6 +6 +6 = 1. Then the Jacobian J is

(D.24)

This method can be extended to bilinear quadrilateral and other elements.

EXAMPLE D .1. For a bilinear quadrilateral element n(e) shown in Fig. D.6, the transformation is given by

X = l(~ , TJ) Xl + 2(~, TJ) X2 + 3(~, TJ) X3 + 4(~, TJ) X4 , (D.25) Y = l(~ , TJ) Yl + 2(~ , TJ) Y2 + 3(~ , TJ) Y3 + 4(~ , TJ) Y4, 418 D. ISOPARAMETRIC ELEMENTS where

Xl Yl1 = ~ [-(1 - TJ) 1 - TJ 1 + TJ -(1 + TJ)] X2 Y2 4 -(1 - ~) -(1 +~) 1 + ~ 1 - ~ [X3 Y3 X4 Y4

and thus, IJI = 9/2.•

y

3 (10,7)

-+----+-----l--4----+--l----1---l--l----1--+-_xo Fig. D.6. A Linear Quadrilateral Element.

D.3. Pascal Triangle

After a suitable subdivision of a region into finite elements is made, the next choice is to represent the element approximation in terms of variational parameters. In view of the Weierstrass approximation theorem, polynomial approximation in each element is the choice of such an approximation. Moreover, polynomials are easy to work with, both algebraically and computationally. The following guidelines must be followed in the choice of a suitable polynomial. D.3. PASCAL TRIANGLE 419

1. The number of terms in the polynomial must be equal to the total number of degrees of freedom (dof) associated with the element. For example, a triangle with 3 nodes, one dof at each node, requires u = Cl + CZX + C3Y . Again, an element with 4 nodes and 2 dofat each node will need an 8-parameter polynomial z U = Cl + CZX + C3Y + C4XZ + C5XY + C6Y + C7XZy + cSxyz. 2. The approximation must have geometric invariance, i.e., there should be no preference for either x or y direction . 3. The choice of the approximation must enable the element to reproduce rigid body motions and stress-strain relation precisely, i.e., the convergence of the method must be required. 4. Higher-order terms must not be retained at the expense of the lower order terms . Thus choose complete polynomials, i.e., polynomials in which all terms up to any given degree are present. The terms of successive degrees of polynomials are conveniently represented by the Pascal Triangle (Fig. D.?). It can be used to write complete polynomials or to choose number of terms equal to the dof of any element. Its form up to a lO-th degree polynomial is given below:

1 x Y xZ xy yZ x 3 xZy xyz y3 x4 x3y xZyz xy3 y4 x5 x 4y x3yz x Zy3 xy4 y5 x6 x5y x4yz x3y3 x Zy4 xy5 y6

Fig. D.? Pascal Triangle. E

Green's Identities

Let thefunctions M(x,y, z), N(x, y,z), and P(x, y, z) be the components of the 3 vector G in R , where (x, y, z) is a point in O. Then, by the divergence theorem

rrr (BM + BN + BP) dn JJo. Bx By Bz J (E.1) = J~o.[Mcos(n ,x)+Ncos(n,y)+pcos(n,z)] ss, where dBdenotes the surface element, Bndenotes the boundary ofn, and cos (n, x), cos(n, y), and cos(n, z) the direction cosines of n. If we take M = u ~~, Bv Bv . N = u By' and P = u Bz' then (E.1) yields

r rr (BU av + au Bv + au BV) dn = Jr r u Bv as _ Jfr ruV2vdn, JJ l« Bx Bx By By Bz Bz Jan Bn Jo. (E.2) which is known as Green's first identity. Moreover, if we interchange u and v in (E.!), we get

rr (BU Bv + Bu Bv + Bu BV) dO = fr r v Bu es _ ffr rvV2udn. ffJo. Bx Bx By By Bz Bz Jan Bn Jo. (E.3) Ifwe subtract (E.2) from (E.3), we obtain Green's second identity: 422 E. GREEN 'S IDENTITIES whichis alsoknown as Green's reciprocity theorem. Thisresultalsoholdsin R2. Note that Green's identities are valideven if the domainn is boundedby finitely manyclosedsurfaces; however, in thatcasethesurfaceintegrals must beevaluated over all surfacesthat makethe boundary of n,and in R2 the line integrals must be evaluated over all paths that makethe boundary of n. F

Gaussian Quadrature

Gaussian quadrature is a very powerful method for numerical integration, which uses unequally spaced intervals. It is defined as follows:

b N 1f( x) dx = L W i f (Xi)' (F.l) a i=1 where a + b+ (b - a)~i Xi = 2 (F .2)

Here ~i are known as the Gauss points on the interval (-1,1), which are the zeros of the Legendre polynomial Pn (x) of degree n in the interval (-1, 1). Although tables for the Gauss points and the weights Wi are readily available (see, e.g., Abramowitz and Stegun 1968), they can be easily generated by Mathematica func­ tion LegendreP [n, x} , For example, the first four Legendre polynomials are: 1 2 1 3 Po(x) = 1, P1(x) = X, P2(x) = 2" (3x - 1), and P3(x) = 2" (5x - 3x) . These polynomials of degree n can also be obtained from Rodrigues' formula n 1 d Pn(x) = --I -d (2x - I)n , (F .3) 2n n. xn or from the recurrence relation

(n + I)Pn+1 (x) - (2n + 1) X Pn(x) +nPn-1 (x) = O. (FA)

The orthogonality and normalization relations with the unit weight function are

1 { 0 ifn =J m , Pn(x)Pm(x) dx = 2 . (F.5) -1 -- if n ee rn. 1 2n+ 1 424 F. GAUSSIAN QUADRATURE

The zeros of each polynomial Pn(x) are real and distinct, and they are located in the interval (-1,1). The weights Wi and the corresponding Gauss points ~i for N = 1,2,3,4 are given in Table F.l.

Table F.I . Data for Gaussian Quadrature.

N 1 o 2.0000000000 2 ±0.5773502692 = ±1/J3 1.0000000000 3 o 0.8888888889 ±0.7745966692 = ±/375 0.5555555556 4 ±0.3399810436 0.6521451549 ±0.8611363116 0.3478548451

Iff is a polynomial of degree n, then formula (F.l) is exact if N is chosen to be 2n + 1. EXAMPLE F .1. Use the Gaussian quadrature with N = 1,2,3, to compute

3 dx 1 (13) 1 = --4 = - In - = 0.206346 (exact value). J1 3x + 3 7 Set x = 2 + u. Then 1 -11 du - -1 3u + 10 '

Now,forN=l: 1=2(11 0)=0.2; for N = 2: 1 = [10 _ 3(0.5~73502692) + 10 + 3(0 .5~73502692)] = 0.206186; 1[5 5] for N = 3: 1 = 9 10 _ 3(0 .7745966692) + 10 + 3(0.7745966692) = 0.206342.•

EXAMPLE F .2 . Use the Gaussian quadrature with N = 4 to compute

1r/ 2 2 1 = x 2 cosxdx = ~ - 2 = 0.4674011 (exact value). lo 4

1l"/2+0 1l"/2-0 1l" Here, Xi = 2 + 2 = "4 (1 + ~ i) ' Thus, 1l" 1l" Xl = - (1 + 6) = - (1 - 0.8611363116) = 0.109063, 4 4 1l" 1l" X2 = "4 (1 + 6) = "4 (1 - 0.3399810436) = 0.518378, 425

11" 11" X3 = 4" (1 +6) = 4" (1 + 0.3399810436) = 1.05242, 11" 11" X4 = 4" (1 + ~4) = 4" (1 + 0.8611363116) = 1.46173. Then f (xd = xi COSXl = 0.0118241, f (xz) = xi cos Xz = 0.233413, f (X3) = xi cos z , = 0.548777, f (X4) = xi COSX4 = 0.23257, and

I = ~ [WI f (xd +Wz f (xz) +W3 f (X3) + W4 f (X4) ]

= ~ [(0.3478548451)(0.0118241) + (0.6521451549)(0.233413) + (0.6521451549)(0.548777) + (0.3478548451)(0.23257)] = 0.467402.•

Now, we consider the double integrals on quadrilateral and triangular regions.

QUADRILATERAL REGIONS. The double integral

(F.6) over a rectangular region {(x , y) : -1 ::; x, y ::; I} is numerically evaluated first by computing the inner integral, and then computer the outer integral, in the same manner as in §F.l. Thus, evaluating the inner integral we get

lIN 1 ~ (~i) 1111 f(x, y) dx = Wi f 11 g(y) dy, (F.7) where Wi and ~i are the weight and the Gauss points, respectively. The outer integral yields

(F.8) where W j and Tlj denote the weight and the Gauss points, respectively. Hence, combining (F.7) and (F.8)we obtain

where the values of M and N are obtained by equating (2m - 1) to the highest power of x and (2n-1) to the highest power of y in the integrand f(x, y). Formula 426 F. GAUSSIAN QUADRATURE

(F.9) can be viewed as a single sum over M x N = K Gauss points with wiwrtype products as the corresponding weights .

TRIANGULAR REGIONS. The integral I6 f(x, y) dxdy over a triangular region t:::, is

lll Lf(x,y)dXdy = l - {2 g (6 ,6 ) 1IJl ld6 dx 2

N (F.1O) = L W i9((6 )i,(6 )i). i=l The Guass points and the weights in Formula (F.lO) are defined in Table F.2 (see Hammer et al., 1956, for details) .

Table F.2. Gauss Points and Weights in a Triangular Region .

w·I Error

2 a (1/3, 1/3) 112 ois,

a (112,0) 1/6 2 3 b (1/2, 112) 1/6 O(h ) c (0, 112) 116

a (113, 113 ) 0.11250 b (a, ~) 0.0661917 c (~ , ~) 0.0661917 (~, 6) 6 d a) 0.0661917 O(h e (y, y) 0.0629696 f (0, y) 0.0629696 g (y,o) 0.0629696

In the above table, a = 0.0597159, f3 = 0.470142, 'Y = 0.101287, 8 0.797427, and (6, 6) denote the triangular coordinates, and h is the step size.

EXAMPLE F .3. Compute (a) I = [11 [11 6xy2 dy dx; t' r": (b) 1= io io 6~~ d6 d6 · 427

SOLUTION. (a) For N = 2. we have

2 2 2 2 1= LL WiWj! (~i)! (~j) = LL WiWj 6~;TJJ i = l j=l i=l j=l = 6 . 2[(0.5773502692)2 + (0.5773502692)2] = 2.666666663.

The exact valueis 8/3 = 2.666666667. (b) For N = 6, we have

I =(0 .1125)G·~) +0.0661917 [(0.0597159)(0.470142)2 + (0.470142) 3 + (0.470142)(0.0597159)2) + 0.0629696 [(0.101287)3 + (0.797427)(0.101287)2 + (0.101287)(0.797427)2] = 0.0166661. • G

Gradient-Based Methods

In this appendix we will show some gradient-based numerical methods for finding minimizers of functions from RN to R with no constraints.

G.!. Method of Steepest Descent

The simplest of the gradient-based methods is the method of steepest descent, N which is based on the fact that if f : R -4 R has all first-order derivatives, then its value decreases the fastest along the direction determined by the negative of

N the gradient, i.e., the fastest descent direction is -1I~j~:~ II at any point x in R . Suppose that f has a minimizer x* at which \7f(x*) = 0 holds necessarily. To find x", we take an initial guess x(1) and calculate \7f(x(1)). If\7f(x(l)) i=- 0, t \7f(X(l)) let t denote any real number. Then x(t) = x(l) - II\7f(x)11 is a parametric equation of the straight line passing through X(l) in the direction determined by \7f(x(l)) Th .. i ] hi I' , 'I . bl f . -11\7f(x)II' e restnction 0 on t 1S me 1S a sing e vana e unction

_ (1) t \7f(x(l)) get) - f(x )- II\7f(x(l))II'

Suppose that t(l) is a real number at which g(t(l)) = ming(t). Let tER

(2) _ (1) t(l)\7f(x(l)) x - x - 11\7f(x(l))11 ' 430 C. GRADIENT-BASED METHODS

Then, by definition, j(x{1)) > j(X(2)), since g(t{1)) < g(O) . If we repeat this process with a new 'initial guess' x(2) and if we again get Vj(x(2)) =I- 0, then N It.genera t es a sequence 0 fpom . t s in . R : x(1) ,x (2) , ... ,x(k) , ... ,whiIC hsa tiISfy j(X(l)) > j(X(2)) > ... > j(x(k)) > ... , Under appropriateconditions on I, we will have lirnk_oo x(k) = x*.

EXAMPLE G .1. Solve the unconstrained program

Minimize j(X1,X2)=2xi+x~-2x1X2-X1-X2, (Xl ,X2)ER2, (G.1) by using the methodof steepest descent. Here we have

For simplicity, let the initial guess be x(O) = [0 OIT. Then

Vj(O, 0) = [-1 _l]T,

Vj~,~ T - II\7j(O,O)1I = [-0.7071 0.7071] ,

1 2 g(t) = j(O + O. 707lt, 0 + 0.7071 t) = '2 t - 1.4142 t + 1.

Solving g'(t(O)) = t(O) - 1.4142 = 0, we get t(O) = 1.4142. Thus,

x(l) = [0 O]T + 1.4142 [0.7071 0.7071t = [1.0 LOt.

Repeating this precess, we obtain

_ \7j(1.0,1.0) = [-0.7071 0.7071f 11\7 j(1.0, 1.0) II ' g(t) = j(1.0 - 0.7071 t, 1.0 +0.7071 t) = 2.5 t2 - 1.4142 t, g'(t) = 5t -1.41421.

Solvingg'(t{1)) = 0, we get ttl) = 0.2828, and

T X(2) = [1.0 1.ojT + 0.2828 [-0.7080 0.70631 = [0.8 1.2]T.

Similarly,

X(3) = [0.8 1.2jT + 0.2828 [0.7071 0.7071jT = [1.0 1.4]T.

The exact solutionis found by directly solving G.2. CONJUGATE GRADIENT METHOD 431 which gives [xi x;]T = [1.0 1.5]T. It can be easily checked that after one more iteration, we have X(4) = x* .•

G.2. Conjugate Gradient Method

A faster way of solving an equation of the type (G.l) is the generalized conjugate gradient method. It can be stated as follows. Let x(O) be an arbitrary initial guess, let r(O) = - \1 f(x(O)), and y(O) = r(O) . Let t(O) be the solution of min f(x(O) + tER ty(O)), and let

X(l) = x(O) + t(O)y(O),

(0) _ [r(1) - r(O)]T r(l) r(l) = -\1f(x(1)), 7 - [r(O)]T r(O) ,

y(l) = r(1) + 7(0)y(0).

Repeat this process with t(1) as the solution of min f(x(1) + t y(1)). When this tER method is applied to aquadraticfunction f(x) = ~xT A x-xTb, with the gradient \1f(x) = Ax - b, the method is called the linear conjugate gradient method, or simply the conjugate gradient method.

EXAMPLE G .2. Solve the unconstrained program

2 Minimize f(XI,X2) = 2xi + x~ - 2XIX2 - Xl - X2, (XI,X2) E R , (G.2) by using the conjugate gradient method. The first step is the same as in the method of steepest descent. We take the same initial guess x(O) = [0 O]T as in the previous example. Then

r(O) = y(O) = _ \1 f(O, 0) = [07071 07071]T II\1f(O,O)II' ., and

X(l) = [0 O]T +1.4142 [0.7071 0.7071]T = [1.0 1.0]T,

r(l) = _ nf(x(1)) = _ \1f(1.0, 1.0) = [-0.7071 0 7071]T v II\1f(1.0,1.0)11 ., [r(1) r(O)JT r(l) 7 (0) - - -10 - [r(O)]T r(O) -.,

y(l) = r(l) + 7(0)y(0) = [0 1.4142]T, 2 g(t) = f(1.0, 1.0 + 1.4142 t) = 2t - 1.4142 t - 1. 432 c. GRADIENT-BASED METHODS

Solving g'(t(1)) = 0, we get t(1) = 0.3535, and

X(2 ) = [1.0 1.0f + 0.3535 [0 1.4142]T = [1.0 1.~9991T.

It can be easilyseenthataftertwoiterations, thismethodproduces a moreaccurate approximation of the exactsolution as comparedwith the previous method. • Bibliography

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A boundary conditions, 3, 6ff, 9, 13ff, 17ff, acceleration due to gravity, 173 20,40,45,47, 55, 58,63, 66ff, adiabatic isentropic pressure, 180 69,72,77, 80ff, 108, 110, 117 advection-diffusion analysis, 239ff 126,135,146,159,172,179, air-like gas, 204 200, 202ff, 205, 207, 214, 229, algorithm, 318ff 231,236,250,255,258,260, angle of twist, 144 270, 281, 313,326, 341, 344, angular momentum, 180 348 Ansys codes, 363ff convection, 169 programming, 225 Dirichlet, 3, 10, 15, 172 apple, 205ff essential, 5ff, 8, 55, 76ff, 159,270 approximation(s) homogeneous, 13 semidiscrete, 213, 228 homogeneous essential, 4, 13 semidiscrete Galerkin , 206 impermeable, 146 semidiscrete Rayleigh-Ritz, 208ff insulated, 164 temporal, 405ff mixed, 3, 223, 235 e-,406 natural, 3, 5ff, 24, 76ff, 136, 159, aquifer, unconfined , 178ff 261,270,326 area coordinates, 97, 186 Neumann, 3 integrals, 186, 188 nonhomogeneou s, 13 of an element, 90, 98, 110 of the first kind, 3 of a triangle, 97 of the second kind, 3 aspect ratio, 285 of the third kind, 3 assembly of element matrices, 112 Robin, 3 axial elastic rod, 324 boundary integral(s), 111 ff, 138 axial vibrations of a plastic rod, 342ff boundary value problem, 9, 12,25,37, axisymmetric heat transfer, 175ff 72,107,244 Poisson's, 20 B two-point, 39, 253, 268 backtracking, 315 nonlinear two-point, 264 beam, 78, 83, 85, 210 blood,320 cantilever, 294ff Euler-Bernoulli, 33, 323ff, 332, 348 C overlapping, 81 cable, 66 beam problem, 87ff can of mushroom soup, 189ff bending moment, 76, 80ff, 82 Cauchy conditions , 3 boundary cavity, unit square, 307, 319ff impermeable, 146 circular bar, 72 insulated, 140 cylinder, 169 440 INDEX

pipe, 158, 259 elastic constants , 270 cladding, 70ff plate, 282, 286 coefficient(s) rod, 47 consistency, 256, 320 solid plane, 231 convection, 166, 189, 266ff elasticity, plane, 269ff convective heat transfer, 137,203 electrodes, 124, 126 film, 50, 64, 70ff, 135 electrostatic field, 124 of permeability, 146, 149, 178 potential, 126 thermal expansion, 54 element computer codes, Ansys, 363ff area of an, 90, 98 Fortran, 374ff length, 98 Mathematica, 351ff volume, 98 Matlab, 350, 368ff elements concrete slab, 169 bilinear rectangular, 9lff, 93, 107ff, coolant, 70 111, 126, 131,134, 170, I82ff, coordinate(s) 187ff, 195, 203, 205, 225, 284, area, 97 288, 350, 398ff, 402 curvilinear, 415ff constant-strain triangular, 270ff cylindrical polar, 347 CST, 270ff, 280, 276, 294, 296, global, 147 cubic, 66 local, 147 elastic, 272 oblique, 96 isoparametric, 409ff of nodes , 160ff linear for heat transfer in fluids, parametric, 31 194ff polar, 23, 175, 202, 347 linear triangular, 89ff, 91, 101ff, rectangular Cartesian, 97 107ff, 119, 125, 133, 144, 153, triangular, 96ff, 197 159ff, 170, 179, 18lff, 184ff, Cramer's rule, 29 195,199,203,215,233,275, cubic element, 66 308, 336, 397ff, 399, 402 current, 126 one-dimensional, 37ff cylinder penalty bilinear rectangular Stokes, elliptic, 152 309ff,320 solid circular, 169 penalty linear triangle power-law cylindrical sleeve, 67 Stokes, 31Iff penalty linear triangular Stokes, D 30Iff dc current , 126ff quadratic, 35, 45ff, 49 deflection of a beam, 81, 83, 85 quadrilateral, 45ff, 108 transverse, 66 two-dimensional, 89ff degree(s) of freedom, 94, 269, 284, 289 energy density, 181, 231 elastic potential, 325ff, 327 discretization, 107 kinetic, 325ff, 327 displacement, 87ff potential, 2, 342 longitudinal, 254 total, 325ff, 328, 336, 342 vector, 335 equality constrained programs, 297ff divergence theorem, 293, 394 equation(s) Bessel's 13 E deformation of a bar, 18 eigenfunction(s), 347ff finite element, 50, 54, 59, 77, 79, eigenvalue(s), 11, 282,328, 347ff 109,146,176, 202ff, 218ff, eigenvector(s), 328 228,232 INDEX 441

fourth-order 14, 37 laminar ideal, 37 fourth -order (beam), 31, 37, 86 laminar power-law, 195 Eule r-Bernoulli, 37, 75ff Newtonian Stokes , 319 Euler-Lagrange, 2ff, 6ff, 24, 58, 74, non-Newtonian, 267 136, 261, 300, 311 of ideal fluids, 152 generalized second-order, 107 power-law non-Newtonian, 319 heat, 194 power-law pressure driven, 256ff Helmholtz, 347 partially filled pipe, 158ff hyperbolic, 344ff shear thinning, 320 Laplace, 124, 131ff, 133, 135, 146, turbulent, 260ff 152 viscous on a moving plate, 238 linear interpolation, 245 fluids linear Stokes, 299ff food, 256 mixing-length, 259ff non-Newtonian, 196,267,316 nonlinear elliptic, 197 power-law, 256ff, 267ff, 319 Navier-Stokes, 8 viscous incompressible, 229 normalized, 173 flux, 49, 52 ordinary differential, 344 food products, 257 parabolic, 346 form partial differential, 7 basis , 9, 12 Poisson 's, 4, IOff, 15, 22, 107ff, bilinear,5ff, 109, 177,202 116, 129ff, 131, 144,167,172, linear, 5ff, 109, 177, 202 205 quasi-weak, 207ff polynomial, 346 variation al, 76ff power-law Stokes, 311 weak,3, 17ff,49, 53,58,66,75 Rayleigh-Ritz, 16 weak variational, 1, 3ff, 49, 108ff, second-order (bar), 24, 37, 175 177 Stokes, 297ff formula wave, 344ff Dai-Yuan , 264 error, 9, 39 Hestenes-Stiefel, 264 approximation, 107 integration by parts, 7, 38, 183,244, discretization, 107, 116 393ff roundoff, 200, Rodrigues' , 423 vector, 38, 242 formulation Euler-Bernoulli beam theory, 75ff penalty, 299ff experimental data, 182 local weak , 108ff semidiscrete, 208, 212 F semidiscrete weak, 229 fuel element, 70ff weak, 3, 17ff, 86 fin weak variational, I, 3ff, 49, 108ff, circular pipe, 222 177 cooling, 243, 248,250 Fortran codes, 374ff parabolic, 71 Fourier's law, 182 rectangular cooling, 68 Fourier sine series , 237 tapered, 56ff forward time-marching process, 406 two-dimensional circular, 164 free axial vibrations of an elastic rod, flexural rigidity, 83 327ff flow discharge, 162ff free in-plane vibrations of an elastic fluid flows , 152ff plate, 334ff, 348 Couette, 256 free vibrations of a Euler elastic beam, laminar, 259 330ff 442 INDEX free vibration mode, 328 linear, 2 frequency,328,348 quadratic, 17 Froude number, 173 furnace, 170 function(s) admissible, 3, 300 G basis, 22 Galerkin finite element method, 37ff Bessel,I70 Gauss integration scheme, 198 bilinear" 91 points, 198, 423 bilinear interpolation, 309 Gaussian quadrature, 181,202, 423ff characteristic, 27 weight, 198 cubic shape, 36 gradient local element , 304 deflection, 76 vector, 144ff Dirac delta, 83 gradient-based methods, 429ff discrete energy, 346 gravitational force, 44, 46 first-order continuity, 412ff Green's identity, 1, 173 gamma, 393 identities , 421ff global, 94 grid of heating cables, 169 global cubic shape, 28ff groundwater seepage, 146 global piecewise linear shape, 36 global shape, 94,96, 106,262,313 H Heaviside unit, 87 half-beam, 216 Hermite cubic, 31ff half-cylinder, 170 Hermite interpolation, 76 Hamiltonian principle, 323ft, 325ff, Hermite shape, 31ff, 37ff, 330 347ff interpolation, 76, 89, 91, 350 head, constant, 146 Lagrange, 297ff, 410 heat conduction, 18ff, 51, 172, 182 Lagrange shape, 25, 34, 37, 270 flux, 50, 68 linear, 412 time-dependent, 217ff local cubic shape, 30 unsteady, 234ff local linear interpolation shape, 26, heat diffusion , 239ff local quadratic shape, 29, 34 heat transfer, 137ff local shape, 26ff, 92 axisymmetric, 175ff linear global shape, 28ff convection, 239ff linear interpolation, 26 generation, 137, 142, 166 linear shape, 178, 30 I nonlinear, 196ff linear triangular shape, 312 steady-state, 137ff nonlinear, 250 supply, 142 one-dimensional shape, 25ff heat treated 2024 aluminum alloy, 253 quadratic, 59,413 heating cables, 169 quadratic shape, 29ff Hooke's law, generalized, 252ff shape, 28ff, 36, 95, 98, 101, 105, human blood, 258 109,244 hydraulic conductivity, 149 stream, 152, 157 stress, 134, 144, 167ff I tent-shaped, 27, 38 industrial oven, 267ff test, 9, 14,49,53,76,108, 138, initial condition(s), 211ff, 214, 234, 238 212,215,228,231,273,344 initial guess, 241 weight, 423ff interface continuity, 114 functional(s), 2, 7, 12ff, 17,58,74, 324ff integrals bilinear, 3 one-dimensional, 416 energy, 74, 324ff two-dimensional, 412ff I NDEX 443 integration tridiagonal, 40 by parts, 7, 38, 183,244, 393ff membrane, elliptical , 168 formulas , 393ff rectangular, 134 Gauss-Legendre, 198ff methodes) numerical time, 22lff classical, 207ff inter-element quantities , 247 conjugate gradient, 58ff, 265ff, internal energy generation, 72 313ff, 318ff, 43 Iff interpolation, 273 Galerkin, 1, 3, 9ff, 10, 17,21, 37ff, bilinear, 93 72, 208ff, 211, 233, 241, 244 finite element, 220 Galerkin finite element , 37ff linear, 245 Gauss elimination, 156 Hermite, 330 Gauss-Legendre integration, 198ff semidiscrete , 230 gradient-based, 429ff iteration(s), 60ff, 198ff, 241, 264 Laplace transform , 208ff McCauly's, 81 J Newton's, 197ff, 24Iff, 248ff, 250, Jacobian matrix, 197ff, 416ff, 418 252 nonlinear conjugate gradient, 26Iff, K 318ff Kronecker delta, 91 numerical time integration, 221ff of separation of variables, 226, 236 L of steepest descent, 59, 263ff, 266, Lagrange multipliers , 297ff 297,429ff Laplacian, 398 penalty, 297ff, 309 length of an element, 28 Rayleigh-Ritz, 1,3, II , 13, 15, 17, linear triangular elements , 89ff, 91, 21,241,26lff lOIff, 107ff, 119, 125, 133, a-scheme, 235ff 144,153, 159ff, 170, 179, trapezoidal, 233 18lff, 184ff, 195, 199,203, weighted residual, 1, 3,207 215,233,275, 308, 336, 397ff, minimizers, 298 399,402 modulus of elasticity, 54, 75 load (force) vector, 39, 50ff, 57, 66, 77ff, moments , 87ff 79, 83ff, 86, 110ff, 117ff, 125, mushroom can, 235 144, 178,275 low carbon steel (annealed), 253 N Newtonian model, 256 M Newton's law of cooling, 137 magnetic field, 238 nodal value, 28 matrix boundary, 308, 313 coefficient, 39, 270 node(s) connectivity, 79, 113ff, 119, 117, global, 25, 48, 78, 89, 129ff, 169, 119,125,140,147,154,189, 277, 296, 343 308,313,338 local, 26, 94, 129ff, 277 global stiffness, 39, 42, 50, 68, 83ff, no-slip wall condition, 259 85, 117ff, 148, 153ff, 288, 291, nuclear reactor, 70 333,340 numerical time integration, 22lff Jacobian, 197ff, 416ff, 418 local stiffness, 4Iff, 51, 57, 66, 77, o 85ff, 110ff, 134, 140, 144, 147, open square cavity, 307 153, 16 ~ 178,275,350 operator, II strain, 273 differential, 272 444 INDEX

p radially symmetric , 175ff parametric coordinates, 31 second-order linear, 45ff Pascal triangle, 418ff transient, 207ff penalty formulati on, 299ff, 311ff two-dimensional, 107ff, 137ff penalty parameter, 309 two-dimensional transient, 228ff permeability coefficient, 178 program piezometric head, 178 equality-constrained, 297ff piston flow, 194 linear, 299 plane elasticity, 269ff nonlinear, 2563, 313 plastic rod, 252 Matiab, 319ff polar coordinates, 23, 175, 202, 347 Mathematica, 320 moment , 73 unconstrained, 263ff, 313ff, 430 polyethelene melt, 199 pumping , 178ff polynomial Legendre, 423 Q potential lines, 150ff quadrant , 180 power-law fluids, 256, 267, 319 quadrilateral elements, 109 Prandtl theory of torsion, 144, 168 quadrilaterals, 107 pressure, 8, 72, 173, 229, 299ff, 311 gradient, 180, 257 R principle of virtual work, 325ff radial distance, 180 prismatic bar, 73 radian frequency of vibrations, 73 rod,37, 44, 46,343 radiation, 250 problem heat transfer, 243ff axisymmetric, 183,203,293 recharge, 178 cavity Stokes, 307ff, 314ff, 319ff rectangle, 139 Dirichlet , I rectangular bar, 165, 167 eigen value, 73, 344ff, 347 residual, 9,17, 211 equality-constrained , 297ff Reynolds number, 173, 259ff heat conduction, 239 river, 146 heat transfer, 202, 346, 398 Rodrigues' formula, 423 homogeneous, 328 membrane, 345ff S minimizat ion, 261 scheme Neumann, I backward difference, 349, 405 Newtonian linear Stokes, 313ff conjugate gradient, 58, 265ff, 313ff, nonlinear heat conduction, 71 318ff, 43lff optimization, 297 Crank-Nicolson, 209, 405 plane strain, 265ff Euler, 344 plane stress, 269ff finite difference, 221 potential flow, 146 forward difference, 405 power-law Stokes, 311ff, 318 Galerkin, 209, 222, 405 Stokes, 309 Gauss integration, 198 Sturm-Liouville , 77 Newmark, 211, 217, 407 two-point boundary value, 39, 253, Runge-Kutta , 344 268 0-, 208ff, 235 unsteady heat conduction, 235ff search,319 problems seepage, 146ff eigenvalue, 222, 328, 347ff shear force, 76ff, 80 nonlinear one-dimensional, 241ff modulus, 73, 144 one-dimensional, 37ff stress, 144 I NDEX 445 silica brick , 267 tractions surface, 281 soup, mushroom, 189ff transient problem s, 207ff specific heat, 180 trapezoidal rule, 221 square cavity, 307, 319ff triangle , 99ff plate, 279, 337 triangular regions, 426 region , 131 twisting of a square bar, 144 unit, 134 stainless steel bar, 253 U rod, 44, 46 unit square, 120 stationary point, 298ff square cavity, 307, 319ff sterlizing process, 189 upper echelon form, 156 strain , 76, 252 strain-displacement relations, 269ff V stress analysis, 252ff variables stress-strain relations, 229ff, 325 primary, 6, 78, 116 support conditions of a beam, 78 secondary, 8, 67, 78, 149 system two dependent, 63ff finite element, 246 vector Galerkin finite element , 197 force (load), 39, 50ff, 57, 66, 77ff, global , 182,251,260 79, 83ff, 86, IlOff, 117ff, 125, nonlinear, 197 144,178, 275 oblique, 96 local load, 41 rectangular Cartesian, 97 velocity mean flow, 196 T potential, I49ff, 152, 157 temperature vectors, 150, 299 ambient, 66, 137, 164,243 vibration analysis, 323ff change, 54 of elastic rods, 329ff, 348 distribution, 56, 64, 67,138,164, of a membrane, 343 170, 200ff, 218, 222, 225, 227, virtual displacement, 272, 274 239, 244, 400 work, 274, 325ff gradient, 164, 167,202 viscosity, 299 temporal approximations, 405ff von Karman constant, 259 theorem, vortex core flow, 179 divergence, 293, 384 vortex-flow temperature separation , 179 gradient, 394 vortex tubes, 179 thermal conductivity, 19,51,63, 70ff, 142, 165ff, 167, 18lff, 189, W 203,244f weak form, 3, 17f[, 137,273 thin-film lubrication, 72 weak formul ation, 86 time-dependent heat conduction, 217ff weak variational form, I, 3ff, 49, 108ff, time integration, numerical, 22lff 177 time-step, 215 weight function, 423 tolerance, 241 weights, 77, 405, 423ff tomato paste, 266 wiggle effect, 240 torsion , 144ff work, 324ff, 328 of a hollow square membrane, 167 Prandtl theory, 144 y torsional vibrations, 73 Young's modulus, 47, 63, 279, 324, 333