Absorbing Boundary Conditions For Corner Regions

Abstract

This thesis contains the work that extends the continued fraction absorbing boundary conditions (CFABC's) to corner regions. We combine the ideas related to optimal discretization of perfectly matched layers (PML's) presented by Asvadurov et al. and the continued fraction expansion of one-way wave equations through finite element discretization given by Guddati to arrive at the new formulation of the CFABC and its extension to corner regions.

It will be shown that CFABC is a special case of the discrete PML, where CFABC is obtained from PML as a result of finite element discretization of the PML via one-point integration with purely imaginary element length. Discretization of the two-dimensional corner region is performed as a tensor product of the two CFABC one-dimensional discretization, which when viewed from the PML approach, is equivalent to discretizing the Helmholtz equation where pure imaginary stretching function and 1 by 1 integration are used. Extension to non-orthogonal corners by the use of parallelogram elements is also performed.

The result is that the dynamic stiffness matrix is independent of frequency, resulting in identical matrix entries in both the wave number-frequency and space-time domain. The dynamic stiffness matrix simply becomes the element stiffness matrix in the space-time domain. This allows for extremely easy finite element implementation for both transient and time-harmonic cases.

An implicit scheme is currently being used with the CFABC. A full explicit scheme is not possible since the mass matrix is singular (the absorbing boundary conditions do not contribute to the mass matrix. The second part of the thesis deals with the exploration of a pseudo-explicit time stepping scheme for the CFABC's. Currently, the implementation is limited to the case of straight computational boundaries. This initial work, together with the computer code developed, will provide a reference for the future improvement of the scheme.