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A quasi-algebraic multigrid approach to fracture problems based on extended finite elements. (English) Zbl 1390.74181

Summary: The modeling of discontinuities arising from fracture of materials poses a number of significant computational challenges. The extended finite element method provides an attractive alternative to standard finite elements in that they do not require fine spatial resolution in the vicinity of discontinuities nor do they require repeated remeshing to properly address propagation of cracks. They do, however, give rise to linear systems requiring special care within an iterative solver method. An algebraic multigrid method is proposed that is suitable for the linear systems associated with modeling fracture via extended finite elements. The new method follows naturally from an energy minimizing algebraic multigrid framework. The key idea is the modification of the prolongator sparsity pattern to prevent interpolation across cracks. This is accomplished by accessing the standard levelset functions used during the discretization process. Numerical experiments illustrate that the resulting method converges in a fashion that is relatively insensitive to mesh resolution and to the number of cracks or their location.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74R10 Brittle fracture

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