Weighted overconstrained least-squares mixed finite elements for static and dynamic problems in quasi-incompressible elasticity

Abstract

The main goal of this contribution is the improvement of the approximation quality of least-squares mixed finite elements for static and dynamic problems in quasi-incompressible elasticity. Compared with other variational approaches as for example the Galerkin method, the main drawback of least-squares formulations is the unsatisfying approximation quality in terms of accuracy and robustness. Here, lower-order elements are especially affected, see e.g. [33]. In order to circumvent these problems, we introduce overconstrained first-order systems with suited weights. We consider different mixed least-squares formulations depending on stresses and displacements with a maximal cubical polynomial interpolation. For the continuous approximation of the stresses Raviart–Thomas elements are used, while for the displacements standard conforming elements are employed. Some numerical benchmarks are presented in order to validate the performance and efficiency of the proposed formulations.

Appendix 1: Interpolation matrices of \(RT_m\) and \(P_k\)

The fourth order plane strain compliance tensor ${\mathbb {C}}^{-1}$ is given in a matrix notation as

$$\begin{aligned} {\mathbb {C}}^{-1}_{ijkl}&= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\mathbb {C}}^{-1}_{1111} &{} {\mathbb {C}}^{-1}_{1112} &{} {\mathbb {C}}^{-1}_{1121} &{} {\mathbb {C}}^{-1}_{1122} \\ {\mathbb {C}}^{-1}_{1211} &{} {\mathbb {C}}^{-1}_{1212} &{} {\mathbb {C}}^{-1}_{1221} &{} {\mathbb {C}}^{-1}_{1222} \\ {\mathbb {C}}^{-1}_{2111} &{} {\mathbb {C}}^{-1}_{2112} &{} {\mathbb {C}}^{-1}_{2121} &{} {\mathbb {C}}^{-1}_{2122} \\ {\mathbb {C}}^{-1}_{2211} &{} {\mathbb {C}}^{-1}_{2212} &{} {\mathbb {C}}^{-1}_{2221} &{} {\mathbb {C}}^{-1}_{2222} \\ \end{array} \right] \nonumber \\&= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \dfrac{-\lambda }{4(\lambda \mu +\mu ^{2})} +\dfrac{1}{2\mu }&{} 0 &{} 0 &{} \dfrac{-\lambda }{4(\lambda \mu + \mu ^{2})} \\ 0 &{} \dfrac{1}{2\mu } &{} 0 &{} 0 \\ 0 &{} 0 &{} \dfrac{1}{2\mu } &{} 0 \\ \dfrac{-\lambda }{4(\lambda \mu + \mu ^{2})} &{} 0 &{} 0 &{} \dfrac{-\lambda }{4(\lambda \mu +\mu ^{2})} +\dfrac{1}{2\mu } \\ \end{array} \right] .\nonumber \\ \end{aligned}$$

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For the approximation of the unknown fields the interpolation matrices related to the stresses ($\varvec{D}$, $\varvec{{S}}^{J}$, ${\hat{\varvec{{S}}}}^{J}$) and the displacements ($\varvec{N}^{I}$, $\varvec{B}^{I}$) are

$$\begin{aligned} \varvec{{S}}^J \!&= \! \left[ \! \begin{array}{c@{}c@{}c@{}c} \psi ^{J}_{1} &{} \psi ^{J}_{2}&{} 0&{} 0 \\ 0&{} 0&{}\psi ^{J}_{1} &{} \psi ^{J}_{2} \\ \end{array} \!\!\right] ^T\!\!, \; {\hat{\varvec{{S}}}}^{J}\!=\! \left[ \! \begin{array}{c@{}c@{}c@{}c} 0 &{} \dfrac{1}{2}\psi ^{J}_{2} &{} -\dfrac{1}{2}\psi ^{J}_{2} &{} 0 \\ 0 &{} -\dfrac{1}{2}\psi ^{J}_{1} &{} \dfrac{1}{2}\psi ^{J}_{1} &{} 0 \\ \end{array} \!\right] ^T \!\!, \nonumber \\ \varvec{N}^{I}&= \left[ \begin{array}{c@{\quad }c} N^{I} &{} 0 \\ 0 &{} N^{I} \\ \end{array} \right] , \; \varvec{B}^{I} = \left[ \begin{array}{c@{\quad }c} N^{I}_{,1} &{} 0 \\ 0.5 N^{I}_{,2} &{} 0.5 N^{I}_{,1} \\ 0.5 N^{I}_{,2} &{} 0.5 N^{I}_{,1} \\ 0 &{} N^I_{,2} \\ \end{array} \right] \quad \text{ and } \quad \; \nonumber \\ \varvec{D}^{T}&= \left[ \begin{array}{c@{\quad }c} \displaystyle \frac{\partial }{\partial _{1}} &{} 0 \\ \displaystyle \frac{\partial }{\partial _{2}} &{} 0 \\ 0 &{}\displaystyle \frac{\partial }{\partial _{1}} \\ 0 &{} \displaystyle \frac{\partial }{\partial _{2}} \\ \end{array} \right] . \end{aligned}$$

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