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Optimal topologies derived from a phase-field method. (English) Zbl 1274.74408
Summary: A topology optimization method allowing for perimeter control is presented. The approach is based on a functional that takes the material density and the strain field as arguments. The cost for surfaces is included in the functional that is minimized. Diffuse designs are avoided by introducing a penalty term in the functional that is minimized. Equilibrium and a volume constraint are enforced via a Lagrange multiplier technique. The extremum to the functional is found by use of the Cahn-Hilliard phase-field method. It is shown that the optimization problem is suitable for finite element implementation and the FE-formulation is discussed in detail. In the numerical examples provided, the influence of surface penalization is investigated. It is shown that the perimeter of the structure can be controlled using the proposed scheme.

MSC:
 74P15 Topological methods for optimization problems in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics
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References:
 [1] Allaire G (2002) Shape optimization by the homogenization method. Springer, New-York · Zbl 0990.35001 [2] Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall 27:1085–1095 [3] Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224 · Zbl 0671.73065 [4] Bendsøe M, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654 · Zbl 0957.74037 [5] Bendsøe MP, Sigmund O (2003) Topology optimization. Theory methods and applications. Springer, New-York [6] Blank L, Garcke H, Sarbu L, Srisupattarawanit T, Styles V, Voigt A (2010) Phase-field approaches to structural topology optimization. Preprint Nr.06/2010, Universität Regensburg, Mathematik · Zbl 1356.49044 [7] Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190:4911–4923 · Zbl 1022.74035 [8] Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM: Control, Optimization and Calculus of Variations 9:19–48 · Zbl 1066.49029 [9] Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Engng 190:3443–3459 · Zbl 1014.74057 [10] Bruns T, Sigmund O, Tortorelli D (2002) Numerical methods for the topology optimization of structures that exhibit snap-through. Int J Numer Methods Eng 55:1215–1237 · Zbl 1027.74053 [11] Burger M, Stainko R (2006) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 45:1447–1466 · Zbl 1116.74053 [12] Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28:258–267 [13] Cherkaev A (2000) Variational methods for structural optimization. Springer, New York · Zbl 0956.74001 [14] Christensen P, Klarbring A (2008) An introduction to structural optimization. Springer, New York · Zbl 1180.74001 [15] Ciarlet PG (1978) The finite element method of elliptic problems. Studies in mathematics and its applications, vol 4. North-Holland, Amsterdam [16] Feng X, Wu H (2008) A posteriori error estimates for finite element approximations of the Cahn–Hilliard equation and the Hele–Shaw flow. J. Comput. Math. 26:767–796 · Zbl 1174.65035 [17] Fleury C, Braibant V (1986) A structural optimization: a new dual method using mixed variables. Int J Numer Methods Eng 22:409–428 · Zbl 0585.73152 [18] Haber R, Jog J, Bendsøe M (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11:1–12 [19] Kohn R, Strang G (1986) Optimal design and relaxation of variational problems, part i–ii. Commun Pure Appl Math 39:113–137, 141–182 · Zbl 0609.49008 [20] Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscipl Optim 41:87–106 · Zbl 1274.74392 [21] Petersson J (1999) Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng 171:123–140 · Zbl 0947.74050 [22] Peterson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41:1417–1434 · Zbl 0907.73044 [23] Sigmund O, Peterson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards mesh-dependence and local minima. Struct Optim 16:68–75 [24] Svanberg K (1987) The method of moving asymptotes- a new method for structural optimization. Int J Numer Methods Eng 24:359–373 · Zbl 0602.73091 [25] Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229:2697–2718 · Zbl 1185.65109 [26] Tartar L (2000) An introduction to the homogenization method in optimal design. In: Optimal shape design. Lecture notes in mathematics, Troia, 1998, vol 1740. Springer, Berlin [27] Wang MY, Zhou S (2004) Synthesis of shape and topology of multi-material structures with a phase-field method. J Comput-aided Mater Des 11:117–138 [28] Wang MY, Zhou S (2006) 3D multi-material structural topology optimization with the generalized Cahn-Hilliard equations. Comput Model Eng Sci 16:83–102 [29] Zhou S, Wang MY (2007) Multimaterial structural topology optimization with generalized Cahn–Hilliard model of multiphase transitions. Struct Multidiscipl Optim 33:89–111 · Zbl 1245.74077
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