A new algorithm for topology optimization using a level-set method

The level-set method has been recently introduced in the field of shape optimization, enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms. However, most of these algorithms use a Hamilton-Jacobi equation to connect the evolution of the level-set function with the deformation of the contours, and consequently they can hardly create new holes in the domain (at least in 2D). In this work, we propose an evolution equation for the level-set function based on a generalization of the concept of topological gradient. This results in a new algorithm allowing for all kinds of topology changes.

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

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

[3]  Samuel Amstutz,et al.  Sensitivity analysis with respect to a local perturbation of the material property , 2006, Asymptot. Anal..

[4]  Ole Sigmund,et al.  A 99 line topology optimization code written in Matlab , 2001 .

[5]  Ted Belytschko,et al.  Structured extended finite element methods for solids defined by implicit surfaces , 2002 .

[6]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[7]  R. Stainko.  An adaptive multilevel approach to the minimal compliance problem in topology optimization , 2005 .

[8]  Martin Berggren,et al.  Shape optimization of an acoustic horn , 2003 .

[9]  G. Allaire,et al.  Optimal design for minimum weight and compliance in plane stress using extremal microstructures , 1993 .

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

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

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

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

[14]  S. Osher,et al.  Level Set Methods for Optimization Problems Involving Geometry and Constraints I. Frequencies of a T , 2001 .

[15]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[16]  F. Murat,et al.  Sur le controle par un domaine géométrique , 1976 .

[17]  M. Burger.  A framework for the construction of level set methods for shape optimization and reconstruction , 2003 .

[18]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[19]  V. Kobelev,et al.  Bubble method for topology and shape optimization of structures , 1994 .

[20]  Oleg Iliev,et al.  On numerical simulation of flow through oil filters , 2004 .

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

[22]  J. Cea.  Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût , 1986 .

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

[24]  Xiaoming Wang,et al.  Incorporating Topological Derivatives into Level Set Methods for Structural Topology Optimization , 2004 .

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

[27]  Niels Olhoff,et al.  Generalized shape optimization of three-dimensional structures using materials with optimum microstructures , 1998 .

[28]  J OsherStanley,et al.  Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum , 2001 .

[29]  G. Allaire,et al.  A level-set method for vibration and multiple loads structural optimization , 2005 .

[30]  Philippe Guillaume,et al.  The Topological Asymptotic for PDE Systems: The Elasticity Case , 2000, SIAM J. Control. Optim..