A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement.
Mots-clés : structural topology optimization, linear elasticity, phase-field method, first order conditions, matched asymptotic expansions, shape calculus, numerical simulations
@article{COCV_2014__20_4_1025_0,
author = {Blank, Luise and Garcke, Harald and Hassan Farshbaf-Shaker, M. and Styles, Vanessa},
title = {Relating phase field and sharp interface approaches to structural topology optimization},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1025--1058},
publisher = {EDP-Sciences},
volume = {20},
number = {4},
year = {2014},
doi = {10.1051/cocv/2014006},
mrnumber = {3264233},
zbl = {1301.49113},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2014006/}
}
TY - JOUR AU - Blank, Luise AU - Garcke, Harald AU - Hassan Farshbaf-Shaker, M. AU - Styles, Vanessa TI - Relating phase field and sharp interface approaches to structural topology optimization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 1025 EP - 1058 VL - 20 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014006/ DO - 10.1051/cocv/2014006 LA - en ID - COCV_2014__20_4_1025_0 ER -
%0 Journal Article %A Blank, Luise %A Garcke, Harald %A Hassan Farshbaf-Shaker, M. %A Styles, Vanessa %T Relating phase field and sharp interface approaches to structural topology optimization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1025-1058 %V 20 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014006/ %R 10.1051/cocv/2014006 %G en %F COCV_2014__20_4_1025_0
Blank, Luise; Garcke, Harald; Hassan Farshbaf-Shaker, M.; Styles, Vanessa. Relating phase field and sharp interface approaches to structural topology optimization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1025-1058. doi : 10.1051/cocv/2014006. http://www.numdam.org/articles/10.1051/cocv/2014006/
[1] , and , Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22 (2012) 1150013. | MR | Zbl
[2] , Optimization by the Homogenization Method. Springer, Berlin (2002). | MR | Zbl
[3] , and , Structural optimization using sensitivity analysis and a level set method. J. Comput. Phys. 194 (2004) 363-393. | MR | Zbl
[4] , Minimal interface criterion for phase transitions in mixtures of Cahn−Hillard fluids. Ann. Inst. Henri Poincaré 7 (1990) 67-90. | Numdam | MR | Zbl
[5] , and , On sharp interface limits of Allen−Cahn/Cahn−Hilliard variational inequalities. Discrete Contin. Dyn. Syst. Ser. S1 (2008) 1-14. | MR | Zbl
[6] , and , Finite Element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 46 (2004) 738-772. | MR | Zbl
[7] and , Topology Optimization. Springer, Berlin (2003). | MR | Zbl
[8] , , and , Non-local Allen-Cahn systems: analysis and a primal dual active set method. IMA J. Numer. Anal. 33 (2013) 1126-1155. | MR | Zbl
[9] , , , , and , 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. | MR
[10] and , Curvature dependent phase boundary motion and parabolic double obstacle problems. IMA J. Math. Appl. 47 (1993) 19-60. | MR | Zbl
[11] and , Design-dependent loads in topology optimization. ESAIM: COCV 9 (2003) 19-48. | Numdam | MR | Zbl
[12] and , 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.
[13] , and , 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. | MR | Zbl
[14] , and , A three layered minimizer in R2 for a variational problem with a symmetric three-well potential. Commun. Pure Appl. Math. 47 (1996) 677-715. | MR | Zbl
[15] and , 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. | MR | Zbl
[16] and , Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006) 1447-1466. | MR | Zbl
[17] , A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5 (2003) 301-332. | MR | Zbl
[18] , and , Incorporating topological derivatives into level set methods. J. Comput. Phys. 194 (2004) 344-362. | MR | Zbl
[19] and , Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258-267.
[20] , Phase-field models for microstructure evolution. Ann. Rev. Mater. Research 32 (2002) 113-140.
[21] , Mathematical Elasticity, Three Dimensional Elasticity, vol. 1. Elsevier (1988). | MR | Zbl
[22] , UMFPACK Version 5.2.0 User Guide. University of Florida (2007).
[23] , and , Computation of geometric pdes and mean curvature flow. Acta Numerica (2005) 139-232. | MR | Zbl
[24] , , , Isogeometric analysis for topology optimization with a phase field model, Arch. Comput. Methods Eng. 19 (2012) 427-465. | MR
[25] and , A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. SFB256, Preprint 195, University Bonn (1999).
[26] , Dynamics of internal layers and diffusive interfaces. Vol. 53 of CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, Philadelphia (1988). | MR | Zbl
[27] and , Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. EJDE (1995) 1-49. | MR | Zbl
[28] , and , Modeling of phase separation in alloys with coherent elastic misfit. J. Statist. Phys. 95 (1999). | MR | Zbl
[29] , The Γ-limit of the Ginzburg-Landau energy in an elastic medium. AMSA 18 (2008) 345-379. | MR | Zbl
[30] , On Cahn−Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 307-331. | MR | Zbl
[31] , and , On anisotropic order parameter models for multi-phase systems and their sharp interface limits. Phys. D 115 (1998) 87-108. | MR | Zbl
[32] , and , A multi phase field concept: numerical simulations for moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60 (1999) 295-315. | MR | Zbl
[33] and , A singular limit for a system of degenerate Cahn−Hilliard equations. Adv. Differ. Eqs. 5 (2000) 401-434. | MR | Zbl
[34] , , , Stress and diffusion induced interface motion: Modelling and numerical simulations. Eur. J. Appl. Math. 18 (2007) 631-657. | MR | Zbl
[35] and , Second order phase field asymptotics for multicomponent systems. Interfaces Free Boundaries 8 (2006) 131-157. | MR | Zbl
[36] and , An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della normale, Scuola Normale Superiore Pisa (2005). | MR | Zbl
[37] . An introduction to continuum mechanics. Math. Sci. Engrg. 158 (2003). | Zbl
[38] and , 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. | MR | Zbl
[39] and , The effect of self-stress on diffusion in solids. Acta Metall. 30 (1982) 1835-1845.
[40] , The gradient theory of phase transitions and minimal interface criterion. Arch. Rat. Mech. Anal. 98 (1987) 123-142. | MR | Zbl
[41] and , Etudes des problèmes d'optimal design. Lect. Notes Comput. Sci. Springer Verlag, Berlin 41 (1976) 54-62. | Zbl
[42] and , Geometric motion for a degenerate Allen−Cahn/Cahn−Hillard system: The partial wetting case. Physica D 209 (2005) 205-235. | MR | Zbl
[43] , and , Mathematical problems in elasticity and homogenization. In vol. 26 of Studies Math. Appl. (1992) 1-398. | MR | Zbl
[44] and , 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. | MR | Zbl
[45] and , Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | MR | Zbl
[46] , and , Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition. Roc. Roy. Soc. London A 429 (1990) 505-532. | MR | Zbl
[47] , Some convergence results in perimeter-controlled topology optimization. Comput. Meth. Appl. Mech. Eng. 171 (1999) 123-140. | MR | Zbl
[48] , Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York (1984). | MR | Zbl
[49] and , Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48 (1992) 249-264. | MR | Zbl
[50] , and , A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: COCV 18 (2012) 229-258. | Numdam | MR | Zbl
[51] and , Design and adaptive finite element software. The finite element toolbox ALBERTA. In vol. 42 of Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2005). | MR | Zbl
[52] , , Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Multidisc Optim. 16 (1998) 68-75.
[53] , Differentiation with respect to domain boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649-687. | MR | Zbl
[54] and , Introduction to shape optimization: shape sensitivity analysis, vol. 10. Springer Ser. Comput. Math. Springer, Berlin (1992). | MR | Zbl
[55] , and , Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys. 229 (2010) 2697-2718. | MR | Zbl
[56] , Optimal control of partial differential equations: theory, methods and applications, vol. 112. Graduate Studies Math. (2010). | MR | Zbl
[57] , 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. | MR | Zbl
[58] and , Howard's algorithm in a phase-field topology optimization approach. Int. J. Numer. Meth. Eng. 94 (2013) 43-59. | MR
[59] and , Phase field: A variational method for structural topology optimization. Comput. Model. Eng. Sci. 6 (2004) 547-566. | MR | Zbl
[60] and , Multimaterial structural topology optimization with a generalized Cahn−Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89-111. | MR | Zbl
[61] and , 3D multi-material structural topology optimization with the generalized Cahn−Hilliard equations. Comput. Model. Eng. Sci. 16 (2006) 83-102.
[62] , Nonlinear Functional Analysis and its Applications, I: Fixed-point theorems. Springer-Verlag (1986). | MR | Zbl
[63] , Nonlinear Functional Analysis and its Applications, IV. Applications Math. Phys. Springer Verlag (1988). | MR | Zbl
[64] , Nonlinear Functional Analysis and its Applications, II/B. Nonlinear Monotone Operators. Springer Verlag (1990). | MR | Zbl
[65] and , Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49-62. | MR | Zbl
Cité par Sources :
