Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations

TL;DR

This paper develops a hybrid stress quadrilateral finite element method for stochastic plane elasticity problems with random material properties and loads, providing uniform error estimates and numerical validation.

Contribution

It introduces a combined stochastic and finite element approach with uniform error bounds for stochastic elasticity equations, enhancing accuracy and robustness.

Findings

Error estimates are uniform with respect to Lamé constant.

Finite element methods effectively handle stochastic fields.

Numerical results validate theoretical predictions.

Abstract

This paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen-Lo$\grave{e}$ve expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by $k-$version/$p-$version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We show that the derived a priori error estimates are uniform with respect to the Lam$\acute{e}$ constant $\lambda\in (0, +\infty)$. Finally, we provide some numerical results.