Subspace Reduction for Stochastic Planar Elasticity

Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the spectrum. If the inputs, domain or material, are perturbed, the cluster breaks and tracing of the eigenpairs become difficult due to possible crossing of the modes. In this paper we have shown that the eigenvalue crossing can occur within clusters not only by perturbations of the domain, but also of material parameters. What is new is that in this setting, the crossing can be controlled; that is, the effect of the perturbations can actually be predicted. Moreover, the basis of the subspace is shown to be a well-defined concept and can be used for instance in low-rank approximation of solutions of problems with static loading. In our industrial model problem, the reduction in solution times is significant.

An Immersed Finite Element Method for Planar Elasticity Interface Problem

This paper presents the P1/CR immersed finite element (IFE) method to solve planar elasticityinterface problem. By adding some stabilisation terms on the edges of interface elements, thestability of the discrete formulation and a priori error estimate in an energy norm are presented.Finally, numerical examples are given to confirm our theoretical results.

Error estimates for a partially penalized immersed finite element method for elasticity interface problems

This article is about the error analysis for a partially penalized immersed finite element (PPIFE) method designed to solve linear planar-elasticity problems whose Lamé parameters are piecewise constants with an interface-independent mesh. The bilinear form in this method contains penalties to handle the discontinuity in the global immersed finite element (IFE) functions across interface edges. We establish a stress trace inequality for IFE functions on interface elements, we employ a patch idea to derive an optimal error bound for the stress of the IFE interpolation on interface edges, and we design a suitable energy norm by which the bilinear form in this PPIFE method is coercive. These key ingredients enable us to prove that this PPIFE method converges optimally in both an energy norm and the usual L2 norm under the standard piecewise H2-regularity assumption for the exact solution. Features of the proposed PPIFE method are demonstrated with numerical examples.