Abstract
The minimization of compliance subject to a mass constraint is the topology optimization design problem of interest. The goal is to determine the optimal configuration of material within an allowed volume. Our approach builds upon the well-known density method in which the decision variable is the material density in every cell in a mesh. In it’s most basic form the density method consists of three steps: 1) the problem is convexified by replacing the integer material indicator function with a volume fraction, 2) the problem is regularized by filtering the volume fraction field to impose a minimum length scale; 3) the filtered volume fraction is penalized to steer the material distribution toward binary designs. The filtering step is used to yield a mesh-independent solution and to eliminate checkerboard instabilities. In image processing terms this is a low-pass filter, and a consequence is that the decision variables are not independent and a change of basis could significantly reduce the dimension of the nonlinear programming problem. Based on this observation, we represent the volume fraction field with a truncated Fourier representation. This imposes a minimal length scale on the problem, eliminates checkerboard instabilities, and also reduces the number of decision variables by over 100 × (two dimensions) or 1000 × (three dimensions).
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References
Amrosio L, Buttazaro G (1993) An optimal-design problem with perimeter penalization. Calc Var Partial Differ Equ 1(1):55–69
Bendsoe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsoe MP, Sigmund O (2003) Topology optimization theory, methods, and applications. Springer, Berlin
Bourdin B (2001) Filters in topology optimization. Int J Num Meth Eng 50:2143–2158
Bruns TE, Tortorelli DA (2001) Topology optimization of nonlinear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:26–27
Bruns TE, Tortorelli DA (2003) An element removal and rein- troduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57(10):1413–1430
Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194:344–364
Chen J, Shapiro V (2008) Optimization of continuous heterogenous models. Heterog Objects Model Appl: Lect Notes Comput Sci 4889:193–213
Frigo M, Johnson SG (2005) The design and implementation of fftw3. Proc IEEE 93(2):216–231
Gomes A, Suleman A (2006) Application of spectral level set methodology in topology optimization. Struct Multi Optim 31:430–443
Guest TBJK, Prevost JH (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Meth Eng 61(2):238–254
Guest JK, Smith Genut LC (2010) Reducing dimensionality in topology optimization using adaptive design variable fields. Int J Numer Meth Eng 81(8):1019–1045
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically-a new moving morphable components based framework. J Appl Mech-Trans ASME 81(8):p081009
Haber RB, Jog CS, Bendsoe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1):1–12
Kang Z, Wang Y (2011) Structural topology optimization based on non-local Shepard interpolation of density field. Comp Meth Appl Mech Eng 200(49-52):3515–3525
Kim YY, Yoon GH (2000) Multi-resolution multi-scale topology optimization - a new paradigm. Int J Solids Struct 37(39):5529–5559
Kolev T (2013) MFEM: Modular finite element methods. http:www.mfem.org
Körner TW (1988) Fourier analysis. Cambridge University Press, Cambridge
Lazarov BS, Sigmund O (2011) Filters in topology optimization based on helmholtz-type differential equations. Int J Numer Meth Eng 86:765–781
Luo Z, Zhang N, Wang Y, Gao W (2013) Topology optimization of structures using meshless density variable approximants. Int J Numer Meth Eng 93(4):443–464
Niordson F (1983) Optimal-design of elastic plates with a constraint on the slope of the thickness function. Int J Solids Struct 19(2):141–151
Norato J, Bendsoe M, Tortorelli D (2007) Topological derivative method for topology optimization. Struct Multidiscip Optim 33:375–386
Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comp Meth App Mech Eng 293:306–327
Poulsen T (2002) Topology optimization in wavelet space. Int J Numer Meth Eng 53:567–582
Qian X (2013) Topology optimization in b-spline space. Comp Meth Appl Mech Eng 265:15–35
Rozvany GIN (2001) Aims, scope, methods, history, and unified terminology of computer aided optimization in structural mechanics. Struct Multidiscip Opt 21(2):90–108
Sayood K (2012) Introduction to data compression. Morgan Kaufmann, Burlington
Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163:489–528
Shapiro ABV, Tsukanov I (2004) Heterogeneous material modeling with distance fields. Comp Aided Geom Des 21:215–232
Sidmund O, Petersson J (1998) Numerical instabilities in topology optimization A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16: 68–75
van Dijk N, Maute K, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48:437–472
Wächter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Programm 106(1):5–57
Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246
Wächter A, Biegler LT (2005) Line search filter methods for nonlinear programming Motivation and global convergence. SIAM J Optim 16(1):1–31
Wang F, Boyan L, Sigmund O (2011) On projection methods, convergence and robust forumations in topology optimization. Struct Multidiscip Optim 43:767–784
Wang Y, Kang Z, He Q (2013) An adaptive refinement approach for topology optimization based on separated density field description. Comput Struct 117:10–22
Wang Y, Kang Z, He Q (2014) Adaptive topology optimization with independent error control for separated displacement and density fields. Comput Struct 135:50–61
Zhang S, Norato JA, Gain AL, Lyu N (2016) A geometry projection method for the topology optimization of plate structures. Struct Multidiscip Optim 54(5, SI):1173–1190
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White, D.A., Stowell, M.L. & Tortorelli, D.A. Toplogical optimization of structures using Fourier representations. Struct Multidisc Optim 58, 1205–1220 (2018). https://doi.org/10.1007/s00158-018-1962-y
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DOI: https://doi.org/10.1007/s00158-018-1962-y