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

  • Author(s): Sagiyama, Koki
  • Advisor(s): Govindjee, Sanjay
  • et al.
Abstract

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 absorbing

boundary conditions which do work well for elastodynamics are in demand. In this work

we investigate three classes of absorbing boundary conditions which we believe are promis-

ing, viz., perfectly matched layers (PMLs), perfectly matched discrete layers (PMDLs), and

high-order absorbing boundary conditions (HOABCs). We first devise a PML formulation on

spherical domains which is particularly suited for the simulation of a large class of MEMS-

resonator systems. What distinguishes our original PML formulation from most existing

PML formulations is that it works with standard numerical solvers such as discontinuous

Galerkin methods on unstructured meshes and that it allows for a natural application of

Neumann 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 degrees

of freedom than any existing PML formulations. We demonstrate the applicability of our

spherical 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 explicit

fourth-order Runge-Kutta method. PMDL methods and HOABC methods are alternatives

to PML methods, which in the context of the scalar wave equation surpass PML methods in

their 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 to

result in unstable systems. Due to its involved nature, we focus in this work on the analysis

of PMDLs/HOABCs for the scalar wave equation and prove several useful identities which

will also be useful in the analysis of PMDLs/HOABCs for elastodynamics.

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