PARAMETRIC SHAPE AND TOPOLOGY OPTIMIZATION WITH RADIAL BASIS FUNCTIONS

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1 PARAMETRIC SHAPE AND TOPOLOGY OPTIMIZATION WITH RADIAL BASIS FUNCTIONS Michael Yu Wang 1 and Shengyin Wang 1 Department of Automation and Computer-Aided Engineering The Chinese University of Hong Kong Shatin, NT, Hong Kong Tel.: ; Fax: yuwang@acae.cuhk.edu.hk (M.Y. Wang) Abstract Keywords: This paper describes a parametric optimization technique for shape and topology optimization. The proposed method is a generalization of the classical method of level sets which are represented with discrete grids. In using radial basis functions (RBFs), the proposed formulation projects the geometric motion of the level sets of an implicit function onto its parametric representation. The resulting method provides a set of new capabilities for general shape and topology optimization, particularly treating it as a parameter problem with the RBF implicit models. Numerical examples are presented in the paper, suggesting the potential of the method. shape and topology optimization; radial basis functions; level set method; parameter design 1. Introduction The finite element (FE) based strategy for structural topology optimization has received wide attention and experienced considerable progress recently since the seminal work of Bendsøe and Kikuchi [4]. A wide range of approaches and techniques have been developed as reviewed in detail in [1, 5]. The method of implicit moving interface using the level set technique has been an emerging scheme for shape and topology optimization [2, 12, 11]. The level set method introduced by Osher and Sethian [8] is a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions. Since interfaces may easily develop sharp corners, break apart, and merge together, the level set method has a wide range of applications, including problems in fluid mechanics, combustion, solids modelling, computer animation and image processing [9].

2 2 Along with its benefits, the discrete representation also limits the utility of the classical level set method. The level set function has no analytical form, and the entire design domain must be discretized into a rectilinear grid for level set processing, often through a distance transform. In applying the classical level set method for structural topology optimization, the implementation requires an appropriate choice of the upwind scheme, extension velocity and reinitialization algorithm [9, 12]. Furthermore, there is no nucleation mechanism in the conventional level set method, if the Hamilton-Jacobi equation is solved under a strict condition for numerical stability [10, 7]. Indeed, new holes cannot be created in the interior of a material region because the Hamilton-Jacobi equation satisfies a maximum principle and reinitialization must be applied to the level set function to ensure its regularity [7]. Although some attempts have been made to incorporate the topological derivatives into the level set schemes [7], it is shown to be difficult to combine the intrinsically discontinuous topological derivatives with the continuous shape derivatives [13]. Hence, the numerical considerations of discrete computation have severely limited the primary advantages of the level set methods in solid optimization. Other attempts have also been made to use an implicit function as a regularization method without invoking the discrete level set system [3]. In this paper, we describe an extension of the level set methodology with a parameterized implicit representation. The proposed scheme retains topological benefits of the implicit representation of evolving surfaces while avoiding the drawbacks of a grided spatial discretization. In other words, we can treat shape and topology optimization as a parameter optimization problem. A parameterized implicit surface can be propagated under an arbitrary speed function. The propagation occurs by changing the implicit s parameters with the use of a common search algorithm in conventional parameter optimization. We employ the popular radial basis functions (RBFs) to provide a parameterized implicit representation of shapes. The proposed RBF-level set method is implemented in the framework of minimum compliance design that has been extensively studied in the topology optimization field. Numerical examples are given to illustrate the success of the method for 2D structures. 2. Topology Optimization with Level Set Model In the level set framework, a surface is represented implicitly through a level set function Φ(x), which is Lipschitz-continuous, and the surface itself is the zero isosurface or zero level set {x R d Φ(x) = 0} (d = 2 or 3) [9]. In shape and topology optimization of a solid, the solid structure is defined as Φ(x) = 0 Φ(x) < 0 Φ(x) > 0 x Ω D x Ω\ Ω x (D\Ω) (1)

3 Parametric Shape and Topology Optimization with Radial Basis Functions 3 where D R d is a fixed design domain in which all admissible shapes Ω (a smooth bounded open set) are included, i.e., Ω D. With this implicit representation, the structural topology optimization process operates on the implicit scalar function Φ(x) defined in Eq. (1) [12, 2], often using a gradient method for the minimization of an objective function J(Φ). In the present study, only a relatively simple unconstrained minimum compliance design problem presented in [2] with a design-independent load is considered, which can be expressed as min: Φ(x) J(Φ) = D (ε(u)) T Cε(u)H( Φ) dω + l D H( Φ) dω (2) where u is the displacement field, ε(u) the strain field, H(Φ) the Heaviside step function of the implicit function Φ(x), l a positive Lagrange multiplier, and the volume V of the admissible design can be written as V = H( Φ) dω (3) D Given a local perturbation of the boundary, ẋ for x Ω, of the admissible domain Ω, the resulting change in the objective function J(Φ) gives rise to the so-called shape derivative [12, 2], written as dj d t = Ω ( (ε(u)) T Cε(u) l ) v n ds (4) where t is the artificial time, and v n the normal velocity representing the perturbation v n (x) = ( Φ) ẋ for x Ω. Following the development in [8], the change in the surface of the solid as a result of the normal velocity v n is captured by changing its underlying level set function via: Φ t + v n Φ = 0 (5) This is known as the Hamilton-Jacobi equation. Hence, moving the boundary of the shape (Φ(x) = 0) along the normal direction is equivalent to transporting Φ by solving the Hamilton-Jacobi equation (5). With the above shape derivative of Eq. (4) and the time derivative of (Eq. 5), it is natural to define the normal velocity v n as a descent change for the minimization of the objective function [12, 2], as v n = G = (ε(u)) T Cε(u) l (6) Thus, by substituting the normal velocity v n of Eq. (6) into Eq. (5), one obtain the level set propagation equation that, when integrated, defines a moving boundary of the solid that evolves under the minimization of the objective function [12, 2].

4 4 3. RBF-Parameterized Implicits In the conventional level set-based topology optimization, a general analytical function for Φ(t, x) is not known. Thus, it must be discretized for level set processing, often through a distance transform. In an Eulerian approach, a numerical procedure for solving the Hamilton-Jacobi PDE is indispensable. This procedure requires appropriate choice of the upwind schemes, extension velocities and reinitialization algorithms, which may limit the utility of the level set method. Some of the limitations are discussed earlier. For example, reinitialization prevents a level set function from nucleation of holes in the interior of material regions [10, 7]. Another major limitation lies in the discrete representation. Therefore, a better method is to retain topological benefits of the implicit representation of a level set model while avoiding the drawbacks of using its discrete samples on a fixed grid or mesh. In the present study, we propose to generalize the level set function Φ(t, x) to include alternative implicit surface representations which provide a free-form representation with parameterization. To this end, a level set method using radial basis functions (RBFs) is developed for structural topology optimization. Radial basis functions are radially-symmetric functions centered at a particular point [6], or knot, which can be expressed as follows: ϕ i (x) = ϕ ( x x i ), x i D (7) where denotes the Euclidean norm on R d [6], and x i the position of the knot. Only a single fixed function form ϕ : R + R with ϕ(0) 0 is used as the basis to form a family of independent functions. There is a large class of possible radial basis functions. Commonly used RBFs include thin-plate splines, polyharmonic splines, Sobolev splines, Gaussians, multiquadrics, compactly supported RBFs [6], and others. Among these common functions, the multiquadric (MQ) spline appears to be the overall best performing RBF, which can be written as ϕ i (x) = x x i 2 + c 2 i (8) where c i is the free shape parameter which is commonly assumed to be a constant for all i in most applications [6]. In the present RBF implicit modelling, MQ is used to define the scalar implicit function Φ (x) with N MQs centered at N knot positions: Φ (x) = N α i ϕ i (x) + p (x) (9) where α i is the weight, or expansion coefficient, of the radial basis function positioned at the i-th knot x i, and p (x) a first-degree polynomial to account

5 Parametric Shape and Topology Optimization with Radial Basis Functions 5 for the linear and constant portions of Φ (x) and to ensure polynomial precision [6]. For three dimensional (3D) problems, p (x) is given by p (x) = p 0 + p 1 x + p 2 y + p 3 z (10) in which p 0, p 1, p 2 and p 3 are the coefficients of the polynomial p (x). Because of the introduction of this polynomial, to ensure a unique solution, the coefficients in Eq. (9) must be subject to a set of orthogonality or side constraints [6]: N N N α i = 0; α i x i = 0; α i y i = 0; Thus, Eq. (9) can be re-written in a vector form as N α i z i = 0 (11) Φ (x) = g T (x)α (12) where g (x) = [ ϕ 1 (x) ϕ N (x) 1 x y z ] T R (N+4) 1 α = [ α 1 α N p 0 p 1 p 2 p 3 ] T R N+4 (13) (14) 4. Radial Basis Function Propagation As aforementioned, in the level set-based topology optimization methods, moving the boundary of the shape along a descent gradient direction to find an optimal shape and topology is equivalent to transporting the scalar implicit function Φ(x) by solving the Hamilton-Jacobi equation (5) and thus the optimal propagation of the front is performed by solving the Hamilton-Jacobi PDE [12, 2]. In the present study with RBF implicit representation, it is assumed that the space and time are separable and the time dependence of the implicit function Φ is due to the generalized expansion coefficients α of the RBFs in Eq. (14). Then, the implicit function Eq. (12) becomes time dependent as follows: Φ = Φ (x, t) = g T (x) α(t) (15) Substituting Eq. (15) into the Hamilton-Jacobi equation (5) yields g T dα dt + v n ( φ) T α = 0 (16) To time advance the initial values α, a collocation method is introduced. In the present Eulerian type approach, all the nodes of the fixed mesh are taken as the fixed knots of the RBF interpolation for the implicit function Φ(x). As an extension, Equation (16) is then applied to each of these knots of the RBF

6 6 interpolation, rather than only the points at the front. The normal velocity v n in Eq. (16) is thus extended as v e n to all these knots in the design domain D. This is illustrated in Fig. 1, where each of the grid point is considered as a knot of the RBF. Figure 1: An extension velocity field for the level set method. According to Eq. (6), a natural extension of the normal velocity can be obtained if the strain field is defined over the entire design domain D by assuming ε(u) = 0, u (D\Ω). Since both the strain energy density inside the design domain and the constraint-related Lagrange multiplier are included, this extension velocity is physically meaningful. Using Euler s method, an approximate solution to Eq. (16) can be found. It should be noted that the ODE may be rather stiff, since its counterpart of the Hamilton-Jacobi PDE has a numerical stability condition known as the Courant-Friedrichs-Lewy (CFL) condition [9]. A small time step is advised. After obtaining the approximate solution at each time step, the time-dependent shape and topology can be updated by using Eq. (15). The reader is referred to [14] for details. 5. Parametric Shape and Topology Optimization We are further interested in the shape evolution of implicit surface representation by a parametric method. To this end, we consider the RBF level set function Φ(x(t), α(t)) = 0; (17) as the solution x with a set of shape parameters α. Assuming the shape changes through a velocity field v n (t), we can find the shape evolution in the form of a parameter change directly from Eq. (16) as ( v n = 1 N ) ϕ i α i + α 0 (18) Φ

7 Parametric Shape and Topology Optimization with Radial Basis Functions 7 The resulting equation couples the changes in parameters α(t) with the speed function v n (t) on the surface of the solid. Now, we can substitute Eq. (18) into Eq. (4). The resulting equation couples the change in parameters α with the resulting change in the objective function of the optimization J. Essentially, we have derived the parametric sensitivity of the objective function with respect to the coefficients α of the RBF implicit surface representation: J α = J α i = J α 0 = [ J, J ] T,... (19) α 1 α 2 G 1 Ω Φ ϕ ids, i = 1, 2,... N (20) G 1 ds (21) Φ Ω where G is given in Eq. (6). Therefore, we have transformed the general shape and topology optimization problem into a parameter optimization problem. This would enable us to use algorithms that are based on sensitivity analysis to derive a search strategy in the parameter space. The costly grid discretization in the classical level set method is thus avoided. Therefore, we may call the method presented here parametric shape and topology optimization. A standard search method is to use the parameter sensitivity to define a search direction and a move step, such as the steepest descent method in mathematical programming. More elaborate techniques such as the method of moving asymptote (MMA) are also feasible. 6. Numerical Examples of a Michell Type Structure In this section, numerical examples in two dimensions are presented to illustrate the present RBF-level set method for structural topology optimization. First, we show a Michell type example with the RBF propagation scheme discussed in Section 4. The Michell type structure, as shown in Fig. 2, is loaded with a concentrated vertical force of P at the center of the bottom edge. The basic parameters are assumed to be L = 4, H = 1, thickness t = 1.0, load P = 1, E = 1 for solid material, E = for void material, ν = 0.3, and the fixed Lagrange multiplier l = 10. It is also assumed that the time step size τ = 10 4 and the mesh size Figure 3 shows extension velocities at each knot and the front for the initial and final designs. The final design is quite similar to a truss structure with pin-like connection at some joints. Next, we illustrate the parametric scheme discussed in Section 5. For this scheme, we employ the inverse multi-quadric (IMQ) spline as the basis func-

8 8 L L H P Figure 2: Definition of the minimum compliance design problem. (a) Initial front and extension velocity (b) Final front and extension velocity Figure 3: Front and extension velocity for the Michell type structure. tion, defined as ϕ i (x) = 1/ x x i 2 + c 2 i (22) where c i = The volume ratio is fixed at 0.4. The rest parameters remain the same. For the parametric scheme, we use the method of moving asymptote (MMA) that is papular for the topology optimization problem solutions. The initial design and the corresponding level set function are shown Fig. 4, while the final optimal solution and its corresponding level set function are shown in Fig. 5. (a) Initial shape (b) Initial level set function Figure 4: The initial design and level set function.

9 Parametric Shape and Topology Optimization with Radial Basis Functions 9 (a) Initial shape (b) Initial level set function Figure 5: The final design and level set function. 7. Conclusions In this study, we focus on level set based methods for structural shape and topology optimization. Radial basis functions are used to define a general implicit representation with weight coefficients as parameters. This enables level set algorithms to avoid a costly discrete computational scheme and use a mathematically more convenient parameter search technique instead. The formulation applies to the general problem of shape and topology optimization with the surface of the solid object evolving under a speed function. Numerical examples of 2D structures are given to show the success of the present method. It is suggested that the introduction of the radial basis functions to the conventional level set methods possesses promising potentials in parametric shape and topology optimization. Any serious application of the parametric optimization techniques with RBF implicit surface representation described here requires careful numerical analysis for stability and convergence, and should be demonstrated on a wide variety of situations. In particular, the capability of hole nucleation needs thorough analysis and verification. We plan to develop the depth of analysis required and investigate general 3D applications. Acknowledgements This research work is supported in parts by the Research Grants Council of Hong Kong SAR (Project Nos. CUHK4164/03E and CUHK416205), a Post-doctoral Fellowship from the Chinese University of Hong Kong (No. 04/ENG/1), and the Natural Science Foundation of China (NSFC) (Grants No and No ), which the authors gratefully acknowledge. References

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