I. Kaljevic et S. Saigal, STOCHASTIC BOUNDARY ELEMENTS FOR 2-DIMENSIONAL POTENTIAL FLOW IN NONHOMOGENEOUS MEDIA, Computer methods in applied mechanics and engineering, 121(1-4), 1995, pp. 211-230
A stochastic boundary element formulation is presented for the treatme
nt of two-dimensional problems of steady-state potential flow in non-h
omogeneous media that involve a random operator in the governing diffe
rential equation. The randomness is introduced through the material pa
rameter of the domain which is described as a non-homogeneous random f
ield. The random field is discretized into a set of correlated random
variables and a perturbation is applied to the differential equation o
f the problem. This leads to differential equations for the unknown po
tential and its first- and second-order derivatives, respectively, eva
luated at the mathematical expectations of the random variables result
ing from the discretization of the random field. An approximate method
is applied for the solution of these equations by expressing the pote
ntial and its derivatives as a sum of functions of descending order. T
hese solutions are introduced into the differential equations and upon
equating similar order terms, a sequence of Poisson's equations is ob
tained for each order. A transformation of the correlated random varia
bles into an uncorrelated set is performed to reduce the number of num
erical operations by retaining a small number of transformed random va
riables. The resulting equations are solved using the boundary element
method to obtain the unknown boundary values of the potentials and th
eir respective first- and second-order derivatives which are then used
to compute the desired response statistics. Quadratic, conforming bou
ndary elements are used in the boundary integration and four-node quad
rilateral cells are used in the domain integration. Strongly singular
terms of the boundary element matrices are obtained indirectly by appl
ying a state of uniform unit potential over the entire contour of the
object. The singular domain integrals are calculated analytically. Dir
ect solution techniques are used to calculate the response variables a
nd their derivatives, respectively. A number of example problems are p
resented and the results are compared with those obtained from Monte C
arlo simulations. A good agreement of the results is observed.