On the stability of direct time-domain boundary element methods for elastodynamics

Numerical stability has been a fundamental challenge in direct time-domain boundary element methods (TD-BEM) for elastodynamics. In this paper, an analytical framework for the evaluation of the critical aspect is presented. By casting a convolution integral-based TD-BEM algorithm in the form of a linear multi-step method with a hybrid amplification matrix and the incorporation of some fundamental characteristics of commonly-used transient Green's functions, a rigorous assessment of the problem is shown to be reducible to a standard spectral analysis in matrix theory by which the stability threshold can be clearly defined. Apt to be relevant to the evaluation of other TD-BEMs, the approach is applied to a regularized time-domain direct boundary element method with optional collocation weights and orders of solution variable projections as illustration. By virtue of the proposed formalism and a systematic parametric study, a proper resolution of the critical aspect for some past schemes as well as the versatility of the generalized TD-BEM algorithm as they pertain to the benchmark finite-domain square-bar and the infinite-domain cavity elastodynamic problems are given as examples.