Chapter 7 - Absorbing boundary conditions

Abstract

This chapter extensively discusses the concept of the Perfectly Matched Layer (PML) technique; the most practical method for truncating an infinite domain for coupled acoustic problems. First, we present a PML layer's construction for a simple model problem and explain its behavior. We then discuss some of the most frequently used stretching functions; then we reveal the deeper connection of the PML technique with Piola transforms, parametric finite elements, energy spaces, and the corresponding exact sequences connecting these spaces at discrete and continuous levels. These observations give rise to the formulation of the generalized approach of applying the PML to any weak form and provide all forms of the required transformations, detailed for Cartesian, cylindrical, and spherical coordinates. Finally, we briefly discuss the instability problems in PMLs, which arise when the solution excites back-propagating modes.