A new procedure for evaluating the time domain boundary influence matrix of unbounded media

The finite element method can only deal with finite domains with well defined boundaries. For dynamic problems involving unbounded media, the boundaries of the finite model distort the real physical behaviour of the problem if they remain untreated. For many problems it is possible to formulate so-called silent boundary conditions which perfectly simulate the effect of the truncated unbounded medium. Unfortunately, most of these conditions are properly formulated in the frequency domain.

The present paper introduces a new procedure which employs these frequency-dependent boundary conditions to calculate the time domain influence matrix of the truncated unbounded medium. This matrix, which was introduced in a previous publication, is used to calculate the reflection-free response of the truncation boundary one time step ahead of the present time station. The known boundary response is then used as a prescribed condition for the finite model. A one-dimensional example with a frequency-dependent boundary condition is presented to examine the effectiveness of the new procedure. Two other silent boundary conditions formulated directly in the time domain are also examined.