Efficient stochastic finite element modeling using parameterized reduced order models

Finite element analysis of complex geometries is limited by several factors, including the time required to develop solid geometries and meshes, and the computational time required to perform the simulation of a system with, potentially, millions of degrees of freedom. As a result, the analysis of complex systems is often relegated to deterministic approaches in which geometries other than the idealized geometry rarely are considered. In order to robustly design a system, though, and to quantify the margins of uncertainty that a manufactured realization might yield, a stochastic sampling of the permissible geometries is needed. For complex geometries, a stochastic sampling can involve dozens, if not hundreds, of dimensions that can vary. Exploring a large sample space on even a modestly large model quickly becomes prohibitively expensive— and nearly impossible when a mesh change is required for each sample point. This paper presents a new methodology for utilizing reduced order models in which the number of necessary full degree of freedom models (and corresponding meshes) is minimized. The benefits of parameterizing the high fidelity model at an elemental level, the reduced order model at a system level, and the eigen-space representation are discussed. An application of the method on a model with a non-trivial number of degrees of freedom is presented.