Stochastic BEM—Random Excitations and Time-Domain Analysis

Abstract

Boundary element formulations for the treatment of boundary value problems in 2-D elasticity with random boundary conditions are presented. It is assumed that random boundary conditions may be described as second order random fields that possess finite moments up to the second order. This permits the use of the mean square calculus for which the operations of integration and mathematical expectations commute. Spatially correlated, time-independent and time-dependent boundary conditions are considered. For time-independent boundary conditions, the stochastic equivalent of Somigliana's identity is used to obtain deterministic integral equations for mathematical expectations and covariances of the response variables, and crosscovariances of the response variables with respect to prescribed boundary conditions. The random field used to describe the boundary conditions is discretized into a finite set of random variables defined at the element nodes. Quadratic, conforming boundary elements are used to arrive at discretized equations for the response statistics of unknown boundary variables. These values may then be used to calculate the response statistics of internal variables and boundary stresses. For time-dependent, spatially correlated boundary conditions, the stochastic equivalent of Stokes's time-domain integral representation is used to obtain deterministic integral equations for the response statistics of unknown boundary variables. An approximate procedure for the calculation of the covariance matrix, that reduces the number of matrix operations and computer storage requirements, is developed. The derivations for the treatment of boundary conditions that are random in time and may be described as an evolutionary white noise are also presented.