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Erschienen in:
Relaxation of p-Growth Integral Functionals Under Space-Dependent Differential Constraints Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

Numerical Approach to a Model for Quasistatic Damage with Spatial BV -Regularization

verfasst von : Sören Bartels, Marijo Milicevic, Marita Thomas

Erschienen in: Trends in Applications of Mathematics to Mechanics

Verlag: Springer International Publishing

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Abstract

We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV -regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove stability of the method and show that a discrete version of a semistable energetic formulation of the rate-independent system holds. Moreover, we present our numerical results for two benchmark problems.

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Vorheriges Kapitel Three Examples Concerning the Interaction of Dry Friction and Oscillations
Nächstes Kapitel Rigidity Effects for Antiferromagnetic Thin Films: A Prototypical Example
Literatur
1.
Alberty, J., Carstensen, C., Funken, S.A., Klose, R.: Matlab implementation of the finite element method in elasticity. Computing 69, 239–263 (2002)MathSciNetCrossRef
2.
Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57 (2016)MathSciNetCrossRef
3.
Bartels, S.: Total variation minimization with finite elements: convergence and iterative solution. SIAM J. Numer. Anal. pp. 1162–1180 (2012)MathSciNetCrossRef
4.
Bartels, S.: Numerical Methods for Nonlinear Partial Differential Equations. Springer, Heidelberg (2015)CrossRef
5.
6.
Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18, 2419–2434 (2009)MathSciNetCrossRef
7.
8.
9.
Bouchitté, G., Mielke, A., Roubíček, T.: A complete-damage problem at small strain. Zeit. Angew. Math. Phys. 60, 205–236 (2009)MathSciNetCrossRef
10.
Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Boundaries 9, 411–430 (2007)
11.
Braides, A., Cassano, B., Garroni, A., Sarrocco, D.: Quasi-static damage evolution and homogenization: a paradigmatic case of non-commutability. Ann. Inst. H. Poincaré Anal. Non Linéaire, online (2014)
12.
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2008)
13.
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)
14.
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)MathSciNetCrossRef
15.
Fiaschi, A., Knees, D., Stefanelli, U.: Young-measure quasi-static damage evolution. IMATI-Preprint 28PV10/26/0 (2010), Pavia (2010)
16.
Fortin, M., Glowinski, R.: Augmented Lagrangian Methods. North-Holland, Amsterdam (1983)
17.
18.
Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33, 1083–1103 (1996)MathSciNetCrossRef
19.
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comp. Maths. Appls. 2, 17–40 (1976)CrossRef
20.
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)CrossRef
21.
Glowinski, R., Le Tallec, P.: Augmented Lagrangians and Operator-Splitting Methods in Nonlinear Mechanics. IAM, Philadelphia (1989)
22.
Glowinski, R., Marroco, A.: Sur l’approximation par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de dirichlet non linéaires. Revue française d’automatique, informatique, recherche opérationelle. Analyse numérique 9, 41–76 (1975)
23.
Goldstein, T., O’Donoghue, B., Setzer, S., Baraniuk, R.: Fast alternating direction optimization methods. SIAM J. Imaging Sci. 7, 1588–1623 (2014)MathSciNetCrossRef
24.
Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)MathSciNetCrossRef
25.
Gurtin, M., Francis, E.: Simple rate-independent model for damage. J. Spacecr. Rocket. 18(3), 285–286 (1981)CrossRef
26.
Hackl, K., Stumpf, H.: Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-static brittle fracture. Int. J. Solids Struct. 30, 1567–1584 (2003)
27.
Halphen, B., Nguyen, Q.: Sur les matériaux standards généralisés. J. Mécanique 14, 39–63 (1975)
28.
Hanke, H., Knees, D.: A phase-field damage model based on evolving microstructure. Asymptot. Anal. 101(3), 149–180 (2017)MathSciNetCrossRef
29.
Heinemann, C., Kraus, C.: Existence of weak solutions for Cahn–Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 321–359 (2011)
30.
Heinemann, C., Kraus, C.: Complete damage in linear elastic materials – modeling, weak formulation and existence results. Calc. Var. Partial Differ. Equ. 217–250 (2015)MathSciNetCrossRef
31.
Hintermüller, M., Rautenberg, C.N., Hahn, J.: Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction. Inverse Prob. 30(5), 055,014 (2014)MathSciNetCrossRef
32.
Kachanov, L.: On creep rupture time (in russian). Izv. Acad. Nauk. SSSR Otd. Techn. Nauk. (8), 26–31 (1958)
33.
Kachanov, L.: Introduction to Continuum Damage Mechanics, 2nd edn. Mechanics of Elastic Stability. Kluwer Academic Publishers, Dordrecht (1990)
34.
35.
Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23(04), 565–616 (2013)MathSciNetCrossRef
36.
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRef
37.
Marigo, J., Maurini, C., Pham, K.: An overview of the modelling of fracture by gradient damage models. Meccanica 51(12), 3107–3128 (2016)MathSciNetCrossRef
38.
Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199(45), 2765–2778 (2010)MathSciNetCrossRef
39.
Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int. J. Numer. Meth. Eng. 83, 1273–1311 (2010)MathSciNetCrossRef
40.
Mielke, A.: Evolution in rate-independent systems (Ch.6). In: Dafermos, C., Feireisl, E. (eds.) Handbook of Differential Equations, Evolutionary Equations, vol. 2, pp. 461–559. Elsevier B.V., Amsterdam (2005)CrossRef
41.
Mielke, A., Roubíček, T.: Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16(2), 177–209 (2006)MathSciNetCrossRef
42.
Mielke, A., Roubíček, T.: Rate-independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015)CrossRef
43.
Mielke, A., Roubíček, T., Zeman, J.: Complete damage in elastic and viscoelastic media and its energetics. Comput. Methods Appl. Mech. Eng. 199, 1242–1253 (2010). Submitted. WIAS preprint 1285MathSciNetCrossRef
44.
Mielke, A., Roubíček, T., Zeman, J.: Complete damage in elastic and viscoelastic media and its energetics. Comput. Methods Appl. Mech. Eng. 199, 1242–1253 (2010)MathSciNetCrossRef
45.
Persson, P.O., Strang, G.: A simple mesh generator in matlab. SIAM Rev. 42, 329–345 (2004)
46.
Rocca, E., Rossi, R.: “entropic” solutions to a thermodynamically consistent pde system for phase transitions and damage. SIAM J. Math. Anal. 74, 2519–2586 (2015)MathSciNetCrossRef
47.
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877–898 (1976)MathSciNetCrossRef
48.
49.
Roubíček, T., Panagiotopoulos, C.G., Mantič, V.: Local-solution approach to quasistatic rate-independent mixed-mode delamination (2015)MathSciNetCrossRef
50.
Roubíček, T., Valdman, J.: Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation. SIAM J. Appl. Math. 76, 314–340 (2016)MathSciNetCrossRef
51.
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MathSciNetCrossRef
52.
Schlüter, A., Willenbücher, A., Kuhn, C., Müller, R.: Phase field approximation of dynamic brittle fracture. Comput. Mech. 54, 1141–1161 (2014)MathSciNetCrossRef
53.
Thomas, M.: Quasistatic damage evolution with spatial BV -regularization. Discrete Contin. Dyn. Syst. Ser. S 6, 235–255 (2013)
54.
Thomas, M., Bonetti, E., Rocca, E., Rossi, R.: A rate-independent gradient system in damage coupled with plasticity via structured strains. In: Düring, B., Schönlieb, C.B., Wolfram, M. (eds.) Gradient Flows: From Theory to Application, vol. 54, pp. 54–69. EDP Sciences (2016)
55.
Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain: existence and regularity results. Zeit. Angew. Math. Mech. 90(2), 88–112 (2010)MathSciNetCrossRef
56.
Weinberg, K., Dally, T., Schuß, S., Werner, M., Bilgen, C.: Modeling and numerical simulation of crack growth and damage with a phase field approach. GAMM-Mitt. 39(1), 55–77 (2016)MathSciNetCrossRef
57.
Wu, C., Tai, X.C.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and higher order models. SIAM J. Imaging Sci. 3, 300–339 (2010)MathSciNetCrossRef
Metadaten
Titel
Numerical Approach to a Model for Quasistatic Damage with Spatial BV -Regularization
verfasst von
Sören Bartels
Marijo Milicevic
Marita Thomas
Copyright-Jahr
2018
Buch
Trends in Applications of Mathematics to Mechanics

Print ISBN: 978-3-319-75939-5

Electronic ISBN: 978-3-319-75940-1

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-75940-1_9