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Structural and Multidisciplinary Optimization

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01.10.2012 | Brief Note

Sensitivity filtering from a continuum mechanics perspective

verfasst von: Ole Sigmund, Kurt Maute

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 4/2012

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Abstract

In topology optimization filtering is a popular approach for preventing numerical instabilities. This short note shows that the well-known sensitivity filtering technique, that prevents checkerboards and ensures mesh-independent designs in density-based topology optimization, is equivalent to minimizing compliance for nonlocal elasticity problems known from continuum mechanics. Hence, the note resolves the long-standing quest for finding an explanation and physical motivation for the sensitivity filter.

Nächster Artikel New analytical benchmarks for topology optimization and their implications. Part I: bi-symmetric trusses with two point loads between supports

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Aage N, Jørgensen M, Andreasen C, Sigmund O (2012, submitted) Interactive topology optimization on hand-held devices

Adalsteinsson D, Sethian J (1995) A fast level set method for propagating interfaces. J Comput Phys 118(2):269–277. doi:10.1006/jcph.1995.1098 MathSciNetMATHCrossRef

Aifantis E (2003) Update on a class of gradient theories. Mech Mater 35(3–6):259–280. doi:10.1016/S0167-6636(02)00278-8 CrossRef

Allaire G (2007) Conception optimale de structures. Springer, Berlin, Heidelberg. doi:10.1007/978-3-540-36856-4

Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetMATHCrossRef

Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var 1:55–69MathSciNetMATHCrossRef

Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidisc Optim 43:1–16. doi:10.1007/s00158-010-0594-7. MATLAB code available online at www.topopt.dtu.dk CrossRef

Askes H, Morata I, Aifantis E (2008) Finite element analysis with staggered gradient elasticity. Comput Struct 86(11–12):1266–1279. doi:10.1016/j.compstruc.2007.11.002 CrossRef

Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRef

Bendsøe MP, Sigmund O (2004) Topology optimization - theory, methods and applications. Springer Verlag, Berlin, Heidelberg

Boardman A (1982) Electromagnetic surface modes. Wiley, Chichester

Borrvall T (2001) Topology optimization of elastic continua using restriction. Arch Comput Methods Eng 8(4):351–385MathSciNetMATHCrossRef

Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190:4911–4928MathSciNetMATHCrossRef

Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetMATHCrossRef

Brooks A, Hughes T (1982) Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations. Comput Methods Appl Mech Eng 32(1–3):199–259MathSciNetMATHCrossRef

Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459MATHCrossRef

Bukshtynov V, Volkov O, Protas B (2011) On optimal reconstruction of constitutive relations. Phys D Nonlinear Phenom 240(16):1228–1244. doi:10.1016/j.physd.2011.04.006 MathSciNetMATHCrossRef

Charpiat G, Maurel P, Pons JP, Keriven R, Faugeras O (2007) Generalized gradients: Priors on minimization flows. Int J Comput Vis 73(3):325–344. doi:10.1007/s11263-006-9966-2 CrossRef

De Gournay F (2006) Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J Control Optim 45(1):343–367. doi:10.1137/050624108 MathSciNetMATHCrossRef

Díaz AR, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45CrossRef

Eringen A (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703. doi:10.1063/1.332803 CrossRef

Eringen A, Edelen D (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248. doi:10.1016/0020-7225(72)90039-0 MathSciNetMATHCrossRef

Guest J, Prevost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254MathSciNetMATHCrossRef

Gutkin M (2000) Nanoscopics of dislocations and disclinations in gradient elasticity. Rev Adv Mater Sci 1(1):27–60MathSciNet

Gutkin M, Aifantis E (1999) Dislocations in the theory of gradient elasticity. Scr Mater 40(5):559–566. doi:10.1016/S1359-6462(98)00424-2 CrossRef

Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on the perimeter. Struct Optim 11(1):1–11CrossRef

Huiskes R (2000) If bone is the answer, then what is the question? J Anat 197:145–156CrossRef

Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3–4):203–226MathSciNetMATHCrossRef

Lazarov B, Sigmund O (2011) Filters in topology optimization as a solution to Helmholtz type differential equation. Int J Numer Methods Eng 86(6):765–781. doi:10.1002/nme.3072 MathSciNetMATHCrossRef

Leblond JB, Perrin G, Deveaux J (1994) Bifurcation effects in ductile metals with damage delocalization. J Appl Mech 61:236–242MATHCrossRef

Mullender MG, Huiskes R, Weinans H (1994) A physiological approach to the simulation of bone remodelling as a self-organizational controll process. J Biomech 11:1389–1394CrossRef

Peerlings R, Geers M, de Borst R, Brekelmans W (2001) A critical comparison of nonlocal and gradient-enhanced softening continua. Int J Solids Struct 38(44–45):7723–7746. doi:10.1016/S0020-7683(01)00087-7 MATHCrossRef

Petersson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41(8):1417–1434MathSciNetMATHCrossRef

Reddy J, Pang S (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103(2):023,511. doi:10.1063/1.2833431 CrossRef

Ru C, Aifantis E (1993) A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech 101:59–68. doi:10.1007/BF01175597 MathSciNetMATHCrossRef

Schevenels M, Lazarov B, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200(49–52):3613–3627. doi:10.1016/j.cma.2011.08.006 MATHCrossRef

Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mechan Struct Mach 25(4):493–524CrossRef

Sigmund O (2001) A 99 line topology optimization code written in MATLAB. Struct Multidisc Optim 21:120–127. doi:10.1007/s001580050176. MATLAB code available online at www.topopt.dtu.dk CrossRef

Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4–5):401–424CrossRef

Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sin 25(2):227–239. doi:10.1007/s10409-009-0240-z CrossRef

Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75CrossRef

Tcherniak D, Sigmund O (2001) A web-based topology optimization program. Struct Multidisc Optim 22(3):179–187CrossRef

Tezduyar T, MS, Ray S, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput Methods Appl Mech Eng 95:221–242MATHCrossRef

Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414. doi:10.1007/BF00253945 MathSciNetMATHCrossRef

Tvergaard V, Needleman A (1995) Effects of nonlocal damage in porous plastic solids. Int J Solids Struct 32(8–9):1063–1077. doi:10.1016/0020-7683(94)00185-Y MATHCrossRef

Wang F, Lazarov B, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43(6):767–784. doi:10.1007/s00158-010-0602-y CrossRef

Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside funtions. Struct Multidisc Optim 41:495–505MathSciNetCrossRef

Zhou M, Rozvany GIN (1991) The COC algorithm, part II: Topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336CrossRef

Titel: Sensitivity filtering from a continuum mechanics perspective
verfasst von: Ole Sigmund
Kurt Maute
Publikationsdatum: 01.10.2012
Verlag: Springer-Verlag
Erschienen in: Structural and Multidisciplinary Optimization / Ausgabe 4/2012
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-012-0814-4

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