Semi-analytical Complex Variable Based Stochastic Finite Element Method

The stochastic finite element method (SFEM) is an approach that allows an analyst to define material, load, and geometry parameters as random variables to represent uncertainty in an engineering problem. The method is then used to estimate the probability of exceeding specified performance thresholds. A necessary ingredient for this analysis is consistent, accurate and efficient algorithms for computing finite element response sensitivities. In this work, the semi-analytical complex variable method (SACVM) is introduced as a method for computing accurate response sensitivities of stochastic models. The SACVM incorporates the complex variable method (CVM) with the semi-analytical method (SAM). It takes advantage of the CVM and the SAM to compute response sensitivities consistently, accurately and efficiently. To date, this approach has not been reported or published in the context of the stochastic finite element method. The SACVM combined with the first-order reliability method (FORM) algorithm becomes the semi-analytical complex variable based stochastic finite element method (SACV-SFEM) and is then applied to various benchmark problems. Specifically, the method is used to evaluate the reliability index of the beam and plate bending problems, steady-state heat conduction problems, linear elastic fracture mechanics problems, material and geometric nonlinear problems. The accuracy and efficiency achieved by the SACV-SFEM approach are compared with the semi-analytical finite difference based stochastic finite element method (SAFD-SFEM), the finite difference based stochastic finite element method (FD-SFEM) and the complex variable based stochastic finite element method (CV-SFEM). The results show that the SACV-SEFM obtains the reliability index in a consistent, accurate, and efficient manner for all the benchmark problems. In the application of heat conduction in electronic packaging problems, the SACV-SFEM can always use one perturbation size to obtain accurate reliability index for all the thermal conductivities whose values may have a large difference. The SAFD-SFEM produces inaccurate shape sensitivities in linear beam and thin plate bending problems, whereas the SACV-SFEM produces accurate shape sensitivities. The FD-SFEM produces inaccurate sensitivities in linear fracture mechanics problems due to the ill-conditioned global stiffness matrix that is used twice in the algorithm. However, the SAFD-SFEM achieves higher accuracy by using the global stiffness matrix only once. In the material and geometrically nonlinear problems, the SACV-SFEM shows higher efficiency than the FD-SFEM and the CV-SFEM, and is more consistent and accurate than the FD-SEFM and the SAFD-SFEM.