Realization of multi-dimensional random field based on Jacobi–Lagrange–Galerkin method in geotechnical engineering

Abstract

Geomaterial has spatial variability, which can be described by the random field theory. A powerful tool for realizing random fields is the Karhunen–Loève series expansion (K–L expansion) where the number of random variables depends on the number of K–L expansion terms rather than on the number of grids of geo-structure model. However, the K–L expansion requires the solution of an integral eigenvalue problem whose analytical form exists only in the special case. Hence, the Galerkin method is usually employed to calculate the approximate solution, which especially for multi-dimensional random field inevitably causes the huge computational cost of multi-fold integrals and the approximate error of the solution. For quickly and accurately simulating the random field with fewer random variables, a Jacobi–Lagrange–Galerkin (JLG) method is proposed where the huge amounts of multi-fold integrals are transformed into simple matrix multiplications which are implemented quickly in a MATLAB environment. Furthermore, the discretization error and its influence factors are discussed to determine the parameters of the JLG method, and the procedure of the JLG method is proposed. Finally, a two-dimensional shallow foundation and a three-dimensional slope are employed to demonstrate computational efficiency, accuracy, and applicability of the JLG method.