A non-intrusive mesh partitioning algorithm for conforming Finite Element simulations on fractured domains
Résumé
This article introduces a novel strategy for assembling the finite element matrices of boundary value problems posed in arbitrary 2d and 3d fractured domains with existing third-party solvers, used in a non intrusive manner. For numerical accuracy, the fractured geometry is represented by a conforming mesh; the discontinuous nature of the solution raises the need to create copies of the degrees of freedom of the finite element space according to the topology of the fracture: duplication across regular regions, triplication at Y junctions\dots This task is difficult to carry out in a general way which is compatible with the non intrusive use of an external finite element solver, as ``degenerate'' configurations associated to physically coincident, yet topologically disjoint nodes are often incompatible with standard softwares. To overcome this issue, our method leverages a constrained graph partitioning algorithm that constructs a minimal partition of the fractured geometry, into a set of non overlapping ``regular'' subdomains. Standard finite element kernels can operate independently on each region, and the resulting local contributions are glued into a global stiffness matrix thanks to a suitable label mapping attached to the partition. We demonstrate the robustness of the method through several 2d and 3d numerical examples featuring intricate crack patterns and multi-screen geometries. The proposed framework is implemented in the open-source library PyMedit.
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