The stability of integral equation time‐domain scattering computations for three‐dimensional scattering; similarities and differences between electrodynamic and elastodynamic computations

Abstract

SUMMARYTime-domain integral equation analyses are prone to instabilities, in a range of applications areas including acoustics, electrodynamics and elastodynamics, and a variety of retrospective averaging schemes have been proposed to improve matters. In this paper, we investigate stability behaviour, in parallel, in electrodynamic and elastodynamic cases. It is observed empirically that the tendency to instability is increased as the treatment becomes more nearly explicit. The timestepping procedure is recast in a recursive matrix formulation, and it is shown that it is the eigenvalues of this large matrix which determine the longterm stability behaviour. This treatment is then extended to cover general averaging schemes, allowing the likely effectiveness of such schemes to be assessed. Simple modelling rules, which for all practical purposes will ensure stability, are presented.

Time Domain Boundary Element Analysis of Wake Fields in Long Accelerator Structures

“…Many previous works [16]- [19] indicate that ( is the maximum mesh size) is a stable condition in many practical cases. In fact, this tendency has been numerically validated in the Walker's eigenvalue analysis [18] for several simple cases. In this choice of time step size, a spherical surface with radius centered at the observation point r contains a lot of neighbor boundary elements, or at least the closest boundary elements.…”

Temporal preconditioners for marching-on-in-time-based time domain integral equation solvers

“…The MOT matrix elements are determined via the semianalytical method described in [1][2][3] and algebraic stability analysis is carried out as described in [16]. …”

Nullspaces of MFIE and CalderÓn Preconditioned EFIE Operators Applied to Toroidal Surfaces

“…Since in , the integral vanishes. This implies that the solution of the following Neumann problem therefore exists and is unique up to a constant in (11) on (12) The field with clearly resides in . Moreover (13) with a loop in that circles hole once and is positively oriented w.r.t [ Fig.…”

“…Equation (74) constitutes a so-called polynomial eigenvalue problem. This problem can be solved by leveraging the companion matrix [12]. The nullspace elements of the static outer and inner MFIE operators constructed in Section II cannot be resolved exactly by the polynomial eigenvalue equation corresponding to the RWG discretization of the TD MFIE operators.…”