Material interpolation schemes, like SIMP, are very popular in topology optimization. They convert the difficult 0-1 problem into a nonlinear programming problem defined over a convex set by involving an interpolation (or penalization) function, usually constructed in rather empirical ways. This paper gives an insight into such methods with the help of the notion of topological sensitivity, and in particular provides some arguments for the choice of the penalization function. A simple algorithm based on these concepts is proposed and illustrated by numerical experiments.
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Allaire G (2002) Shape optimization by the homogenization method. In: Applied mathematical sciences, vol 146. Springer, New York
Allaire G (2007) Conception optimale de structures. In: Mathématiques & applications (Berlin), vol 58. Springer, Berlin (with the collaboration of Marc Schoenauer (INRIA) in the writing of Chapter 8)
Allaire G, de Gournay F, Jouve F, Toader A-M (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34:59–80
MATH
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393
MathSciNetMATHCrossRef
Ammari H, Kang H (2007) Polarization and moment tensors. In: Applied mathematical sciences, vol 162. Springer, New York (with applications to inverse problems and effective medium theory)
Amstutz S (2006a) Sensitivity analysis with respect to a local perturbation of the material property. Asympt Anal 49(1–2):87–108
MathSciNetMATH
Amstutz S (2006b) Topological sensitivity analysis for some nonlinear PDE system. J Math Pures Appl (9) 85(4):540–557
Amstutz S, Andrä H (2006) A new algorithm for topology optimization using a level-set method. J Comput Phys 216(2):573–588
MathSciNetMATHCrossRef
Bendsoe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
CrossRef
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
CrossRef
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654
CrossRef
Bendsøe MP, Sigmund O (2003) Theory, methods and applications. In: Topology optimization. Springer, Berlin
Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1): 344–362
MathSciNetMATHCrossRef
Dieudonné J (1969) Foundations of modern analysis. Academic, New York (enlarged and corrected printing: Pure Appl Math 10-I)
MATH
Eschenauer H, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51
CrossRef
Garreau S, Guillaume P, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39(6):1756–1778 (electronic)
MathSciNetMATHCrossRef
Guzina BB, Bonnet M (2004) Topological derivative for the inverse scattering of elastic waves. Q J Mech Appl Math 57(2):161–179
MathSciNetMATHCrossRef
Hinze M, Pinnau R, Ulbrich M, Ulbrich S (2009) Optimization with PDE constraints. In: Mathematical modelling: theory and applications, vol 23. Springer, New York
Murat F, Simon J (1974) Quelques résultats sur le contrôle par un domaine géométrique. Publ. Laboratoire d’Analyse Numérique, Université Paris 6, pp 1–46
Nazarov SA, Sokołowski J (2003) Asymptotic analysis of shape functionals. J Math Pures Appl (9) 82(2):125–196
Novotny AA, Feijóo RA, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Methods Appl Mech Eng 192(7–8):803–829
MATHCrossRef
Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. In: Applied mathematical sciences, vol 153. Springer, New York
Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1):12–49
MathSciNetMATHCrossRef
Rietz A (2001) Sufficiency of finite exponent in simp (power law) methods. Struct Multidisc Optim 21:159–163
CrossRef
Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidisc Optim 21:120–127
CrossRef
Sokołowski J, Żochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272 (electronic)
MathSciNetMATHCrossRef
Sokołowski J, Zolésio J-P (1992) Shape sensitivity analysis. In: Introduction to shape optimization. Springer series in computational mathematics, vol 16. Springer, Berlin
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Titel
Connections between topological sensitivity analysis and material interpolation schemes in topology optimization
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