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Dynamic and Weighted Stabilizations of the L-scheme Applied to a Phase-Field Model for Fracture Propagation

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Numerical Mathematics and Advanced Applications ENUMATH 2019

Abstract

We consider a phase-field fracture propagation model, which consists of two (nonlinear) coupled partial differential equations. The first equation describes the displacement evolution, and the second is a smoothed indicator variable, describing the crack position. We propose an iterative scheme, the so-called L-scheme, with a dynamic update of the stabilization parameters during the iterations. Our algorithmic improvements are substantiated with two numerical tests. The dynamic adjustments of the stabilization parameters lead to a significant reduction of iteration numbers in comparison to constant stabilization values.

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References

  1. E. L. Allgower and K. Georg. Numerical continuation methods: an introduction. Springer, 1990.

    Book  Google Scholar 

  2. D. Arndt, W. Bangerth, T. C. Clevenger, D. Davydov, M. Fehling, D. Garcia-Sanchez, G. Harper, T. Heister, L. Heltai, M. Kronbichler, R. M. Kynch, M. Maier, J.-P. Pelteret, B. Turcksin, and D. Wells. The deal.II library, version 9.1. J. Numer. Math., 27(4):203–213, 2019.

    Google Scholar 

  3. M. K. Brun, T. Wick, I. Berre, J. M. Nordbotten, and F. A. Radu. An iterative staggered scheme for phase field brittle fracture propagation with stabilizing parameters. Comp. Meth. Appl. Mech. Engrg., 361:112752, 2020.

    Article  MathSciNet  Google Scholar 

  4. T. Gerasimov and L. D. Lorenzis. A line search assisted monolithic approach for phase-field computing of brittle fracture. Comp. Meth. Appl. Mech. Engrg., 312:276–303, 2016.

    Article  MathSciNet  Google Scholar 

  5. F. List and F. A. Radu. A study on iterative methods for solving Richards’ equation. Comput. Geosci., 20(2):341–353, 2016.

    Article  MathSciNet  Google Scholar 

  6. A. Mesgarnejad, B. Bourdin, and M. Khonsari. Validation simulations for the variational approach to fracture. Comp. Meth. Appl. Mech. Engrg., 290:420–437, 2015.

    Article  MathSciNet  Google Scholar 

  7. C. Miehe, F. Welschinger, and M. Hofacker. Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Engrg., 83:1273–1311, 2010.

    Article  MathSciNet  Google Scholar 

  8. I. S. Pop, F. Radu, and P. Knabner. Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math., 168(1–2):365–373, 2004.

    Article  MathSciNet  Google Scholar 

  9. M. Wheeler, T. Wick, and W. Wollner. An augmented-Lagangrian method for the phase-field approach for pressurized fractures. Comp. Meth. Appl. Mech. Engrg., 271:69–85, 2014.

    Article  Google Scholar 

  10. T. Wick. An error-oriented Newton/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation. SIAM J. Sci. Comput., 39(4):B589–B617, 2017.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

TW is supported by the German Research Foundation, Priority Program 1748 (DFG SPP 1748) under the project No. 392587580. CE is supported by the German Research Foundation, via Priority Program 1648 (DFG SPP 1648) under the grant No. EN-1042/2-2 and via EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. ISP is supported by the Research Foundation-Flanders (FWO), Belgium through the Odysseus programme (project G0G1316N).

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Correspondence to Thomas Wick .

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Engwer, C., Pop, I.S., Wick, T. (2021). Dynamic and Weighted Stabilizations of the L-scheme Applied to a Phase-Field Model for Fracture Propagation. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_117

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