1582 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 3, MARCH 2009 Level Set-Based Topology Optimization for Electromagnetic Systems Hokyung Shim , Vinh Thuy Tran Ho , Semyung Wang ,and Daniel A. Tortorelli Department of Mechatronics, Gwangju Institute of Science and Technology, Gwangju, 500-712, Korea Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaingn, Urbana, IL 61801 USA This research presents a level set-based topology optimization for electromagnetic problems dealing with geometrical shape derivatives and topological design. The shape derivative is computed by an adjoint variable method to avoid numerous sensitivity evaluations. A level set function interpolated into a fixed initial domain is evolved by using the Hamilton–Jacobi equation. The moving free boundaries (dynamic interfaces) represented in the level set model determine the optimal shape via the topological changes. In order to improve efficiency of level set evolution, a radial basis function (RBF) is introduced. The optimization technique is illustrated with 2-D example captured from 3-D level set configuration and the resulting optimum shape is compared to conventional topology optimization. Index Terms—Gradient-based optimization, level set (LS) method, radial basis function, shape derivative, topology optimization. I. INTRODUCTION I N shape optimization, the design variables directly control the exterior and interior boundary shapes of the structures. Shape optimizations allow more explicit representations of any features. However, such a boundary representation often has severe limitations: computational cost due to mesh re-genera- tion, tendency of convergence to local minima, and inconvenient compatibility for complex geometrical problems. Furthermore, the shape optimization [1] can not create new holes, which is aimed at improving the existent designs. However, topology optimization [2] focuses on obtaining an initial conceptual design. It does not require a sophisticated ini- tial design, but rather only requires enough geometric informa- tion to define the boundary conditions. The topology optimiza- tion technique is now sufficiently mature and can be extended to various physical systems [3], [4]. However, the topology opti- mization has typical difficulties, such as gray areas and checker- board patterns [5]. Optimized topologies with those problems are hard to be manufactured in industry. In order to overcome the problems that occur in conventional shape and topology optimization, the level set (LS) method [6], [7] has become an attractive tool for optimization techniques in mechanical structure designs. However, it has yet to be intro- duced in electromagnetic field. In this paper, the LS-based topology optimization is presented for electromagnetic problems and compared with the conven- tional topology optimization based on the solid isotropic mi- crostructure with penalization (SIMP). A mathematical model for electromagnetic system is formulated for a general optimiza- tion problem with one objective function subject to specified constraint. II. CONVENTIONAL LEVEL SET METHOD OF IMPLICITLY MOVING FREE BOUNDARIES Let and denote an isosurface with the moving free boundary and level set function, respectively. A process of Manuscript received October 07, 2008. Current version published February 19, 2009. Corresponding author: S. Wang (e-mail: smwang@gist.ac.kr) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2009.2012748 the optimization is described by letting the level set function dynamically change in time, such that the movable model at an arbitrary time, , is expressed as (1) where is an arbitrary single value, and is a set of points in an -dimensional real space. By differentiating both sides of (1) with respect to the param- eter and applying the chain rule, the Hamilton–Jacobi equa- tion is obtained (2) where is the velocity vector for each position and is often referred to as velocity function of the level set evolution. By introducing a normal velocity, , perpendicular to the boundary, , (2) becomes (3) where , and . Here, it is assumed that there is no movement of the boundary with respect to the tangential velocity, . In the conventional level set method (LSM) [6], the level set function, , evolves at each time step with a normal velocity field by solving (3) of a first-order partial differential equation (PDE). In general, the velocity in the Hamilton–Jacobi equation that advances the level set function is obtained from the sensitivity analysis of the objective function, and the descent direction of the objective is determined by the steepest descent method. A highly robust and accurate computational method, called the “upwind scheme [8],” is employed on the basis of the finite difference approach. III. ADVANCED LEVEL SET METHOD USING RADIAL BASIS FUNCTION Each radial basis function (RBF), , is a radially symmetric function centered at position [9]. The multiquadric spline is used here with the RBFs. It is written as (4) 0018-9464/$25.00 © 2009 IEEE