Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part II. Formulation in the time domain

A variable-order, infinite ‘‘wave-envelope’’ element scheme is formulated for transient, unbounded acoustical problems. The transient formulation which is local in space and time is obtained by applying an inverse Fourier transformation to a time-harmonic wave-envelope model whose formulation is described in a companion article. This procedure yields a coupled system of second-order differential equations which can be integrated in time to yield transient pressure histories at discrete nodal points. Far-field transient pressures can also be obtained at adjusted times.The method can be applied quite generally to two dimensional and three-dimensional problems and is compatible with a conventional finite element model in the near field. The utility of the method is confirmed by the presentation of transient solutions for axisymmetric and fully three-dimensional test problems. An implicit time integration scheme is used and computed results are compared to analytic solutions and to solutions obtained from alternative numerical schemes. Close correspondence is demonstrated and the scheme is shown to be stable for the problems which are presented. CPU times for a large three-dimensional problem are shown to compare favorably with those required for an equivalent transient boundary element computation