Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Abstract

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.

Appendix A: Shape functions

1.1 Appendix A.1: Quadrilateral finite elements

The shape functions in Eq. (34) are given for $\xi _1=\xi _4=-1,~\xi _2=\xi _3=+1,~\eta _1=\eta _2=-1,~\eta _3=\eta _4=+1$ as follows, a being the node number according to Fig. 1. For Q4, they are defined as $N_a(\xi ,\eta )=\frac{1}{4}(1+\xi _a \xi )(1+\eta _a \eta )$. For Q9, with $\xi _8=-1$, $\xi _5=\xi _7=\xi _9=0$, $\xi _6=+1,~\eta _5=-1,~\eta _6=\eta _8=\eta _9=0$ and $\eta _7=+1$, they are given as

$$\begin{aligned}&\text {vertex nodes:}&N_a&=\frac{1}{4}\xi \eta (\xi +\xi _a)(\eta +\eta _a),\\&\text {edge nodes 5 and 7: }&N_a&=\frac{1}{2}\eta (1-\xi ^2)(\eta +\eta _a),\\&\text {edge nodes 6 and 8: }&N_a&=\frac{1}{2}\xi (\xi +\xi _a)(1-\eta ^2),\\&\text {central node:}&N_9&=(1-\xi ^2)(1-\eta ^2),\\ \end{aligned}$$

while for Q16, with $\xi _{11}=\xi _{12}=-1$, $\xi _5=\xi _{10}=\xi _{13}=\xi _{16}=-\frac{1}{3}$, $\xi _6=\xi _{9}=\xi _{14}=\xi _{15}=+\frac{1}{3}$, $\xi _{7}=\xi _{8}=+1$, $\eta _{5}=\eta _{6}=-1$, $\eta _{7}=\eta _{12}=\eta _{13}=\eta _{14}=-\frac{1}{3}$, $\eta _{8}=\eta _{11}=\eta _{15}=\eta _{16}=+\frac{1}{3}$ and $\eta _{9}=\eta _{10}=+1$, they are given as

$$\begin{aligned}&\text {vertex nodes:}\\&\quad N_a =\frac{81}{256}(1+\xi _a\xi )(1+\eta _a\eta )\left( \frac{1}{9}-\xi ^2\right) \left( \frac{1}{9}-\eta ^2\right) ,\\&\text {edge nodes with } \xi _a=\pm 1 \text { and } \eta _a=\pm \frac{1}{3}:\\&\quad N_a = \frac{243}{256}(1-\xi ^2)\left( \eta ^2-\frac{1}{9}\right) \left( \frac{1}{3}+3\xi _a\xi \right) (1+\eta _a\eta ),\\&\text {edge nodes with } \xi _a=\pm \frac{1}{3} \text { and } \eta _a=\pm 1:\\&\quad N_a = \frac{243}{256}(1-\eta ^2)\left( \xi ^2-\frac{1}{9}\right) \left( \frac{1}{3}+3\eta _a\eta \right) (1+\xi _a\xi ),\\&\text {internal nodes:}\\&\quad N_a =\frac{729}{256}(1-\xi ^2)(1-\eta ^2)\left( \frac{1}{3}+3\eta _a\eta \right) \left( \frac{1}{3}+3\xi _a\xi \right) . \end{aligned}$$