An efficient finite element method for embedded interface problems

Abstract

We focus on developing a computationally efficient finite element method for interface problems. Finite element methods are severely constrained in their ability to resolve interfaces. Many of these limitations stem from their inability in independently representing interface geometry from the underlying discretization. We propose an approach that facilitates such an independent representation by embedding interfaces in the underlying finite element mesh. This embedding, however, raises stability concerns for existing algorithms used to enforce interfacial kinematic constraints. To address these stability concerns, we develop robust methods to enforce interfacial kinematics over embedded interfaces. We begin by examining embedded Dirichlet problems – a simpler class of embedded constraints. We develop both stable methods, based on Lagrange multipliers,and stabilized methods, based on Nitsche’s approach, for enforcing Dirichlet constraints over three-dimensional embedded surfaces and compare and contrast their performance. We then extend these methods to enforce perfectly-tied kinematics for elastodynamics with explicit time integration. In particular, we examine the coupled aspects of spatial and temporal stability for Nitsche’s approach.We address the incompatibility of Nitsche’s method for explicit time integration by (a) proposing a modified weighted stress variational form, and (b) proposing a novel mass-lumpingprocedure.We revisit Nitsche’s method and inspect the effect of this modified variational form on the interfacial quantities of interest. We establish that the performance of this method, with respect to recovery of interfacial quantities, is governed significantly by the choice for the various method parameters viz.stabilization and weighting. We establish a relationship between these parameters and propose an optimal choice for the weighting. We further extend this approach to handle non-linear,frictional sliding constraints at the interface. The naturally non-symmetric nature of these problems motivates us to omit the symmetry term arising in Nitsche’s method.We contrast the performance of the proposed approach with the more commonly used penalty method. Through several numerical examples, we show that with the pro-posed choice of weighting and stabilization parameters, Nitsche’s method achieves the right balance between accurate constraint enforcement and flux recovery - a balance hard to achieve with existing methods. Finally, we extend the proposed approach to intersecting interfaces and conduct numerical studies on problems with junctions and complex topologies.