A DOMAIN METHOD FOR SHAPE DESIGN ... - Science Direct

COMPUTER METHODS NORTH-HOLLAND

IN APPLIED

MECHANICS

AND ENGINEERING

57 (1986) 1-15

A DOMAIN METHOD FOR SHAPE DESIGN SENSITIVITY ANALYSIS OF BUILT-UP STRUCTURES* Kyung K. CHOI and Hwal G. SEONG Center for Computer Aided Design and Department of Mechanical Engineering, Iowa City, IA 52242, U.S.A.

Received February

The University of Iowa,

1985

A numerical method of obtaining accurate shape design sensitivity information for built-up structures is developed and demonstrated through examples. The method utilizes and basic character of the finite element method that gives accurate information not on the boundary, but in the domain. The method is shown to drastically simplify the derivation of shape design sensitivity formulas for complex built-up structures.

1. Introduction Shape design sensitivity analysis for several structural components has been treated in [l], where sensitivity information is explicitly expressed as boundary integrals, using integration by parts and boundary and/or interface boundary conditions to obtain identities for the transformation of domain integrals to boundary integrals. Numerical calculation of design sensitivity information in terms of the resulting boundary integrals thus requires stresses, strains, and/or normal derivatives of state and adjoint variables on the boundary. Accurate evaluation of this information on the boundary is crucial. When a numerical method, such as the finite element method, is used for analysis, one has to check the accuracy of finite element results for state and adjoint variables on the boundary. It is well known [2] that results of finite element analysis on the boundary may not be satisfactory for systems with nonsmooth loads and for interface problems. Since the adjoint load for an average stress constraint on an element fiP is a concentrated load on L$,, problems with the accuracy of adjoint variables might be expected on the boundary. Since butlt-up structures are combinations of a variety of structural components, with interface conditions that are generalizations of the interface problem of linear elasticity in [l], one might expect problems with the accuracy of state and adjoint variables on interface boundaries. There are several methods one might consider to overcome this difficulty. One approach is to use a finite element method that gives accurate results on the boundary. A second approach is to use a different numerical method, such as the boundary element method [3,4]. In the finite element method, the unknown function, e.g., the displacement, is approximated by trial functions that do not satisfy the governing equations, but usually satisfy kinematic boundary * Research supported 0045-7825/86/$3.50

0

by NASA Grant No. NAG-1-215. 1986, Elsevier Science Publishers

B.V. (North-Holland)

2

K.K.

Choi, H.G.

Seong, Domain method for shape design

conditions. Nodal parameters zi, e.g., nodal displacements, are then determined by approximate satisfaction of both differential equations and nonkinematic boundary conditions in a domain integral mean sense. On the other hand, in the boundary element method, approximating functions satisfy the governing equations in the domain, but not the boundary conditions. Nodal parameters are determined by approximate satisfaction of boundary conditions in a weighted boundary integral sense. An important advantage of the boundary element method in shape design sensitivity analysis is that it better represents boundary conditions and is usually more accurate in determining stress at the boundary. Another method to be investigated is the use of a domain method to best utilize the basic character of finite element analysis. In the domain method, sensitivity information is expressed as domain integrals, instead of boundary integrals. Details of the domain method will be developed and illustrated here for a plane-stress interface problem and a simple box built-up structure. 2. Plane-stress interface problem Consider a thin elastic solid that is composed of two different materials and subjected to simple tension, as shown in Fig. 1. The finite element configuration, dimensions, material properties of each body, and loading conditions are shown in Fig. 1. Body i occupies domain @, i = 1,2, AB is the interface boundary y, and r” and r2 are the clamped and loaded boundaries, respectively. The design variable b controls the position of the interface boundary y, while the overall dimensions of the structure are fixed. The variational equation of elasticity for the interface problem is [l]

(1) Y b

E'z2.3 Gpa 2/‘=0.3

I I

.ON

E217.6

Gpa

vco.3

Fig. 1. Plane-stress

interface problem.

3

K. K. Choi, H. G. Seong, Domain method for shape design

where z = [zl, z21t is a displacement vector, uii and 8 are stress and strain tensors, respectively, and T = [T I, T*] is the traction vector. In (l), 2 is the space of kinematically admissible displacements, i .e . ,

z = {z E [H’(L!‘)]’ x [H’(n’)]‘: zilfp = zilfp , i=1,2,xEyandz’=0,i=1,2,xEr0},

(2)

where H’(&?‘) is the Sobolev space of order 1 over fli, i = 1,2 [6]. The variational equation (1) may be viewed as the principle of virtual work, and the finite element method as an application of the Galerkin method to the variational equation for the approximate solution of the associated boundary value problem. Consider the von Mises yield stress functional, averaged over finite element aP,

(3) where g(~(z)) is the von Mises yield stress, defined as g@(z)) = (a%” and mP is a characteristic

+ U**U**+ 3a1*a1* - &7**)1’* ,

(4)

function on flP, defined as [l, 51

(5) For numerical comparison, two methods are used for shape design sensitivity analysis; the boundary method of [l, 51 and the domain method developed here. For the boundary method, if 0, C 0’ and the boundary rP of flP intersects y, one can use the result of [5, Eq. (42)] with limits of summation running from 1 to 2 and an appropriate modification of the generalized Hooke’s Law. For the adjo@t equation, one can use [5, Eq. (35)]. On the other hand, if the closure of flP is such that flP C a’, then the third integral on the right of [5, Eq. (42)] becomes zero. In [5, Eq. (42)] n is the outward unit normal to 0’. One can obtain similar results for the case J$ C 0”. For the domain method, since the externally loaded boundary is not moving, taking the variation of (1) [6, Lemma 3.2.11, one obtains i I=1

j-

Ial,$ [a’j(z’)~~j(Z)+ d(z)dj(Z’)]

d0

b-1

V da + 2 1 lfll [ 2 I=1

i,j=l

a”(z)a”(i)]div

V dJ2

4

K.K. Choi, H.G. Seong, Domain method for shape design

where V= [V”, V’]’ is the design velocity field, Using i = z’ + Vz’V and t = 2’ + V?V= equation (6) can be rewritten as

a”‘(z) E@(i)]div V d&2= 0

for all E E Z

.

0 [l],

(7)

It can be verified that i

a’j(+ij(VZlfV)

= $

i,j=t

~ij(~)(V~~V “I-WV;,

(8)

i,j==l

and L

i

[cri~(z)(V~j’V) f oy(z)(vzj’v)]

,

q-1

(9)

where subscripts on .zi and I$ = [Vf , Vf]’ denote derivatives with respect to variable xi. Using the above results, (7) becomes

for all Z E Z .

(10)

Note that (10) is a variational equation for i E Z [l]. Next, if ft C f2l, one may take the material derivative of (3) [5, 61 to obtain

=

,$ 1,/--l

g,y(z)[gi’(i)

- t~~(V~‘V)lrn, dJZ


(11) It can be shown that o’j(Vz’V) = 2

Dijk’(Vz:‘V + Vz”‘v,)

(12)

k,l=l

and

(13) where Dijkr is the elastic modulus tensor, which satisfies 2 d@)

=

c k,l=l

Di’k’ei’(z),

i,

j,

k,

(14)

IJsing the above results, (11) becomes

dfi -

( Ia,,$ [ i t,3-1

k,l=l

g,.(r)Di’kJ(Vzkt~)]

mp da

(15) Define the adjoint equation

as [l, 51

Then, one obtains the sensitivity formula as [l, 51

- i

I=1

+

1 In, [ i

i,j=l

II

0’

a”(z)8(

A)]div V dJn

g(z) div Vm, dCI -

mp div V d0 ,

where h is the solution of (16). One can obtain a similar result for the case ‘(zpC $2”.

(17)


6

For numerical computations, the finite element method is used to approximate the state and adjoint equations (1) and (16), respectively. In order to compute the design sensitivity $j of (17), one must define a design velocity field V that satisfies regularity properties defined in [ 1,6], in terms of variations in the design variable 6. To have a continuous design velocity field, one may define V’=+b, and v’=

20-x, 20-b

V2=0,

’

V2=0,

(18)

ona’,

(1%

ona’.

The finite element model shown in Fig. 1 contains 32 elements, 121 nodal points, and 233 degrees of freedom. The 8-noded isoparametric element is employed for design sensitivity analysis. For the boundary method, stresses and strains are obtained at Gauss points and extrapolated to the boundary to obtain accurate results on the boundary [7]. Define $i and $i as the functional values for the initial design b and modified design b + 6b, respectively. Let A&, = $i - I,!I~and let I& be the predicted difference from (17). The ratio $ilAlcb times 100 is used as a measure of accuracy, i.e., 100% means that the predicted change is exactly the same as the actual change. Notice that this accuracy measure will not give meaningful information when A&, is very small compared to I/J;, because the difference AI+~~ may lose precision due to the subtraction (cl: - 1+9:. Numerical results with a 3% design change, i.e., 6b = O.O3b, are shown in Table 1 for the boundary method and in Table 2 for the domain method. Due to symmetry, sensitivity results for only the lower half of the structure are given. One can see from these results that the domain method gives excellent results, whereas the accuracy of the boundary method is not acceptable. Table 1 Boundary method for interface problem Element number 1 2 5 6 9 10 13 14 17 18 21 22 25 26 29 30

393.01304 364.37867 388.07514 402.26903 386.43461 407.14612 388.59634 379.04276 441.68524 424.05820 424.19015 378.85433 407.71528 387.87307 400.61014 394.61705

393.17922 364.76664 388.36215 402.83406 386.84976 407.48249 388.95414 379.25247 442.25032 425.22910 424.70840 378.97497 408.23368 387.32342 400.60112 394.00702

0.16618 0.38796 0.28701 0.56503 0.41515 0.33637 , 0.35780 0.20971 0.56507 1.17089 0.51825 0.12064 0.51840 -0.54962 -0.00903 -0.61003

0.20403 0.67218 0.56684 0.42080 -0.08520 0.14159 -0.53089 -1.90134 - 13.85905 - 13.63066 -0.21408 0.76770 0.49780 -0.48837 0.01423 -0.57794

122.8 173.3 197.5 74.5 -20.5 42.1 - 148.4 -906.6 -2452.6 -1164.1 -41.3 636.4 96.2 88.9 - 157.7 94.7

K. K. Choi, N. G. Seong, Domain method for shape design

7

Table 2 Domain method for interface problem

1 2 5 6 9 10 13 14 17 18 21 22 25 26 29 30

393.01304 364.37867 388.07514 402.26903 386.43461 407.14612 388.59634 379.04276 441.68524 424.05820 424.19015 378.85433 407.71528 387.87307 400.61014 394.61705

393.17922 364.76664 388.36215 402.83406 386.84976 407.48249 388.95414 379.25247 442.25032 425.22910 424.70840 378.97497 408.23368 387.32342 400.60112 394.~702

0.16618 ‘0.38796 0.28701 0.56503 0.41515 0.33637 0.35780 0.20971 0.56507 1.17089 0.51825 0.12064 0.51840 -0.54962 -0.00903 -0.61003

0.17954 0.37840 0.28671 0.59634 0.41515 0.33637 0.37548 0.20159 0.57069 1.12871 0.53919 0.06396 0.51710 -0.56083 -0.00298 -0.58529

108.0 97.5 99.9 105.5 100.6 109.6 104.9 96.1 101.0 96.4 104.0 53.0 99.7 102.0 33.0 95.9

For elements 22 and 29 the predicted values are less accurate than others. However, the magnitude of actual diffences A& for those elements are smaller than others, so A$, may lose precision. Several comments should be made about advantages and disadvantages of this domain method. A disadvantage is that one must define a velocity field in the domain that satisfies regularity properties [ 1,6]. There is no unique way of defining domain velocity fields for a given normal velocity field (V’n) on the boundary [6]. Also, numerical evaluation of the sensiti~ty result of (17) is more complicated and inefficient than evaluation of [5, Eq. (42)], since (17) requires domain integration over the entire domain, whereas [5, Eq. (42)] requires integration over only the variable boundary. On the other hand, there are several advantages associated with the domain method, in addition to numerical accuracy. Note that, in the derivation of (17), interface and boundary conditions are not required to transform domain integrals to boundary integrals using integration by parts. Thus, for a mean stress ~nctional, one does not have to treat the special case in which rP intersects y, as was required in [5]. The result of (17) is valid for both cases. The biggest advantage of the domain method is obtained in built-up structures that are made of combinations of a variety of structural components, with interface conditions that are generalizations of the present plane-stress interface problem. When one applies the domain method, interface conditions are not required to obtain shape design sensitivity formulas. This greatly simplifies the derivation, since one simply adds contributions from each component. As for numerical accuracy, results of finite element analysis on interface boundaries are often unsatisfactory [2] for built-up structures, due to abrupt changes of boundary conditions. Using the domain method and careful finite element analysis, one may avoid stress evaluation at interfaces and obtain accurate sensitivity results, as shown in Table 2.

8


3. Simple box spatial built-up structure problem Consider a simple box, in which five plane elastic solid plates are welded together, attached to a wall as shown in Fig. 2. A dist~buted line load is applied on top of the two side plates and on the end plate. Assume that the design variables in this problem are the length b, , width b,, and height b, of the box. That is, it is assumed that the five plane elastic plates remain plane and orthogonal to each other after design variation, Note that b = [b,, b,, bJt is a vector of shape design variables, since variation of b cause domain variations for each plane elastic solid. Let @, i = 1,2,3,4,5, denote the plane elastic solids shown in Fig. 2. The state variables for this built-up structure consist of in-plane displacements for each plane elastic solid. For adjacent plates, components of kinematically admissible displacements that are tangent to the common interface boundary are equal at the interface. Kinematically admissible displacements are equal to zero on r”. The variational equation of the simple box, as in the plane-stress interface problem, is

where r2 is the boundary on which a distributed line load T = [T ‘, T2jt is applied and 2 is the space of kinematically admissible displacements. Consider the von Mises yield stress functional, averaged over finite element fzP C a’, as

(21) where g(a(z)) is the von Mises yield stress defined in (4) and mp is a characteristic function on finite element OP. In this problem, unlike the plane-stress interface problem, variation of the design variable vector [b, , b,, bJt will move the externally loaded boundary. Assume that traction is constant along I’2 and independent of position. Then, T’ = 0 and, by taking the variation of the right side of (20), one obtains [6]

Fig. 2. Simple box problem.


[%f)l’ =

j-r2 [i T’i”] dr + j-r2 &T’i’)‘n(Vh)] dr + (;:

i=l

Pr’~,(p,))

1n2

The last four terms on the right of (22) denote corner terms due to movement of points p1 and p2 [8]. In these four terms, notation Ink denotes that 5’ from Rk is used for evaluation of these terms. Define the adjoint equation as

a(&

i) = 1 /nq [ i$1 g,ij(Z)aij( i)],,,

By applying the same procedure sensitivity formula as t,h;=

da

(23)

for all h E 2 .

as in the plane-stress

interface

problem,

one obtains the

glj-Jo, ,$ [c+“(z)(VA”I$) + gij( h)(Vz”l/;.)] da ~,I--1

-

I

r2 $

T’(VA”V) dT+

lrz[ $‘(Tlh’)‘n](V’n)

1-l

+@‘h’Vz-(~,))~~~+(i:

dr + (2

i=l

i=l

TiAiVT(pl,) 1n2

i=l

~i%(~,,)~,,c($

T’hiVT(p,))ln5. i=l

(24)

Note that (24) can be obtained by simply adding contributions from each component. In (24), since shape design variables are given as [b, , b,, b3]‘, one can define the velocity field to be linear on each plate. Thus, div V is c.onstant on each plate. For numerical calculations, an 8-noded isoparametric element model with 320 elements, 993 nodes, and 1886 degrees of freedom is used. For numerical calculations, Young’s modulus and

10

EC.I(. Chd, H.G. Seong, ~#rnai~ method far shape design

Dimensions of the structure at the nominal design are b, = b, = b, = 8 in. and thickness of all plates is 0.1 in. The uniform

Poisson’s ratio are 1.0 X lo7 psi and 0.316, respectively.

externai load is 4.77lbiin. In Tables 3 and 4, sensitivity accuracy results are given for arbitra~ly selected elements. The

results of Tables 3 and 4 are for 3% changes in b, and b,, respectively. Due to symmetry, results for only one side are given. The results given show excellent agreement between predictions $; Table 3 Domain method for simple box problem, 3 percent perturbation of length b, Element numbera 1 4 9 12 17 20 25 28 33 36 41 44 49 52 57 60 65 68 73 76 81 84 89 92 97 100 105 108 113 116 121 124 129 132 137 140 145 148 153 156

100.63443 33.64000 84.75172 31.40171 71.55644 29.10066 63.92617 25.62709 58.13858 21.28009 52.16973 18.05108 43.7~~9 20409424 28.21065 31.15718 86.28385 29.26721 75.61336 26.96837 65.59970 24.58127 59.01335 21.34630 53.42710 17.77022 47.01331 16.13709 37.36988 20.66130 21.39204 34.36406 171.97848 119.63305 129.44261 120.84447 105.70496 115.08585 93.64369 104.97155

103.78530 35.95122 86.95996 33.73828 73.28717 31.38476 65.38354 27.63564 59.36943 22.79503 53.22842 18.91286 44.61852 20.33228 28.74663 31.30541 89.52932 31.49583 78.01060 29.18948 67.47185 26.74014 60.55661 23.26564 54.71222 19.23274 48.10625 16.92466 38.29384 20.81388 22.86005 34.46283 174.92077 122.23507 131.12998 123.52891 107.07836 117.32766 94.81903 106.62767

3.15087 2.31122 2.20825 2.33657 1.73073 2.28411 1.45737 2.00855 1.23085 1.51493 1.05869 0.86178 0.91043 0.23804 0.53597 0.14823 3.24548 2.22862 2.39724 2.22111 1.87214 2.15887 1.54326 1.91934 1.28512 1.46252 1.09294 0.78757 0.92397 0.15259 0.46801 0.09877 2.94229 2.60202 1.68736 2.68444 1.37340 2.24181 1.17534 1.65611

3.07549 2.28113 2.14499 2.30087 1.67965 2.25004 1.42286 1.98508 1.21307 1.50010 1.04999 0.84558 0.92291 0.22397 0.59021 0.15058 3.17833 2.19866 2.34965 2.18423 1.83760 2.12160 1.52421 1.89200 1.28203 1.44528 1.10137 0.77243 0.93633 0.14922 0.47243 0.11078 2.83083 2.52531 1.63050 2.61374 1.30398 2.19249 1.12334 1.61890

97.6 98.7 97.1 98.5 97.0 98.5 97.6 98.8 98.6 99.0 99.2 98.1 101.4 94.1 110.1 101.6 97.9 98.7 98.0 98.3 98.2 98.3 98.8 98.6 99.8 98.8 100.8 98.1 101.3 97.8 100.9 112.2 96.2 97.1 96.6 97.4 94.9 97.8 95.6 97.8

K.K. Choi, H.G. Seong, Domain method for shape design Table 3 (cont.) Domain method for simple box problem, 3 percent perturbation of length b, Element 161 164 169 172 177 180 18.5 188 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287

84.56467 93.15356 76.84401 81.23386 68.59733 70.12211 55.10083 60.89820 43.06239 53.52694 52.27702 38.16057 42.60581 47.24021 42.77325 29.52699 44.39629 39.79717 31.97706 24.68556 45.66577 35.13625 23.97411 22.73317

85.55652 94.28181 77.65623 81.91789 69.21429 70.41920 55.48157 60.90101 43.02243 53.50754 52.27824 38.16603 42.62436 47.26791 42.82741 29.53403 44.47833 39.82967 32.01814 24.73067 45.78814 35.16403 23.98394 22.80974

0.99185 1.12825 0.81222 0.68403 0.61696 0.29709 0.38073 0.00281 -0.03996 -0.01940 0.00122 0.00546 0.01855 0.02770 0.05416 0.00704 0.08203 0.03250 0.04108 0.04511 0.12237 0.02779 0.00983 0.07657

0.94271 1.09605 0.75988 0.65822 0.53167 0.28300 0.05267 0.00516 0.06560 -0.00629 0.00657 0.00782 0.02982 0.03331 0.06086 0.01035 0.08544 0.03146 0.04544 0.04968 0.12677 0.02644 0.01220 0.08197

95.0 97.1 93.6 96.2 86.2 95.3 13.8 184.1 - 164.2 32.4 537.8 143.2 160.7 120.3 112.4 147.0 104.2 96.8 110.6 110.1 103.6 95.1 124.1 107.1

a Element numbers: l-64 for the top; 65-128 for the bottom; 129-256 for the sides; 257-320 for the end. Table 4 Domain method for simple box problem, 3 percent perturbation of height b, Element numbera

*:

*z

1 4 9 12 17 20 25 28 33 36 41 44 49 52 57 60 65

100.63443 33.64000 84.75172 31.40171 71.55644 29.10066 63.92617 25.62709 58.13858 21.28009 52.16973 18.05108 43.70809 20.09424 28.21065 31.15718 86.28385

97.71343 32.57416 82.13030 30.42382 69.20184 28.22262 61.74584 24.88163 56.11622 20.67423 50.35392 17.49795 42.24665 19.32151 27.40639 29.77414 82.99484

A$P -2.92100 -1.06584 -2.62142 -0.97790 -2.35460 -0.87803 -2.18033 -0.74546 -2.02236 -0.60586 -1.81580 -0.55313 -1.46144 -0.77274 -0.80426 -1.38304 -3.28901

*; -3.02560 - 1.10330 -2.71628 - 1.01214 -2.43981 -0.90871 -2.25854 -0.77144 -2.09421 -0.62693 - 1.87975 -0.57285 -1.51226 -0.80181 -0.83223 -1.43646 -3.39641

(+;/A$,

x lOO)%

103.6 103.5 103.6 103.5 103.6 103.5 103.6 103.5 103.6 103.5 103.5 103.6 103.5 103.8 103.5 103.9 103.3

11


12

Table 4 (cont.) Domain method for simple box problem, 3 percent perturbation of height b, Element number” 68 73 76 81 84 89 92 97 100 105 108 113 116 121 124 129 132 137 140 145 148 153 156 161 164 169 172 177 180 185 188 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287

29.26721 75.61336 26.96837 65.59970 24.58127 59.01335 21.34630 53.42710 17.77022 47.01331 16.13709 37.36988 20.66130 21.39204 34.36406 171.97848 119.63305 129.44261 120.84447 105.70496 115.08585 93.64369 104.97155 84.56467 93.15356 76.84401 81.23386 68.59733 70.12211 55.10083 60.89820 43.06239 53.52694 52.27702 38.16057 42.60581 47.24021 42.77325 29.52699 44.39629 39.79717 31.97706 24.68556 45.66577 35.13625 23.97411 22.73317

28.14580 72.71050 25.93788 63.06451 23.64699 56.72553 20.54120 51.35160 17.10474 45.18366 15.52403 35.91812 19.83577 20.56917 32.93564 167.71438 115.99074 126.59456 117.04449 103.25986 111.66459 91.44357 102.10465 82.59761 90.80305 75.13751 79.31704 67.23706 68.56864 54.25866 59.58460 42.96419 52.52667 50.89411 37.12849 42.41664 46.63766 41.67905 28.63060 43.91085 39.66406 31.34971 23.63504 45.01216 35.28374 23.72807 21.53773

-1.12141 -2.90286 -1.03049 -2.53519 -0.93428 -2.28782 -0.80510 -2.07550 -0.66548 - 1.82965 -0.61306 -1.45176 -0.82552 -0.82287 -1.42842 -4.26410 -3.64231 -2.84805 -3.79998 -2.44509 -3.42127 -2.20111 -2.86690 -1.96706 -2.35051 -1.70650 -1.91682 -1.36027 - 1.55346 -0.84218 -1.31360 -0.09821 -1.00027 -1.38291 -1.03208 -0.18917 -0.60255 -1.09419 -0.89639 -0.48544 -0.13312 -0.62736 -1.05052 -0.65361 0.14750 -0.24604 - 1.19544

-1.15766 -2.99854 -1.06359 -2.61931 -0.96399 -2.36378 -0.83036 -2.14435 -0.68615 -1.89021 -0.63284 - 1.49927 -0.85490 -0.84901 - 1.48260 -4.41063 -3.75305 -2.95053 -3.91629 -2.53832 -3.51647 -2.28507 -2.94150 -2.04206 -2.41224 -1.77203 -1.97041 - 1.41412 -1.60041 -0.87811 - 1.35720 -0.10655 - 1.04084 - 1.42855 - 1.06297 -0.21486 -0.62371 -1.12847 -0.93170 -0.52915 -0.13907 -0.64641 - 1.09766 -0.70665 0.14909 -0.25508 - 1.24756

103.2 103.3 103.2 103.3 103.2 103.3 103.1 103.3 103.1 103.3 103.2 103.3 103.6 103.2 103.8 103.4 103.0 103.6 103.1 103.8 102.8 103.8 102.6 103.8 102.6 103.8 102.8 104.0 103.0 104.3 103.3 108.5 104.1 103.3 103.0 113.6 103.5 103.1 103.9 109.0 104.5 103.0 104.5 108.1 101.1 103.7 104.4

a Element numbers: l-64 for the top; 65-128 for the bottom; 129-256 for the sides; 257-320 for the end.

K. K. Choi, H.G.

Seong, Domain method for shape design

Table 5 Boundary method for simple box problem, 3 percent perturbation of height b, Element number* 1 4 9 12 17 20 25 28 33 36 41 44 49 52 57 60 129 132 137 140 145 148 153 156 161 164 169 172 177 180 185 188 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287

A% 100.634 33.640 84.752 31.402 71.556 29.101 63.926 25.627 58.139 21.280 52.170 18.051 43.708 20.094 28.211 31.157 171.978 119.633 129.443 120.844 105.705 115.086 93.644 104.972 84.565 93.154 76.844 81.234 68.597 70. I22 55.101 60.898 433.062 53.527 52.277 38.161 42.606 47.240 42.773 29.527 44.396 39.797 31.977 24.686 45.666 35.136 23.974 22.733

97.713 32.574 82.130 30.424 69.202 28.223 61.746 24.882 56.116 20.674 50.354 17.498 42.247 19.322 27.406 29.774 167.714 115.991 126.595 117.044 103.260 111.665 91.443 102.105 82.598 90.803 75.138 79.317 67.237 68.569 54.259 59.585 42.964 52.527 50.894 37.128 42.417 46.638 41.679 28.631 43.911 39.664 31.350 23.635 45.012 35.284 23.728 21.538

-2.921 -1.066 -2.621 -0.978 -2.355 -0.878 -2.180 -0.745 -2.022 -0.606 -1.816 -0.553 -1.461 -0.773 -0.804 -1.383 -4.264 -3.642 -2.848 -3.800 -2.445 -3.421 -2.201 -2.867 -1.967 -2.351 - 1.707 -1.917 -1.360 - 1.553 -0.842 -1.314 -0.098 -1.000 -1.383 - 1.032 -0.189 -0.603 -1.094 -0.896 -0.485 -0.133 -0.627 -1.051 -0.654 0.147 -0.246 -1.195

-1.231 -0.825 -1.806 -0.739 - 1.741 -0.611 -1.571 -0.417 -1.841 -0.176 -2.765 -0.138 -2.394 -0.862 -0.505 -2.373 - 10.550 -1.050 - 17.597 -1.404 -17.666 -1.395 -17.710 -1.103 -16.821 -0.212 - 16.429 0.382 -1.233 0.053 -11.464 -0.459 -9.016 -0.566 -0.622 -0.414 -7.311 -0.656 -0.609 -0.408 -6.642 -0.278 -0.369 -0.710 -6.351 0.048 -0.125 -0.954

a Element numbers: l-64 for the top; 65-128 for the bottom; 129-256 for the sides; 257-320 for the end.

42.1 77.4 68.9 75.6 73.9 69.6 72.0 56.0 91.0 29.1 152.3 24.9 163.8 111.5 62.8 171.6 247.4 28.8 617.9 36.9 722.5 40.8 804.6 38.5 855.1 9.0 962.7 -19.9 90.7 -3.4 1361.2 34.9 9180.0 56.6. 44.9 40.1 3865.0 108.8 55.7 45.5 1368.3 208.6 58.8 67.6 971.7 32.4 50.6 79.8

13

14

151.I(. Choi, H. 6. Seong, Domain method

for

shape design

and actual changes A$,, except in elements 185,188,257,259,261,263,265, and 271 of Table 3. However, one may note that those elements are in the low stress region and AJIP are small compared to others, and may not be accurate. For nume~cal comparison, the boundary method of [l, 53 is applied to this problem for a 3% change in the height b,. Results are given in Table 5. Comparing with the results of Table 4, one sees that the domain method gives vastly superior results.

4. Conclusion Results presented in this paper indicate that the domain method of shape design sensitivity analysis for built-up structures has a promising future. Substantial improvement in numerical accuracy is obtained, as noted in numerical results. Moreover, the domain method offers striking simplification in the derivation of shape design sensitivity formulas for built-up structures; one simply adds contributions from individual components. This gives a method for the systematic organization of shape design sensitivity analysis of built-up structures. That is, one can derive shape design sensitivity formulas for each standard component type (rod, beam, plate, plane elastic solid, three-dimensional elastic solid, etc.). The result will then be standard formulas that can be used for many structural types, by simply adding cont~butions from each component. This suggests a ‘design component method’ for characterizing both conventional and shape design of built-up structures by assembling component contributions to system design sensitivity expressions. That is, one can define a library of basic structural components that may be assembled to form a built-up structure. For this, the variational design sensitivity analysis fo~ulation outlined in this paper and developed in more detail in [l, 5,6] can be used to define the contribution from each component to design sensitivity analysis of the overall built-up structure.

Acknowledgment The authors wish to acknowledge suggestions of Dr. Robert L. Benedict of Goodyear Tire & Rubber Co. that contributed to results presented herein.

References [l] K.K. Choi and E.J. Haug, Shape design sensitivity analysis of elastic structures, J. Structural Mech. Il(2) (1983) 231-269. (21 I. BabuSka and AK. Aziz, Survey lectures on the mathematical foundations of the finite element method, in: AK. Aziz, ed., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Academic Press, New York, 1972) l-359. [3] P.K. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, New York, 1981). [4] CA. Mota Soares, H.C. Rodrigues and ILK. Choi, Shape optimal structural design using boundary eIement and minims compliance techniques, ASME J. Mechanics, Transmission, Automation Design 106 (4) (1984) 516-521.


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[5] K.K. Choi, Shape design sensitivity analysis of displacement and stress constraints, J. Structural Mech. 13 (1) (1985) 27-41. [6] E.J. Haug, K.K. Choi and V. Komkov, Design Sensitivity Analysis of Structural Systems (Academic Press, New York, 1985). [7] R.J. Yang and K.K. Choi, Accuracy of finite element based design sensitivity analysis, J. Structural Mech. 13 (2) (1985) 222-239. [8] J.P. Zolesio, Gradient des couts governes par des problemes de Neumann poses sur des ouverts anguleux en optimisation de domain, CRMA Report 1116, University of Montreal, Canada, 1982.