Compressed polynomial chaos basis for time-domain stochastic finite element simulation of uncertain seismic wave propagation through uncertain solids

This paper mathematically demonstrates that the number of polynomial chaos (PC) terms needed as per conventional tensor computations for representing the quantities of interest (QoIs) in a time-domain, intrusive stochastic finite element analysis involving non-Gaussian material parameter(s) and non-stationary Gaussian forcing can be substantially reduced. The reduction results from the special orthogonality properties of Hermite PC and allows for removal of redundant PC basis terms from the conventionally derived PC basis set. A compressed PC basis can be utilised for the QoIs which can further allow for removal of corresponding block rows from the forcing vector as well as blocks rows and columns from the stiffness, mass, and damping matrices. The resulting drop in size in the system of ordinary differential equations (ODEs) can lead to significant computational benefit, in particular, in solving large scale problems. It is shown that the larger the dimension of the PC expansion for the forcing random process compared to that of the material parameter random field(s), the larger is the drop in size of the system of ODEs. The benefit of using the compressed PC basis for simulating large scale geotechnical engineering problems is demonstrated with a site response analysis of a three-dimensional canyon.