Applied Mathematics and MechanicsPublished by Shanghai University .( English Edition, Vol 22. No 11, Nov 2001) .Shanghai, ChinaArticle ID: 0253-482720001)11-1325-08ANALYTIC SENSITIVITY ANALYSIS FORSHAPE OPTIMIZATION *ZHANG De-xin (張德欣)'，JIANG Yun-zheng (江允正)，CAI Jian(蔡謇)3(1.703 Research Institute，the Seventh Academy, China State ShipbuildingCorporation, Harbin 10001, P R China;2. Yantai University，Yantai 264005, P R China;3. 468 Factory, Jiangjin, Chongqing 400226，P R China)(Communicated by CHIEN Wei-zang)Abstract: Analytic sensitivity unalysis technology for the boundary element method ( BEM )is presented, combined with a shape optimization system for structural analysis. A shapeoptimization was done for an elastomer under planar stress，based on this new algorithm. Amulti object problem was studied as on ilustrative example for the programmer， using .weighted summing method . The result is feasible .Key words : BEM; dcsign sensitivity analysis; shape optimization: square plate with a holeCLC number: U224Document code: AIntroductionShape optimization for elastomer is an important measure to relieve stress contrition . preventbreaking，improve load-bearing capacity . Structural repetition analysis ，sensitivity analysis andoptimal algorithm are three important links of shape optimization. In the course of theoptimization, sensitivity analysis plays a determinative role to get a higher precision .Most existing sensitivity analysis systems are based on the finite- element method. A detailedexposition of sensitivity analysis in FEM have been writen in Ref.[1] by Wang, et al. InRef.[2] Ramakrshnan and Francavilla give a finite- -element method with sensitivity analysis,using implicit derivation to formulate sensitivity equations. The BEM can reduce one dimensionof the problem compared with the FEM, so it is very important to formulate analytic sensitivityanalysis equation. Those equations are presented in this paper，referring to the equations of FEM .Using the method, we can simplify the problem and improve the efficiency. The process isdetermined by the parameters on the boundary, that make shape optimization simpler and moreprecise .The analyic sensitivity analysis of BEM developed in this paper is applied in an example .Correctness of the algorithm is certified by compavism with existing results .中國煤化工Received date: 1999-12-18; Revised date: 2001-06JMYHCNMHGBiography: ZHANG De- xin ( 1964 - ), Professor13251326ZHANG De-xin, JIANG Yun-zheng and CAI Jian1 Basic Theory and Formula of BEMConsider a linear isotropic elastic body 0 (Its boundary is noted as r )，the integralequation on the boundary can be writtenCluis +f. Piundr=|, uiP:dr+ | uib;dQ,(1)JrJin whichCu(P) =( 4r(1 -叭- {2(1-a)-專sin2a}一sin'a4r(1一D)- sin2a4r(1-)- {2(1-a)- -sin2a }here,「= Tμ+ Tp, Tu is boundary displacement, and Ip is superficial force on the boundary.Q. P∈r，Ua(P'Q') is basic solution of displacement， Pix(P'Q") is superficial forcecorresponding to Un(P'Q' ).The basic solution of planar stress problem and the superficial force coke:U:(PQ)= [(3-v)nou +(1+v)r.r.],4πEPi(P'Q') =-{[(1-1)8x +2(1+p)r.r.sJ-(1-)r.nma-r.m)}. (2)where E and v are modules of elasticity and Posang' s ratio respectively; r is the distance betweenp and Q. The basic solution of planar stain problem can be got by replacement of E and v.Consider a quadratic clement as example，we discrete border into N elements . Nodal pointsdistribute at two ends of each element and at middle point, so the number of nodal points are 2N .Using the zero dimension method, these deformation, superficial force of the quadratic elementare .x(()= 2中(e)x，U;(t) = 2中.(&)Uj，P;(&) = Q,(&)Pg，(3)φ(ξ=六ξ(ξ-1)，φ2(ξ)=(1-E)(1+E)，中(&)=六ξ(∈+1).Eq.(1) can also be expressed in matrix form asCi{U}' + [P']{U}dr=|. [U' ]{P}dr+ |。 [U']{b }d2.(4)In which{P}={例}，{U}={}, (0)={%},[P°]=Pi P2J[U°] =lU2i UiJExpanding and merging , those expressions give中國煤化工Zl@]}{(Sa)FYHCNMHGwhereSensitivity Analysis for Shape Optimization1327[φ' =|[φ' 0{}.l。如小{U)i=Similarly we can get{P}=D]'{P}',(5b){X}= 2[φ]{x}'.(5c)Substitution of Eqs. (5a), (5b), and (5c) into Eq.(4), gives[H]{U}= [G]{P}，<6)where[H} =之"[p" }[h]d&，[G] =2"1v][h]JdE, .here! h] is a matrix including function φ'，Eq. (6) can be dispersed into blocks and rearranged ,we can get[A]{Y}= {F}.(7)in which{F}= [M]}{Y}， {Y}={} {)={"},[A] =[- G1 H2][- Hu G12]l_ G2 H2小.[M」=l_H2，C,小2 Design Sensitivity Analysis of BEMIn calculation of shape optimization, partial derivatives of objective function and constrainfunction are often required.As to many problem， the most of these functions consist of superficial force anddisplacement. Solving equation [ A]{Y} = {F} for displacement and superficial force， thestress can be derived. The stress can be got using the known superficial force and displacement ,that is so-called stress restoration . The sensitivity of these stress to the No. I design variabale maybe obtained by conduction implicit derivation to X[. We can find out that stress sensitivitydepends on sensitivity of superficial force ，displacement and geometric parameters .2.1 Sensitivity of displacement and superficial forceConsider the implicit derivative of Eq . (7) with respect to design variable X, the sensitivityequation of unknown displacement and superficial force can be written as[A]{Y}.u = {r},，(8){r}. = {F},r-[A],{Y}， {F}, = {M],;{Y}+ [M]{W},.2.2 Sensitivity of subsidiary parameter of BEMAccording to Eq . (8), we know that derivation of mat;ired in order中國煤化工to calculate these matrixes, that isTYHCNMHG[H].r =[P'].I[h]J + lP"][hIJ,IJdE,(9)1328ZHANG De-xin, JIANG Yun- zheng and CAI JianlG]1= I[[U*].I[h]J +[U'][hjJrJdε.(10)In order to calculate Eqs.(9) and (10), the basic solution [ U* ]，the superficial force. [P'], and their derivations are needed:[P°] =- K;(CR-4[P]q + R-2[D][Nl),(11)[U'] =- K2(r[1] + R-2[P]),(12)[P'l, = - K[4CR-'R.[P}q + R-4C([P],1q + [P1q,I) -2R-3R .[D][N] + R-2([D],I[N]+ [D][N],)」,(13)lU'], = K2[r.i[I]-2R~'R[P] + R-2[P],i],(14)whereK=二K2 =日，r= (3- v)ln-3.5， C= ;24πr4πER-q = D.n; + D2n2， R= (D} + D吃)|/2， D,= x{- d,R = R-'(DDj.1 + D2D2.s)， q.1 = Di,mn1 + Din1,1 + D2.mn2 + D2n2.1.n\ = JXz.e， n2 = J-X,e， n1.1 =-J2J,1X2.e + J.' X2.e，n2,I = J-2J.X1,e - JX1.g， r.t =-(3- v)R-'R.，DD\D2.[D| - D:][P] =.D2D、吃」[D]=lD2 D,」n1n2]n1.I"2.1][N] =[N], =2D); DI,uD.ID2 + D\D2.1][P],r =lD.rD2 + D:D2.c2D2D2.1 I{DJ.1 =D1.u - D2.1]D2,uD1,1n; is the component of unit vector, x，d; are the distances from original point to experimentalpoint and to loaded point respectively .2.3 Sensitivity analysis of boundary stressOn border Iσ1 = Pi,(15)012二P2,(16)022 = σ11 +-,2e22，(17)in whiche22 =duzduz dξ中國煤化工(18)ddξ dsd42 =夕照u;.MYHCNMHG(19)dξF 臺(tái)dξjSensitivity Analysis for Shape Optimization1329Consider the derivative of form Eq.(15) to Eq.(17) with respect to Xl . The sensitivity ofthese stress respect to the No. l design variable can be gotσi1,I = Pi.r,(20)σ12,1 = P2,1,(21)E022.1 = - rσ11,1 + 2e22.1，(22)wheree22.=臺(tái)ddh(V2,- J-2J.iU2,「dx」{dx;J= d(w).+ d()小.dξ),2.4 Calculate J using linear element! J1=irdε =dξdr。v+正一(需)+(霜小(號(hào)到)(2號(hào)可)dξ =ae，which givesI J l=方2.5 Calculate 川and nh the component of elemental vector川= Jye，ln = Jx.c，J =亦幾=今，y,= φ1y1 +中2Yi+1，in whichφ(() = -(1- ξ)，φ2(ξ) = (1 + &),y,e = (中:yi +更2yi+1),ε =更1.e9: + φ2.&Yi+1 = (yi+1 - y;),| 2 =六(yint- y;)，So吃=(x,-1 - x;).2.6 Calculate[D;=戈一d,D2=y-d;d and dr represent the distance from original point to loaded point associated with the x中國煤化工coordinate and y coordinate respectively. In the same way.MYHCNMHCD= x-山= (φ1x +Ox")-d)=(號(hào)x +z-xj+ 2(x r)E，1330ZHANC De- xin, JIANG Yun-zheng and CAI JianD%=(號(hào)y+主川-州+號(hào)("-y)e.2.7 Calculate four integral equations as following using Gauss integration的= f. oP'dr;=[ (1-日)P°氣d=年(1-0)PdE.Similarly的=”,(1+E)P'd&，g=J_" (1-6)U"dE，弱=」.11 + )U'dξ.The formula being applied isf($)dξ = 2 wf(&).3 Calculation of ExampleUsing optimal programmer compiled in this paper, a square plate with a hole is studied as anexample . The configure of the square plate with a hole and the load case are shown in Fig.I. Inabove P = 50MPa, E = 200GPa, 。= 0.25.Note that the square plate is symmetric, we take one out of four as optimal model showed inFig. 2. BEM is conducted using planar linear element.P，8000p5Q.00 t00:50 112.T4100200300400 s00:'so1000x 1000Fig.1 Configure and load caseFig.2 Discretization of boundary elementWhen the square plate is subjected to be tension. the point of interior hole on the border caneasily be broken for the maximum tangential stress . Therefore ，shape optimization to the interiorhole is needed, in order to minimize the maximum tangential stress. In this paper, elipse graphis used to describe the border of interior hole, so we can select the two axes of ellipse as designvariables，they are x[4 and yg. The boundary equation of interior hole is5 =;中國煤化工Consider not to increase the weight of the squareYHCN M H Qdel of shapeoptimization can be written asSensitivity Analysis for Shape Optimization133 1min: G(1) = F(x) = x; | o(x) 1/[σ],s.tC(i) = ([σ]- [σ{(x)})/[σ}≥0 (i = 2,3,..,6j = i+5)，G(7) = 1 - X(1)/500 ≥0,G(8) = 1 - X(2)/500 ≥0,G(9) = X(1) ≥0,G<10) = X(2)≥0,G(11) = nX(1)x(2)/(rR2)-1≥0.Using shape optimization programmer with analytic sensitivity analysis to calculate thisexample, we get the result shown in Table 1. Calculating the same problem by means of semi-analytic and differential sensitivity analysis respectively we get the results shown in Table 2. It isshown that the maximum value to tangential stress on the border of interior hole it reduces by 8percent and 20 percent using analytic sensitivity analysis than using semi- analytic sensitivityanalysis and differential time by 25 percent and 35 percent .Table 1 Analytic sensitivity analysis resultsthe final resultsX(1:2)= 140.615 700 0046.717 050 00GX(1:11)= 298 .338 600 000.581 920 300.593 623 100.629 590 900.728 420 900.974 751 800.296 921600.532 829 50140.615 700 000.375 142 00OPT. T-Stress are:83.615 94081.275 38074.081 83054.315 830- 5.049 633reanaliestimes: 249Sens. analies times: 38Table 2 Semi analytic senssitivity analysis resultsX(1:2)= 115.234 900 0041 475 510 00GX{1:11)=321.111 700 000.552 810 800.562 322 800.616 525500.677 147 300.985 634 900.423 825 500.585 244 90115.234 900 0041.475 510 000.000 493 26OPT. T- Stress are :89. .437 84087.535 44076.694 90064. .570 5402.873 024reapplysis times: 302Sens. analyze times: 434 Conclusions中國煤化工1) This paper has studied general formula of analy,MYHCNMHGBEMandcompiled computer programmer ，and conducted optimization to a square plate with a hole inside1332ZHANG De-xin, JIANG Yun- zheng and CAI Jianas an example. Its accuracy is very high compared the result with others. In the field ofSensitivity analysis of BEM, in order to reduce the calculating fee and not to reduce the accuracytoo much approximate analytic method is feasible . At present, this method is under development .2) In design of shape optimization, BEM has many advantages comparing with FEM .Applying BEM in shape optimization can reduce the number of dimensions by one， reducecalculating work and the complexity of calculation. This method can also combine with existingshape optimization programmer easily, therefore it has a good prospect.References:[! ] Wang S, Sun Y, Gallagher R H. Sensitivity analysis in shape optimization of continuum structures[J]. Comput & Structures ,1985 , 20<5):855 - 867.[ 2] Ramakrishnan C V, Francavilla A. Structural shape optimization using penalty functions[J]. JStruct Mech , 1975 ,3(4) :403 - 422.[ 3] Barone M R. Caulk D A. Optimal arrangement of holes in a two dimensional heat conductor by aspecial boundary integral methods[J]. Internat J Numer Methods Engrg . 1982 , 18(5):675 - 685.[ 4 ] Banerjee P K. Butterfield R. Boundary Element Methods in Engineering Science[M]. New York.N Y: McGraw-Hill Book Co, New York,N Y. 1981.[ 5] Eizadian D. Optimization of the shape of bidimcnsional structures by the boundary integral equationmethod[ D]. France: the National Institute of Applied Science of Lyon, 1984.[ 6 ] Eizadian D. Trompette shape optimization of bidimensional structures by the boundary elementmethod[ A]. Conf on CADICAM Robotics and Automation in Design[C]. Tucson Ariz , 1985.[ 7] Trompette Ph, Marcelin J L, Lallemaud C. Shape opimization of axisymmeric structures[ M].New York: N Y. The Optimum Shape Plenum Press , 1985.283 - 296.i 8 ] Zochowski A, Mizukami K. A comparison of BEM and FEM in minimum weight design[ A]. In:C A Brebbia Ed. Boundary Element[ C]: Heidelberg: Wcst Germay: Springer- Verlag, 1983 .901 -911.[ 9 ] Rodrigues H C. Mota Soares C A. Shape opimization of shafts[ A]. In: 3rd Nat Cong of Theoretand Appl Mech[ C]. Lisbon Porugal ( in Portuguese) , 1983.[ 10 ] A Chaudouet- Miranda F. Elyafi Recent. Optimum design using BEM tcchnique[ A]. [n: A CruseEd. Advances in Boundary Element Aralysis Methods[C] . Department of Engineering Mechanics,San Antonio TX: Southwest Research Institute, 78284,U S A.[11] LIU Hong-qui, XIA Ren-wei . Senstivity analysis on shape optimization bascd on adjoint systcmequations with two-cass variables [ J]. Computational Structural Mechanics und Applications ，1993, 10(3):7- 14.中國煤化工MYHCNMHG