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 interafces.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-lumping procedure.We revisit Nitsche's method and inspect the effect of this modified variational iv 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 proposed 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.v To Amma (mom) and Nannagaru (dad), vi

High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation

“…Here, an alternative approach is considered by means of the Nitsche approach [29,30]. This approach was extensively used in the context of both Meshless [45,46] and X-FEM [30,31,32,47] recently. In this case a Heaviside enrichment is used across the material interfaces, and the displacement jump is weakly canceled.…”

“…Stress is also put on the optimal convergence of the approach for material interfaces, which was not obtained in [25]. This is why a Heaviside enrichment is considered for material interfaces, together with the use of Nitsche's method [29,30,31,32] to cancel the displacement jump. This contribution is organized as follows: In a first part, the X-FEM is recalled.…”

Essential boundary and interface conditions in the meshless analysis of shells.

“…In [28] Nitsche's method is derived from a Augmented Lagrange standpoint for an idealized di usion problem. In [15], it is used for jump and Dirichlet problems in linear elasticity, with a more precise way to evaluate the ux on the interface and to determine the stabilization parameter in each element.…”

“…If the approximations are similar (same material properties and similar particle spacing), the result obtained for one boundary can be used on the others and the interfaces. A local approach for the eigenvalue problem is available for Finite Elements [15], where the mesh dependent Numerical results show that the method is e cient, with similar error and convergence properties as Lagrange Multipliers and more stable than the Penalty method. Flexibility was demonstrated as curved interfaces, folded shells and discretization with di erent particle density were accurately represented.…”

“…It has been applied with eXtended Finite Elements (XFEM) for EBCs and ICs in single-eld problems [15,25,28], to enforce weak discontinuities between elastic materials [51,52,2,1]; and applied to non conforming embedded FE meshes for elastostatics [53,54]. B-splines and NURBs-base FEs share similar features with meshless approximations and can also take advantage of Nitsche's Method, as in [16] for 2nd and 4th order problems, [46,33,49,50] for elasticity, [27,26] for thin plates and [22,23] for thin shells.…”