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Topology optimization using a reaction-diffusion equation. (English) Zbl 1230.74151
Summary: This paper presents a structural topology optimization method based on a reaction-diffusion equation. In our approach, the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction-diffusion equation. The distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction-diffusion equation, so free generation of new holes is allowed without the use of the topological gradient method. Our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model. The evolution of the density field is based on the implicit finite element method. As numerical examples, compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method’s effectiveness and utility.

MSC:
74P15 Topological methods for optimization problems in solid mechanics
74P05 Compliance or weight optimization in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
76R99 Diffusion and convection
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[1] Murray, J.D., Mathematical biology, (1993), Springer-Verlag Berlin · Zbl 0779.92001
[2] Abraham, E.R., The generation of plankton patchiness by turbulent stirring, Nature, 391, 577-580, (1998)
[3] Peters, N., Turbulent combustion, (2000), Cambridge University Press Cambridge · Zbl 0955.76002
[4] Ross, J.; Müller, S.C.; Vidal, C., Chemical waves, Science, 240, 460-465, (1988)
[5] Tyson, J.J.; Keener, J.P., Singular perturbation theory of traveling waves in excitable media (a review), Physica D: nonlinear phenom., 32, 3, 327-361, (1988) · Zbl 0656.76018
[6] Turing, A.M., The chemical basis of morphogenesis, Philos. trans. R. soc. London B, 237, 641, 37-72, (1952) · Zbl 1403.92034
[7] Fisher, R.A., The wave of advance of advantageous genes, Ann. eugenics, 7, 355-369, (1937) · JFM 63.1111.04
[8] Bendsøe, M.P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. methods appl. mech. engrg., 71, 197-224, (1988) · Zbl 0671.73065
[9] Suzuki, K.; Kikuchi, N., A homogenization method for shape and topology optimization, Comput. methods appl. mech. engrg., 93, 291-318, (1991) · Zbl 0850.73195
[10] Sigmund, O., On the design of compliant mechanisms using topology optimization, Mech. based des. struct. machines, 25, 4, 493-524, (1997)
[11] Nishiwaki, S.; Frecker, M.I.; Min, S.; Kikuchi, N., Topology optimization of compliant mechanisms using the homogenization method, Int. J. numer. methods engrg., 42, 3, 535-559, (1998) · Zbl 0908.73051
[12] Sigmund, O., Materials with prescribed constitutive parameters: an inverse homogenization problem, Int. J. solids struct., 31, 17, 2313-2329, (1994) · Zbl 0946.74557
[13] Silva, E.C.N.; Fonseca, J.S.O.; Kikuchi, N., Optimal design of piezoelectric microstructures, Comput. mech., 19, 5, 397-410, (1997) · Zbl 0889.73053
[14] Choi, J.S.; Yoo, J., Design and application of layered composites with the prescribed magnetic permeability, Int. J. numer. methods engrg., 82, 1-25, (2010) · Zbl 1183.74045
[15] Bendsøe, M.P., Optimal shape design as a material distribution problem, Struct. optim., 1, 193-202, (1989)
[16] Bendsøe, M.P.; Sigmund, O., Topology optimization: theory, methods and applications, (2003), Springer-Verlag Berlin · Zbl 0957.74037
[17] Diaz, A.R.; Sigmund, O., Checkerboard patterns in layout optimization, Struct. optim., 10, 40-45, (1995)
[18] Sigmund, O.; Petersson, J., Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. optim., 16, 68-75, (1998)
[19] Haber, R.B.; Jog, C.S.; Bendsøe, M.P., A new approach to variable-topology shape design using a constraint on perimeter, Struct. optim., 11, 1-12, (1996)
[20] Sethian, J.A.; Wiegmann, A., Structural boundary design via level-set and immersed interface methods, J. comput. phys., 163, 489-528, (2000) · Zbl 0994.74082
[21] Wang, M.Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput. methods appl. mech. engrg., 192, 227-246, (2003) · Zbl 1083.74573
[22] Eschenauer, H.A.; Kobelev, V.V.; Schumacher, A., Bubble method for topology and shape optimization of structures, Struct. optim., 8, 42-51, (1994)
[23] Allaire, G.; Gournay, F.; Jouve, F.; Toader, A.M., Structural optimization using topological and shape sensitivity via a level set method, Control cybern., 34, 1, 59-80, (2005) · Zbl 1167.49324
[24] Wei, P.; Wang, M.Y., Piecewise constant level set method for structural topology optimization, Int. J. numer. methods engrg., 78, 379-402, (2009) · Zbl 1183.74222
[25] Lie, J.; Lysaker, M.; Tai, X.C., A variant of the level set method and applications to image segmentation, Math. comput., 75, 255, 1155-1174, (2006) · Zbl 1096.35034
[26] Yamada, T.; Izui, K.; Nishiwaki, S.; Takezawa, A., A topology optimization method based on the level set method incorporating a fictitious interface energy, Comput. methods appl. mech. engrg., 199, 2876-2891, (2010) · Zbl 1231.74365
[27] Wang, S.L.; Sekerka, R.F.; Wheeler, A.A.; Murray, B.T.; Coriell, S.R.; Braun, R.J.; McFadden, G.B., Thermodynamically-consistent phase-field models for solidification, Physica D, 69, 189-200, (1993) · Zbl 0791.35159
[28] Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Physica D, 63, 410-423, (1993) · Zbl 0797.35175
[29] Jacqmin, D., Calculation of two-phase navier – stokes flows using phase-field modeling, J. comput. phys., 155, 96-127, (1999) · Zbl 0966.76060
[30] Bourdin, B.; Chambolle, A., Design-dependent loads in topology optimization, ESAIM: control, Optim. calculus variations, 9, 19-48, (2003) · Zbl 1066.49029
[31] Zhou, S.; Wang, M.Y., Multimaterial structural topology optimization with a generalized cahn – hilliard model of multiphase transition, Struct. multi. optim., 33, 89-111, (2007) · Zbl 1245.74077
[32] Zhou, S.; Wang, M.Y., 3D multi-material structural topology optimization with the generalized cahn – hilliard equations, Comput. model. engrg. sci., 16, 2, 83-101, (2006)
[33] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 2, 258-267, (1958)
[34] Takezawa, A.; Nishiwaki, S.; Kitamura, M., Shape and topology optimization based on the phase field method and sensitivity analysis, J. comput. phys., 229, 2697-2718, (2010) · Zbl 1185.65109
[35] Allen, S.M.; Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta metallurgica, 27, 1085-1095, (1979)
[36] Belegundu, A.D.; Chandrupatla, T.R., Optimization concepts and applications in engineering, (1999), Prentice-Hall New Jersey · Zbl 0941.90074
[37] Yamasaki, S.; Nomura, T.; Kawamoto, A.; Sato, K.; Izui, K.; Nishiwaki, S., A level set based topology optimization method using the discretized signed distance function as the design variables, Struct. multi. optim., 41, 685-698, (2010) · Zbl 1274.74420
[38] Allaire, G., Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation, (2007), Oxford University Press New York · Zbl 1120.65001
[39] Choi, K.K.; Kim, N.H., Structural sensitivity analysis and optimization 1: linear systems, (2005), Springer Berlin
[40] Bastos, J.P.A.; Sadowski, N., Electromagnetic modeling by finite element methods, (2003), Marcel Dekker Inc. New York
[41] Choi, J.S.; Yoo, J., Structural optimization of ferromagnetic materials based on the magnetic reluctivity for magnetic field problems, Comput. methods appl. mech. engrg., 197, 4193-4206, (2008) · Zbl 1194.74268
[42] Brauer, J.R., Simple equations for the magnetization and reluctivity curves of steel, IEEE trans. magn., 11, 81, (1975)
[43] Choi, J.S.; Yoo, J., Simultaneous structural topology optimization of electromagnetic sources and ferromagnetic materials, Comput. methods appl. mech. engrg., 198, 2111-2121, (2009) · Zbl 1227.78027
[44] Borrvall, T.; Petersson, J., Topology optimization using regularized intermediate density control, Comput. methods appl. mech. engrg., 190, 4911-4928, (2001) · Zbl 1022.74035
[45] Lee, J.; Seo, J.H.; Kikuchi, N., Topology optimization of switched reluctance motors for the desired torque profile, Struct. multi. optim., 42, 783-796, (2010)
[46] Petersson, J., Some convergence results in perimeter-controlled topology optimization, Comput. methods appl. mech. engrg., 171, 123-140, (1999) · Zbl 0947.74050
[47] Warren, J.A.; Kobayashi, R.; Lobkovsky, A.E.; Carter, W.C., Extending phase field models of solidification to polycrystalline materials, Acta materialia, 51, 6035-6058, (2003)
[48] Salon, S.J., Finite element analysis of electrical machines, (1995), Kluwer Academic Publishers Norwell, MA
[49] Michaleris, P.; Tortorelli, D.A.; Vidal, C.A., Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity, Int. J. numer. methods engrg., 37, 2471-2499, (1994) · Zbl 0808.73057
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