Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty

The present paper proposes a method for stochastic problems that have uncertainty in the boundary geometry. The method is developed by applying the spectral stochastic approach to the boundary element method and is called the spectral stochastic boundary element method (SSBEM). In the SSBEM, the uncertainty in the boundary geometry is represented by the Karhunen-Loève expansion. It is shown that, by utilizing material derivative, variation of boundary element matrices associated with the geometrical fluctuation of the boundary can be approximated by the Taylor expansion. The solution is represented by a stochastic process expressed in the form of polynomial chaos expansion. The stochastic equation is then projected on a homogeneous chaos space. This procedure reduces a stochastic equation to an ordinary linear matrix equation that can be solved by conventional schemes. The SSBEM can estimate not only mean values and variances of the solutions but also their probability density functions. In order to examine the performance, the SSBEM is applied to two-dimensional elastostatic and elastodynamic problems with geometrical boundary uncertainty. Computation results of the SSBEM exhibit good agreement with those obtained by Monte Carlo simulation. The efficiency of the SSBEM is verified by comparison of their computation times.