Analysis of Dual Boundary Element in Shape Optimization

Transcription

1 IOSR Joural of Mechaical ad Civil Egieerig (IOSR-JMCE) e-issn: ,p-ISSN: X, Volume 13, Issue 3 Ver. IV (May- Ju. 016), PP Aalysis of Dual Boudary Elemet i Shape Optimizatio G. Gomes a,*, A. Portela a, B. Emidio a, I. Lustosa b a UiversityofBrasilia, Campus Darcy Ribeiro, zip , Brasilia/Brazil b Federal Istitute of Techology ad Sciece of Piau Campus Floriao, zip , Floriao/Brazil Abstract: The umerical aalysis with dual boudary elemet method for shape optimal desig i twodimesioal liear elastic structures is proposed. The desig objective is to miimize the structural compliace, subject to a area costrait. Sesitivities of objective ad costrait fuctios, derived by meas of Lagragea approach ad the material derivative cocept with a adjoit variable techique, are computed through aalytical expressios that arise from optimality coditios. The dual boudary elemet method, used for the discretizatio of the state problem, applies the stress equatio for collocatio o the desig boudary ad the displacemet equatio for collocatio o other boudaries. The perturbatio field is described with liear cotiuous elemets, i which the positio of each ode is defied by a desig variable. Cotiuity alog the desig boudary is assured by forcig the ed poits of each discotiuous boudary elemet to be coicidet with a desig ode. The optimizatio problem is solved by the modified method of feasible directios available i the PYOPT program ad the accuracy ad efficiecy of the aalysis is assessed through two examples of a plate with a hole, makig this formulatio ideal for the study of shape optimal desig of structures. Keywords:dual boudary elemet, shape optimizatio, feasible directios, pyopt software I. Itroductio Shape optimal desig is a importat class of structural desig problems i which the shape of a structural compoet is to be determied, subject to a give set of costraits. Sesitivity aalysis plays a fudametal role i the determiatio of the shape optimal desig. A uified theory of cotiuum shape desig sesitivity aalysis for elastic structures was developed by Choi et al. [1] ad Haug et al. [] by meas of a variatioal formulatio ad material derivative cocept. The major advatage of this theory is that the optimality coditios, derived o a cotiuum approach, lead to explicit desig sesitivity expressios. The success of this formulatio relies o the accurate computatio of desig sesitivities which, however, deped o boudary stresses that are ofte difficult to evaluate. I geeral, umerical methods must be used for shape optimal desig of egieerig structures. The fiite elemet method has a log ad well- documeted history i shape optimal desig [3] to [6]. A itrisic feature of the fiite elemet method, commo to all displacemet based formulatios, is the iaccuracy obtaied i the computatio of boudary stresses, as reported by Yag et al. [7] ad Choi et al. [8]. ]. I order to overcome this difficulty, mixed formulatios of the fiite elemet method have bee preseted by MotaSoares et al. [9] ad Rodrigues [10]. Aother drawback of the fiite elemet method is the eed for cotiuously remeshig, i order to elimiate the error geerated by mesh distortio durig the shape redesig process, as reported by Rodrigues [10]. Despite these drawbacks, the fiite elemet method has bee the most widely used techique for shape optimal desig. The boudary elemet method (BEM) is a well-established umerical techique i the egieerig commuity; see Brebbia et al. [11]. Its formulatio i elastostatics ca be based either o Betti s reciprocity theorem, Rizzo [1], or simply based o the classical work theorem, Portela [13]. The BEM has also bee applied for shape optimal desig, sice it overcomes the difficultiespoited out for the fiite elemet method, as reported by MotaSoares et al. [14]. Despite its good performace i shape optimal desig, the stadard BEM still lacks some accuracy i the evaluatio of boudary stress, see MotaSoares et al. [15]. Thus, there is a clear eed for a alterative modellig strategy for the aalysis of shape optimal desig of structures. I this work the BEM is formulated by meas of two idepedet boudary itegral equatios; the displacemet ad the stress boudary itegral equatios (DBEM). The use of the stress boudary itegral equatio, first reported by Portela et al. [16], allows high accuracy i the computatio of boudary stresses, sice this equatio is idepedet of the displacemet boudary itegral equatio ad furthermore oe approximatio is itroduced i its derivatio. The preset paper is cocered with the applicatio of the DBEM to the aalysis of shape optimal desig of two-dimesioal liear elastic structures. The optimizatio problem is preseted, the boudary itegral equatios are de- fied ad the modellig strategy is discussed. Determiig the optimal shape of the desig boudary, a iterative aalysis is performed usig the Modified Method of Feasible Directios (MMFD) DOI: / Page

2 available i the pyopt program [17]. I each iteratio of the aalysis, i which alog the desig boudary the state problem is modeled with straight quadratic boudary elemets ad the perturbatio field is described with liear elemets, the DBEM is applied for the evaluatio of the objective ad costrait fuctios, as well as their respective gradiets. The effective treatmet of the improper itegrals of the boudary itegral equatios is a matter of fudametal importace i the DBEM. For curved boudary elemets, the atural defiitio of ordiary fiite-part itegrals is applied to regularize the improper itegrals. For straight boudary elemets, aalytic itegratio is carried out. Fially, umerical results are obtaied for a plate with a hole, for differet loadig coditios, ad for the graphical visualizatio are used the MATLAB tool. It is demostrated that the preset modellig strategy ca be used for shape optimal desig of structures accurately ad efficietly. II. Shape Optimal Desig The desig objective is to fid the shape of a tractio-free regular boudary such that, uder a area costrait, the structure exhibits miimum compliace. Let Ω be the domai of a two-dimesioal liear elastic structure ad Γ its boudary subdivided i Γ u ad Γ t. Cosider that the structure has costraied displacemets, defied o Γ u ad is loaded by a system of tractios, applied o Γ t. The desig boudary Γ d, assumed as regular, is defied o Γ t, as presetedi Figure 1. Figure1:Two-dimesioalelasticstructurewhere -domai; u -boudarywithfixeddisplacemets u = 0 ; t -boudarywithappliedtractios; d -desigboudary t = 0 ; u -displacemets; t -tractios; xi - cartesiacoordiates. The liear elastic problem is defied as G Gu u,, 0 j ii i i 1 (1) withthe boudarycoditios u u o () i u ad t t o (3) i i t where u i ad ti j arerespectivelythedisplacemetadthetractiocompoets; arethestresscompoets; arethecompoetsoftheuitoutwardormaltotheboudary; G j istheshearmodulusad ispoisso scoefficiet.itisfurtherassumedthat isaciematicallyfixedboudary,thatis ui 0 o u ad d isatractio-freeboudary,thatis ti 0 o d. The objective fuctio is the compliace of the structure, give by d DOI: / Page

3 1 0, ui, ti ti ui d, iwhich u i ad t i arethestatevariables,obtaiedfromthesolutioofthestateproblem,equatios1to3.thedesigoptimizatioproble misexpressedasfidig d suchthatthecompliace 0 ismiimizeduderthecoditiothatthetotalareaofthestructure 1 doesotexceedaprescribedvaluea.thiscostraitcoditiocabeexpressedas (5) 1 1 d xiid A, iwhich x i represetcartesiacoordiates.theumericalsolutioofthisoliearoptimizatioproblem,derivedbymeasoflagrageaapproachadthematerialderivativecoceptwithaadjoit variabletechique,requirestheevaluatioofthefirstvariatioofboththeobjectiveadcostraitfuctioalfordesigse sitivityaalysis.thefirstvariatioofboththecompliaceadtheareacostraitisrespectivelygiveby 0 ad 1 d d W d d i which W is the strai eergy desity ad is the ormal perturbatio of the desig boudary. Sice is a tractio-free boudary, the strai eergy desity ca be defied as W 11 I I E I E (8) E E 1 iplaestraiad E E iplaestress,iwhich E isthemodulusofelasticity; I where isthepricipalstressad isthecorrespodigpricipalstrai. I Theormalperturbatioofthedesigboudary isrepresetedifigure.atapoitofthedesigboudary,adesigvariable b isdefiedastheormofthepositiovectorofthatpoitwithrespecttoapredefiedorigio,asshowifigure.thus,fora variatioofthedesigvariable b,thecorrespodigormalperturbatioofthedesigboudaryisgivebytheorthogoalprojectioof b otothedirectiooftheuitoutwardormaltotheboudary,thatis b. b (9) i i (6) (7) (4) d d Figure : Normal perturbatio of the desig boudary where b - desig variable; b - variatio of the desig variable; - uit outward ormal to the boudary; - ormal perturbatio of the desig DOI: / Page

4 boudary; x i - cartesia coordiates; O - origi of desig variables. 1 = t j x i x CVP Sk x, xuk xdx III. Boudary Itegral Equatios Theboudaryitegralequatios,owhichtheBEMisbased,arethedisplacemetadthestressboudaryitegra lequatios.thepresetatiooftheboudaryitegralequatiosfollowsportelaetal.[18].itheabseceofbodyforces,the boudaryitegralrepresetatioofthedisplacemetcompoetsu i,ataiteralpoit X,isgiveby,, (10) u X T X x u x d x U X x t x d x i j j where i ad j deote cartesia compoets; T X, x adu, X x represet the Kelvi tractio ad displacemet fudametal solutios, respectively, at a boudary poit x. The distace betwee the poits Xad x is deoted by r. The itegrals i equatio (10) are regular, provided r 0. As the iteral poit approaches the boudary, that is as X x, the distace r teds to zero ad, i the limit, the fudametal solutios exhibit sigularities; they are a strog sigularity of order1 r it ad a weak sigularity of order l 1 r i U. Assumig cotiuity of the displacemets at x, this limitig process produces, i the first itegral of equatio (10), a jump term o the displacemet compoets ad a improper itegral. For a boudary poit, equatio (10) ca ow be writte as c x u x CVP T ( x, x) u (x)d (x) U ( x, x) t (x)d (x) j j j wherecvp stads for the Cauchy pricipal-value itegral, ad the coefficiet c (x ) is give by / for a smooth boudary at the poit x ( is the Kroecker delta). Itheabseceofbodyforces,thestresscompoetsσareobtaiedbydifferetiatio ofequatio(10), followedbytheapplicatio of Hooke slaw; they aregiveby X S X, xu xdx D X, xt xdx (1) k k k k I this equatio, S k (X, x) ad D k (X, x) are liear combiatios of derivatives of T (X, x) ad U (X, x), respectively. The itegrals i equatio (1) are regular, provided r 0. As the iteral poit approaches the boudary, that is X x, the distace r teds to zero ad S k exhibits a hypersigularity of the order 1/r, while D k exhibits a strog sigularity of the order 1/r. Assumig cotiuity of both strais ad tractios at x, the limitig process produces improper itegrals ad jump terms i strais ad tractios, i the first ad secod itegrals of equatio (1), respectively. For a poit o a smooth boudary, these jump terms are equivalet to boudary stresses. Hece, equatio (1) ca ow be writte as 1 x S x, xu xdx HPV D x, xt xdx (13) k k k k wherehpvstads for the Hadamard [19] pricipal value itegral. O a smooth boudary, the tractio compoets, t j, are give by, x HVP D x x t x d x i k k whereideotestheicompoetoftheuitoutwardormaltothe boudary,atthepoitx.equatios(11)ad(14)costitutethebasisofthebem.oatractiofreeboudary,asithecaseofthedesigboudary,theseequatiosaresimplified;thedisplacemetadthestressequatio saregiveby (11) (14) DOI: / Page

5 c x u x CVP T x, x u x d x 0 j j d ad x HPV S x, x u x d x 0 i k k d respectively, where deotes the tractio-free boudary. d = BothCauchyadHadamardpricipalvalueitegralsarefiitepartsofimproperitegrals,seeLiz[0]adPortela[1].Pricipalvalueitegrals,computedthroughfiitepartitegrals,wereitroduceditheboudaryelemetmethodbyPortelaetal.[16]. (15) (16) IV. ModeligStrategy The ecessary coditios for the existece of pricipal-value itegrals, assumed ithederivatioofthe boudary itegralequatios,imposespecialrestrictiosothe boudarydiscretizatio.cosiderthat boththegeometryad boudarystate variablesaredescribed byapiecewisecotiuouslydifferetiableapproximatio.thus,cauchyadhadamardpricipalvalueitegralsareequivalettofiite-partitegralsoffirstadsecodorder,respectively. Cosider that collocatio is always doe at the boudary elemet odes. Uder this circumstace, the fiitepart itegral of first order, i the displacemet equatio (15), requires cotiuity of the displacemet compoets at the odes: ay cotiuous or discotiuous boudary elemet satisfies this requiremet. I the stress equatio (16), the fiite-part itegral of secod order requires cotiuity of the displacemet derivatives at the odes, o a smooth boudary: discotiuous quadratic boudary elemets implicitly have the ecessary smoothess, sice the odes are iteral poits of the elemet. For the sake of efficiecy ad to keep the simplicity of the stadard boudary elemets, alog the desig boudary the preset formulatio uses discotiuous quadratic elemets to model the elastic field ad cotiuous liear elemets to describe the perturbatio field. The geeral modellig strategy developed i the preset paper, schematically represeted i Figure 3, ca be summarized as follows: Discotiuous quadratic elemets are used to model the desig boudary, as well as other boudaries o which remeshig is allowed; All the remaiig boudaries of the body are modeled with cotiuous quadratic elemets; The stress equatio (14) is applied for collocatio at the odes of the desig boudary; The displacemet equatio (11) is applied for collocatio at the odes of the remaiig boudaries; Cotiuous liear elemets are used to model the ormal perturbatio field; O the desig boudary, each quadratic elemet of the state problem is forced to be coicidet with a liear perturbatio elemet. Therefore, straight quadratic discotiuous elemets are cosidered i this modellig strategy to describe the state problem o the desig boudary, as show i Figure 3. Figure 3: Modelligstrategy. DOI: / Page

6 V. ComputatioofGradiets Whe the discretizatio of the desig boudary, defied i the previous sectio, is cosidered, that is the use of straight discotiuous quadratic elemets to model the elastic field ad cotiuous liear elemets to model the ormal perturbatio field, as represeted i Figure 3, all the itegrals i equatios (6) ad (7) are most effectively computed by direct aalytic itegratio, which is preseted i the followig. Cosider a straight discotiuous quadratic boudary elemet, with the odes positioed arbitrarily at the poits, 0 3 ad. The shape fuctiosof thiselemet are give by: N1 N 1 1 N Cosideralsoacotiuousliearboudaryelemet,withtheodespositioedatthepoitsξ= 1adξ=+1.Forthisele met,theshapefuctiosaregiveby: 1 1 M1 1 M 1 For a elemet of the desig boudary, the gradiets of both the compliace ad the area costrait, respectively equatios (6) ad (7), are represeted by: 1 1 e 0 W d Wi N1 M j J d j E (17) e 1 ad 1 (18) e 1 d M j J d j e 1 wherewiadvjareodalvariablesadj(ξ)isthejacobiaofthecoordiatetrasformatio.sicetheseelemetsar eassumedstraight, J = l iwhichlrepresetstheelemetlegth.thus,equatios17ad18cabeitegratedaalyticallytoleadto, respectively: e l (19) W1 1 W 1 5 W3 3 E ad e l (0) 1 1 The improper itegrals that arise i the boudary itegral equatios are easily hadled by the classical sigularity-subtractio techique, Davis ad Rabiowitz [], which leads to the atural defiitio of ordiary fiite-part itegrals. I the viciity of a collocatio ode the regular part of the itegrad is expressed as a Taylor s expasio. If a sufficiet umber of terms of the expasio are subtracted from the regular part of the itegrad ad the added back, the sigularity ca be isolated. The origial improper itegral is thus trasformed ito the sum of a regular itegral ad a itegral of the sigular fuctio. This latter itegral is the evaluated aalytically, while stadard Gaussia quadrature is used for umerical itegratio of the regular itegral. Portela [1] has show that this procedure is geeral ad applicable to ay type of boudary elemet, i which the ecessary coditios for the existece of the fiite-part itegrals are implicitly satisfied. Ithispapertheormalperturbatiofieldismodeledwithapiecewiseliearapproximatio,asshowiFigure3.Cosequetly,iordertobecompatiblewiththe assumedliearapproximatioof theperturbatiofield,the desigboudaryismodeledaspiecewisestraight,whetheelasticfieldisregarded. Therefore, forpiece-wisestraightboudaries,alltheitegrals resultig from equatios(15)ad(16)aremosteffectivelycarriedoutbydirectaalyticitegratio,whichispresetediappedix. DOI: / Page

7 VI. PYOPT Softwarecharacteristics The pyopt is a ope-source software Pytho-based package for formulatig ad solvig oliear costraied optimizatio problems.differet type of ope-source ad licesed optimizers that solve the geeral oliear optimizatio problem have bee itegrated ito the package, but i this work has bee used the MMFD optimizer, a extesio of the method of feasible directios, which utilizes the directio-fidig subproblem from the Method of Feasible Directios [3] to fid a search directio but does ot require the additio of a large umber of slack variables associated with iequality costraits. The operatigstructure ofpyoptfollows thestepsi solvigoptimizatio problems, foud o [17]: 1. Optimizatio Problem Defiitio: Solvig geeral costraied oliear optimizatio, that is mi f(x) w.r.t. x s.t.g_j(x) = 0, j = 1,...,m_e g_j(x) <= 0, j = m_e + 1,..., m x_i_l<= x_i<= x_i_u, i = 1,..., where: x is the vector of desig variables f(x) is a oliear fuctio g(x) is a liear or oliear fuctio is the umber of desig variables m_e is the umber of equality costraits m is the total umber of costraits (umber of equality costraits: m_i = m - m_e). Optimizatio Class: Istatiatethe optimizatio problem with the followig settigs Objective Fuctio Template Objective value (f) Array of costrait values (g) Desig Variables (cotiuous, iteger ad discrete) Sigle Desig Variable A Group of Desig Variables Costraits (iequality ad equality) Sigle Costrait A Group of Costraits 3. Optimizer Class: Istatiate a Optimizer (e.g. CONMIM, MMFD). Here, we have bee used the MMFD algorithm. 4. Optimizig: Solvig the Optimizatio Problem (opt method) Output (iitial values, specific solutio ad file output) History (stores all fuctio evaluatios) 5. Parallel Processig Gradiet-based optimizers Parallel Objective Aalysis (default) Parallel Gradiets Populatio-based Parallel Objective Aalysis Static Process Maagemet Dyamic Process Maagemet 1.1. Modellig the optimizatio problem The two examplesof a plate, cosideredi the ext sectio,followigthe stepsgive above, ad the step 5was based o thegradietoptimizer. Thus,the optimizatio problemwith pyopt package cosists of twosteps: 1. Settig up Set a Objective Fuctio: i this case, the objective fuctio is the compliace of the plate give by equatio (4); DOI: / Page

8 The objective fuctio takes i the desig variables array ad returs a value. The array, i this case, it has five ad four desig variables, respectively for each example; Set a array of Costraits: i this case, the objective fuctio is subject to a area costrait give by equatio (5), i which the maximum admissible area is equal to the iitial area of the plate; Iitializes the optimizatio problem: opt_prob = pyopt.optimizatio( Optimizatio Problem with MMFD, objfuc); Fially, we complete the setupby addig, separately, the objective fuctio, the desigvariablesad the array of costraits: opt_prob.addobj( f ) : add objective fuctio ame opt_prob.addvar( dv ame, dv type,lower boud,upper boud,dv value) : add oe-by-oe desig variables (dv) opt_prob.addcogroup( g ame costr,total costr, type costr ) : add a group of costraits. Solver Iitializes the optimizer: i this case, we have used the MMFD (Modified Method of Feasible Directios) algorthm, that is mmfd = pyopt.mmfd() Defie the Parallel Processig: i this case, the parallel-gradiet has bee used because the sesitivities are provided by the user. Here, the sesitivities are give by equatios (19) ad (0), respectively. mmfd(opt_prob, ses_type=gradfuc) VII. NumericalResults Two applicatios of shape optimal desig will be studied i this sectio. Cosider a ifiite square plate of side h with a cetral tractio-free square hole of side a, loaded with biaxial uiform tractio, t1 ad t, as preseted i Figure 4a. The objective fuctio is the compliace of the plate, subject to a area costrait, i which the maximum admissible area is equal to the iitial area of the plate. Whe the cetral hole is cosidered as the desig boudary, the aalytical solutio of this problem is give by Baichuk [4]; the optimal shape of the hole is a circumferece, whe t1 = t, ad it is a ellipse with a semi-axis ratio equal to the ratio of the applied tractios, whe t1 t. It was cosidered i both applicatios with the ratio a/h = 0.15, i plae stress with the elastic costats E = 00 GPa ad ν = 0.3. Sice the problem is symmetric, oly 1/4 of the plate was cosidered i the aalysis ad a mesh with 16 quadratic boudary elemets was set up, i which 4 discotiuouselemets were used alog the desig boudary, as preseted i Figure 4.b. (a) (b) Figure 4: a) Ifiite plate with a square hole (a/h = 0.15); b) Mesh with of 1/4 the plate. For the first applicatio a loadig ratio give by t1 / t = 1 was cosidered. Results of the shape optimal desig problem were obtaied with five desig variables that are schematically represeted i Figure 5. Figure6showstheiitialadfialBEMmeshofthedesigboudary, which theaalyticalsolutiois a circumferece.here, the boudary desig has bee elarged for better viewig. DOI: / Page

9 Figure 5: Desig variables: b1 to b5 i the case t 1/t =1; b1 to b4 i the case t 1/t =0,75 Figure 6: Iitial ad fial BEM mesh of the desig boudary(t 1/t =1) The desig variables of the fial shape with their respective values, which is the radius of the circumferece, as well as the ratios betwee the mea value ad each desig variable, are preseted i Table 1. Figure 7 shows the radius ad the ormalized desig variables, obtaied with the pyopt program ad compared with the referece radius, which we ote that the maximum deviatio from the mea value is 0.4%. Table 1: Radius ad ormalized desigvariables DesigVariable T 1/t =1.0 t 1/t =0.75 Radius b j/5 j Radius a/bi b b b b b DOI: / Page

10 Figure 7: Radius ad ormalized desig variables (t 1/t =1) The evolutio of the desig boudary durig the optimizatio process is represeted i Figure 8. Figure 9 shows the evolutio of the compliace, ormalized by its iitial value, as well as the percetage error of the pricipal stresses, computed at BEM odes of the desig boudary. It ca be see that the error is slightly higher at the mid-side ode of each elemet tha it is at the ed odes. This is due to the straightess of BEM quadratic elemets assumed alog the desig boudary, i order to be compatible with the liearapproximatio deied for the perturbatio field. Figure 8: Evolutio of the desig boudary(t 1/t =1). DOI: / Page

11 Figure 9: Evolutio of the compliace ad Error of pricipal stress at BEM odes of the desig boudary (t 1/t =1) A loadig ratio give by (t 1/t =0.75) was cosidered i the secod applicatio. Results of the shape optimal desig problem were obtaied for this case with four desig variables, schematically represeted i Figure 5. The aalytical solutio of this problem is a ellipse with a semi-axis ratio equal to the ratio betwee the applied tractios. Table 1 ad Figure 10 show the ratios betwee the mior semi-axis of the ellipse a/, ad each desig variable bi. It ca be see that the error obtaied is less tha 0.8%. Figure 10:Radius adormalized desig variables (t 1/t =0.75) The evolutio of the desig boudary durig the optimizatio process is represeted i Figure 11. Figure 1 shows the evolutio of the compliace, ormalized by its iitial value. The ormalized compliace decreases mootoically util a miimum value is reached. This behavior is i cotrast with the oe preset i the results reported by Leal et al. [5], obtaied by a mixed formulatio of the fiite elemet method. DOI: / Page

12 Figure 11: Evolutio of the desig boudary(t 1/t =0.75) Figure 1: Evolutio of the compliace (t 1/t =0.75) VIII. FialRemarks I this paper, the boudary elemet method is applied to the shape optimal desig of two-dimesioal liear elastic structures. The desig objective is to miimize the structural compliace, subject to a area costrait. To solve this optimizatio problem, a iterative aalysis is carried out usig the modified method of feasible directios available i the program PYOPT. For each iteratio of the aalysis, i which alog the desig boudary the state problem is modeled with straight quadratic boudary elemets ad the perturbatio field is described with liear elemets, the dual boudary elemet method is applied for the evaluatio of the objective ad costrait fuctios, as well as their respective gradiets. The reliability of the whole desig strategy lies i the accuracy of the computatio of boudary stresses. The dual boudary elemet method icorporates two idepedet boudary itegral equatios: oe is the stress boudary itegral equatio, used for collocatio o the desig boudary ad the other is the displacemet boudary itegral equatio, used for collocatio o other boudaries. The use of the stress boudary itegral equatio, discretized with discotiuous quadratic elemets to satisfy the ecessary coditios for the existece of fiite-part itegrals, al- lows a efficiet ad accurate computatio of stresses o the desig boudary. This discretizatio strategy ot oly automatically satisfies the ecessary coditios for the existece of the fiite-part itegrals, which occur aturally i the stress boudary itegral equatio, but also circumvets the problem of collocatio at kiks ad corers. This feature costitutes a practical advatage of the preset formulatio of the dual boudary elemet method over the fiite elemet method ad other boudary elemet formulatios. DOI: / Page

13 Accurate results of shape optimal desig were obtaied for the problem of a ifiite plate with a hole. The accuracy ad efficiecy of the implemetatio described herei make this formulatio very effective for the study of shape optimal desig of structures. Fially, it is importat to highlight that the effectivity was also achieved with the use of the PYOPT program, sice its object-orieted features has facilitated the commuicatio ad iteractio betwee the formulatios. Appedix Cosider a discotiuous quadratic boudary elemet of geeral shape Γ e that cotais the collocatio ode. The local parametric coordiate ξ is defied, as usual, i the rage 1 ξ +1 ad the collocatio ode ξ is mapped oto ξ via the cotiuous elemet shape fuctios. The displacemet compoets u j are approximated i the local coordiate by meas of the odal values u ad the discotiuous elemet shape fuctios. The first order fiite-part itegral of equatio 15 ca be expressed i the local coordiate as, j 1 1 f CVP T x, xu j xdx u jcvp d where (1) e f is aregular fuctio, give by the product of the fudametal solutio, a shape fuctio ad the Jacobia of the coordiate trasformatio, multiplied by the term ξ ξ. The itegral of the right had side of equatio(1) ca be trasformed with the aid of the first term of a Taylor s expasio ofthe fuctio f, aroud the collocatio ode, to give f f f d CVP d d f CVP () Now, the first itegral of the right had side is regular ad the secod oe ca be itegrated aalytically to give: 1 d 1 CVP l (3) 1 1 I equatio, the existece of the fiite-part itegral requires the Hldercotiuity of f at the collocatio ode. For the discotiuous elemet, this requiremet is automatically satisfied because the odes are iteral poits of theelemet, where f is cotiuously differetiable. The secod order fiite-part itegral of equatio 16 ca be expressed i the local parametric coordiate as g HPV S x x u x d x u HPV d where 1 k k, k k (4) 1 e g k isaregularfuctioas f withthejacobiamultipliedbytheterm DOI: / Page. The itegral of the right had side of equatio 4 cabe trasformed withtheaidofthefirstadsecodtermsofataylor sexpasioofthedesity fuctio collocatio ode, to 1 1 k k k k d 1 k g g g g HPV d d g HPV g CPV k where g d 1 deotes the first derivative of g. At the collocatio ode the fuctio k k g k, i the eighborhood of the (5) g k is required to have cotiuity of its secod derivative or, at least, ahlder-cotiuous first derivative, for the fiite-part itegrals to exist. This requiremetisalsoautomaticallysatisfiedbythediscotiuouselemet,sice the odes are iteral poits of the elemet. Now, i equatio 5, the first itegraloftherighthadsideisregularadthethirditegralisideticalwith the oe give i equatio 3. The secod itegral of the right had side of equatio 5 ca be itegrated aalytically to give:

14 HPV 1 d 1 1 (6) Equatiosad5aretheordiarydouble-sidedfirstadsecodorderfiite-part itegrals respectively, as defied by Kutt [3]. Ithispapertheormalperturbatiofieldismodeledwithapiece-wiseliear approximatio,asshowifigure 3. Cosequetly,iordertobecompatible with the assumed liear approximatio of the perturbatio field the desig boudary is modeled as piece-wise straight, whe the elastic field is regarded. For piece-wise straight boudaries, all the itegrals i equatios ad 5 are most effectively carriedoutbydirectaalyticitegratio,whichispresetedi the followig. Cosider a straight discotiuous quadratic boudary elemet with odes positioed at poits ξ = /3, ξ = 0 ad ξ = +/3 as doe before. For this elemet, the itegral of equatio 15 is represeted by: e 1, j j, i 1 hu (7) CPV T x x u x d x u CPV T x N J d whereu deotes the odal displacemet compoets ad J(ξ) is the Jacobia of the coordiate trasformatio. Because of the assumed straightess of the elemet, J = l/, where l represets the elemet legth ad the matrix h is give by N CVP d h (8) CVP The first order fiite-part itegrals are itegrated aalytically to give: 3 1 N1 3 1 d l (9) CVP ad CVP N 3 The itegral of equatio 16 is represeted by 1 N 1 1 d l 9 1 (30) d l (31) 1 e 1 k, k k k, 1 hu (3) HVP S x x u x d x u HVP S x N J d where h the matrix is give by: 1 E N h S d (33) 4 1 HVP l 1 The matrix S, is give by S where 1 ad are the compoets of the uit outward ormal to the elemet. The secod-order fiite-part itegrals of equatio 33 are itegrated aalytically to give: (34) DOI: / Page

15 HPV HPV ad HPV 1 N1 d 3 1 l N d 9 l 1 1 = (35) 1 (36) d 3 1 l 1 N (37) Equatio 33 shows that the terms of the matrix h are iversely proportioal to the elemet legth l. This property is computatioally advatageous because it ca lead to diagoally domiat systems of algebraic equatios.higher accuracy will be obtaied if the same order of approximatio is used both for the perturbatio field ad BEM aalysis, o the desig boudary. IX. Refereces [1]. Choi, K.K., Haug, E.J., 1983, Shape Desig Sesitivity Aalysis of Elastic structures, J. Struct. Mech., 11, []. Haug, E.J., Choi, K.K, Komkov, V., 1986, Desig Sesitivity Aalysis of Structural Systems, Academic Press, New York. [3]. Chu, Y.W., Haug, E.J., 1978, Two-dimesioal Shape Optimal Desig,It. J. Numer. Methods Eg., 13, [4]. Choi, K.K., Haug, E.J., Hou, J.W., Sohoi, V.N., 1983, Psheichy sliearizatio Method for Mechaical System Optimizatio, J. Mech. Des. ASME, 105, [5]. Haug, E.J., Choi, K.K., Hou, J.W., Yoo, Y.M., 1984, A Variatioal Method for Shape Optimal Desig of Elastic Structures, i New Directios i Optimal Structural Desig, E. Atreket al. (Eds.), Wiley, New York, 1984, pp [6]. Kikuchi, N., Chug, K.Y., Torigaki, T., Taylor, J.E., 1986, Adaptive Fiite Elemet Methods for Shape Optimizatio of Liearly Elastic Structures, Comp. Methods Appl. Mech. Eg., 57, [7]. Yag, R.J., Choi, K.K., 1985, Accuracy of Fiite Elemet Based Shape De- sig Sesitivity Aalysis, J. Struct. Mech., 13, [8]. Choi, K.K., Seog, H.W., 1986, A Domai Method for Shape Desig of Built-up Structures, Comp. Methods Appl. Mech. Eg., 57, [9]. MotaSoares, C.A., Leal, R.P., 1987, Mixed Elemets i the Sesitivity Aalysis of Structures, Eg. Opt., 11, [10]. Rodrigues, H.C., 1988, Shape Optimal Desig of Elastic Bodies Usig a Mixed Variatioal Formulatio, Comput. Meth. Appl. Mech. Eg., 69, [11]. Brebbia, C.A., Domiguez, J., 1989, Boudary Elemets - a Itroductory Course, Computatioal Mechaics Publicatios, Southampto, UK. [1]. Rizzo, F. J., 1967, A Itegral Equatio Approach to Boudary Value Problems of Classical Elastostatics, Quarterly of Applied Mathematics, 5, [13]. Portela, A., 1980, Theoretical Basis of Boudary Solutios for Liear Theory of Structures, i New Developmets i Boudary Elemet Method, C.A.Brebbia (Ed.), Proc. of the Secod Iteratioal Semiar o Recet Advaces i Boudary Elemet Methods, Uiversity of Southampto. [14]. MotaSoares, C.A., Rodrigues, H.C., Choi, K.K., 1984, Shape Optimal Structure Usig Boudary Elemets ad Miimum Compliace Techiques, J. Mech. Tras. Automat. Des. ASME, 106, [15]. MotaSoares, C.A., Leal, R.P., Choi, K.K., 1987, Boudary Elemets i Shape Optimal Desig of Structural Compoets, Computer Aided Optimal Desig: Structural ad Mechaical Systems, C.A. MotaSoares (Ed.), Spriger Verlag, Berli. [16]. Portela, A., Aliabadi, M.H., Rooke, D.P., 199, The Dual Boudary Elemet Method: Effective Implemetatio for Crack Problems, It. J. Nu- mer. Methods Eg., 33, [17]. Perez R.E., Jase P.W., ad Martis J.R.R.A. (01) pyopt: A Pytho- Based Object-Orieted Framework for Noliear Costraied Optimizatio,Structures ad Multidiscipliary Optimizatio, 45(1): [18]. Portela, A., Aliabadi, M.H., Rooke, D.P., 1993, Dual BoudaryElemet Aalysis of Liear Elastic Crack Problems, i Advaced Formulatios i Boudary Elemet Methods, Computatioal Mechaics Publicatios, Southampto, U.K.. [19]. Hadamard, J., 193, Lectures o Cauchy s Problem i Liear Partial Differetial Equatios, New Have, Yale Uiversity Press. [0]. Liz, D.P., 1985, O the Approximate Computatio of Certai Strogly Sigular Itegrals, Computig, 35, [1]. Portela, A., 199, Dual Boudary Elemet Icremetal Aalysis of Crack Growth, Ph.D. thesis, Wessex Istitute of Techology, Southampto, Portsmouth Uiversity, U. K.. []. Davis, P.J., Rabiowitz, P., 1984, Methods of Numerical Itegratio, Academic Press, Lodo. [3]. Vaderplaats, G.N., 1987, ADS A Fortra Program for Automated Desig Sythesis, Versio.01, Egieerig Desig Optimizatio. [4]. Baichuk, N.V., 1983, Problems ad Methods of Optimal Structural Desig, Pleum Press, New York. [5]. Leal, R.P., MotaSoares, C.A., 1993, Mixed Elemets i Shape Optimal Desig of Structures Based o Compliace, i Advaced Techiques i the Optimum Desig of Structures, S. Heradez (Ed.), Computatioal Mechaics Publicatios, Southampto. DOI: / Page