Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixed finite element methods for linear Cosserat equations

Wietse Marijn Boon, Omar Duran, Jan Martin Nordbotten

Abstract

We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.

Mixed finite element methods for linear Cosserat equations

Abstract

We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.
Paper Structure (15 sections, 14 theorems, 103 equations, 7 figures)

This paper contains 15 sections, 14 theorems, 103 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and problem definition
  3. General notation
  4. The linearized Cosserat equations
  5. Connection to linear elasticity
  6. Problem definition
  7. Strongly coupled mixed finite elements
  8. Weakly coupled mixed-finite elements
  9. The degenerate linearized Cosserat equations
  10. Well-posedness of the degenerate linearized Cosserat equations
  11. Weakly coupled mixed finite element approximation
  12. Numerical verification
  13. Verification of expected convergence rates and stability for l>0
  14. Verification of expected convergence rates and stability for nearly incompressible materials
  15. Verification of robustness for degenerate l

Key Result

Corollary 2.4

\newlabelcor2.30 Whenever Assumption a2.1 holds, $A_\sigma^{-1}$ is well defined, symmetric positive definite, and has upper and lower bounds (given by $\left(A_\sigma^-\right)^{-1}$ and $\left(A_\sigma^+\right)^{-1}$, respectively). Similarly, if Assumption a2.2a (or a2.2b) holds, $A_\omega^{-1}$

Figures (7)

  • Figure 1: Examples of grid used in the numerical validation. Left: Grid for Section \ref{['sec5.1']} and \ref{['sec5.2']}. Right: Grid for Section \ref{['sec5.3']}.
  • Figure 2: Convergence data for the methods as summarized in Theorem \ref{['th3.9']} and \ref{['th4.5']}, for spatially constant $\ell\in\left\{{1,10}^{-2},{10}^{-4}\right\}$ and $\lambda_\sigma=1$. The error is calculated according to equations \ref{['eq3.20']} and \ref{['eq4.17a']}, for the strongly coupled and weakly coupled methods, respectively. Left panel shows convergence results for $k=0$, right panel for $k=1$.
  • Figure 3: Observed improved convergence rates seen using the norms stated in equations \ref{['eq5.IC-norms1']} and \ref{['eq5.IC-norms2']} for spatially constant $\ell\in\left\{{1,10}^{-2},{10}^{-4}\right\}$ and $\lambda_\sigma=1$. Left panel shows convergence results for $k=0$, right panel for $k=1$.
  • Figure 4: Convergence data for the methods as summarized in Theorem \ref{['th3.9']} and \ref{['th4.5']}, for spatially constant $\lambda_\sigma\in\left\{{1,10}^2,{10}^4\right\}$ and $\ell=1$. The error is calculated according to equations \ref{['eq3.20']} and \ref{['eq4.17a']}, for the strongly coupled and weakly coupled methods, respectively. Left panel shows convergence results for $k=0$, right panel for $k=1$. Note that as the methods are fully robust with respect to $\lambda_\sigma$, the lines are overlapping within the resolution of the figure.
  • Figure 5: Super-convergence data for the strongly coupled methods as summarized in Theorem \ref{['th3.9']}. for spatially constant $\lambda_\sigma\in\left\{{1,10}^2,{10}^4\right\}$ and $\ell=1$. Super-convergence appears in the rotation and couple-stress variables, and is calculated according to \ref{['eq3.27']}. Left panel shows convergence results for $k=0$, right panel for $k=1$. Note that as the methods are fully robust with respect to $\lambda_\sigma$, the lines are overlapping within the resolution of the figure.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Corollary 2.4: Invertibility of material tensors
  • Example 2.5: Isotropic Cosserat material
  • Remark 2.6
  • Remark 2.7
  • Corollary 3.1: Well-posedness with Dirichlet conditions
  • Definition 3.2: Strongly coupled spaces
  • Remark 3.3
  • Definition 3.4: Bounded cochain projection
  • Theorem 3.5: Optimal approximation
  • Proof 1
  • ...and 26 more