Domain Optimization Analysis in Linear Elastic Problems * (Approach Using Traction Method)

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1 Domain Optimization Analyi in Linear Elatic Problem * (Approach Uing Traction Method) Hideyuki AZEGAMI * and Zhi Chang WU *2 We preent a numerical analyi and reult uing the traction method for optimizing domain in term of which linear elatic problem are defined. In thi paper we conider the application of the traction method, which wa propoed a a olution to domain optimization problem in elliptic boundary value problem. The minimization of the mean compliance i conidered. Uing the Lagrange multiplier method, we obtain the hape gradient function for thee domain optimization problem from the optimality criteria. In thi proce we conider variation in the urface force acting on the boundary and variation in the tiffne function and the body force ditributed in the domain. We obtain olution for an infinite plate with a hole and a rectangular plate clamped at both end. Key Word: Optimum Deign, Computer Aided Deign, Numerical Analyi, Computational Mechanic, Finite Element Method, Domain Optimization, Linear Elatic Problem, Gradient Method, Traction Method. Introduction Domain optimization problem in linear elatic continua auming a geometrical domain hape a a deign variable are common problem that arie in the deign proce for olid tructure. In thi paper, we propoe a numerical analyi method applicable to thee problem. In order to optimize the domain, one practical approach i to provide a ubtitute model with a finite number of degree of freedom a a continuum model before formulation of the optimization problem. Baed on thi model, a hape optimization problem, in which the deign variable are defined in a finite-dimenional vector pace, can be analyzed numerically uing the mathematical programming technique in the ame way a common ize optimization problem. A method in which deign variable are defined by nodal coordinate on a deign boundary in a finite element model ha been examined ince the early 970 ().Uingthi approach, the ocillation of the deign boundary i oberved during the optimization proce. Method to * Received 5th June Japanee original: Tran. Jpn. Soc. Mech. Eng., Vol. 60, No. 578, A (994), pp (Received 3rd December, 993) * Department of Mechanical Engineering, Toyohahi Univerity of Technology, - Hibarigaoka, Tempaku-cho, Toyohahi 44 Japan *2 Department of Mechanical Engineering, Ehime Univerity, 3 Bunkyo-cho, Matuyama 79, Japan overcome thi deficiency are baed on the adaptive finite element method (2). An alternate way of defining deign variable i to ue the degree of freedom of the B-pline function repreenting the deign boundary (3). Thi method, however, i not effective in locating an optimal olution in problem with a large number of deign variable becaue of the large number of dimenion in the deign pace. An alternate approach to the optimization of geometrical domain i to decribe the problem uing a ditributed mapping function, the derivative of which with repect to hape variation correpond to a velocity field. The governing equation of the velocity field derived by applying the gradient method for ditributed parameter ytem i then olved. Uing thi approach, a enitivity function, which we call a hape gradient function, can be derived theoretically a a coefficient function of the velocity field. A numerical analyi method i formulated to olve the governing equation. We propoed a traction method for olving domain optimization problem in which elliptic boundary value problem are defined. In thi method, the velocity field are obtained a olution of peudoelatic problem in peudolinear elatic continua defined on the deign domain and loaded with peudoditributed external force, or traction, in proportion to the hape gradient function in the deign domain under contraint on diplacement of the invariable boundarie.

2 Serie A,Vol. 39, No. 2, P(x) 2 f(x) C(x) u(x) h(x) X x=t(x) V(x) Θ Fig. Linear elatic problem Thi olution i called the traction method. The peudolinear elatic problem can be analyzed uing any numerical analyi technique applicable to linear elatic problem, uch a the finite element method or boundary element method. In thi paper, we apply the traction method to the optimization of domain in linear elatic continua. Conidering the mean compliance minimization problem, we preent the following: () the theoretical derivation of the optimum criteria and the hape gradient function auming that a boundary loaded with an external boundary force can be varied and that the body force and elatic tiffne can have nonuniform ditribution; (2) the validity of the numerical analyi method baed on the traction method uing the derived hape gradient function. The notation ued in the linear elatic problem and domain variation i preented firt. Uing thi notation, we derive the mean compliance minimization formula and deduce the optimum criteria and hape gradient function. Then the traction method i introduced to olve thee problem. Finally a dicuion baed on the numerical examination i preented. 2. Linear Elatic Problem Let u define the notation ued in the linear elatic problem. A linear elatic continuum i defined in an open domain R n, n = 2, 3, with a boundary. We conider that a coercive diplacement h ditributed on,avolumeforcef in and a traction P on the boundary 2 = \ yield a diplacement u in. R i the et of real number and \ indicate the ubtraction between et. For the elatic continuum, the variational form, or the weak form, of the equilibrium equation i expreed a a(v, w) =l(w) l h (w) v = u h H w H, () D Fig. 2 Domain variation where the bilinear form a(, ) and the linear form l( )andl h ( ) are defined by a(v, w) = C ijkl v k,l w i,j dx (2) l(w) = f i w i dx + P i w i d (3) 2 l h (w) = C ijkl h k,l w i,j dx, (4) and the kinetically admiible et of diplacement by H = {v H () v = 0 on }. (5) The Hooke elaticity C, orc ijkl, (L ()) n4,ha ymmetry and poitive definitene given by C ijkl = C jikl = C ijlk = C klij (6) α >0 : C ijkl ξ kl ξ ij αξ ij ξ ij, ξ ij = ξ ji ξ R n2 in. (7) The function defined above are aumed to be f H (), h H () andp H /2 ( 2 ). In thi paper, boldface vector notation and tenor notation with ubcript are ued. In the tenor notation, the Eintein ummation convention and gradient notation ( ),i = ( )/ x i are ued. L () and H m () denote the bounded Lebegue functional pace and the Sobolev pace, repectively. 3. Domain Variation The domain variation can be denoted by the material derivative method (4)(5) a preented in the previou paper (6). Let a domain be variable in an admiible domain D with a partially mooth boundary a hown in Fig. 2. The change of domain to domain can be decribed uing a one-to-one mapping T (X) defined in the cloed domain D by T (X) : D X x() D, (8)

3 274 JSME International Journal where nonmooth point on the boundary of D, if they exit, are fixed to avoid changing the admiible domain D. Thevariable denote a hitory of the variation. If the domain variation i contrained on a ubdomain or ubboundary Θ of D which include thee nonmooth point, the mapping T (X) igivenbythe identical mapping I(X) a T (X) =I(X) X Θ D. (9) The coordinate ytem X i called the Lagrange coordinate ytem, or the material coordinate ytem, and x the Euler coordinate ytem, or the real coordinate ytem. Infiniteimal variation of the domain i given by T + (X) =T (X)+ V + O( ), (0) where O( ) i defined by O( ) 0( 0). In Eq.(0), the velocity field V i defined by the Euler derivative of T (X) a T T (x) =V (x) () V C Θ = {V C ( D) n i V i =0on D, V = 0 in Θ D}. (2) The notation z y(x) indicate the mapping relation x y(x) z(y(x)) and C k ( D) denote the et of k cla continuou function. The derivative of a ditributed parameter can be expreed in two different way uing the Lagrange expreion φ (X), X, and the Euler expreion φ (x), x. Thee are related to T (X) a φ (X) =φ T (X) X. (3) Subtituting Eq.(0), we can define the Euler derivative, or the material derivative, φ and a hape derivative φ. Thee are related by φ = φ + φ,i V i, (4) where φ and φ are defined a φ = lim 0 φ = lim 0 (φ+ φ ) (5) (φ + φ ). (6) The derivative of functional are obtained a follow. In the cae of a functional J of a ditributed function φ over a domain : J = φ dx. (7) The derivative J i given uing Eq.(0) a J = φ dx + φ v n d = ( φ + φ V i,i )dx, (8) where v n = n i V i. In the cae of a functional J of a ditributed function φ over a boundary : J = φ d. (9) The derivative J i given by J = {φ +(φ,in i + φ κ)v n } d = ( φ + φ κv n ) d, (20) where κ denote the curvature when i a twodimenional domain and a half of the mean curvature when i a three-dimenional domain *3. 4. Mean Compliance Minimization We conider linear elatic continua with the mean compliance a an objective functional. A hown below, thee are elf-adjoint problem uch that the enitivity function, or the hape gradient function, are given with only the tate variable function that i the diplacement. 4 Formulation The mean compliance minimization problem i formulated a follow. We aume that the ditributed function C, f, h and P are determined uniquely. A example, we conider the following condition. Fixing in pace: ( )=( ),i V i ; ( ) = 0 in (2) ( )=0; ( ) Fixing +( ),i V i = 0 in in material: ( )=0; ( ) +( ),i n i v n = 0 (22) on ( )+( )V i,i = 0 ; ( ) Covariation + {( )V i },i = 0 in with material: ( )+( )κv n = 0 ; (23) ( ) + {( ),i n i +( )κ}v n = 0 on When we aume a contraint of the magnitude of the domain m = dx, (24) the mean compliance minimization problem i decribed a follow. Problem: Given ditributed function C, f, h and P C ( D) that are determined uniquely with repect to the domain variation defined in Eq. () and (2) and *3 Corrected original paper by adding a half of

4 Serie A,Vol. 39, No. 2, a magnitude limit for the domain M R +, find = T () that minimize *4 *5 the mean compliance, l(u) l h (u), (25) ubject to the equilibrium equation a(v, w) =l(w) l h (w) v = u h H w H, (26) and the domain magnitude contraint m M 0, (27) where R + i a et of poitive real number. 4 2 Optimality criteria Applying the Lagrange multiplier method, we derive the optimality criteria. Uing w a the Lagrange multiplier for Eq.(26) and Λ a the Lagrange multiplier for the inequality (27), the above problem can be rendered into the tationalization problem of the Lagrange functional *6 L = l(u) l h (u) a(v, w)+l(w) l h (w) +Λ(m M) w H Λ R +. (28) The derivative L of the Lagrange functional L with repect to the domain variation i derived uing the *4 Corrected original paper in Eq. (25) by adding l h (u) *5 The minimization problem of the mean compliance i equivalent to the maximization problem of a potential energy, that i max R n min u h H a(u, u) l(u). *6 2 Corrected original paper in Eq. (28) by changing v to u in the firt and the econd term on the right ide formula in ection 3 a follow *7. L = (f i u i + f i u i) dx + f i u i n j V j d + {P i u i + P i u i 2 +(P i,j n j u i + P i u i,j n j + P i u i κ)n k V k } d (C ijklh k,l u i,j + C ijkl h k,lu i,j +C ijkl h k,l u i,j) dx C ijkl h k,l u i,j n m V m d (C ijklv k,l w i,j + C ijkl v k,lw i,j +C ijkl v k,l w i,j) dx C ijkl v k,l w i,j n m V m d + (f iw i + f i w i) dx + f i w i n j V j d + {P i w i + P i w i 2 +(P i,j n j w i + P i w i,j n j + P i w i κ)n k V k } d (C ijklh k,l w i,j + C ijkl h k,lw i,j +C ijkl h k,l w i,j) dx C ijkl h k,l w i,j n m V m d + Λn i V i d + Λ(m M) = l(v ) l h (v ) a(v, w)+l(w ) l h (w ) a(v, w )+ Λ(m M)+l G (V ), (29) where l G (V )= {f i (v i + w i ) C ijkl (v k,l w i,j + h k,l v i,j +h k,l w i,j )+Λ}n m V m d + {f i (v i + w i ) C ijkl (v k,lw i,j + h k,l v i,j +h k,l w i,j )+C ijkl (h k,l v i,j + h k,l w i,j)} dx + {P i v i + P i w i +(P i,j n j v i + P i v i,j n j 2 +P i v i κ + P i,j n j w i + P i w i,j n j +P i w i κ)n k V k } d. (30) Under the aumption that C, f, h and P are determined uniquely, we recognize that l G (V ) i a linear form of the velocity field V given by l G (V )= G i V i dx. (3) The coefficient vector function G i the enitivity of the velocity field to the objective functional and i called the hape gradient function. Now we derive expreion for G for the cae in which C, f and h are *7 Corrected original paper in Eq. (29) by changing v to u in the term on the upper five line

5 276 JSME International Journal fixed in pace and h = 0 in D. When P i fixed in pace, that i P = 0, G i given by G = γ ((f i u i + f i w i C ijkl u k,l w i,j + Λ)n) +γ 2 ((P i,j n j u i + P i u i,j n j + P i u i κ +P i,j n j w i + P i w i,j n j + P i w i κ)n), (32) where γ (φ) i the trace operator projecting φ defined in to it boundary. When P i fixed in the material, Ṗ = 0, and G = γ ((f i u i + f i w i C ijkl u k,l w i,j + Λ)n) +γ 2 ((P i u i,j n j + P i u i κ + P i w i,j n j +P i w i κ)n). (33) If P covarie with the material, then Ṗ + P κv n = 0, and G = γ ((f i u i + f i w i C ijkl u k,l w i,j + Λ)n) +γ 2 ((P i u i,j n j + P i w i,j n j )n). (34) In the three cae above, G i given a a ditributed function of the normal vector on the boundary. From Eq.(29), the tationality condition of the Lagrange function L are a(v, w )=l(w ) l h (w ) w H (35) a(v, w) =l(v ) l h (v ) v H (36) l G (V )=0 V C Θ (37) Λ(m M) = 0 (38) m M 0. (39) Under thee condition, Eq.(35) i the equation governing v that agree with Eq.(). Uing Eq.(36), we can determine w from the relation v = w. (40) In general, w i called an adjoint function and Eq.(36) an adjoint equation. Since the tate equation and the adjoint equation agree, then thi problem i known a a elf-adjoint problem. Equation (38) and (39) are part of the Kuhn-Tucker condition relating to the inequality condition of Eq.(27). The Lagrange multiplier Λ can be determined uing thee condition. Conidering the relation v = w determined from Eq.(35) and Λ determined from Eq. (38) and (39), the derivative of the Lagrange function L, which agree with the derivative of the objective function with repect to the velocity field V, can be expreed a L = l G (V ). (4) 5. Traction Method Since the derivative of the Lagrange functional L i obtained a a linear form of the velocity field V with a coefficient function of G, we can apply the traction method to the compliance minimization problem. Firt we decribe the traction method conciely (4). We conider the k-th domain variation with the velocity field V (k). The traction method i ued to determin V (k) by a(v (k), w) = l (k) G (w) V (k) C Θ w C Θ.(42) We can confirm that a domain variation with velocity field V (k) decreae the Lagrange functional L a follow. When the tate equation (35) and the Kuhn- Tucker condition (38) and (39) are atified, the perturbational expanion of the Lagrange functional L around the k-th mapping T (k)(x) igivenby L (k) = l (k) G ( V (k) )+O( ). (43) Subtituting Eq.(42) into Eq.(43) and conidering the poitive definitene of a(v, w) baed on Eq.(7), then α >0: a(ξ, ξ) α ξ 2 ξ H ( ). (44) The following relation i obtained for a ufficiently mall value of. L (k) = a( V (k), V (k) ) < 0 (45) Thi equation how that varying the domain with velocity field V (k) obtained from Eq.(42) decreae the Lagrange functional L, or the objective function, in cae in which the convexity and boundne are enured. Equation (42) indicate that the velocity field V (k) i obtained a a diplacement of the peudoelatic body defined in due to the loading of a peudoexternal force in proportion to G under contraint on the diplacement of the invariable ubdomain or ubboundarie a hown in Fig. 3. To olve the peudoelatic problem, we can ue any numerical analyi method applicable to linear elatic problem, uch a the finite element or the boundary element method. In thi work, the FEM wa ued. The Lagrange multiplier Λ that atifie the Kuhn- Tucker condition (38) and (39) i determined a follow. Since Λn contribute to the peudoforce G a a uniform boundary force, the relationhip among the variation of the uniform boundary force Λn, thevariation of the velocity field V and the variation of the magnitude of the domain m i obtained by elatic deformation analyi baed on the following equation

6 Serie A,Vol. 39, No. 2, σ -G(x) V(x) σ Fig. 3 Traction method Fig. 5 Infinite plate problem with a hole: elatic deformation (left) / domain variation (right) Λn(x) V(x) Fig. 6 Plate with optimized hole: finite element meh (left) / train energy denity (right) Fig. 4 Uniform traction with a uniform boundary force Λn a hown in Fig. 4. a( V, w) = Λ n i w i d V C Θ w C Θ (46) m = n V d (47) Baed on the linearity between V and m for mall deformation, we derive the following renewal procedure for Λ and V. max[ 0, Λ (old) + Λ m (old) M ] m (Λ (old) > 0, m (old) M 0) Λ (new) = Λ m (old) M (48) m (Λ (old) =0,m (old) M>0) 0 (Λ (old) =0,m (old) M 0) V (new) = V (old) Λ (new) Λ (old) Λ 6. Numerical Analye (49) We preent reult of ome numerical analye for baic problem uing the traction method and hape gradient function derived in ection 4. 6 Infinite plate with a hole A a domain optimization problem for which the analytical olution i known, we applied the traction method to the plane Fig. 7 Plate problem clamped at both end: elatic deformation (left) / domain variation (right) tre in an infinite plate with a hole loaded with a traction of : 2 perpendicularly at infinite ditance a hown in Fig. 5. The optimal hape of the hole i known to be an ellipe with axe of : r in the cae of perpendicular loading of : r (7). For the numerical analyi, we choe the firt quadrant with a uitable radiu conidering the ymmetry of the problem. The traction on the outer arc wa determined from the analytical olution for a plate with a circular hole. The velocity field wa analyzed under the contraint on the outer arc. Since the hole i traction-free, the hape gradient function G wa calculated uing only the γ ( ) term in Eq.(32), where f = 0. In the numerical analye, we ued eight-node ioparametric element. The reult obtained by tarting with a circular hole i hown in Fig. 6. Thi reult agree with the analytical olution for an elliptical hole with axe of : 2. We confirmed that the reult obtained for other initial hape converged to the ame hape. The iteration number, which depend on the value of, wa5in the cae hown in Fig Plate clamped at both end To confirm the validity of the traction method uing hape gradient function when the loaded boundary varie, we applied it to a imple plate clamped at both end

7 278 JSME International Journal hape with the analytical olution in the cae of an infinite plate with a hole, and by comparion between the mean compliance of converged reult obtained uing different hape gradient function in the cae of a plate clamped at both end with varying load boundarie. Fig. 8 Optimized plate clamped at both end loaded with traction fixed in pace P = 0 or fixed in material Ṗ = 0: finite element meh (left) / train energy denity (right) Reference () Zienkiewicz, O. C. and Campbell, J. S., Shape optimization and equential linear programming, Optimum Structural Deign - Theory and Application -, edited by Gallagher, R. H. and Zienkiewicz, O. C., (973), John Wiley & Son, Fig. 9 Optimized plate clamped at both end loaded with traction covarying with material Ṗ + P κv n = 0: finite element meh (left) / train energy denity (right) a hown in Fig. 7. The traction force P applied on the upper and lower boundarie wa aumed to have a downward and uniform ditribution at the initial boundary. Then, the traction fixed in pace at P = 0 agree with that fixed in the material at Ṗ = 0. In thi cae, the traction i applied uniformly on the boundarie while the domain varie. In contrat, when we aume that the traction covarie with the material, i.e., Ṗ + P κv n = 0, the magnitude of the traction varie in invere proportion to the expanion ratio of the applied boundary. The optimized reult are hown in Fig. 8 for the cae in which the traction i fixed in pace, P = 0, and that in which it i fixed in the material Ṗ = 0. Figure9howthecaeinwhichthetractioncovarie with the material, Ṗ + P κv n = 0. The hape gradient function G are calculated uing Eq.(33) and (34), repectively. In both cae, we confirmed that the mean compliance of the converged reult obtained uing the correct hape gradient function are maller than thoe obtained uing other hape gradient function. The reult verify the validity of the traction method uing hape gradient function derived in ection 4. (2) Kikuchi, N., Chung, K. Y., Torigaki, T. and Taylor, J. E., Adaptive finite element method for hape optimization of linear elatic tructure, The Optimum Shape, Automated Structural Deign, edited by Bennett, J. A. and Botkin, M. E., (985), Plenum Pre, (3) Braibant, V. and Fleury, C., Shape optimal deign uing B-pline, Comput. Method Appl. Mech. Engrg., 44, (984), (4) Sokolowki, J. and Zoléio,J.P.,Introduction to Shape Optimization, Shape Senitivity Analyi, (99), Springer-Verlag. (5) Arora, J. S., An expoition of the material derivative approach for tructural hape enitivity analyi, Comput. Method Appl. Mech. Engrg., 05 (993), (6) Azegami, H., Solution to Domain Optimization Problem, (in Japanee), Tran. of Jpn. Soc. of Mech. Eng., 60, 574, A (994), (7) Kritenen, E. S. and Maden, N. F., On the optimum hape of fillet in plate ubject to multiple in-plane loading, International J. Numerical Method Eng., 0 (976), Concluion In thi tudy, we derived hape gradient function for mean compliance minimization problem theoretically, allowing variation of the loaded boundary, the body force and the elatic tiffne. The validity of the traction method uing the derived hape gradient function wa confirmed by the agreement of the converged