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

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Trends in Applications of Mathematics to Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 27))

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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References

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  3. Bartels, S.: Total variation minimization with finite elements: convergence and iterative solution. SIAM J. Numer. Anal. pp. 1162–1180 (2012)

    Article  MathSciNet  Google Scholar 

  4. Bartels, S.: Numerical Methods for Nonlinear Partial Differential Equations. Springer, Heidelberg (2015)

    Book  Google Scholar 

  5. Bartels, S., Milicevic, M.: Alternating direction method of multipliers with variable step sizes (2017). URL https://aam.uni-freiburg.de/agba/prof/preprints/BarMil17-pre.pdf. Cited 13 Mar 2017

  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)

    Article  MathSciNet  Google Scholar 

  7. Bonetti, E., Schimperna, G.: Local existence for Frémond’s model of damage in elastic materials. Contin. Mech. Thermodyn. 16(4), 319–335 (2004). DOI 10.1007/s00161-003-0152-2. URL http://dx.doi.org/10.1007/s00161-003-0152-2

  8. Bonetti, E., Schimperna, G., Segatti, A.: On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differ. Equ. 218(1), 91–116 (2005). DOI 10.1016/j.jde.2005.04.015. URL http://dx.doi.org/10.1016/j.jde.2005.04.015

  9. Bouchitté, G., Mielke, A., Roubíček, T.: A complete-damage problem at small strain. Zeit. Angew. Math. Phys. 60, 205–236 (2009)

    Article  MathSciNet  Google Scholar 

  10. Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Boundaries 9, 411–430 (2007)

    Google Scholar 

  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)

    Google Scholar 

  12. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2008)

    Google Scholar 

  13. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Fiaschi, A., Knees, D., Stefanelli, U.: Young-measure quasi-static damage evolution. IMATI-Preprint 28PV10/26/0 (2010), Pavia (2010)

    Google Scholar 

  16. Fortin, M., Glowinski, R.: Augmented Lagrangian Methods. North-Holland, Amsterdam (1983)

    Google Scholar 

  17. Frémond, M.: Non-Smooth Thermomechanics. Springer, Berlin (2002)

    Book  Google Scholar 

  18. Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33, 1083–1103 (1996)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  20. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  Google Scholar 

  21. Glowinski, R., Le Tallec, P.: Augmented Lagrangians and Operator-Splitting Methods in Nonlinear Mechanics. IAM, Philadelphia (1989)

    Google Scholar 

  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)

    Google Scholar 

  23. Goldstein, T., O’Donoghue, B., Setzer, S., Baraniuk, R.: Fast alternating direction optimization methods. SIAM J. Imaging Sci. 7, 1588–1623 (2014)

    Article  MathSciNet  Google Scholar 

  24. Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)

    Article  MathSciNet  Google Scholar 

  25. Gurtin, M., Francis, E.: Simple rate-independent model for damage. J. Spacecr. Rocket. 18(3), 285–286 (1981)

    Article  Google Scholar 

  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)

    Google Scholar 

  27. Halphen, B., Nguyen, Q.: Sur les matériaux standards généralisés. J. Mécanique 14, 39–63 (1975)

    Google Scholar 

  28. Hanke, H., Knees, D.: A phase-field damage model based on evolving microstructure. Asymptot. Anal. 101(3), 149–180 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  32. Kachanov, L.: On creep rupture time (in russian). Izv. Acad. Nauk. SSSR Otd. Techn. Nauk. (8), 26–31 (1958)

    Google Scholar 

  33. Kachanov, L.: Introduction to Continuum Damage Mechanics, 2nd edn. Mechanics of Elastic Stability. Kluwer Academic Publishers, Dordrecht (1990)

    Google Scholar 

  34. Knees, D., Negri, M.: Convergence of alternate minimization schemes for phase-field fracture and damage. Accepted in M3AS (2015). URL http:// cvgmt. sns. it/ media/ doc/ paper/ 2832/ KneesNegri.pdf

  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)

    Article  MathSciNet  Google Scholar 

  36. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)

    Article  MathSciNet  Google Scholar 

  37. Marigo, J., Maurini, C., Pham, K.: An overview of the modelling of fracture by gradient damage models. Meccanica 51(12), 3107–3128 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  41. Mielke, A., Roubíček, T.: Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16(2), 177–209 (2006)

    Article  MathSciNet  Google Scholar 

  42. Mielke, A., Roubíček, T.: Rate-independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015)

    Book  Google Scholar 

  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 1285

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  45. Persson, P.O., Strang, G.: A simple mesh generator in matlab. SIAM Rev. 42, 329–345 (2004)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  47. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877–898 (1976)

    Article  MathSciNet  Google Scholar 

  48. Roubíček, T., Thomas, M., Panagiotopoulos, C.: Stress-driven local-solution approach to quasistatic brittle delamination. Nonlinear Anal. Real World Appl. 22, 645–663 (2015). DOI 10.1016/j.nonrwa.2014.09.011. URL http://dx.doi.org/10.1016/j.nonrwa.2014.09.011

  49. Roubíček, T., Panagiotopoulos, C.G., Mantič, V.: Local-solution approach to quasistatic rate-independent mixed-mode delamination (2015)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  51. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  Google Scholar 

  52. Schlüter, A., Willenbücher, A., Kuhn, C., Müller, R.: Phase field approximation of dynamic brittle fracture. Comput. Mech. 54, 1141–1161 (2014)

    Article  MathSciNet  Google Scholar 

  53. Thomas, M.: Quasistatic damage evolution with spatial BV -regularization. Discrete Contin. Dyn. Syst. Ser. S 6, 235–255 (2013)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was carried out within the project Finite element approximation of functions of bounded variation and application to models of damage, fracture, and plasticity within the DFG Priority Programme SPP 1748 “Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.”

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Authors and Affiliations

  1. Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Freiburg i. Br., Germany

    Sören Bartels & Marijo Milicevic

  2. Weierstrass-Institute for Applied Analysis and Stochastics, Berlin, Germany

    Marita Thomas

Authors
  1. Sören Bartels
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  2. Marijo Milicevic
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  3. Marita Thomas
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Correspondence to Marita Thomas .

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Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Augusto Visintin

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Bartels, S., Milicevic, M., Thomas, M. (2018). Numerical Approach to a Model for Quasistatic Damage with Spatial BV -Regularization. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_9

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