Infinite elements in the time domain using a prolate spheroidal multipole expansion

Many phenomena in acoustically loaded structural vibrations are better understood in the time domain, particularly transient radiation, shock, and problems involving non-linearities, cavitation, and bulk structural motion. In addition, the geometric complexity of structures of interest drives the analyst toward domain-discretized solution methods, such as Þnite elements or Þnite di¤erences, and large numbers of degrees of freedom. In such methods, e¦cient numerical enforcement of the Sommerfeld radiation condition in the time domain becomes di¦cult. Although a great many methodologies for doing so have been demonstrated, there seems to exist no consensus on the optimal numerical implementation of this boundary condition in the time domain. Here, we present theoretical development of several new boundary operators for conventional Þnite element codes. Each proceeds from successful domain-discretized, projected Þeld-type harmonic solutions, in contrast to boundary integral equation operators or those derived from analyses of outgoing waves. We exploit the separable prolate-spheroidal co-ordinate system, which is su¦ciently general for a large variety of problems of naval interest, to obtain Þnite element-like operators (matrices) for the boundary points. Use of this co-ordinate system results in element matrices that can be analytically inverse transformed from the frequency to the time domain, without imposing continuity requirements on the solution above those imposed by the underlying partial di¤erential equation. In addition, use of element-like boundary operators does not alter the banded structure of assembled system matrices. Results presented here include theoretical derivation of the inÞnite elements, resolution of the Fourier inversion issues, and element matrices for the boundary operators which introduce no new continuity requirements on the ßuid Þeld variable. The simplest inÞnite elements are veriÞed in a coupled threedimensional context against DAA2 andHelmholtz integral equation results