A simple impedance-infinite element for the finite element solution of the three-dimensional wave equation in unbounded domains

This paper is concerned with the development of an efficient and accurate impedance-infinite element that can be used either in the frequency- or directly in the time-domain for the modeling and solution of problems described by the scalar three-dimensional wave equation in infinite or semi-infinite domains. The infinite domain is truncated and the effect of the truncated infinite region is simulated by the introduction of an absorbing boundary condition prescribed on the truncation boundary. A systematic procedure for the construction of a family of such conditions of increasing accuracy and complexity is developed with explicit formulas given for approximations up to second order. A central feature of this high-order approximation is that it can be expressed, within the context of a finite element formulation, as a set of local infinite elements located at the boundary of the computational domain, with each element defined by a pair of symmetric, time-invariant, stiffness and damping matrices. This makes it possible to incorporate readily the new local boundary element into finite element software developed for purely interior regions, for applications involving steady-state harmonic or transient excitations. Whereas the theory has been developed formally for arbitrary smooth boundary surfaces, here details are provided for ellipsoidal and spherical boundaries. Thus far, only the latter has been implemented and tested in problems involving cavities and rigid scatterers of spherical and cubic shape. Numerical experiments in both the frequency- and time-domain attest to the efficacy and accuracy of the proposed new element.