Topological constraints in 2D structural topology optimization

One of the straightforward definitions of structural topology optimization is to design the optimal distribution of the holes and the detailed shape of each hole implicitly in a fixed discretized design domain. However, typical numerical instability phenomena of topology optimization, such as the checkerboard pattern and mesh dependence, all take the form of an unexpected number of holes in the optimal result in standard density-type design methods, such as SIMP and ESO. Typically, the number of holes is indirectly controlled by tuning the value of the radius of the filter operator during the optimization procedure, in which the choice of the value of the filter radius is one of the most opaque and confusing issues for a beginner unfamiliar with the structural topology optimization algorithm. Based on the soft-kill bi-directional evolutionary structural optimization (BESO) method, an optimization model is proposed in this paper in which the allowed maximal number of holes in the designed structure is explicitly specified as an additional design constraint. The digital Gauss-Bonnet formula is used to count the number of holes in the whole structure in each optimization iteration. A hole-filling method (HFM) is also proposed in this paper to control the existence of holes in the optimal structure. Several 2D numerical examples illustrate that the proposed method cannot only limit the maximum number of holes in the optimal structure throughout the whole optimization procedure but also mitigate the phenomena of the checkerboard pattern and mesh dependence. The proposed method is expected to provide designers with a new way to tangibly manage the optimization procedure and achieve better control of the topological characteristics of the optimal results.

[1]  Yongwu Rong,et al.  Digital topological method for computing genus and the Betti numbers , 2010 .

[2]  H. P. Mlejnek,et al.  Some aspects of the genesis of structures , 1992 .

[3]  Ole Sigmund,et al.  On projection methods, convergence and robust formulations in topology optimization , 2011, Structural and Multidisciplinary Optimization.

[4]  Jan G. Korvink,et al.  Self-consistent adjoint analysis for topology optimization of electromagnetic waves , 2018, J. Comput. Phys..

[5]  Bruno Dehez,et al.  Topology Optimization Method Based on the Maxwell Stress Tensor for the Design of Ferromagnetic Parts in Electromagnetic Actuators , 2011, IEEE Transactions on Magnetics.

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

[7]  Shinji Wakao,et al.  Topology Optimization Based on Regularized Level-Set Function for Solving 3-D Nonlinear Magnetic Field System With Spatial Symmetric Condition , 2016, IEEE Transactions on Magnetics.

[8]  Y. Xie,et al.  A simple evolutionary procedure for structural optimization , 1993 .

[9]  J. Petersson,et al.  Topology optimization of fluids in Stokes flow , 2003 .

[10]  Y. Xie,et al.  Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials , 2009 .

[11]  O. Sigmund.  Morphology-based black and white filters for topology optimization , 2007 .

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

[13]  Ping Zhang,et al.  Topology optimization of unsteady incompressible Navier-Stokes flows , 2011, J. Comput. Phys..

[14]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[15]  George I. N. Rozvany,et al.  A critical review of established methods of structural topology optimization , 2009 .

[16]  James K. Guest,et al.  Achieving minimum length scale in topology optimization using nodal design variables and projection functions , 2004 .

[17]  B. Bourdin.  Filters in topology optimization , 2001 .

[18]  Sang-Hoon Lee,et al.  Level set based robust shape and topology optimization under random field uncertainties , 2010 .

[19]  O. Sigmund,et al.  Minimum length scale in topology optimization by geometric constraints , 2015 .

[20]  O. Sigmund,et al.  Topology optimization of channel flow problems , 2005 .

[21]  Yi Min Xie,et al.  Evolutionary Structural Optimization , 1997 .

[22]  Takayuki Yamada,et al.  A topology optimization method based on the level set method incorporating a fictitious interface energy , 2010 .

[23]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[24]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[25]  O. Sigmund,et al.  Filters in topology optimization based on Helmholtz‐type differential equations , 2011 .

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

[27]  K. Saitou,et al.  Multi-material topology optimization using ordered SIMP interpolation , 2016, Structural and Multidisciplinary Optimization.

[28]  Ole Sigmund,et al.  Length scale and manufacturability in density-based topology optimization , 2016, Archive of Applied Mechanics.

[29]  Ole Sigmund,et al.  Design of multiphysics actuators using topology optimization - Part I: One-material structures , 2001 .

[30]  Anders Clausen,et al.  Efficient topology optimization in MATLAB using 88 lines of code , 2011 .

[31]  Weihong Zhang,et al.  Note on spatial gradient operators and gradient-based minimum length constraints in SIMP topology optimization , 2019, Structural and Multidisciplinary Optimization.

[32]  G. Paulino,et al.  PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab , 2012 .

[33]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[34]  G. Rozvany.  Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics , 2001 .

[35]  Shing-Tung Yau,et al.  Computational Conformal Geometry , 2016 .

[36]  K. Maute,et al.  Topology optimization of fluid domains: kinetic theory approach , 2008 .

[37]  Ole Sigmund,et al.  Manufacturing tolerant topology optimization , 2009 .