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Solution to Boundary Shape Optimization Problems Hideyuki Azegami Department of Complex Systems Science, Graduate School of Information Science, Nagoya University, Furo-cho, Chigusa-ku, Nagoya, 464-8601, Japan EMail: [email protected] Abstract This paper presents a numerical analysis method of nonparametric boundary shape optimization problems with respect to boundary value problems of partial differential equations. The nonparametric boundary variation can be formulated by selecting a one parameter family of continuous oneto-one mappings from an original domain to variable domains. The shape gradient with respect to domain variation can be evaluated by the adjoint variable method. However, the direct application of the gradient method often results in oscillating shapes. It has been known that the oscillating phenomenon is caused by a lack of smoothness of the shape gradient. To make up the irregularity, a smoothing gradient method and its concrete numerical procedure called traction method have been presented by the author and coworkers. However, in the previous papers, numerical procedure of the traction method did not have been illustrated. This paper presented a generalized description of the traction method and gave a precise algorithm of the traction method.

1

Introduction

Shape optimization problems with respect to geometrical boundary shapes of elastic bodies, heat transfer fields, flow fields, sound pressure fields, etc. can be generalized as boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. Numerical solutions for the boundary value problems itself can be obtained by well developed numerical analysis methods such as the finite

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element method or the boundary element method. When CAD date describing boundary shapes or coefficients of basis vectors giving typical domain variations are chosen as design variables, these shape optimization problems come down to parametric optimization problems defined in design spaces of finite dimension with the same number of design variables. With respect to these parametric optimization problems, the mathematical programming techniques can be applicable. However, with respect to nonparametric shape optimization problems in which degrees of freedom for domain variation are infinite theoretically and those in discredited formulation are degrees of freedom of nodes on finite element model, it is not easy to solve by the mathematical programming techniques because of enormous number of degrees of freedom that is equivalent to the dimension of design space. The nonparametric boundary variation can be formulated by selecting a one parameter family of continuous one-to-one mappings as the design variable that is defined in an original domain and yields variable domains.1, 2 The shape gradient that is defined with an inner product giving variation of an objective functional with domain variation under satisfaction of state equation of the boundary value problem can be evaluated by the adjoint variable method.3 However, the direct application of the gradient method that is moving boundary to the outside in proportion to the negative value of the shape gradient in minimization problems often results in oscillating shapes.4 It is known that the oscillating phenomenon is caused by a lack of smoothness of the shape gradient.5, 6 To overcome this irregularity, the author and coworkers introduced an idea of a smoothing gradient method.5 The traction method previously proposed by the author and coworkers7–9 was a concrete numerical procedure of the smoothing gradient method. In the traction method, domain variation that minimize objective functional are obtained as solution of pseudolinear elastic problem of continua defined on design domain and loaded with pseudo-distributed traction in proportion to the shape gradient on the design domain. For numerical analysis of the pseudo-linear elastic problems the finite-element method or the boundary-element method can be applied. In the previous papers, numerical procedure of the traction method has not been illustrated sufficiently. This paper presents a generalized description of the traction method and gives a precise algorithm of the traction method.

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Domain variations

Let Ω ⊂ Rd (R denotes the set of real number) (d = 2, 3) be a given bounded domain and ∂Ω = Γ be its boundary. One way to generate the small variations of Ω is to use a one-parameter family of one-to-one mappings T s ≡ {Tsi }di=1 : Rd → Rd (0 ≤ s < ) where s represents history of domain

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variation and  is a small positive number and T s : Ω  X → x ∈ Ωs T −1 s

(1)

: Ωs  x → X ∈ Ω

(2)

To keep one-to-one property, the following conditions are required.10 1,∞ (i) T s and T −1 (Rd ))d or (C 1 (Rd ))d for all s ∈ [0, ). s belong to (W 1 d (ii) The mappings s → T s (x) and s → T −1 s (x) belong to (C ([0, ))) for d all x ∈ R .

The notations (W m,p ( · ))d and (C m ( · ))d for all integers m ≥ 0 and any number p satisfying 1 ≤ p ≤ ∞ denote the Sobolev space of d-th dimensional functions of p-th power integrable in the sense of the Lebesgue measure until m-th derivatives and the set of continuous functions until m-th derivatives respectively defined on ( · ). A derivative of T s with respect to s defined by ∂T s −1 (T s (x)) ∂s is called velocity of domain variation. V (x) ≡

3

x ∈ Ωs

(3)

Boundary shape optimization problems

For simplicity, let us consider an elliptic boundary value problem of the second order related to a real-valued scalar state function. This problem is described in the strong form as − ∇ · A(x)∇u(x) + c(x)u(x) = f(x) x ∈ Ω u(x) = u0 (x) x ∈ Γ0 A(x)∇u(x) · n(x) = g(x) x ∈ Γ \ Γ¯0

(4) (5) (6)

where u0 , A ≡ {Aij }di,j=1 = AT (( · )T denotes the transpose), c, f and g are given functions defined in Rd . n ≡ {ni }di=1 denotes the outer normal vector. ( · ) \ ( · ) denotes subtraction between sets. ( ¯· ) denotes colosed set of ( · ). For ellipticity, ∃α > 0 : c(x) ≥ α and z · A(x)z ≥ α|z|2

∀z ∈ Rd

∀x ∈ Rd

(7)

The weak form is given by a(u, v) = l(v)

u − u0 ∈ U

u0 ∈ U0

∀v ∈ U

(8)

where the bilinear form a( · , · ) and the linear form l( · ) are defined by  a(u, v) ≡ (∇u · A∇v + cuv) dx (9) Ω  l(v) ≡ fv dx + gv dΓ (10) Ω

¯0 Γ \Γ

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and the admissible sets U and U0 are given by   ¯0 U0 = u ∈ H 1 (Ω )| u(x) = 0, x ∈ Γ \ Γ    u dx = 0 if U = u ∈ H 1 (Ω )| u(x) = 0, x ∈ Γ0 , Ω

 dΓ = 0

(11) (12)

Γ0

 d×d Let A ∈ L∞ (Rd ) , c ∈ L∞ (Rd ), f and g ∈ H 0 (Rd ) for existance of one unique solution. A boundary shape optimization problem to minimize an objective functional J0 (u) under q constraint conditions with functionals {Jm (u)}qm=1 :   Jm (u) ≡ φm (u) dx + ϕm (u) dΓ (m = 0, 1, 2, · · · , q) (13) Ω

Γ

can be formulated by min J0 (u) such that

Ω⊂Rd

a(u, v) = l(v)

u − u0 ∈ U

u0 ∈ U0

[0] ≤ 0 (m = 1, 2, · · · , q) and Jm (u) − Jm

∀v ∈ U (14)

[0]

where {Jm }qm=1 are given real numbers. For simplicity, let the coefficient functions of u0 , A, c, f and g be fixed ¯0 ∩ Γ \ Γ0 and in Rd during domain variations and the velocity V = 0 at Γ the singular points on Γ . Applying the adjoint variable method for state equation and the Lagrange multiplier method for constraint conditions with respect to functionals, the material derivative of the objective functional J˙0 with respect to s is obtained as a linear form with velocity V by

  q Λm Gm n · V dΓ G0 + (15) J˙0 (u) = Gn, V  ≡ Γ

m=1

Gm = φm (u) + (∇ · nϕm + ϕm κ) − ∇u · A∇vm − cuvm + fvm + ∇ · ngvm + g∇ · nvm + gvm κ (m = 0, 1, 2, · · · , q) (16) where the adjoint variables {vm }qm=0 and the Lagrange multipliers {Λm }qm=1 are determined by the adjoint equations and the Kuhn-Tucker conditions:   dφm  dϕm  a(u , vm ) = u dx + u dΓ ∀u ∈ U (m = 0, 1, 2, · · · , q) Ω du Γ du (17) [0] [0] Λm Jm (u) − Jm = 0, Λm ≥ 0, Jm (u) − Jm ≤ 0 (m = 1, 2, · · · , q) (18) κ denotes the d − 1 times of the mean curvature. Since Gn is a coefficient with respect to velocity V in an inner product form giving the material

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derivative of the objective functional under satisfaction of the state and adjoint equations and the Kuhn-Tucker conditions for the constraint conditions, Gn is called shape gradient of the boundary shape optimization problem given by eqn. (14). G is called shape gradient density. {Gm}qm=0 are called the shape gradient densities with respect to {Jm }qm=0 respectively.

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Gradient method in Hilbert space

Since the shape gradient was derived, a gradient method can be applicable to reshaping algorithm. Before applying to the boundary shape optimization problem, let us show definition of the gradient method in Hilbert space. Let Φ be a real Hilbert space with scalar product ( · , · )Φ and norm  · Φ . Let J : Φ → R be a real valued functional. Let us concider to find a unique solution φ∗ ∈ Φ such that J(φ∗ ) = min J(φ)

(19)

φ∈Φ

If J is differentiable in Φ, the gradient of J, denoted by GJ , is defined by a unique element of Φ which satisfies11 (GJ , ϕ)Φ = lim

ζ→0

1 (J(φ + ζϕ) − J(φ)) ζ

∀ϕ ∈ Φ

(20)

The gradient method in Φ is to determine increment of φ, denoted by ∆φ, as a unique element of Φ which satisfies b(∆φ, ϕ) = −(GJ , ϕ)Φ

∀ϕ ∈ Φ

(21)

where b( · , · ) is a uniformly bounded and coercive bilinear form in Φ that satisfies12 ∃α, β > 0 : b(φ, φ) ≥ αφ2Φ ∀φ ∈ Φ and b(φ, ϕ) ≤ βφΦ ϕΦ ∀φ, ϕ ∈ Φ

(22)

Then, it is guaranteed that ζ∆φ decreases the functional: J(φ + ζ∆φ) = J(φ) + (GJ , ζ∆φ)Φ + o(ζ) = J(φ) − b(∆φ, ζ∆φ) + o(ζ) ≤ J(φ) − αζ∆φ2 + o(ζ) (23) where ζ is a small positive number and o( · ) is the Landau functional, i.e., limζ→0 ζ1 o(ζ) = 0. Indeed, the second term on the right side of the inequality is strictly negative and the third term can be made very small.

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Traction method

To apply the gradient method in Hilbert space to the boundary shape optimization problems, it is required to select an appropriate Hilbert space. However, the design variable T s and its derivative V belong to (W 1,∞(Rd ))d , which is a Banach space not a Hilbert space. This indicates that it is not possible to apply the gradient method in Hilbert space to boundary shape optimization problems directly. A well-advised idea is to select a Hilbert space which includes (W 1,∞ (Rd ))d and to find a domain variation V that belongs to (W 1,∞ (Rd ))d . Such a Hilbert space can be found in (H 1 (Rd ))d = (W 1,2 (Rd ))d that can be defined by 

 D = V ∈ (H 1 (Rd ))d  V (x) = 0,   x ∈ Γ¯0 ∩ Γ \ Γ0 and singular points on Γ ,  rigid motion constraints and design constraints if necessary (24) One of the most familiar coercive bilinear forms in (H 1 (Rd ))d is that used in linear elastic continuum problem restricting rigid motions:  a ˆ(u, v) ≡ Cijkluk,l vi,j dx (25) Ω

 d×d×d×d where {Cijkl}dijkl=1 ∈ L∞ (Rd ) denotes an elastic stiffness tensor of positive definite. In tensor notation with dimension d, the Einstein summation convention and gradient notation ( · ),i = ∂( · )/∂xi are used. Using a( · , · ) for b( · , · ) in eqn. (22), a concrete solution can be presented for ˆ determining velocity V ∈ D by a ˆ(V , y) = − Gn, y

∀y ∈ D

(26)

and reshaping with ∆sV for a given small positive number ∆s where · , ·  is defined in eqn. (15). This solution has been called the traction method.5, 7, 8 Whether or not the solution V in eqn. (26) belongs to D∩(W 1,∞ (Rd ))d depends on smoothness of shape gradient function. The necessary smoothness for boundary and given coefficient functions was discussed in a previous paper5 using the regularity theorem for elliptic boundary value problems. Domain reshaped by this solution has smoother boundary for one-time differentiability than that obtained by a direct gradient method moving boundary in proportion to the shape gradient.5

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Algorithm of traction method

Since the traction method is based on the gradient method, an iteration approach is applied. In this section, notations J0 (Ω ) and {Jm (Ω )}qm=1 are

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used for an objective and constraint functionals at domain Ω respectively. {Gm}qm=0 are used for the shape gradient densities with respect to {Jm }qm=0 as defined by eqn. (16). (i) Solve state equation (eqn. (8)) with respect to domain Ω and evaluate objective and constraint functionals {Jm (Ω )}qm=0 . (ii) Solve adjoint equations (eqn. (17)) with respect to domain Ω if the adjoint equations are different from the state equation and evaluate shape gradient densities {Gm }qm=0 . (iii) Solve velocity {V m }qm=0 ∈ D with respect to domain Ω using {Gm }qm=0 by a ˆ(V m , y) = − Gm n, y

∀y ∈ D

(27)

where ˆa( · , · ) and D are defined by (25) and (24) respectively. (iv) Determine Lagrange multipliers {Λm }qm=1 by ⎡ ⎤⎧ ⎫

G1 n, ∆sV 1  · · · G1 n, ∆sV q  ⎪ ⎬ ⎨ Λ1 ⎪ ⎢ ⎥ .. . . . .. .. .. −⎣ ⎦ . ⎪ ⎭ ⎩ ⎪ Λq

Gq n, ∆sV 1  · · · Gq n, ∆sV q  ⎫ ⎧ [0] ⎪ ⎪ n, ∆sV  + J (Ω ) − J

G ⎪ ⎪ 1 0 1 1 ⎬ ⎨ . . = . ⎪ ⎪ ⎪ ⎭ ⎩ G n, ∆sV  + J (Ω ) − J [0] ⎪ q q 0 q

(28)

assuming incremental parameter ∆s = 1 and {Jm (Ω )}qm=1 = 0. When there exists inactive constraint condition m ∈ {m ∈ {1, 2, · · · , q} | Gm n, ∆sV 0  < 0}, m-th diagonal element of the matrix in (eqn. (28)), that is Gm n, ∆sV m , is multiplied by sufficiently large number M0 > 01 , that is given as an input parameter, before solving eqn. (28). If the solution of eqn. (28) satisfies all {Λm ≥ 0}qm=1 , go to the next step. If there exists negative Lagrange multiplier Λm < 0, resolve eqn. (28) after m-th diagonal element is multiplied by sufficiently large number M0 > 0. If all {Λm ≥ 0}qm=1 were not satisfied after repeating q times, output a warning code and exit. When the matrix in (eqn. (28)) { Gi n, ∆sV j }ij becomes singular due to existence of equivalent constraint conditions, try to exclude active constraint condition one by one by multiplying diagonal element by sufficiently large number M0 > 0 until the matrix becomes regular. If a regular condition could be found out, go to the next step. Otherwise, output a warning code and exit. 1 For

example M0 ≈ 103.

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(v) Create domain Ω∆s by mapping I + ∆sV: Ω → Ω∆s (I denotes the q identical mapping) in which V = V 0 + m=1 Λm V m . Estimate the P absolute maximum principle stress ε (V (x)) by velocity V (x) for all x ∈ Ω . Determine incremental parameter ∆s by ∆s =

maxx∈Ω

ε0 |εP (V (x))|

(29)

where ε0 > 0 is a limit value, that is given as an input parameter, for the maximum value of the absolute maximum principle stress during one shape variation2. (vi) Check inequality constraint conditions {Jm (Ω ) ≤ 0}qm=1 . If all the inequality conditions are satisfied, go to Step (viii). Otherwise, determine {Λm }qm=1 by eqn. (28) using ∆s evaluated in Step(v) and {Jm (Ω )}qm=1 . If there exists inactive constraint condition m ∈ {m ∈ [0] {1, 2, · · · , q} | Gm n, ∆sV 0  + Jm (Ω ) − Jm < 0}, resolve eqn. (28) after m-th diagonal element is multiplied by sufficiently large number M0 > 0. (vii) Remake domain Ω∆s by mapping I + ∆sV : Ω → Ω∆s in which q V = V 0 + m=1 Λm V m using ∆s evaluated in Step (v) and {Λm }qm=1 evaluated in Step (vi), renew ∆s by   ε0 , ∆s → ∆s (30) min maxx∈Ω |εP (V (x))| and remake once more Ω∆s by the mapping I + ∆sV : Ω → Ω∆s if ∆s was changed. (viii) Solve state equation with respect to renewed domain Ω∆s and evaluate objective and constraint functionals {Jm (Ω∆s )}qm=0 . (ix) Judge suitability of ∆s with respect to nonlinearity of objective functional J0 (Ω∆s ) and active constraint functionals {Jm (Ω∆s )| Λm > 0 (m = 1, 2, · · · , q)} by the Armijo criterion:133 Jm (Ω∆s ) − Jm (Ω )
0}, replace Ω∆s by Ω and return to Step (ii) under condition of V 0 = 0 and skipping Step (iv) and Step (v) until to complete one regular shape variation. If the number of iteration to return to Step (ii) without regular shape variation exceeded a limit number, that is given as an input parameter, output a warning code and exit.

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Application

A volume minimization problem under constraint on mean compliance of a knuckle joint of automotive suspension was demonstrated with a program developed in ADVENTURE project4 which is a free release software for PC cluster using balancing domain decomposition method(Figure 1). Finite element model consists of 283,681 nodes and 176,650 tetrahedral second order elements was analyzed using a PC cluster consisting of one Pentium3 with 800MHz (1.5GB) dual processors and four Pentium3 with 800MHz (1GB) dual processors. It took 25 hours in total calculation under 35 times 4 http://adventure.q.t.u-tokyo.ac.jp/

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of regular shape variation. The volume and the maximum Mises stress decreased 26.5% and 18.1% respectively compared with the initial shape.

References 1. J. Cea. Problems of shape optimization. In E. J. Haug and J. Cea, editors, Optimization of Distributed Parameter Structures, volume 2, pages 1005–1048. Sijthoff & Noordhoff, Alphen aan den Rijn, 1981. 2. J. P. Zol´esio. The material derivative (or speed) method for shape optimization. In E. J. Haug and J. Cea, editors, Optimization of Distributed Parameter Structures, volume 2, pages 1089–1151. Sijthoff & Noordhoff, Alphen aan den Rijn, 1981. 3. E. J. Haug, K. K. Choi, and V. Komkov. Design Sensitivity Analysis of Structural Systems. Academic Press, Orland, 1986. 4. M. H. Imam. Three-dimensional shape optimization. Int. J. Num. Meth. Engrg., 18:661–673, 1982. 5. H. Azegami, S. Kaizu, Shimoda M., and E. Katamine. Irregularity of shape optimization problems and an improvement technique. In S. Hernandez and C. A. Brebbia, editors, Computer Aided Optimization Design of Structures V, pages 309–326. Computational Mechanics Publications, Southampton, 1997. 6. B. Mohammadi and O. Pironneau. Applied shape optimization for fluids. Clarendon Press, Oxford, 2001. 7. H. Azegami. A solution to domain optimization problems. Trans. of Jpn. Soc. of Mech. Engs., Ser. A, 60:1479–1486, 1994. (in Japanese). 8. H. Azegami, M. Shimoda, E. Katamine, and Z. C. Wu. A domain optimization technique for elliptic boundary value problems. In S. Hernandez, M. El-Sayed, and C. A. Brebbia, editors, Computer Aided Optimization Design of Structures IV, Structural Optimization, pages 51–58. Computational Mechanics Publications, Southampton, 1995. 9. H. Azegami and Z. C. Wu. Domain optimization analysis in linear elastic problems (approach using traction method). JSME International Journal, Ser. A, 39:272– 278, 1996. 10. J. Sokolowski and J. P. Zol´esio. Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer–Verlag, New York, 1991. 11. O. Pironneau. Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York, 1984. 12. J. Cea. Numerical methods of shape optimal design. In E. J. Haug and J. Cea, editors, Optimization of Distributed Parameter Structures, volume 2, pages 1049–1088. Sijthoff & Noordhoff, Alphen aan den Rijn, 1981. 13. L. Armijo. Minimization of functions having lipschitz-continuous first partial derivatives. Pacific Journal of Mathematics, 16:1–3, 1966. 14. Y. Asaga, K. Takeuchi, H. Azegami, and S. Yoshimura. Large-scale shape optimization analysis using the domain decomposition method. 7(2):841–844, 2002. (in Japanese).