The a posteriori analysis of the discretization error and the modeling error is studied for a compliance cost functional in the context of the optimization of composite elastic materials and a two-scale linearized elasticity model. A mechanically simple, parametrized microscopic supporting structure is chosen and the parameters describing the structure are determined minimizing the compliance objective. An a posteriori error estimate is derived which includes the modeling error caused by the replacement of a nested laminate microstructure by this considerably simpler microstructure. Indeed, nested laminates are known to realize the minimal compliance and provide a benchmark for the quality of the microstructures. To estimate the local difference in the compliance functional the dual weighted residual approach is used. Different numerical experiments show that the resulting adaptive scheme leads to simple parametrized microscopic supporting structures that can compete with the optimal nested laminate construction. The derived a posteriori error indicators allow to verify that the suggested simplified microstructures achieve the optimal value of the compliance up to a few percent. Furthermore, it is shown how discretization error and modeling error can be balanced by choosing an optimal level of grid refinement. Our two scale results with a single scale microstructure can provide guidance towards the design of a producible macroscopic fine scale pattern.
Accepté le :
DOI : 10.1051/m2an/2017004
Mots-clés : Elastic shape optimization, two-scale optimization, nested laminates, homogenization, a posteriori error estimates, adaptive meshes
@article{M2AN_2018__52_4_1457_0,
author = {Conti, Sergio and Geihe, Benedict and Lenz, Martin and Rumpf, Martin},
title = {A posteriori modeling error estimates in the optimization of two-scale elastic composite materials},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1457--1476},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017004},
mrnumber = {3875293},
zbl = {1456.65159},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017004/}
}
TY - JOUR AU - Conti, Sergio AU - Geihe, Benedict AU - Lenz, Martin AU - Rumpf, Martin TI - A posteriori modeling error estimates in the optimization of two-scale elastic composite materials JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1457 EP - 1476 VL - 52 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017004/ DO - 10.1051/m2an/2017004 LA - en ID - M2AN_2018__52_4_1457_0 ER -
%0 Journal Article %A Conti, Sergio %A Geihe, Benedict %A Lenz, Martin %A Rumpf, Martin %T A posteriori modeling error estimates in the optimization of two-scale elastic composite materials %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1457-1476 %V 52 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017004/ %R 10.1051/m2an/2017004 %G en %F M2AN_2018__52_4_1457_0
Conti, Sergio; Geihe, Benedict; Lenz, Martin; Rumpf, Martin. A posteriori modeling error estimates in the optimization of two-scale elastic composite materials. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1457-1476. doi : 10.1051/m2an/2017004. http://archive.numdam.org/articles/10.1051/m2an/2017004/
[1] , , and , Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27–68. | DOI | MR | Zbl
[2] and , Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur. J. Mech. A Solids 12 (1993) 839–878. | MR | Zbl
[3] , Shape optimization by the homogenization method. Vol. 146 of Appl. Math. Sci. Springer Verlag, New York (2002). | DOI | MR | Zbl
[4] and , Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions. Quart. Appl. Math. 51 (1993) 675–699. | DOI | MR | Zbl
[5] , , , , and , On shape optimization with stochastic loadings. In Constrained Optimization and Optimal Control for Partial Differential Equations. edited by , , , , , , and . Vol. 160 of International Series of Numerical Mathematics, Chap. 2. Springer, Basel (2012) 215–243. | DOI | MR
[6] , Optimal bounds and microgeometries for elastic two-phase composites. SIAM J. Appl. Math. 47 (1987) 1216–1228. | DOI | MR | Zbl
[7] and , Shape and topology optimization for periodic problems. I. The shape and the topological derivative. Struct. Multidiscip. Optimiz. 40 (2010) 381–391. | DOI | MR | Zbl
[8] and , Shape and topology optimization for periodic problems. II. Optimization algorithm and numerical examples. Struct. Multidiscip. Optim. 40 (2010) 393–408. | DOI | MR | Zbl
[9] and , Optimization of bodies with locally periodic microstructure. Mech. Adv. Materials Structures 19 (2012) 290–301. | DOI
[10] , and , Weighted marking for goal-oriented adaptive finite element methods. SIAM J. Numer. Anal. 49 (2011) 2451–2469. | DOI | MR | Zbl
[11] , and , Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optimiz. 39 (2000) 113–132. | DOI | MR | Zbl
[12] and , A feed-back approach to error control in finite element methods: Basic analysis and examples. Comput. Mech. 5 (1997) 434–446.
[13] , , , and , An analytical model to predict optimal material properties in the context of optimal structural design. Trans. ASME J. Appl. Mech. 61 (1994) 930–937. | DOI | MR | Zbl
[14] , Optimization of structural topology, shape and material. Springer Verlag, Berlin (1995). | DOI | MR | Zbl
[15] , and , Topology and generalized layout optimization of elastic structures. In Topology design of structures (Sesimbra, 1992). Vol. 227 of NATO Adv. Sci. Inst. Ser. E Appl. Sci. Kluwer Acad. Publ., Dordrecht (1993) 159–205. | MR
[16] and , Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71 (1988) 197–224. | DOI | MR | Zbl
[17] and , A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optimiz. Appl. 44 (2009) 3–25. | DOI | MR | Zbl
[18] and , Homogenization of multiple integrals. Vol. 12 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York (1998). | MR | Zbl
[19] , , and , Advanced numerical methods for pde constrained optimization with application to optimal design in navier stokes flow. In Constrained Optimization and Optimal Control for Partial Differential Equations, edited by , , , , , , and . Vol. 160 of International Series of Numerical Mathematics. Springer Basel (2012) 257–275. | DOI | MR | Zbl
[20] and , Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optimiz. 23 (1991) 17–49. | DOI | MR | Zbl
[21] , Variational methods for structural optimization. In Vol. 140 of Applied Mathematical Sciences. Springer Verlag, New York (2000). | DOI | MR | Zbl
[22] , The finite element method for elliptic problems. North-Holland Publishing Company (1978). | MR | Zbl
[23] and , An Introduction to Homogenization. Oxford University Press, Oxford (1999). | MR | Zbl
[24] , , and , Two-stage stochastic optimization meets two-scale simulation. In Trends in PDE Constrained Optimization, edited by , , , , , , and . Vol. 165 of Inter. Series Numer. Math. Springer International Publishing (2014) 193–211. | DOI | MR
[25] and , Shapes and Geometries: Analysis, Differential Calculus and Optimization. In Vol. 4 of Advances in Design and Control SIAM (2001). | MR | Zbl
[26] , A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. | DOI | MR | Zbl
[27] and , The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. | DOI | MR | Zbl
[28] and , The heterogeneous multi-scale method for homogenization problems. In Multiscale Methods in Science and Engineering. Vol. 44 of Lect. Notes Comput. Sci. Eng. Springer Berlin Heidelberg (2005) 89–110. | MR | Zbl
[29] , and , Heterogeneous multiscale method: A general methodology for multiscale modeling. Phys. Rev. B 67 (2003) 1–4.
[30] and , Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 18 (2005) 121–156. | MR | Zbl
[31] and , Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94 (1986) 307–334. | DOI | MR | Zbl
[32] and , A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems – Series S 9 (2016) 1377–1392. | DOI | MR | Zbl
[33] and , Microstructures of composites of extremal rigidity and exact estimates of the associated energy density. Loffe Physicotechnical Institute (1987) 1115.
[34] and , Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. I. The confocal ellipse construction. J. Mech. Phys. Solids 43 (1995) 933–947. | DOI | MR | Zbl
[35] , The elastic moduli of heterogeneous materials. Trans. ASME Ser. E. J. Appl. Mech. 29 (1962) 143–150. | DOI | MR | Zbl
[36] , , and , Multidisciplinary free material optimization. SIAM J. Appl. Math. 70 (2010) 2709–2728. | DOI | MR | Zbl
[37] and , The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. Numer. Math. 113 (2009) 601–629. | DOI | MR | Zbl
[38] , and , Homogenization of differential operators and integral functionals. Springer Verlag, Berlin (1994). | MR | Zbl
[39] , and , Topology design with optimized, self-adaptive materials. Internat. J. Numer. Methods Engrg. 37 (1994) 1323–1350. | DOI | MR | Zbl
[40] and , Stability of finite element models for distributed-parameter optimization and topology design. Comput. Methods Appl. Mech. Eng. 130 (1996) 203–226. | DOI | MR | Zbl
[41] and , Checkerboard instabilities in topological shape optimization algorithms. In Proc. of the Conference on Inverse Problems, Control and Shape Optimization (PICOF’98) Carthage (1998).
[42] , and , One-shot methods in function space for pde-constrained optimal control problems. Optimiz. Methods Software 29 (2014) 376–405. | DOI | MR | Zbl
[43] , The one-shot method: function space analysis and algorithmic extension by adaptivity. Dissertation, RWTH Aachen (2013).
[44] and , A priori error estimates for finite element discretizations of a shape optimization problem. ESAIM: M2AN 47 (2013) 1733–1763, 11. | DOI | Numdam | MR | Zbl
[45] and , Matrix-Valued L1-Optimal Controls in the Coefficients of Linear Elliptic Problems. Z. Anal. Anwend. 32 (2013) 433–456. | DOI | MR | Zbl
[46] and , Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Arch. Rational Mech. Anal. 102 (1988) 331–350. | DOI | MR | Zbl
[47] , and , Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optimiz. Appl. 55 (2013) 769–802. | DOI | MR | Zbl
[48] , , and , Multiscale approach for optimal design in conductivity of composite materials. SIAM J. Numer. Anal. 53 (2015) 1325–1349. | DOI | MR | Zbl
[49] and , Effective characteristics of composite materials and the optimal design of structural elements. Adv. Mech. 9 (1986) 3–8. | MR | Zbl
[50] , The Theory of Composites. Cambridge University Press (2002). | MR | Zbl
[51] , , and , Adaptive sqp method for shape optimization. In Numerical Mathematics and Advanced Applications 2009, edited by , , and . Springer Berlin Heidelberg (2010) 663–673. | MR | Zbl
[52] , , and , Adaptive finite element method for shape optimization. ESAIM: COCV 18 (2012) 1122–1149, 10 | Numdam | MR | Zbl
[53] and , Calcul des variations et homogénéisation. In Homogenization methods: theory and applications in physics (Bréau-sans-Nappe, 1983), Vol. 57 of Collect. Dir. Etudes Rech. Elec. France. Eyrolles, Paris (1985) 319–369. | MR
[54] and , Adaptive modeling of composite structures: Modeling error estimation. Inter. J. Comput. Civil and Structural Eng. 1 (2000) 1–16.
[55] and , Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms. J. Comput. Phys. 164 (2000) 22–47. | DOI | MR | Zbl
[56] , A posterior error estimates for the heterogenoeous mulitscale finite element method for elliptic homogenization problems. SIAM Multiscale Mod. Simul. 4 (2005) 88–114. | DOI | MR | Zbl
[57] and , A post-treatment of the homogenization method for shape optimization. SIAM J. Control Optim. 47 (2008) 1380–1398. | DOI | MR | Zbl
[58] and , On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Engrg. 176 (1999) 313–331. New advances in computational methods. Cachan (1997) | DOI | MR | Zbl
[59] and , Topology optimal design of thermoelastic structures using a homogenization method. Shape design and optimization. Control Cybernet. 23 (1994) 553–563. | MR | Zbl
[60] , Numerische Näherungsverfahren für elliptische Randwertprobleme: Finite Elemente und Randelemente, edited by , Wiesbaden (2003). | Zbl
[61] and , Layout optimization using the homogenization method. In Optimization of large structural systems, Vol. I, II (Berchtesgaden, 1991). Vol. 231 of NATO Sciences Serie E. Kluwer Acad. Publ., Dordrecht (1993) 157–175. | MR
[62] , Estimations fines des coefficients homogénéisés. In Ennio De Giorgi colloquium (Paris). Vol. 125 of Res. Notes in Math. Pitman, Boston, MA (1985) 168–187. | MR | Zbl
[63] , Modelling error estimation and adaptive modelling of perforated materials. Inter. J. Numer. Methods Engrg. 59 (2004) 1587–1604. | DOI | MR | Zbl
[64] and , Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optimiz. 47 (2008) 509–534. | DOI | MR | Zbl
[65] , Two-dimensional grained composites of extreme rigidity. J. Appl. Mech. 61 (1994) 390–394. | DOI | Zbl
[66] , An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. Ph. D. Thesis, Carnegie Mellon University (2002).
[67] and , On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming. Math. Program. 106 (2006) 25–57. | DOI | MR | Zbl
[68] , Goal-oriented adaptivity for optimization of elliptic systems subject to pointwise inequality constraints: Application to free material optimization. PAMM 10 (2010) 669–672. | DOI
Cité par Sources :
