Level Set-Based Topological Shape Optimization of Nonlinear Heat Conduction Problems Using Topological Derivatives

Abstract A level set-based topological shape-optimization method is developed to relieve the well-known convergence difficulty in nonlinear heat-conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, the solution of the Hamilton–Jacobi equation leads the initial implicit boundary to an optimal one according to the normal velocity determined from the descent direction of the Lagrangian. Topological derivatives are incorporated into the level set-based framework to improve convergence of the optimization process as well as to avoid the local minimum resulting from the intrinsic nature of the shape-design approach.

[1]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[2]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[3]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[4]  Ole Sigmund,et al.  Topology synthesis of large‐displacement compliant mechanisms , 2001 .

[5]  Jan Sokolowski,et al.  On the Topological Derivative in Shape Optimization , 1999 .

[6]  J. Sethian,et al.  Structural Boundary Design via Level Set and Immersed Interface Methods , 2000 .

[7]  O. Sigmund,et al.  Stiffness design of geometrically nonlinear structures using topology optimization , 2000 .

[8]  J. Cea,et al.  The shape and topological optimizations connection , 2000 .

[9]  T. E. Bruns,et al.  Topology optimization of non-linear elastic structures and compliant mechanisms , 2001 .

[10]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[11]  Seonho Cho,et al.  Design sensitivity analysis and topology optimization of displacement-loaded non-linear structures , 2003 .

[12]  R. Feijóo,et al.  Topological sensitivity analysis , 2003 .

[13]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[14]  Topological shape optimization of geometrically nonlinear structures using level set method , 2004 .

[15]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[16]  M. Burger,et al.  Incorporating topological derivatives into level set methods , 2004 .

[17]  Seonho Cho,et al.  Topological Shape Optimization of Heat Conduction Problems using Level Set Approach , 2005 .

[18]  Seonho Cho,et al.  Topological shape optimization of geometrically nonlinear structures using level set method , 2005 .

[19]  Seung-Hyun Ha,et al.  Topological shape optimization of power flow problems at high frequencies using level set approach , 2006 .