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2016, 9(5): 1377-1392. doi: 10.3934/dcdss.2016055

A posteriori error estimates for sequential laminates in shape optimization

1. 

Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universitat Bonn, Endenicher Allee 60, 53111 Bonn, Germany, Germany

Received  December 2014 Revised  July 2015 Published  October 2016

A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density.
Citation: Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055
References:
[1]

G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method,, Numerische Mathematik, 76 (1997), 27. doi: 10.1007/s002110050253.

[2]

G. Allaire and R. V. Kohn, Optimal design for minimum weight and compliance in plane stress using extremal microstructures,, European J. Mech. A Solids, 12 (1993), 839.

[3]

G. Allaire and R. V. Kohn, Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions,, Quart. Appl. Math., 51 (1993), 675.

[4]

G. Allaire, Shape Optimization by the Homogenization Method,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-1-4684-9286-6.

[5]

M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites,, SIAM J. Appl. Math., 47 (1987), 1216. doi: 10.1137/0147082.

[6]

G. Buttazzo and G. D. Maso, Shape optimization for dirichlet problems: Relaxed formulation and optimality conditions,, Applied Mathematics and Optimization, 23 (1991), 17. doi: 10.1007/BF01442391.

[7]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,, Claredon Press, (1998).

[8]

M. P. Bendsøe, Optimization of Structural Topology, Shape, and Material,, Springer-Verlag, (1995). doi: 10.1007/978-3-662-03115-5.

[9]

R. Becker, E. Estecahandy and D. Trujillo, Weighted marking for goal-oriented adaptive finite element methods,, SIAM Journal on Numerical Analysis, 49 (2011), 2451. doi: 10.1137/100794298.

[10]

R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept,, SIAM J. Control Optim., 39 (2000), 113. doi: 10.1137/S0363012999351097.

[11]

C. Brandenburg, F. Lindemann, M. Ulbrich and S. Ulbrich, 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, (2012), 257. doi: 10.1007/978-3-0348-0133-1_14.

[12]

R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples,, Computational Mechanics, 5 (1997), 434.

[13]

C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems. I. The shape and the topological derivative,, Struct. Multidiscip. Optim., 40 (2010), 381. doi: 10.1007/s00158-009-0378-0.

[14]

C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems. II. Optimization algorithm and numerical examples,, Struct. Multidiscip. Optim., 40 (2010), 393. doi: 10.1007/s00158-009-0377-1.

[15]

C Barbarosie and A.-M. Toader, Optimization of bodies with locally periodic microstructure,, Mechanics of Advanced Materials and Structures, 19 (2012), 290. doi: 10.1080/15376494.2011.642939.

[16]

O. Benedix and B. Vexler, A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints,, Computational Optimization and Applications, 44 (2009), 3. doi: 10.1007/s10589-008-9200-y.

[17]

D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford University Press, (1999).

[18]

S. Conti, B. Geihe, M. Rumpf and R. Schultz, Two-stage stochastic optimization meets two-scale simulation,, in Trends in PDE Constrained Optimization. Part II, (2014), 193. doi: 10.1007/978-3-319-05083-6_13.

[19]

Ph. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland Publishing Company, (1978).

[20]

W. Dörfler, A convergent adaptive algorithm for poisson's equation,, SIAM J. Numer. Anal., 33 (1996), 1106. doi: 10.1137/0733054.

[21]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. doi: 10.4310/CMS.2003.v1.n1.a8.

[22]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems,, in Multiscale Methods in Science and Engineering, (2005), 89. doi: 10.1007/3-540-26444-2_4.

[23]

W. E and B. Engquist and Z. Huang, Heterogeneous multiscale method: A general methodology for multiscale modeling,, Physical Review B, 67 (2003), 1. doi: 10.4310/CMS.2003.v1.n1.a8.

[24]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems,, J. Amer. Math. Soc., 18 (2005), 121. doi: 10.1090/S0894-0347-04-00469-2.

[25]

G. A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity,, Arch. Rational Mech. Anal., 94 (1986), 307. doi: 10.1007/BF00280908.

[26]

L. V. Gibiansky and A. V. Cherkaev, Microstructures of composites of extremal rigidity and exact estimates of the associated energy density,, Ioffe Physicotechnical Institute, 1115 (1987).

[27]

Z. Hashin, The elastic moduli of heterogeneous materials,, Trans. ASME Ser. E. J. Appl. Mech., 29 (1962), 143. doi: 10.1115/1.3636446.

[28]

J. Haslinger, M. Kočvara, G. Leugering and M. Stingl, Multidisciplinary free material optimization,, SIAM Journal on Applied Mathematics, 70 (2010), 2709. doi: 10.1137/090774446.

[29]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numerische Mathematik, 113 (2009), 601. doi: 10.1007/s00211-009-0244-4.

[30]

Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials,, J. Mech. Phys. Solids, 11 (1963), 127. doi: 10.1016/0022-5096(63)90060-7.

[31]

A. J. Hoffman, H. W. Wielandt, The variation of the spectrum of a normal matrix,, Duke Math. J., 20 (1953), 37. doi: 10.1215/S0012-7094-53-02004-3.

[32]

F. Jouve and E. Bonnetier, Checkerboard instabilities in topological shape optimization algorithms,, Proceedings of the Conference on Inverse Problems, (1998).

[33]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). doi: 10.1007/978-3-642-84659-5.

[34]

L. Kaland, The One-Shot Method: Function Space Analysis and Algorithmic Extension by Adaptivity,, RWTH Aachen, (2013).

[35]

L. Kaland, J. C. De Los Reyes and N. R. Gauger, One-shot methods in function space for PDE-constrained optimal control problems,, Optimization Methods and Software, 29 (2014), 376. doi: 10.1080/10556788.2013.774397.

[36]

R. V. Kohn and R. Lipton, Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials,, Arch. Rational Mech. Anal., 102 (1988), 331. doi: 10.1007/BF00251534.

[37]

P. Kogut and G. Leugering, Matrix-valued $L^1$-optimal controls in the coefficients of linear elliptic problems,, Z. Anal. Anwend., 32 (2013), 433. doi: 10.4171/ZAA/1493.

[38]

B. Kiniger and B. Vexler, A priori error estimates for finite element discretizations of a shape optimization problem,, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1733. doi: 10.1051/m2an/2013086.

[39]

K. A. Lurie and A. V. Cherkaev, Effective characteristics of composite materials and the optimal design of structural elements,, Adv. in Mech., 9 (1986), 3.

[40]

D. Leykekhman, D. Meidner and B. Vexler, Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints,, Computational Optimization and Applications, 55 (2013), 769. doi: 10.1007/s10589-013-9537-8.

[41]

P. Morin, R. H. Nochetto, M. S. Pauletti and M. Verani, Adaptive SQP method for shape optimization,, in Numerical Mathematics and Advanced Applications 2009, (2009), 663. doi: 10.1007/978-3-642-11795-4_71.

[42]

P. Morin, R. H. Nochetto, M. S. Pauletti, M. Verani, Adaptive finite element method for shape optimization,, ESAIM: Control, 18 (2012), 1122. doi: 10.1051/cocv/2011192.

[43]

F. Murat and L. Tartar, Calcul des variations et homogénéisation,, in Homogenization Methods: Theory and Applications in Physics (Bréau-sans-Nappe, (1983), 319.

[44]

M. Ohlberger, A posterior error estimates for the heterogenoeous mulitscale finite element method for elliptic homogenization problems,, SIAM Multiscale Mod. Simul., 4 (2005), 88. doi: 10.1137/040605229.

[45]

J. T. Oden and K. Vemaganti, Adaptive Modeling of Composite Structures: Modeling Error Estimation,, International Journal for Computational Civil and Structural Engineering, 1 (2000), 1.

[46]

J. T. Oden, K. S. Vemaganti, 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. doi: 10.1006/jcph.2000.6585.

[47]

S. Prudhomme and J. T. Oden, On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors,, New advances in computational methods (Cachan, 176 (1999), 313. doi: 10.1016/S0045-7825(98)00343-0.

[48]

L. Tartar, Estimations fines des coefficients homogénéisés,, in Ennio De Giorgi colloquium (Paris, (1983), 168.

[49]

K. Vemaganti, Modelling error estimation and adaptive modelling of perforated materials,, Internat. J. Numer. Methods Engrg., 59 (2004), 1587. doi: 10.1002/nme.929.

[50]

B. Vexler and W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints,, SIAM Journal on Control and Optimization, 47 (2008), 509. doi: 10.1137/070683416.

[51]

W. Wollner, Goal-oriented adaptivity for optimization of elliptic systems subject to Pointwise inequality constraints: Application to free material optimization,, PAMM, 10 (2010), 669. doi: 10.1002/pamm.201010325.

show all references

References:
[1]

G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method,, Numerische Mathematik, 76 (1997), 27. doi: 10.1007/s002110050253.

[2]

G. Allaire and R. V. Kohn, Optimal design for minimum weight and compliance in plane stress using extremal microstructures,, European J. Mech. A Solids, 12 (1993), 839.

[3]

G. Allaire and R. V. Kohn, Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions,, Quart. Appl. Math., 51 (1993), 675.

[4]

G. Allaire, Shape Optimization by the Homogenization Method,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-1-4684-9286-6.

[5]

M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites,, SIAM J. Appl. Math., 47 (1987), 1216. doi: 10.1137/0147082.

[6]

G. Buttazzo and G. D. Maso, Shape optimization for dirichlet problems: Relaxed formulation and optimality conditions,, Applied Mathematics and Optimization, 23 (1991), 17. doi: 10.1007/BF01442391.

[7]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,, Claredon Press, (1998).

[8]

M. P. Bendsøe, Optimization of Structural Topology, Shape, and Material,, Springer-Verlag, (1995). doi: 10.1007/978-3-662-03115-5.

[9]

R. Becker, E. Estecahandy and D. Trujillo, Weighted marking for goal-oriented adaptive finite element methods,, SIAM Journal on Numerical Analysis, 49 (2011), 2451. doi: 10.1137/100794298.

[10]

R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept,, SIAM J. Control Optim., 39 (2000), 113. doi: 10.1137/S0363012999351097.

[11]

C. Brandenburg, F. Lindemann, M. Ulbrich and S. Ulbrich, 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, (2012), 257. doi: 10.1007/978-3-0348-0133-1_14.

[12]

R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples,, Computational Mechanics, 5 (1997), 434.

[13]

C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems. I. The shape and the topological derivative,, Struct. Multidiscip. Optim., 40 (2010), 381. doi: 10.1007/s00158-009-0378-0.

[14]

C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems. II. Optimization algorithm and numerical examples,, Struct. Multidiscip. Optim., 40 (2010), 393. doi: 10.1007/s00158-009-0377-1.

[15]

C Barbarosie and A.-M. Toader, Optimization of bodies with locally periodic microstructure,, Mechanics of Advanced Materials and Structures, 19 (2012), 290. doi: 10.1080/15376494.2011.642939.

[16]

O. Benedix and B. Vexler, A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints,, Computational Optimization and Applications, 44 (2009), 3. doi: 10.1007/s10589-008-9200-y.

[17]

D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford University Press, (1999).

[18]

S. Conti, B. Geihe, M. Rumpf and R. Schultz, Two-stage stochastic optimization meets two-scale simulation,, in Trends in PDE Constrained Optimization. Part II, (2014), 193. doi: 10.1007/978-3-319-05083-6_13.

[19]

Ph. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North-Holland Publishing Company, (1978).

[20]

W. Dörfler, A convergent adaptive algorithm for poisson's equation,, SIAM J. Numer. Anal., 33 (1996), 1106. doi: 10.1137/0733054.

[21]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. doi: 10.4310/CMS.2003.v1.n1.a8.

[22]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems,, in Multiscale Methods in Science and Engineering, (2005), 89. doi: 10.1007/3-540-26444-2_4.

[23]

W. E and B. Engquist and Z. Huang, Heterogeneous multiscale method: A general methodology for multiscale modeling,, Physical Review B, 67 (2003), 1. doi: 10.4310/CMS.2003.v1.n1.a8.

[24]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems,, J. Amer. Math. Soc., 18 (2005), 121. doi: 10.1090/S0894-0347-04-00469-2.

[25]

G. A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity,, Arch. Rational Mech. Anal., 94 (1986), 307. doi: 10.1007/BF00280908.

[26]

L. V. Gibiansky and A. V. Cherkaev, Microstructures of composites of extremal rigidity and exact estimates of the associated energy density,, Ioffe Physicotechnical Institute, 1115 (1987).

[27]

Z. Hashin, The elastic moduli of heterogeneous materials,, Trans. ASME Ser. E. J. Appl. Mech., 29 (1962), 143. doi: 10.1115/1.3636446.

[28]

J. Haslinger, M. Kočvara, G. Leugering and M. Stingl, Multidisciplinary free material optimization,, SIAM Journal on Applied Mathematics, 70 (2010), 2709. doi: 10.1137/090774446.

[29]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numerische Mathematik, 113 (2009), 601. doi: 10.1007/s00211-009-0244-4.

[30]

Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials,, J. Mech. Phys. Solids, 11 (1963), 127. doi: 10.1016/0022-5096(63)90060-7.

[31]

A. J. Hoffman, H. W. Wielandt, The variation of the spectrum of a normal matrix,, Duke Math. J., 20 (1953), 37. doi: 10.1215/S0012-7094-53-02004-3.

[32]

F. Jouve and E. Bonnetier, Checkerboard instabilities in topological shape optimization algorithms,, Proceedings of the Conference on Inverse Problems, (1998).

[33]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). doi: 10.1007/978-3-642-84659-5.

[34]

L. Kaland, The One-Shot Method: Function Space Analysis and Algorithmic Extension by Adaptivity,, RWTH Aachen, (2013).

[35]

L. Kaland, J. C. De Los Reyes and N. R. Gauger, One-shot methods in function space for PDE-constrained optimal control problems,, Optimization Methods and Software, 29 (2014), 376. doi: 10.1080/10556788.2013.774397.

[36]

R. V. Kohn and R. Lipton, Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials,, Arch. Rational Mech. Anal., 102 (1988), 331. doi: 10.1007/BF00251534.

[37]

P. Kogut and G. Leugering, Matrix-valued $L^1$-optimal controls in the coefficients of linear elliptic problems,, Z. Anal. Anwend., 32 (2013), 433. doi: 10.4171/ZAA/1493.

[38]

B. Kiniger and B. Vexler, A priori error estimates for finite element discretizations of a shape optimization problem,, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1733. doi: 10.1051/m2an/2013086.

[39]

K. A. Lurie and A. V. Cherkaev, Effective characteristics of composite materials and the optimal design of structural elements,, Adv. in Mech., 9 (1986), 3.

[40]

D. Leykekhman, D. Meidner and B. Vexler, Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints,, Computational Optimization and Applications, 55 (2013), 769. doi: 10.1007/s10589-013-9537-8.

[41]

P. Morin, R. H. Nochetto, M. S. Pauletti and M. Verani, Adaptive SQP method for shape optimization,, in Numerical Mathematics and Advanced Applications 2009, (2009), 663. doi: 10.1007/978-3-642-11795-4_71.

[42]

P. Morin, R. H. Nochetto, M. S. Pauletti, M. Verani, Adaptive finite element method for shape optimization,, ESAIM: Control, 18 (2012), 1122. doi: 10.1051/cocv/2011192.

[43]

F. Murat and L. Tartar, Calcul des variations et homogénéisation,, in Homogenization Methods: Theory and Applications in Physics (Bréau-sans-Nappe, (1983), 319.

[44]

M. Ohlberger, A posterior error estimates for the heterogenoeous mulitscale finite element method for elliptic homogenization problems,, SIAM Multiscale Mod. Simul., 4 (2005), 88. doi: 10.1137/040605229.

[45]

J. T. Oden and K. Vemaganti, Adaptive Modeling of Composite Structures: Modeling Error Estimation,, International Journal for Computational Civil and Structural Engineering, 1 (2000), 1.

[46]

J. T. Oden, K. S. Vemaganti, 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. doi: 10.1006/jcph.2000.6585.

[47]

S. Prudhomme and J. T. Oden, On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors,, New advances in computational methods (Cachan, 176 (1999), 313. doi: 10.1016/S0045-7825(98)00343-0.

[48]

L. Tartar, Estimations fines des coefficients homogénéisés,, in Ennio De Giorgi colloquium (Paris, (1983), 168.

[49]

K. Vemaganti, Modelling error estimation and adaptive modelling of perforated materials,, Internat. J. Numer. Methods Engrg., 59 (2004), 1587. doi: 10.1002/nme.929.

[50]

B. Vexler and W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints,, SIAM Journal on Control and Optimization, 47 (2008), 509. doi: 10.1137/070683416.

[51]

W. Wollner, Goal-oriented adaptivity for optimization of elliptic systems subject to Pointwise inequality constraints: Application to free material optimization,, PAMM, 10 (2010), 669. doi: 10.1002/pamm.201010325.

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