The Shifted Interface/Boundary Method for Embedded Domain Computations

Abstract

Numerical computations involving complex geometries have a wide variety of applications in both science and engineering, including the simulation of fractures, melting and solidification, multiphase flows, biofilm growth, etc. Classical finite element methods rely on computational grids that are adapted (fitted) to the geometry, but this approach creates fundamental computational challenges, especially when considering evolving interfaces/boundaries. Embedded methods facilitate the treatment of complex geometries by avoiding fitted grids in favor of immersing the geometry on pre-existing grids.The first part of this thesis work introduces a new embedded finite element interface method, the shifted interface method (SIM), to simulate partial differential equations over domains with internal interfaces. Our approach belongs to the family of surrogate/approximate interface methods and relies on the idea of shifting the location and value of interface jump conditions, by way of Taylor expansions. This choice has the goal of preserving optimal convergence rates while avoiding small cut cells and related problematic issues, typical of traditional embedded methods. In this part, SIM is applied to internal interface computations in the context of the mixed Poisson problem, also known as the Darcy flow problem and is extended to linear isotropic elasticity interface problems. An extensive set of numerical tests is provided to demonstrate the accuracy, robustness and efficiency of the proposed approach.In the second part of the thesis, we propose a new framework for linear fracture mechanics, based on the idea of an approximate fracture geometry representation combined with approximate interface conditions. The approach evolves from SIM, and introduces the concept of an approximate fracture surface composed of the full edges/faces of an underlying grid that are geometrically close to the true fracture geometry. Similar to SIM, the interface conditions on the surrogate fracture are modified with Taylor expansions to achieve a prescribed level of accuracy. This shifted fracture method (SFM) does not require cut cell computations or complex data structures, since the behavior of the true fracture is mimicked with standard integrals on the approximate fracture. Furthermore, the energetics of the true fracture are represented within the prescribed level of accuracy and independently of the grid topology. We present a general computational framework and then apply it in the specific context of cohesive zone models, with an extensive set of numerical experiments in two and three dimensions.In the third and final part, we develop a shifted boundary method (SBM), originated from Main and Scovazzi (2018), for the thermoelasticity equations. SBM requires to shift the location and value of both Dirichlet and Neumann boundary conditions to surrogate boundaries with Taylor expansions. In such a way, an opti- mal convergence rate can be preserved for both temperature and displacement. An extensive of numerical examples in both two and three dimensions are presented in this part to demonstrate the performance of SBM.