Absorbing Boundary Conditions for Scalar and Elastic Waves in the Time-Domain

Absorbing boundary conditions are a requisite element of many computational wave prop-agation problems. With our main motivation being the anchor loss simulations of Micro-electromechanical Systems (MEMS) in three dimensions, efficient time-domain absorbingboundary conditions which do work well for elastodynamics are in demand. In this workwe investigate three classes of absorbing boundary conditions which we believe are promis-ing, viz., perfectly matched layers (PMLs), perfectly matched discrete layers (PMDLs), andhigh-order absorbing boundary conditions (HOABCs). We first devise a PML formulation onspherical domains which is particularly suited for the simulation of a large class of MEMS-resonator systems. What distinguishes our original PML formulation from most existingPML formulations is that it works with standard numerical solvers such as discontinuousGalerkin methods on unstructured meshes and that it allows for a natural application ofNeumann boundary conditions on traction-free surfaces. It is also of significant impor-tance in large three-dimensional problems that our formulation has fewer number of degreesof freedom than any existing PML formulations. We demonstrate the applicability of ourspherical PML formulation to large problems via a simulation of a three-dimensional double-disk resonator in the time-domain using a discontinuous Galerkin method and an explicitfourth-order Runge-Kutta method. PMDL methods and HOABC methods are alternativesto PML methods, which in the context of the scalar wave equation surpass PML methods intheir overall behavior. Unfortunately, their mathematical properties are not as well under-stood in the context of elastodynamics and, at least in a certain setting, they are known toresult in unstable systems. Due to its involved nature, we focus in this work on the analysisof PMDLs/HOABCs for the scalar wave equation and prove several useful identities whichwill also be useful in the analysis of PMDLs/HOABCs for elastodynamics.