no. 4
Convergence analysis of a local stationarity scheme for rate-independent systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 4, pp. 1223-1253

This paper is concerned with an approximation scheme for rate-independent systems governed by a non-smooth dissipation and a possibly non-convex energy functional. The scheme is based on the local minimization scheme introduced in Efendiev and Mielke [J. Convex Anal. 13 (2006) 151–167], but relies on local stationarity of the underlying minimization problem. Under the assumption of Mosco-convergence for the dissipation functional, we show that accumulation points exist and are so-called parametrized BV-solutions of the rate-independent system. In particular, this guarantees the existence of parametrized BV-solutions for a rather general setting. Afterwards, we apply the scheme to a model for the evolution of damage.

DOI : 10.1051/m2an/2022034
Classification : 65J08, 65J15, 65M60, 74H15, 74R05
Keywords: Rate independent systems, parametrized BV-solutions, unbounded dissipation, damage evolutions, finite elements, semismooth Newton methods 
@article{M2AN_2022__56_4_1223_0,
     author = {Sievers, Michael},
     title = {Convergence analysis of a local stationarity scheme for rate-independent systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1223--1253},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {56},
     number = {4},
     doi = {10.1051/m2an/2022034},
     mrnumber = {4444528},
     zbl = {1508.65055},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2022034/}
}
TY  - JOUR
AU  - Sievers, Michael
TI  - Convergence analysis of a local stationarity scheme for rate-independent systems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2022
SP  - 1223
EP  - 1253
VL  - 56
IS  - 4
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2022034/
DO  - 10.1051/m2an/2022034
LA  - en
ID  - M2AN_2022__56_4_1223_0
ER  - 
%0 Journal Article
%A Sievers, Michael
%T Convergence analysis of a local stationarity scheme for rate-independent systems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2022
%P 1223-1253
%V 56
%N 4
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2022034/
%R 10.1051/m2an/2022034
%G en
%F M2AN_2022__56_4_1223_0
Sievers, Michael. Convergence analysis of a local stationarity scheme for rate-independent systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 4, pp. 1223-1253. doi: 10.1051/m2an/2022034

[1] J. Alberty, C. Carstensen, S. A. Funken and R. Klose, Matlab implementation of the finite element method in elasticity. Computing 69 (2002) 239–263. | MR | Zbl

[2] S. Almi and S. Belz, Consistent finite-dimensional approximation of phase-field models of fracture. Ann. Mat. Pura Appl. 198 (2019) 1191–1225. | MR | Zbl

[3] H. W. Alt, Linear Functional Analysis: An Application-Oriented Introduction. Springer (2016). | MR | Zbl

[4] H. Attouch, Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program (1984). | MR | Zbl

[5] F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Methods Appl. Sci. 18 (2008) 125–164. | MR | Zbl

[6] S. Bartels, M. Milicevic and M. Thomas, Numerical approach to a model for quasistatic damage with spatial BV-regularization. In: Trends in Applications of Mathematics to Mechanics. Springer (2018) 179–203. | MR | Zbl

[7] G. Bonfanti, A vanishing viscosity approach to a two degree-of-freedom contact problem in linear elasticity with friction. Annali dell’Universita di Ferrara 42 (1996) 127–154. | MR | Zbl

[8] B. J. Dimitrijevic and K. Hackl, A method for gradient enhancement of continuum damage models. Tech. Mech. 28 (2008) 43–52.

[9] M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13 (2006) 151–167. | MR | Zbl

[10] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 2006 (2006) 55–91. | MR | Zbl

[11] A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Part. Differ. Equ. 22 (2005) 129–172. | MR | Zbl

[12] K. Gröger, A W 1 , p -estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679–687. | MR | Zbl

[13] R. Herzog, C. Meyer and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. J. Math. Anal. Appl. 382 (2011) 802–813. | MR | Zbl

[14] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865–888. | MR | Zbl

[15] D. Knees, Convergence analysis of time-discretisation schemes for rate-independent systems. ESAIM: Control Optim. Calc. Var. 25 (2019) 65. | MR | Zbl | Numdam

[16] D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23 (2013) 565–616. | MR | Zbl

[17] D. Knees, R. Rossi and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains. Nonlinear Anal. Real World Appl. 24 (2015) 126–162. | MR | Zbl

[18] A. Mainik, A rate-independent model for phase transformations in shape-memory alloys. Ph.D. thesis, Universität Stuttgart (2004).

[19] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain. J. Nonlin. Sci. 19 (2009) 221–248. | MR | Zbl

[20] J. A. C. Martins, M. D. P. Monteiro Marques and F. Gastaldi, On an example of non-existence of solution to a quasistatic frictional contact problem. Eur. J. Mech. A Solids 13 (1994) 113–133. | MR | Zbl

[21] J. A. C. Martins, F. M. F. Simões, F. Gastaldi and M. D. P. Monteiro Marques, Dissipative graph solutions for a 2 degree-of-freedom quasistatic frictional contact problem. Int. J. Eng. Sci. 33 (1995) 1959–1986. | MR | Zbl

[22] C. Meyer and M. Sievers, Finite element discretization of local minimization schemes for rate-independent evolutions. Calcolo 56 (2019) 1–38. | MR | Zbl

[23] C. Meyer and L. M. Susu, Analysis of a viscous two-field gradient damage model II: Penalization limit. Zeitschrift für Analysis und ihre Anwendungen 38 (2019) 439–474. | MR | Zbl

[24] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application. Springer, New York (2015). | MR

[25] A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (2014) 67–135. | MR | Zbl

[26] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Continuous Dyn. Syst. A 25 (2009) 585–615. | MR | Zbl

[27] A. Mielke, R. Rossi and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. (JEMS) 18 (2016) 2107–2165. | MR | Zbl

[28] M. Negri, Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim. Calc. Var. 20 (2014) 983–1008. | MR | Zbl | Numdam

[29] M. Negri and R. Scala, A quasi-static evolution generated by local energy minimizers for an elastic material with a cohesive interface. Nonlinear Anal. Real World Appl. 38 (2017) 271–305. | MR | Zbl

[30] M. Sievers, A numerical scheme for rate-independent systems – analysis and realization. Ph.D. thesis, Technische Universität Dortmund (2020).

[31] U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47 (2008) 1615–1642. | MR | Zbl

Cité par Sources :