On volumetric locking in a hybrid symmetric interior penalty method for nearly incompressible linear elasticity on polygonal meshes

Abstract

A hybrid version of a symmetric interior penalty method for nearly incompressible linear elasticity is investigated. When a lifting term is used in the method, the method can be free of volumetric locking. On the other hand, when the lifting term is not used, an interior penalty parameter has to be taken to be of order $\lambda$, the first Lamé parameter, as $\lambda$ tends to infinity, in order to guarantee the coercivity of the bilinear form in the method. Taking the interior penalty parameter to be of order $\lambda$ leads to volumetric locking phenomena when piecewise linear functions are employed to compute approximate solutions.

Appendix A: An analytical representation of \(\widetilde{C}_{*,i}^K\)

We derive an analytical representation of $\widetilde{C}_{*,i}^K$ defined by (68) in the case when $k=1$ and K is a triangle without hanging nodes on $\partial K$.

Let $e_j\,(j=1,\,2,\,3)$ denote the sides of K, and the unit outward normal to K on $e_j$. The constant $\widetilde{C}_{*,i}^K$ is given as the largest eigenvalue of the following generalized eigenvalue problem:

$$\begin{aligned} \frac{1}{4|K|} \left[ \begin{array}{lll} m_1^2 {\varPhi }&{}\quad m_1m_2{\varPhi }&{}\quad m_1m_3{\varPhi }\\ m_2m_1 {\varPhi }&{}\quad m_2^2{\varPhi }&{}\quad m_2m_3{\varPhi }\\ m_3m_1 {\varPhi }&{}\quad m_3m_2{\varPhi }&{}\quad m_3^2{\varPhi }\end{array} \right] \mathbf {u}= \lambda \left[ \begin{array}{lll} {\varPsi }&{}\quad O &{}\quad O \\ O &{}\quad {\varPsi }&{}\quad O \\ O &{}\quad O &{}\quad {\varPsi }\end{array} \right] \mathbf {u}, \end{aligned}$$

(99)

where, for an arbitrary fixed i, $m_j:=n^{e_j}_i|e_j|$ $(j=1,\,2,\,3)$,

$$\begin{aligned} {\varPhi }:= \left[ \begin{array}{ll} 1 &{}\quad 1 \\ 1 &{}\quad 1 \end{array} \right] , \quad {\varPsi }:= \frac{1}{6} \left[ \begin{array}{ll} 2 &{}\quad 1 \\ 1 &{}\quad 2 \end{array} \right] , \end{aligned}$$

and $\{\lambda ,\, \mathbf {u}\} \in \mathbb {R}\times (\mathbb {R}^6{\setminus }\{\mathbf {0}\})$ denotes an eigenpair. The maximum eigenvalue of (99) is given by $\frac{1}{|K|}\sum _{j=1}^3 m_j^2$, and hence

$$\begin{aligned} \widetilde{C}_{*,i}^K = \frac{1}{|K|} \sum _{j=1}^3 ( n^{e_j}_i|e_j| )^2. \end{aligned}$$

This yields (70).