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High-order QMC nonconforming FEMs for nearly incompressible planar stochastic elasticity equations

J. Dick, T. Le Gia, W. McLean, K. Mustapha, T. Tran

TL;DR

This work extends high-order quasi-Monte Carlo uncertainty quantification to nearly incompressible planar stochastic elasticity with spatially varying Lamé parameters. It introduces a locking-free symmetric nonconforming Galerkin FEM capable of handling inhomogeneous coefficients, and develops two high-order QMC strategies (tensor-product and unified) to approximate infinite-dimensional parametric integrals, with regularity results that are independent of the locking parameter $\Lambda$. The authors establish truncation and discretization error bounds, prove $\Lambda$-robust convergence of the FEM, and derive $N^{-1/(p+q)}$ or $N^{-\min(1/p,1/q)}$ QMC rates under appropriate summability assumptions. Numerical experiments across four examples confirm the theoretical rates and demonstrate superior performance of the nonconforming scheme in the nearly incompressible regime. Overall, the paper provides a rigorous framework for efficient, locking-free UQ in stochastic elasticity with variable Lamé parameters, supported by provable error estimates and practical numerical validation.

Abstract

In a recent work (Dick et al, arXiv:2310.06187), we considered a linear stochastic elasticity equation with random Lamé parameters which are parameterized by a countably infinite number of terms in separate expansions. We estimated the expected values over the infinite dimensional parametric space of linear functionals ${\mathcal L}$ acting on the continuous solution $\vu$ of the elasticity equation. This was achieved by truncating the expansions of the random parameters, then using a high-order quasi-Monte Carlo (QMC) method to approximate the high dimensional integral combined with the conforming Galerkin finite element method (FEM) to approximate the displacement over the physical domain $Ω.$ In this work, as a further development of aforementioned article, we focus on the case of a nearly incompressible linear stochastic elasticity equation. To serve this purpose, in the presence of stochastic inhomogeneous (variable Lamé parameters) nearly compressible material, we develop a new locking-free symmetric nonconforming Galerkin FEM that handles the inhomogeneity. In the case of nearly incompressible material, one known important advantage of nonconforming approximations is that they yield optimal order convergence rates that are uniform in the Poisson coefficient. Proving the convergence of the nonconforming FEM leads to another challenge that is summed up in showing the needed regularity properties of $\vu$. For the error estimates from the high-order QMC method, which is needed to estimate the expected value over the infinite dimensional parametric space of ${\mathcal L}\vu,$ we %rely on (Dick et al. 2022). We are required here to show certain regularity properties of $\vu$ with respect to the random coefficients. Some numerical results are delivered at the end.

High-order QMC nonconforming FEMs for nearly incompressible planar stochastic elasticity equations

TL;DR

This work extends high-order quasi-Monte Carlo uncertainty quantification to nearly incompressible planar stochastic elasticity with spatially varying Lamé parameters. It introduces a locking-free symmetric nonconforming Galerkin FEM capable of handling inhomogeneous coefficients, and develops two high-order QMC strategies (tensor-product and unified) to approximate infinite-dimensional parametric integrals, with regularity results that are independent of the locking parameter . The authors establish truncation and discretization error bounds, prove -robust convergence of the FEM, and derive or QMC rates under appropriate summability assumptions. Numerical experiments across four examples confirm the theoretical rates and demonstrate superior performance of the nonconforming scheme in the nearly incompressible regime. Overall, the paper provides a rigorous framework for efficient, locking-free UQ in stochastic elasticity with variable Lamé parameters, supported by provable error estimates and practical numerical validation.

Abstract

In a recent work (Dick et al, arXiv:2310.06187), we considered a linear stochastic elasticity equation with random Lamé parameters which are parameterized by a countably infinite number of terms in separate expansions. We estimated the expected values over the infinite dimensional parametric space of linear functionals acting on the continuous solution of the elasticity equation. This was achieved by truncating the expansions of the random parameters, then using a high-order quasi-Monte Carlo (QMC) method to approximate the high dimensional integral combined with the conforming Galerkin finite element method (FEM) to approximate the displacement over the physical domain In this work, as a further development of aforementioned article, we focus on the case of a nearly incompressible linear stochastic elasticity equation. To serve this purpose, in the presence of stochastic inhomogeneous (variable Lamé parameters) nearly compressible material, we develop a new locking-free symmetric nonconforming Galerkin FEM that handles the inhomogeneity. In the case of nearly incompressible material, one known important advantage of nonconforming approximations is that they yield optimal order convergence rates that are uniform in the Poisson coefficient. Proving the convergence of the nonconforming FEM leads to another challenge that is summed up in showing the needed regularity properties of . For the error estimates from the high-order QMC method, which is needed to estimate the expected value over the infinite dimensional parametric space of we %rely on (Dick et al. 2022). We are required here to show certain regularity properties of with respect to the random coefficients. Some numerical results are delivered at the end.
Paper Structure (13 sections, 9 theorems, 103 equations, 2 figures, 5 tables)

This paper contains 13 sections, 9 theorems, 103 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Estimations of the expected value
  3. Regularity properties
  4. Regularity in ${\boldsymbol{x}}$
  5. Regularity in random coefficients
  6. A truncated problem and error estimates
  7. Nonconforming FEM
  8. Convergence analysis
  9. Numerical experiments
  10. Example 1.
  11. Example 2.
  12. Example 3.
  13. Example 4.

Key Result

Theorem 2.1

Let $\boldsymbol \chi=(\chi_j)_{j\ge 1}$ and $\boldsymbol \varphi = (\varphi_j)_{j\ge 1}$ be two sequences of positive numbers such that $\sum_{j=1}^\infty \chi_j^p$ and $\sum_{j=1}^\infty \varphi_j^q$ being finite for some $0<p,\,q<1.$ Let $\boldsymbol \chi_{s_1}=(\chi_j)_{1\le j\le s_1}$, $\boldsy for all ${\boldsymbol{r}}=({\boldsymbol{y}},{\boldsymbol{z}})\in U_1\times U_2$, $\boldsymbol \alph

Figures (2)

  • Figure 1: The numbers in blue give the values of $j=\rho(k,l)$ if $n=4$.
  • Figure 2: Instance of $\lambda$ from Example 2 with $\alpha=2$, $\Lambda=1$ and $s_1=253$.

Theorems & Definitions (10)

  • Remark 1.1
  • Theorem 2.1: DickEtAl2022
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 5.2