Structural Shape Optimization Using Moving Mesh Method

Excerpt from the Proceedings of the COMSOL Users Conference 2007 Grenoble

Structural Shape Optimization Using Moving Mesh Method Zhenyu Liu*, Jan G. Korvink Department of Microsystems Engineering, University of Freiburg, Germany *Corresponding author: Georges-Köhler-Allee, building 103 DG, D-79110 Freiburg, Germany email address: [email protected] Abstract: To implement structural shape optimization an efficient way to represent the shape of the structural boundary and follow its evolution using specified design variables is needed. In this paper, the moving mesh method is used to implement the structural shape optimization. The optimization design variables are the nodal points along the structural boundary. To update the mesh nodal positions, a moving mesh equation with normal velocity boundary conditions is used to calculate the updated nodal position inside the design domain. Remeshing is implemented to avoid mesh distortion if the initial shape of a structure is far away from the optimal one. A stabilization term based on the boundary curvature information is used to regularize the shape sensitivity. Compliance minimization structural optimization examples illustrate the effect of this moving mesh based shape optimization method. All relevant procedures can be implemented using the commercial FEM software Comsol. Keywords: shape optimization, moving mesh, curvature regularization

1. Introduction The essence of mechanism design is to improve the performance for a specific objective function and manufacturability with a certain process. The structural layout design includes the optimization of structural topology and shape [1]. The optimal topology can be obtained using topology optimization without guessing the initial topology. Shape optimization determines the optimal geometry for a given topology of the structure. Compared with shape optimization, topology optimization has the advantages of providing optimal topology and shape simultaneously. However, shape optimization as a separate or post-processing is still necessary after the topology optimization. In [2], topology optimization is implemented using both the nodal density and level set method. In this paper, we review the shape optimization methods which are commonly used for the design of

mechanisms with specified constraints. A moving mesh shape optimization method is then presented to implement structural shape optimization with area constraint. Details of numerical implementation and possible extensions of this method are also discussed.

2. Shape Optimization Method Shape optimal design has been extensively studied since the publication of the first work by Zienkiewiz and Campvbell [3]. To implement shape optimization, an efficient way to represent the shape of the structural boundary, and follow its evolution using specified control variables is needed. There are two approaches which are widely used in engineering optimization. The first is the Lagrangian approach in which a list of points are sequentially numbered along the structural boundary so that the shape of the structure can be expressed using a smoothed interpolation function, and the boundary can be parameterized in a uniform manner. Thus shape optimization can be implemented by moving the boundary points explicitly. The second approach is the Eulerian approach in which the initial design domain is embedded into a large and regular fictitious domain. The shape optimization can then be implemented by evolving a design surface explicitly. The advantages of the Eulerian approach are simplified mesh generation and no mesh tangling or distortion because of the change of structural boundary. However, the design domain includes both the real material subdomain and weak fictitious subdomain. The numerical solution of the physical problem is influenced by the ratio of real and fictitious material properties. On the contrary, the Lagrangian approach evolves the real material boundary, therefore the numerical solution of the physical problem is more reliable than the Eulerian approach. In this paper, we choose the Lagrangian approach to implement structural shape optimization by using the finite element method. When using FEM, the geometrical domain to be


optimized can be discretized by cubic B-spline curves. The positions of some of the spline control points are selected as optimization parameters. Another possibility, which is used in this paper, is to use boundary mesh nodes as design variables. During the optimization procedure, mesh nodal position need to be updated on the structural boundary and inside the design domain. Because the shape sensitivity or nodal moving velocity is merely defined on the boundary, additional velocity extension for nodes inside the design domain has to be defined properly, otherwise the mesh can become over distorted. Remeshing of the computational domain is also an important step if the initial shape of the structure is far away from the optimal one. The moving mesh method is a dynamic mesh adaptation method. The numerical solution of the moving mesh is based on a moving mesh PDE. The mesh topology is kept unchanged but the mesh nodal points are moved throughout the region to best approximate the updated structural boundary. This is essentially a Lagrangian formulation to update the mesh following the change of the computational domain. Currently, the moving mesh method is not as popular as the h-method in structural shape optimization. Nevertheless, it is suited for shape optimization when compared with the h-method. On one hand, the moving mesh method is different from the h-method, which requires complicated data structures if coarsing is also required. On the other hand, the moving mesh method is naturally consistent with the evolution of structural boundaries by smoothly moving the position of mesh nodes. In the authors' opinion, the reason that the moving mesh method is less popular than other shape optimization method is mainly because the numerical implementation of this method is relatively complicated and the numerical solution of the moving mesh equation needs a high-level transient solver in order to keep the stability of the numerical solution. We use Comsol ALE module to directly implement a moving mesh shape optimization in this paper.

3. Fundament of Shape Optimization A shape optimization can be formulated as a minimization problem with respect to the shape

of a domain Ω in Rd. If y(Ω) is the solution of a boundary value problem in Ω (1) Ly (Ω) = 0 and J(Ω,y(Ω)) is objective functional, then we consider the minimization problem J (Ω * , y (Ω * )) = Min J (Ω, y (Ω)) (2) Ω∈U ad

where uad is a set of admissible domains in Rd. The main step of the shape optimization is to obtain a computable descent direction and a sequence of approximate domain so that the objective functional is minimized. In the rest of this section, we briefly discuss some essential parts of the shape optimization problem. 3.1 Objective Functional Shape optimization for a 2D compliant minimization problem can be written as 1 ⎫ Min : J (Ω, ε (Ω)) = ∫ ( ε T Dε ) dΩ ⎪ 2 Ω ⎪ (3) s.t. ∇ ⋅ ( Dε (Ω)) = f ⎬ ⎪ * ∫ 1dΩ = Vol ⎪ Ω ⎭ where the design domain is represented by Ω, the linear elastic equlibrium equation is used to calculate the strain tensor ε, D is the material elastivcity matrix. The area constraint condition is presented which is typically used in compliant minimization design. The optimization problem (3) can be solved by using a numerical optimization method with equality constraint [4]. In this paper, we combine the objective functional and constraints together by appointing the Lagrangian formulation, and then derive a corresponding sensitivity. 3.2 Sensitivity Analysis The Lagrangian objective functional which includes both the optimization objective and constraint can be expressed as J (Ω, ε (Ω), λ , γ ) = (4) 1 T * ∫Ω ( 2 ε Dε )dΩ + λ ( ∫Ω 1dΩ − Vol ) + γ ∫Γ 1ds where λ is the lagrangian multiplier for the area constraint, and γ>0 is a penalty parameter for the length of the structural boundary Γ. Here we will denote with Γ that part of the boundary of Ω which is free to deform, with κ the curvature of v Γ, and with n the outer unit normal of Γ thus Vn


v

= V ⋅ n . From classical calculus, we know that the extrema of optimization objective function are attained at the position where J'=0. To obtain the boundary sensitivity which evolve the shape of a structure, we need to calculate the variation of the objective functional. We refer to [5] for a description of this topic and numerous applied examples. Here we merely give the results of shape differential calculus. The shape derivative of the functional J(Ω) in the direction of the v vector field V is defiend as the limit v 1 (5) dJ (Ω;V ) = lim ( J (Ω t ) − J (Ω)) t →0 t For the functional J (Ω) = ∫ φdx , the shape derivaΩ

tive of J is v v dJ (Ω;V ) = ∫ div (φV )dx = ∫ φVn ds Ω

(6)

Γ

λ = − ∫ΓVds / ∫Γ 1ds

(12)

3.4 Regularization of Sensitivity The numerical solution of the displacement usually presents singularities at the corners of the structural boundaries or at the changes of boundary conditions type. It results in that the sensitivity is not accurate anymore as the stress vector is not well defined. From a numerical point of view, this leads to extreme normal boundary velocities, and may produce unacceptable mesh distortions (figure 1). To circumvent this problem, we arbitrarily set the shape sensitivity to zero on the corner of the shape and the cross point of the changes of boundary conditions (figure 2).

For the functional J ( Γ) = ∫ ϕds , the shape derivaΩ tive of J is v dJ ( Γ;V ) = ∫ (∂V ϕ + κϕ )Vn ds

(7)

Γ

For the functional J (Ω, y (Ω)) = ∫ ∇y (Ω) dx , the Ω 2

shape derivative of J is 2

dJ (Ω; y (Ω)) = ∫ ∇y Vn ds Γ

(8)

Therefore, the shape derivative of the Lagrangian functional (4) is dJ (Ω, ε (Ω), λ , γ ) (9) 1 = ∫ ( ε T Dε + λ + γκ ) ⋅ Vn ds Γ 2 In the case that 1 (10) Vn = −( ε T Dε + λ + γκ ) 2 is defined on the deformable boundary of Ω, the value of the Lagrangian functional is decreased.

Figure 1. The normal sensitivity on the structural boundaries. The black line means movable boundary and blue line means fixed boundary. The positions where two blue lines on the left side have zero displacement boundary conditions, and the position where the blue line on the right-bottom corner has vertical input force.

3.3 Area Constraints The Lagrangian multiplier λ is an area preserving parameter in the optimization problem. During the shape optimization, it can be expressed explicitly. Here, we use the gradient projection method which was proposed in [6]. Supposing that an initial structural shape satisfies the area constraint condition, this constraint condition should be satisfied all the time with the structural boundary moving: d (11) ( (V + λ )ds ) = 0 dt ∫Γ Therefore we have

Figure 2. Regularized normal sensitivity velocity on the structural boundaries.


3.5 Stopping Criterion A typical convergence criterion for stopping the optimization loop would be to check that the shape sensitivity is small enough in some appropriate form. In this paper, two criterions are used to stop the optimization algorithm, one is the square norm of the sensitivity vector and the other is the maximum value of the mesh displacement on the boundary. In order to avoid the influence of the numerical discretization error caused by the distorted mesh, we re-mesh the design domain with high quality mesh and then re-start shape optimization. The whole procedure is stoppoed only if the movement of the boundary nodes is still small enough after mesh regeneration. In the next subsection we will discuss more issues influencing of the mesh quality.

Figure 3. Optimized shape and deformed mesh using moving mesh equation using the initial shape and normal velocity in figure (2). The color bar expresses the quality of finite element mesh.

3.6 Mesh Smoothing and Remeshing One of the difficulties that may arise with the moving mesh method is the irregular or nonsmooth boundaries and low quality of finite element mesh inside the design domain. With the use of the penalty boundary length [1], the smoothness of the structural boundary is regularized by tuning the penalty number. For the distorted mesh inside the design domain, one can use the anisotropic instead of isotropic, moving mesh equation to obtain a so-called body-fitted mesh. Another easy way is to relocate the nodal position using the smoothing method: 1 nd (13) ∑ xi n d i =1 where nd is the number of adjacent nodes associated with the node Xn. The smoothing process is applied to every internal nodes of the mesh while the nodes on the structural boundary are kept fixed (figure 4). If the optimal shape is far away from an initial shape, mesh smoothing may not be enough to keep good quality of the modified mesh. Mesh regeneration provides a way to obtain a high-quality mesh combined with a good boundary re-parameterization (figure 5).

Figure 4. Smoothed finite element mesh based on the mesh in figure 3. The color bar expresses the quality of finite element mesh.

Xn =

4. Numerical Examples

Figure 5. Re-meshed design domain. The color bar expresses the quality of finite element mesh.

In this section, we demonstrate the effect of the moving mesh shape optimization method using 2D compliant minimization examples.


4.1 Comsol PDE Setup

5. Conclusions

In this paper, the Comsol basic module is used to implement the shape optimization. For the short cantilever beam example, we first add the Moving Mesh (ALE) PDE from Comsol Multiphysics → Deformed Mesh. Then in the Frame (ale) we choose plane stress from the Comsol Multiphysics → structural mechanics. To implement the curvature smoothness, two weak boundary PDEs from Comsol Multiphysics → PDE Modes → Weak Form Boundary are added in the Frame (ale).

This paper presents a procedure to implement compliance minimization structure shape optimization using the moving mesh method via Comsol. All the optimization procedures can be implemented using the Comsol GUI if the computational domain does not need to be remeshed. Comsol Script is used when the design domain needs to be remeshed regularly in order to preserve the domain mesh quality. A good mesh quality will improve not only the accuracy of the forward physical problem but also the quality of sensitivity. The method we presented can be extended to the shape optimization problem in which the forward physical problem is controlled by the static PDEs with self-adjoint elliptic operators.

4.2 Cantilever Beam Examples Two cantilever beam examples which have same topology are used to demonstrate the effect of this method. In the first example, only the boundary of hole is movable. In the second example, both the hole and out-frame boundaries are movable. The sensitivity is regularized using the function flc1hs. The optimal shapes with area constraint are shown in figure (6) and figure (7).

Figure 6. Shape optimization of the inside boundary in which the initial shape is shown in figure (1).

6. References 1 M.P. Bendsoe, O. Sigmund, Topology Optimization Theory, Methods and Applications, Springer, (2003) 2 B. Lemke, Z. Liu, J.G. Korvink, Structural topology optimization using Comsol, Comsol conference (2006). 3 O.C. Zienkiewicz, J.S. Campell, Shape optimization and sequential linear programming, Optimum Structureal Design, Wiely, (1973) 4 J. Nocedal, S. J. Wright, Numerical Optimization. Springer (1999) 5 J. Sokolowski, J. Zolesio, Introduction to SOape optimization, Springer_verlag, 1992 6 H.K. Zhao, T. Chan, B. Merriman, S. Osher, A variational level set approach to multiphase motion. J. Comp. Phys. 127, 179-195 (1996) 7 G. Allaire, O. Pantz, Structural optimization with FreeFem++, 32, 173-181 (2006) 8 G. Dogan, P. Morin R.H. Nochetto M. Verani, Discrete gradient flows for shape optimization and applications, Comput. Methods Appl. Mech. Engrg. 196, 3898-3914 (2007) 9 http://www.comsol.com

7. Acknowledgements Figure 7. Shape optimization of both the inside and out frame boundary in which the initial shape is shown in figure (1).

This work is supported by German Research Foundation (DFG) 1883/9-1. The authors would like to thank Comsol support for their valuable help.

Structural Shape Optimization Using Moving Mesh Method

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