Weakly non-local arbitrarily-shaped absorbing boundary conditions for acoustics and elastodynamics theory and numerical experiments
In this dissertation we discuss the performance of a family of local and weakly non-local in space and time absorbing boundary conditions, prescribed on trun cation boundaries of elliptical and ellipsoidal shape for the solution of two- and three-dimensional scalar wave equations, respectively, in both the time- and frequency-domains. The elliptical and ellipsoidal artiﬁcial boundaries are de rived as particular cases of general arbitrarily-shaped convex boundaries for which the absorbing conditions are developed. From the mathematical per spective, the development of the conditions is based on earlier work by Kalli vokas et al [72–77]; herein an incremental modiﬁcation is made to allow for the spatial variability of the conditions’ absorption characteristics. From the appli cations perspective, the obtained numerical results appear herein for the ﬁrst time. It is further shown that the conditions, via an operator-splitting scheme, lend themselves to easy incorporation in a variational form that, in turn, leads to a standard Galerkin ﬁnite element approach. The resulting wave absorbing ﬁnite elements are shown to preserve the sparsity and symmetry of standard ﬁnite element schemes in both the time- and frequency-domains. Herein, we also extend the applicability of elliptically-shaped truncation boundaries to semi-inﬁnite acoustic media. Numerical experiments for transient and time harmonic cases attest to the computational savings realized when elongated scatterers are surrounded by elliptically- or ellipsoidally-shaped boundaries, as opposed to the more commonly used circular or spherical truncation geome tries in either the full- or half-space cases (near-surface scatterers). Lastly, we treat the two-dimensional elastodynamics case based on a Helmholtz decomposition of the displacement vector ﬁeld. The decomposi tion allows for scalar wave equations to be written for the scalar and vector potential components. Thus, absorbing conditions similar to the ones writ ten for acoustics can be used for the elastodynamics case. The stability of the elastodynamics conditions for time-domain applications remains an open question.