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Relating phase field and sharp interface approaches tostructural topology optimization

Published online by Cambridge University Press:  05 August 2014

Luise Blank ,
Harald Garcke ,
M. Hassan Farshbaf-Shaker  and
Vanessa Styles
Luise Blank
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany. [email protected]; [email protected]
Harald Garcke
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany. [email protected]; [email protected]
M. Hassan Farshbaf-Shaker
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany; [email protected]
Vanessa Styles
Affiliation:
Department of Mathematics, University of Sussex, Brighton, BN1 9QH, UK; [email protected]
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Abstract

A phase field approach for structural topology optimization which allows for topologychanges and multiple materials is analyzed. First order optimality conditions arerigorously derived and it is shown via formally matched asymptoticexpansions that these conditions converge to classical first order conditions obtained inthe context of shape calculus. We also discuss how to deal with triple junctions wheree.g. two materials and the void meet. Finally, we present severalnumerical results for mean compliance problems and a cost involving the least square errorto a target displacement.

Keywords

Structural topology optimizationlinear elasticityphase-field methodfirst order conditionsmatched asymptotic expansionsshape calculusnumerical simulations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1025 - 1058
Copyright
© EDP Sciences, SMAI, 2014

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References

Abels, H., Garcke, H. and Grün, G., Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22 (2012) 1150013. Google Scholar
G. Allaire, Optimization by the Homogenization Method. Springer, Berlin (2002). Google Scholar
Allaire, G., Jouve, F. and Toader, A.-M., Structural optimization using sensitivity analysis and a level set method. J. Comput. Phys. 194 (2004) 363–393. Google Scholar
Baldo, S., Minimal interface criterion for phase transitions in mixtures of Cahn−Hillard fluids. Ann. Inst. Henri Poincaré 7 (1990) 67–90. Google Scholar
Barrett, J.W., Garcke, H. and Nürnberg, R., On sharp interface limits of Allen−Cahn/Cahn−Hilliard variational inequalities. Discrete Contin. Dyn. Syst. Ser. S1 (2008) 1–14. Google Scholar
Barrett, J.W., Nürnberg, R. and Styles, V., Finite Element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 46 (2004) 738–772. Google Scholar
M.P. Bendsoe and O. Sigmund, Topology Optimization. Springer, Berlin (2003). Google Scholar
Blank, L., Garcke, H., Sarbu, L. and Styles, V., Non-local Allen-Cahn systems: analysis and a primal dual active set method. IMA J. Numer. Anal. 33 (2013) 1126–1155. Google Scholar
L. Blank, H. Garcke, L. Sarbu, T. Srisupattarawanit, V. Styles and A. Voigt, Phase-field approaches to structural topology optimization. Constrained Optim. Opt. Control for Partial Differ. Eqs., edited by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, S. Ulbrich. In vol. 160, Int. Ser. Numer. Math. (2012) 245–255. Google Scholar
Blowey, J.F. and Elliott, C.M., Curvature dependent phase boundary motion and parabolic double obstacle problems. IMA J. Math. Appl. 47 (1993) 19–60. Google Scholar
Bourdin, B. and Chambolle, A., Design-dependent loads in topology optimization. ESAIM: COCV 9 (2003) 19–48. Google Scholar
B. Bourdin and A. Chambolle, The phase-field method in optimal design, in vol. 137 of IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials (2006) 207–215. Google Scholar
Bronsard, L., Garcke, H. and Stoth, B., A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretization for the geometric evolution problem. SIAM J. Appl. Math. 60 (1999) 295–315. Google Scholar
Bronsard, L., Gui, C. and Schatzman, M., A three layered minimizer in R2 for a variational problem with a symmetric three-well potential. Commun. Pure Appl. Math. 47 (1996) 677–715. Google Scholar
Bronsard, L. and Reitich, R., On singular three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. Arch. Rat. Mech. Anal. 124 (1993) 355–379. Google Scholar
Burger, M. and Stainko, R., Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006) 1447–1466. Google Scholar
Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5 (2003) 301–332. Google Scholar
Burger, M., Hackl, B. and Ring, W., Incorporating topological derivatives into level set methods. J. Comput. Phys. 194 (2004) 344–362. Google Scholar
Cahn, J.W. and Hilliard, J.E., Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. Google Scholar
Chen, L.Q., Phase-field models for microstructure evolution. Ann. Rev. Mater. Research 32 (2002) 113–140. Google Scholar
P.G. Ciarlet, Mathematical Elasticity, Three Dimensional Elasticity, vol. 1. Elsevier (1988). Google Scholar
T.A. Davis, UMFPACK Version 5.2.0 User Guide. University of Florida (2007). Google Scholar
K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric pdes and mean curvature flow. Acta Numerica (2005) 139–232. CrossRefGoogle Scholar
Dedè, L., Borden, M.J., Hughes, T.J.R., Isogeometric analysis for topology optimization with a phase field model, Arch. Comput. Methods Eng. 19 (2012) 427–465. Google Scholar
C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. SFB256, Preprint 195, University Bonn (1999). Google Scholar
P.C. Fife, Dynamics of internal layers and diffusive interfaces. Vol. 53 of CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, Philadelphia (1988). Google Scholar
P.C. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. EJDE (1995) 1–49. Google Scholar
Fratzl, P., Penrose, O. and Lebowitz, J.L., Modeling of phase separation in alloys with coherent elastic misfit. J. Statist. Phys. 95 (1999). Google Scholar
Garcke, H., The Γ-limit of the Ginzburg-Landau energy in an elastic medium. AMSA 18 (2008) 345–379. Google Scholar
Garcke, H., On Cahn−Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 307–331. Google Scholar
Garcke, H., Nestler, B. and Stoth, B., On anisotropic order parameter models for multi-phase systems and their sharp interface limits. Phys. D 115 (1998) 87–108. Google Scholar
Garcke, H., Nestler, B. and Stoth, B., A multi phase field concept: numerical simulations for moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60 (1999) 295–315. Google Scholar
Garcke, H. and Novick-Cohen, A., A singular limit for a system of degenerate Cahn−Hilliard equations. Adv. Differ. Eqs. 5 (2000) 401–434. Google Scholar
Garcke, H., Nürnberg, R., Styles, V., Stress and diffusion induced interface motion: Modelling and numerical simulations. Eur. J. Appl. Math. 18 (2007) 631–657. Google Scholar
Garcke, H. and Stinner, B., Second order phase field asymptotics for multicomponent systems. Interfaces Free Boundaries 8 (2006) 131–157. Google Scholar
M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della normale, Scuola Normale Superiore Pisa (2005). Google Scholar
M.E. Gurtin. An introduction to continuum mechanics. Math. Sci. Engrg. 158 (2003). Google Scholar
Hlavacek, I. and Necas, J., On inequalities of Korn’s type, I. Boundary value problems for elliptic systems of partial differential equations. Arch. Rat. Mech. Anal. 36 (1970) 312–334. Google Scholar
Larché, F.C. and Cahn, J.W., The effect of self-stress on diffusion in solids. Acta Metall. 30 (1982) 1835–1845. Google Scholar
Modica, L., The gradient theory of phase transitions and minimal interface criterion. Arch. Rat. Mech. Anal. 98 (1987) 123–142. Google Scholar
Murat, F. and Simon, S., Etudes des problèmes d’optimal design. Lect. Notes Comput. Sci. Springer Verlag, Berlin 41 (1976) 54–62. Google Scholar
Novick-Cohen, A. and Peres Hari, L., Geometric motion for a degenerate Allen−Cahn/Cahn−Hillard system: The partial wetting case. Physica D 209 (2005) 205–235. Google Scholar
O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization. In vol. 26 of Studies Math. Appl. (1992) 1–398. Google Scholar
Osher, S.J. and Santosa, F., Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2011) 272–288. Google Scholar
Osher, S.J. and Sethian, J.A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. Google Scholar
Owen, N., Rubinstein, J. and Sternberg, P., Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition. Roc. Roy. Soc. London A 429 (1990) 505–532. Google Scholar
Petersson, J., Some convergence results in perimeter-controlled topology optimization. Comput. Meth. Appl. Mech. Eng. 171 (1999) 123–140. Google Scholar
O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York (1984). Google Scholar
Rubinstein, J. and Sternberg, P., Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48 (1992) 249–264. Google Scholar
Penzler, P., Rumpf, M. and Wirth, B., A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: COCV 18 (2012) 229–258. Google Scholar
A. Schmidt and K.G. Siebert, Design and adaptive finite element software. The finite element toolbox ALBERTA. In vol. 42 of Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2005). Google Scholar
Sigmund, O., Petersson, J., Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Multidisc Optim. 16 (1998) 68–75. Google Scholar
Simon, J., Differentiation with respect to domain boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649–687. Google Scholar
J. Sokolowski and J.P. Zolesio, Introduction to shape optimization: shape sensitivity analysis, vol. 10. Springer Ser. Comput. Math. Springer, Berlin (1992). Google Scholar
Takezawa, A., Nishiwaki, S. and Kitamura, M., Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys. 229 (2010) 2697–2718. Google Scholar
F. Tröltzsch, Optimal control of partial differential equations: theory, methods and applications, vol. 112. Graduate Studies Math. (2010). Google Scholar
van der Waals, J.D., The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), Vol. 1. Verhaendel. Kronik. Akad. Weten. Amsterdam (1983); Engl. translation by J.S. Rowlinson. J. Stat. Phys. 20 (1979) 197–244. Google Scholar
Wallin, M. and Ristinmaa, M., Howard’s algorithm in a phase-field topology optimization approach. Int. J. Numer. Meth. Eng. 94 (2013) 43–59. Google Scholar
Wang, M.Y. and Zhou, S.W., Phase field: A variational method for structural topology optimization. Comput. Model. Eng. Sci. 6 (2004) 547–566. Google Scholar
Wang, M.Y. and Zhou, S.W., Multimaterial structural topology optimization with a generalized Cahn−Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89-111. Google Scholar
Wang, M.Y. and Zhou, S.W., 3D multi-material structural topology optimization with the generalized Cahn−Hilliard equations. Comput. Model. Eng. Sci. 16 (2006) 83–102. Google Scholar
E. Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-point theorems. Springer-Verlag (1986). Google Scholar
E. Zeidler, Nonlinear Functional Analysis and its Applications, IV. Applications Math. Phys. Springer Verlag (1988). Google Scholar
E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B. Nonlinear Monotone Operators. Springer Verlag (1990). Google Scholar
Zowe, J. and Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49–62. Google Scholar