Transient Infinite Elements for Acoustics and Shock

Many phenomena in acoustically loaded structural vibrations are better understood in the time domain, particularly transient radiation, shock, and problems involving nonlinearities 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-discretised, projected field-type harmonic solutions, in contrast to boundary integral equation operators or those derived from algebraic functions. We exploit the separable prolate-spheroidal coordinate 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 coordinate system results in element matrices that can be analytically inverse transformed from the frequency to the time domain, using appropriate approximations, without altering the Hilbert space in which the approximate solution resides. The inverse transformation introduces some additional theoretical issues involving time delays and Stieltjes-type integrals, which are easily resolved. In addition, use of element-like boundary operators does not alter the banded structure of the system matrices, which is of enormous importance for efficient solution of large problems. Results presented here include theoretical derivation of the new “infinite elements”, the approximations for certain problematic frequency-domain terms, resolution of the inversion issues, and element matrices for the boundary operators which introduce no new continuity requirements on the fluid field variable.