Topology optimization is concerned with the identification of optimal shapes of deformable bodies with respect to given target functionals. The focus of this paper is on a topology optimization problem for a time-evolving elastoplastic medium under kinematic hardening. We adopt a phase-field approach and argue by subsequent approximations, first by discretizing time and then by regularizing the flow rule. Existence of optimal shapes is proved both at the time-discrete and time-continuous level, independently of the regularization. First order optimality conditions are firstly obtained in the regularized time-discrete setting and then proved to pass to the nonregularized time-continuous limit. The phase-field approximation is shown to pass to its sharp-interface limit via an evolutive variational convergence argument.
Keywords: Topology optimization, elastoplasticity, first-order conditions
@article{COCV_2022__28_1_A47_0,
author = {Almi, Stefano and Stefanelli, Ulisse},
title = {Topology optimization for quasistatic elastoplasticity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022037},
mrnumber = {4448808},
zbl = {1495.74053},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022037/}
}
TY - JOUR AU - Almi, Stefano AU - Stefanelli, Ulisse TI - Topology optimization for quasistatic elastoplasticity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022037/ DO - 10.1051/cocv/2022037 LA - en ID - COCV_2022__28_1_A47_0 ER -
%0 Journal Article %A Almi, Stefano %A Stefanelli, Ulisse %T Topology optimization for quasistatic elastoplasticity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022037/ %R 10.1051/cocv/2022037 %G en %F COCV_2022__28_1_A47_0
Almi, Stefano; Stefanelli, Ulisse. Topology optimization for quasistatic elastoplasticity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 47. doi: 10.1051/cocv/2022037
[1] , Shape optimization by the homogenization method. Vol. 146 of Applied Mathematical Sciences. Springer-Verlag, New York (2002). | MR | Zbl | DOI
[2] and , Topology optimization for incremental elastoplasticity: a phase-field approach. SIAM J. Control Optim. 59 (2021) 339–364. | MR | Zbl | DOI
[3] and , Topology optimization. Theory, methods and applications. Springer-Verlag, Berlin (2003). | MR
[4] , , and , Sharp interface limit for a phase field model in structural optimization. SIAM J. Control Optim. 54 (2016) 1558–1584. | MR | Zbl | DOI
[5] , , and , Relating phase field and sharp interface approaches to structural topology optimization. ESAIM: COCV 20 (2014) 1025–1058. | MR | Zbl | Numdam
[6] , , and , Elastoplastic topology optimization and cyclically loaded structures via direct methods for shakedown. Struct. Multidisc. Optim. 64 (2021) 189–217. | MR | DOI
[7] and , Design-dependent loads in topology optimization. ESAIM: COCV 9 (2003) 19–48. | MR | Zbl | Numdam
[8] and , Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006) 1447–1466. | MR | Zbl | DOI
[9] , , , , and , Graded-material design based on phase-field and topology optimization. Comput. Mech. 64 (2019) 1589–1600. | MR | Zbl | DOI
[10] , and , Optimal control of static elastoplasticity in primal formulation. SIAM J. Control Optim. 54 (2016) 3016–3039. | MR | Zbl | DOI
[11] , A -estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679–687. | MR | Zbl | DOI
[12] and , Plasticity, Interdisciplinary Applied Mathematics. Springer, New York (2013). | MR | Zbl | DOI
[13] and , On the existence of optimal shapes in contact problems-perfectly plastic bodies. Comput. Mech. 1 (1986) 293–299. | Zbl | DOI
[14] , and , Shape optimization in contact problems. 1. Design of an elastic body. 2. Design of an elastic perfectly plastic body. Analysis and Optimization of Systems, Springer (1986) 29–39. | Zbl | DOI
[15] , and , Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. J. Math. Anal. Appl. 382 (2011) 802–813. | MR | Zbl | DOI
[16] , and , C-stationarity for optimal control of static plasticity with linear kinematic hardening. SIAM J. Control Optim. 50 (2012) 3052–3082. | MR | Zbl | DOI
[17] , Shape optimization of elastoplastic bodies obeying Hencky's law. Appl. Mater. 31 (1986) 486–499. | MR | Zbl
[18] , Shape optimization of an elastic-perfectly plastic body. Appl. Mater. 32 (1987) 381–400. | MR | Zbl
[19] , Shape optimization of elastoplastic axisymmetric bodies. Appl. Math. 36 (1991) 469–491. | MR | Zbl | DOI
[20] , Optimization of elastic-plastic geometrically non-linear lightweight structures under stiffness and stability constraints. J. Civil Eng. Manag. 10 (2004) 97–106.
[21] and , Optimal plastic design of frames using evolutionary structural optimization. Int. J. Civil Eng. 9 (2011) 175–170.
[22] , Evolution variational inequalities and multidimensional hysteresis operators. Technical Report 432, Weierstrass Institute for Applied Analysis and Stochastics (WIAS) (1998). | MR | Zbl
[23] , and , Elasto-plastic shape optimization using the level set method. SIAM J. Control Optim. 56 (2018) 556–581. | MR | Zbl | DOI
[24] , Evolution in rate-independent systems. In Vol. 2 of Handbook of Differential Equations, Evolutionary Equations. Edited by and . Elsevier (2005) 461–559. | MR | Zbl | DOI
[25] and , Rate-independent systems. Vol. 193 of Applied Mathematical Sciences. Springer, New York (2015). Theory and application. | MR | Zbl
[26] , and , -limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008) 387–416. | MR | Zbl | DOI
[27] and , Un esempio di -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285–299. | MR | Zbl
[28] , Topology optimization of 2D-frame structures with path-dependent response. Internat. J. Numer. Methods Eng. 57 (2003) 1471–1501. | Zbl | DOI
[29] , and , A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: COCV 18 (2012) 229–258. | MR | Zbl | Numdam
[30] , Optimal control for nonconvex rate-independent evolution processes. SIAM J. Control Optim. 47 (2008) 2773–2794. | MR | Zbl | DOI
[31] , Approximation of rate-independent optimal control problems. SIAM J. Numer. Anal. 47 (2009) 3884–3909. | MR | Zbl | DOI
[32] and , Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer Ser. Comput. Math. 16. Springer-Verlag, Berlin (1992). | MR | Zbl
[33] , Optimal control of quasi-static plasticity with linear kinematic hardening, Part I: Existence and discretization in time. SIAM J. Control Optim. 50 (2012) 2836–2861 + loose erratum. | MR | Zbl | DOI
[34] , Optimal control of quasistatic plasticity with linear kinematic hardening II: regularization and differentiability. Z. Anal. Anwend. 34 (2015) 391–418. | MR | Zbl | DOI
[35] , Optimal control of quasistatic plasticity with linear kinematic hardening III: optimality conditions. Z. Anal. Anwend. 35 (2016) 81–118. | MR | Zbl | DOI
Cité par Sources :
