A SMOOTHING METHOD FOR SHAPE OPTIMIZATION: TRACTION ...

September 11, 2006 18:8 WSPC/IJCM-j050

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International Journal of Computational Methods Vol. 3, No. 1 (2006) 21–33 c World Scientific Publishing Company

A SMOOTHING METHOD FOR SHAPE OPTIMIZATION: TRACTION METHOD USING THE ROBIN CONDITION

HIDEYUKI AZEGAMI Department of Complex Systems Science Graduate School of Information Science Nagoya University, Furo-cho, Chigusa-ku, Nagoya City Aichi Prefecture, 464-8601, Japan [email protected] KENZEN TAKEUCHI Quint Corporation 1-14-1 Fuchu-cho, Fuchu, Tokyo, 183-0055, Japan [email protected] Received 7 October 2004 Revised 25 March 2005 Accepted 20 April 2005 This paper presents an improved version of the traction method that was proposed as a solution to shape optimization problems of domain boundaries in which boundary value problems of partial differential equations are defined. The principle of the traction method is presented based on the theory of the gradient method in Hilbert space. Based on this principle, a new method is proposed by selecting another bounded coercive bilinear form from the previous method. The proposed method obtains domain variation with a solution to a boundary value problem with the Robin condition by using the shape gradient. Keywords: Optimum design; numerical analysis; finite-element method; adjoint variable method; gradient method.

1. Introduction The use of computer-aided design and simulation techniques in place of experiments has contributed to time and labor savings in the product development process. Techniques for optimal shape design have received considerable attention with the aim of achieving even greater savings. Optimal shape design with parametric design variables has come into wide use recently through commercial programs based on mathematical programming techniques combined with simulation methods. However, with respect to nonparametric optimal shape design, the degrees of freedom for boundary shape variation can be infinite theoretically, and in the case of discretization by the finite element method, they are the degrees of 21


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H. Azegami & K. Takeuchi

freedom of the nodes on the boundaries in the formulation. It is not easy to solve such shape design problems by mathematical programming techniques because of the enormous number of degrees of freedom equivalent to the dimensions of the design space. Nonparametric boundary shape optimization has long been of interest to mathematicians. Differentiability with respect to boundary variation for the first eigenvalue of a membrane was shown at the beginning of the twentieth century by Hadamard [Hadamard (1968); Sokolowski and Zolésio (1991)]. He assumed the boundary was smooth and defined boundary variation as the distance of movement in the normal direction. Theoretical expansion to domains with piecewise smooth boundaries was demonstrated in 1980’s by selecting a one-parameter family of continuous one-to-one mappings from an original domain to variable domains [Cea (1981a), (1981b); Zolésio (1981a), (1981b); Sokolowski and Zolésio (1991)]. The shape gradient with respect to domain variation can be evaluated by the adjoint variable method [Haug et al. (1986); Choi and Kim (2005)]. However, direct application of the gradient method often results in oscillating shapes [Imam (1982)]. It is known that oscillation is caused by a lack of smoothness of the shape gradient [Azegami et al. (1997); Mohammadi and Pironneau (2001)]. To avoid oscillation without reducing the degrees of freedom for boundary shape variation, a method using the Laplace operator on the boundary was proposed and has been used for shape optimization problems of flow fields [Mohammadi and Pironneau (2001); Jameson (2003)]. However, in using a finite element model, an additional procedure is required to reconstruct the mesh to fit the reshaped boundary. To compensate for the lack of smoothness of the shape gradient, the traction method has been developed by the authors and coworkers by applying the gradient method in Hilbert space [Azegami (1994); Azegami et al. (1995); Azegami and Wu (1996); Azegami et al. (1997); Azegami (2000), (2004)]. In the traction method, domain variation that minimizes the objective functional is obtained as a solution to a boundary value problem of a linear elastic continuum defined in the design domain and loaded with traction in proportion to the negative shape gradient on the design boundary. In other words, the negative shape gradient is used for the Neumann condition on the design boundary. However, there are some cases where shape variations stop without converging to the objective functionals, particularly in the case of slender linear elastic continua under free support except at both ends of the shape variation. This paper presents an improved version of the traction method. The new method has been devised by selecting another bounded coercive bilinear form in Hilbert space. The proposed method uses the shape gradient to obtain domain variation as a solution to a boundary value problem involving the Robin condition. Application of the proposed method to a three-dimensional bar problem with a notch resulted in smooth convergence, whereas the problem could not be solved with the previous method.


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A Smoothing Method for Shape Optimization

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2. Shape Optimization Problems 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. Domain variations can be defined as follows. 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 Ts ≡ {Tsi }di=1 : Rd → Rd (0 ≤ s < ) where s represents the history of domain variation, is a small positive number and Ts : Ω X → x ∈ Ωs , T−1 s

(1)

: Ωs x → X ∈ Ω.

(2)

To keep the one-to-one property, the following conditions are required [Sokolowski and Zolésio (1991)]. 1,∞ (1) Ts and T−1 (Rd ))d or (C 1 (Rd ))d for all s ∈ [0, ). s belong to (W 1 d d (2) The mappings s → Ts (x) and s → T−1 s (x) belong to (C ([0, ))) for 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 dth dimensional functions of the pth power integrable in the sense of the Lebesgue integral and the set of continuous functions, respectively, until m-order derivatives defined on (·). A derivative of Ts with respect to s defined by V(x) ≡

∂Ts −1 (Ts (x)) ∂s

x ∈ Ωs ,

(3)

is called the velocity of domain variation. 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) u(x) = u0 (x)

x ∈ Ω,

(4)

x ∈ Γ0 ⊂ Γ,

A(x)∇u(x) · n(x) = g(x)

(5)

¯ 0, x ∈ Γ\ Γ

(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 ¯ denotes a closed set of (·). For ellipticity, subtraction between sets. (·) ∃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

u 0 ∈ U0

∀v ∈ U,

(8)


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

(9) (10)

¯0 Γ\Γ

Ω

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

(11)

dΓ = 0 .

(12)

Γ0

Let A ∈ (L∞ (Rd ))d×d , c ∈ L∞ (Rd ), f and g ∈ L2 (Rd ) for the existence 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 minΩ⊂Rd J0 (u) such that a(u, v) = l(v) u − u0 ∈ U

u 0 ∈ U0

∀v ∈ U,

and

[0] Jm

≤ 0 (m = 1, 2, . . . , q), (14) Jm (u) − [0] q where Jm m=1 are given real numbers. For simplicity, let the coefficient functions of u0 , A, c, f , and g be fixed in Rd ¯ 0 ∩ Γ\Γ0 and the singular during domain variations and the velocity V = 0 at Γ points on Γ. Applying the adjoint variable method for the state equation and the Lagrange multiplier method for the constraint conditions with respect to the 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 J˙0 (u) = Gn, V ≡ Λm Gm n · VdΓ, G0 + (15) Γ

m=1

Gm = φm (u) + (∇ · nϕm + ϕm κ) − ∇u · A∇vm − cuvm + f vm + ∇g · nvm + g∇vm · n + gvm κ {vm }qm=0

(m = 0, 1, 2, . . . , q), {Λm }qm=1

and the Lagrange multipliers where the adjoint variables determined by the adjoint equations and the Kuhn-Tucker conditions: dφm dϕm u dx + u dΓ ∀u ∈ U (m = 0, 1, 2, . . . , q), a(u , vm ) = Ω du Γ du [0] ) = 0, Λm (Jm (u) − Jm

Λm ≥ 0,

[0] Jm (u) − Jm ≤0

(m = 1, 2, . . . , q).

(16) are

(17) (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 derivative of the objective functional under the condition that the state and adjoint equations and


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the Kuhn-Tucker conditions for the constraints are satsified, Gn is called the shape gradient of the boundary shape optimization problem given by Eq. (14). G is called the shape gradient density. {Gm }qm=0 are called the shape gradient densities with respect to {Jm }qm=0 , respectively. 3. Gradient Method in Hilbert Space Since the shape gradient has been derived, we will now consider a reshaping algorithm that uses the gradient method. Although the gradient method was generally defined for a functional on Hilbert space [Cea (1981b)], we will extend the method for a functional defined on Banach space. Let X be a Banach space and J : X → R be a real valued functional. Let us find a unique solution x∗ ∈ X such that J(x∗ ) = min J(x).

(19)

x∈X

If J is differentiable at x in X, the gradient of J, denoted by GJ , is defined by an element of dual space X of X which satisfies 1 (J(x + ζh) − J(x)) ζ→0 ζ

(GJ , h)X ×X = lim

∀h ∈ X.

(20)

In the gradient method, a coercive bilinear form bY (·, ·) in a Hilbert space Y with norm · Y is required that satisfies ∃α > 0 : bY (y, y) ≥ αy2Y

∀y ∈ Y.

(21)

The gradient method is defined as to determine y ∈ X ∩ Y by bY (y, h) = −(GJ , h)X ×X

∀h ∈ X ∩ Y,

(22)

and renew x in the direction y. Then, it is guaranteed that ζy decreases the functional: J(x + ζy) = J(x) + (GJ , ζy)X ×X + o(ζ) = J(x) − bY (y, ζy) + o(ζ) ≤ J(x) − αζy2Y + 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. 4. Previously Proposed Traction Method The traction method has been proposed for determining velocity V ∈ D by a ˆ(V, y) = −Gn, y

∀y ∈ D,

(24)


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and reshaping with ∆sV for a given small positive number ∆s, where D and a bilinear form a ˆ(·, ·) are defined as (25) D = V ∈ (W 1,∞ (Ω))d | constraints on shape variation , a ˆ(u, v) ≡ Cijkl uk,l vi,j dx, (26) Ω

in which {Cijkl }dijkl=1 ∈ (L∞ (Rd ))d×d×d×d denotes an elastic stiffness tensor that is positive definite and ·, · is defined in Eq. (15). In tensor notation with dimension d, the Einstein summation convention and gradient notation (·),i = ∂(·)/∂xi are used. Equation (24) indicates that velocity V is determined by solving the displacement of a pseudo-elastic body defined in domain Ω by the loading of a pseudo-external force −Gn on the boundary Γ under constraints on displacement for shape variation (see Fig. 1) [Azegami (2004)]. The traction method can be implemented through the use of the finite element method or boundary element method by applying the analogy between linear elastic deformation induced by loading an external force and shape deformation induced by loading the shape gradient −Gn. This method is considered to be an application of the gradient method in Hilbert space to boundary shape optimization problems [Azegami (2000)]. In this method, the set D comprising the design variable Ts and its derivative V was included in the Hilbert space (H 1 (Ω))d and the coerciveness of the bilinear forms in D are insured by including constraints on rigid motions in the constraints on shape variation. Whether or not the solution V in Eq. (24) belongs to (W 1,∞ (Rd ))d with constraints on shape variation depends on the smoothness of the shape gradient function. The necessary smoothness for the boundary and given coefficient functions was discussed in a previous paper [Azegami et al. (1997)] using the regularity theorem for elliptic boundary value problems. A design domain reshaped by this solution has a smoother boundary for one-time differentiability than that obtained by the direct gradient method in which the boundary is moved in proportion to the shape gradient.

Fig. 1. Schematic illustration of previously proposed traction method.


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5. Proposal of Traction Method Using the Robin Condition In Eq. (24), the shape gradient Gn : Γ → Rd was used as the Neumann condition. In this paper, the authors propose the use of the Robin condition, that is the use of a coercive bilinear form a ˆ(·, ·) + α·, ·, as a ˆ(V, y) + α(V · n)n, y = −Gn, y

∀y ∈ D,

(27)

where α > 0 is assumed and ·, · is defined in Eq. (15). The coerciveness of the bilinear forms in D are insured at α > 0 regardless of the constraints on rigid motions in the constraints on shape variation. Equation (27) means determining velocity V by solving a displacement of a pseudo-elastic body defined in domain Ω accompanied with a distributed spring of coefficient α connecting the boundary Γ of the pseudo-elastic body to settled points in the normal direction by the loading of a pseudo-external force Gn on the boundary Γ under constraints on displacement for shape variation (see Fig. 2). This method can be also implemented through the use of the finite element method by adding the stiffness matrices of the distributed spring for each boundary element to the global stiffness matrix.

6. Numerical Examples To demonstrate the performance of the new traction method, it was applied to mean compliance minimization problems of slender linear elastic continua under volume constraints. The mean compliance minimization problem of a linear elastic continuum loaded with nonzero external boundary force P ≡ {Pi }di=1 on sub-boundary ΓP

Fig. 2. Schematic illustration of traction method using the Robin condition.


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under a constraint on displacement u ≡ {ui }di=1 on sub-boundary Γ0 can be formulated by minΩ⊂Rd ˆl(u) such that ˆ ∀v ∈ U ˆ , and a ˆ(u, v) = ˆ l(v) u ∈ U dx − M [0] ≤ 0

(28)

Ω

ˆ for displacewhere the bilinear form a ˆ(·, ·), the linear form ˆl(·) and admissible set U ment are defined by Eq. (26) and ˆl(v) ≡ Pi vi dΓ, (29) ΓP

ˆ ≡ v ∈ (H 1 (Ω))d |v|Γ0 = 0 , U

(30)

respectively. M [0] is a limited volume. The shape gradient densities G0 for the mean compliance ˆ l(u) and G1 for the volume are obtained as G0 = −Cijkl uk,l ui,j ,

(31)

G1 = 1,

(32)

respectively corresponding to Eq. (16). The value of Eq. (31) can be evaluated as the double strain energy density. Figure 3 illustrates the boundary conditions for a linear elastic problem of a three-dimensional continuum with a notch and for shape variation in a mean

(a) Constraint on displacement and loading condition.

(b) Constraint on shape variation.

Fig. 3. Boundary conditions for a mean compliance minimization problem of a three-dimensional linear elastic continuum with a notch and constraint on shape variation.


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compliance minimization problem. The final shapes obtained by the traction method using the Robin condition with a distributed spring of normalized stiffness r, which is calculated by r = Kb /Ka where Ka and Kb are the average values of sets of numbers on diagonal elements corresponding to the nodes on the variable sub-boundary of the stiffness matrices for a ˆ(V, y) and for α(V · n)n, y in Eq. (27), after n iterations are shown in Fig. 4. The case of r = 0 indicates the previously proposed traction method. Figure 5 presents the iteration histories of the mean compliance.

(a) Initial

(e) r = 0.01 n = 78

(b) r = 0 n = 100

(f) r = 0.1 n = 17

(c) r = 0.001 n = 100

(d) r = 0.001 n = 680

(g) r = 1 n = 18

(h) r = 10 n = 16

Fig. 4. Final shapes obtained by traction method using the Robin condition with a distributed spring of normalized stiffness r after n iterations.


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Ratio of Mean Compliance to Initial Value

1.00 r=0 r = 0.001 r = 0.01 r = 0.1 r=1 r = 10

0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0

20

40 60 Number of Iterations n

80

100

Fig. 5. Iteration histories for a mean compliance minimization problem of a three-dimensional linear elastic continuum with a notch.

The optimum shape for the mean compliance minimization problem is obviously the bar with a uniform cross section. From the results shown in Figs. 4 and 5, it is observed that the result obtained by the previously proposed traction method did not reach the optimum shape, while the results obtained by the traction method using the Robin condition reached the optimum shape. Rapid convergence was obtained for r ≥ 1. To demonstrate the applicability to multiply connected domains, two types of mean compliance minimization problems were analyzed. Figure 6 illustrates the boundary condition for a linear elastic problem of a three-dimensional continuum with holes that is referred to as the MBB (Messerschmitt-Bölkow-Blohm) beam problem and the final shapes obtained by the previously proposed traction method and the traction method using the Robin condition. With respect to shape variation, it was assumed that sub-boundaries loaded with the external force and constrained on the displacement were fixed and that the top, bottom, and side planes were constrained in the out-of-plane direction. The limited volume was given as being 40% of the initial shape. After 100 iterations, the mean compliance was obtained with 166.9% of the initial shape by the previously proposed traction method, while it was obtained with 161.3% of the initial shape by the traction method using the Robin condition. Figure 7 illustrates the results of a mean compliance minimization problem of a linear elastic continuum of the Michell truss type. With regard to the shape variation, it was assumed that sub-boundaries loaded with the external force and constrained on the displacement were fixed and that the side planes were constrained in the out-of-plane direction. The limited volume was given as being 30% of the initial shape. After 100 iterations, the mean compliance was obtained with 317.2% of the initial shape by the previously proposed traction method, whereas it was obtained with 215.4% of the initial shape by the traction method using the Robin condition.


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(a) Initial shape.

(b) Final shape by previous traction method.

(c) Final shape by new traction method.

Fig. 6. Results of mean compliance minimization problem on the MBB beam.

(a) Initial shape.

(b) Final shape by previous traction method.

(c) Final shape by new traction method.

Fig. 7. Results of mean compliance minimization problem on a linear elastic continuum of the Michell truss type.


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7. Conclusion This paper has presented an improved version of a numerical analysis method, called the traction method, for solving nonparametric boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The principle of the traction method was presented based on the theory of the gradient method in Hilbert space. Based on this principle, a new traction method was proposed by using the Robin condition with the shape gradient instead of the Neumann condition used in the previous traction method. This method can be implemented through the use of the finite element method by adding stiffness matrices of the distributed spring for each boundary element to the global stiffness matrix. From the numerical results obtained for a mean compliance minimization problem of a slender linear elastic continuum with a notch and subject to a volume constraint, it was observed that the new traction method provided smooth convergence, whereas the problem could not be solved with the previous method.

References Azegami, H. [1994] A solution to domain optimization problems. Trans. of Jpn. Soc. of Mech. Engs., Ser. A, 60: 1479–1486. (in Japanese) Azegami, H., Shimoda, M., Katamine, E. and Wu, Z. C. [1995] A domain optimization technique for elliptic boundary value problems. Computer Aided Optimization Design of Structures IV, Structural Optimization, eds. Hernandez, S., El-Sayed, M. and Brebbia, C. A., Computational Mechanics Publications, Southampton, 51–58. Azegami, H. and Wu, Z. C. [1996] Domain optimization analysis in linear elastic problems (Approach using traction method). JSME International Journal, Ser. A, 39: 272–278. Azegami, H., Kaizu, S., Shimoda, M. and Katamine, E. [1997] Irregularity of shape optimization problems and an improvement technique. Computer Aided Optimization Design of Structures V, eds. Hernandez, S. and Brebbia, C. A., Computational Mechanics Publications, Southampton, 309–326. Azegami, H. [2000] Solution to boundary shape identification problems in elliptic boundary value problems using shape derivatives. Inverse Problems in Engineering Mechanics II, eds. Tanaka, M. and Dulikravich, G. S., Elsevier, Tokyo, 277–284. Azegami, H. [2004] Solution to Boundary Shape Optimization Problems, High Performance Structures and Materials II, eds. Brebbia, C. A. and de Wilde, W. P., WIT Press, Southampton, 589–598. Cea, J. [1981a] Problems of shape optimization. Optimization of Distributed Parameter Structures, 2, eds. Haug, E. J. and Cea, J., Sijthoff and Noordhoff, Alphen aan den Rijn, 1005–1048. Cea, J. [1981b] Numerical methods of shape optimal design. ibid., 1049–1088. Choi, K. K. and Kim, N. H. [2005] Structural Sensitivity Analysis and Optimization, 1–2, Springer, New York. Hadamard, J. [1968] Mémoire sur le probléme d’analyse relatif a l’équilibre des plaques élastiques encastrees par M. Jacques Hadamard. Mémoire des savants etragers. Oeuvres de J. Hadamard, CNRS, Paris, 515–629. Haug, E. J., Choi, K. K. and Komkov, V. [1986] Design Sensitivity Analysis of Structural Systems, Academic Press, Orland.


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Imam, M. H. [1982] Three-dimensional shape optimization. Int. J. Num. Meth. Engrg., 18: 661–673. Jameson, A. [2003] Aerodynamic shape optimization using the adjoint method, citeseer.ist.psu.edu/jameson03aerodynamic.html. Mohammadi, B. and Pironneau, O. [2001] Applied shape optimization for fluids, Clarendon Press, Oxford. Pironneau, O. [1984] Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York. Sokolowski, J. and Zolésio, J. P. [1991] Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer–Verlag, New York. Zolésio, J. P. [1981a] The material derivative (or speed) method for shape optimization, Optimization of Distributed Parameter Structures, 2, eds. Haug, E. J. and Cea, J., Sijthoff and Noordhoff, Alphen aan den Rijn, 1089–1151. Zolésio, J. P. [1981b] Domain variational formulation for free boundary problems, ibid., 1152–1194.