抄録
In the transient boundary element method (BEM) analysis, we know that there is instability of solutions. In this paper, we analyze the 2-dimensional scalar wave equation for a multi-reflecting model with several boundary conditions, while varying ratios between boundary element size ΔR and time element size Δt.
For this analysis we consider two linearization methods to deal with singular integration. One is to use conventional space-time elements, and another is a new method which uses what we call space-time planar elements. The use of the space-time planar element is introduced to remove from the singular integration the contribution to a singular node from its adjacent nodes at each instant. In order to evaluate the solution's stability, we use the connection matrix, which represents the contribution from past source terms to the field being considered on the boundary.
In order to achieve a stable solution, we came to the following conclusions. First we find that the use of the space-time planar element is valid. The other has to do with the influence from the difference between boundary conditions. We find the instability under the Dirichlet condition problem at all nodes, while the Neumann condition problem at all nodes is stable. Usually, no problems are created with only the Neumann condition, and the certain Dirichlet conditions can be replaced by Neumann conditions because of symmetry. This type of problem leads to stability given the correct choice of ΔR and Δt which must satisfy the condition cΔt_??_ΔR.
