Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer. (English) Zbl 1398.74447
Summary: A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping \(\tilde{z}=\sqrt{z}\) in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.
MSC:
| 74S20 | Finite difference methods applied to problems in solid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 74A45 | Theories of fracture and damage |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 65M85 | Fictitious domain methods for initial value and initial-boundary value problems involving PDEs |
| 35R35 | Free boundary problems for PDEs |
| 49M25 | Discrete approximations in optimal control |
Keywords:
free discontinuity; free boundary; crack-tip; Mumford-Shah energy; augmented immersed interface method; fast/Poisson solver; irregular domainSoftware:
IIMPACKReferences:
| [1] | Ambrosio, L., Fusco, N. & Pallara, D, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Zbl0957.49001 MR1857292 · Zbl 0957.49001 |
| [2] | Andersson, J. & Hayk Mikayelyan, The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford-Shah functional in the plain, Preprint, arXiv.org, 2015, pp. 1–13. |
| [3] | Bonacini, M. & Morini, M., Stable regular critical points of the Mumford–Shah functional are local minimizers, Ann. Inst. H. Poincar’e Anal. Non Lin’eaire 32 (2015), 533–570.Zbl1316.49026 MR3353700 · Zbl 1316.49026 |
| [4] | Bonnet, A. & David, G., Cracktip is a global Mumford–Shah minimizer, Ast’erisque (2001), no. 274, vi+259.Zbl1014.49009 MR1864620 · Zbl 1014.49009 |
| [5] | Dal Maso, G. & Toader, R., A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002), 101–135.Zbl1042.74002 MR1897378 · Zbl 1042.74002 |
| [6] | David, G., Singular sets of minimizers for the Mumford-Shah functional, Progress in Mathematics, vol. 233, Birkh”auser Verlag, Basel, 2005.Zbl1086.49030 MR2129693 |
| [7] | De Philippis, G. & Figalli, A., Higher integrability for minimizers of the Mumford-Shah functional, Arch. Ration. Mech. Anal. 213(2014), 491–502.Zbl1316.49044 MR3211857 · Zbl 1316.49044 |
| [8] | Francfort, G. A. & Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46(1998), 1319–1342.Zbl0966.74060 MR1633984 · Zbl 0966.74060 |
| [9] | Griffith, A.-A., The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Ser. A Math. Phys. Eng. Sci. 221(1921), 582–593. |
| [10] | Lemenant, A., A rigidity result for global Mumford–Shah minimizers in dimension three, J. Math. Pures Appl.(9) 103 (2015), 1003–1023.MR3318177 |
| [11] | Li, Z., IIMPACK: A collection of fortran codes for interface problems, Anonymous ftp at ftp.ncsu.edu under the directory:/pub/math/zhilin/Package andhttp://www4.ncsu.edu/ zhilin/IIM, last updated: 2008. |
| [12] | Li, Z. & Ito, K., The immersed interface method, Frontiers in Applied Mathematics, vol. 33, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006, Numerical solutions of PDEs involving interfaces and irregular domains.Zbl1122.65096 MR2242805 |
| [13] | Xia, J. & Li, Z., Effective matrix-free preconditioning for the augmented immersed interface method, preprint, 2014. |
| [14] | Mumford, D. & Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), 577–685.Zbl0691.49036 MR0997568 IntroductionMathematical formulationEuler–Lagrange conditionsAsymptotics at the crack-tipThe =z transformThe numerical methodSet up the problemSet up an initial free boundarySolve the Laplace equation on an irregular domainA preconditioning strategyA new iterative scheme for the free boundary problemA summary of the algorithmNumerical experimentsAccuracy check of the Laplacian solver on irregular domains with both Dirichlet and Neumann boundary conditionsCrack simulations, measurements at (x_*,y_*) and observations |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
