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Infinite elements in the time domain using a prolate spheroidal multipole expansion
Author(s) -
Cipolla J. L.,
Butler M. J.
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19981115)43:5<889::aid-nme433>3.0.co;2-l
Subject(s) - finite element method , mathematics , mathematical analysis , boundary value problem , discretization , boundary element method , physics , thermodynamics
Abstract 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 finite elements or finite differences, and large numbers of degrees of freedom. In such methods, efficient numerical enforcement of the Sommerfeld radiation condition in the time domain becomes difficult. 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 finite element codes. Each proceeds from successful domain‐discretized, projected field‐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 sufficiently general for a large variety of problems of naval interest, to obtain finite 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 differential 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 infinite elements, resolution of the Fourier inversion issues, and element matrices for the boundary operators which introduce no new continuity requirements on the fluid field variable. The simplest infinite elements are verified in a coupled three‐dimensional context against DAA2 and Helmholtz integral equation results. © 1998 John Wiley & Sons, Ltd.