A Boundary Element Formulation in Time Domain for Viscoelastic Solids

Viscoelastic solids may be effectively treated by the boundary element method (BEM) in the Laplace domain. However, calculation of transient response via the Laplace domain requires the inverse transform.
Since all numerical inversion formulas depend heavily on a proper choice of their parameters, a direct evaluation in the time domain seems to be preferable. On the other hand, direct calculation of viscoelastic solids in the time domain requires the knowledge of viscoelastic fundamental solutions.
Such solutions are simply obtained in the Laplace domain with the elastic±viscoelastic correspondence principle, but not in the time domain. Due to this, a quadrature rule for convolution integrals, the
`convolution quadrature method' proposed by Lubich, is applied. This numerical quadrature formula determines their integration weights from the Laplace transformed fundamental solution and a linear multistep method. Finally, a boundary element formulation in the time domain using all the advantages of
the Laplace domain formulation is obtained. Even materials with complex Poisson ratio, leading to time-dependent integral free terms in the boundary integral equation, can be treated by this formulation.
Two numerical examples, a 3D rod and an elastic concrete slab resting on a viscoelastic half-space, are presented in order to assess the accuracy and the parameter choice of the proposed method