# Bilinear Shape Functions

^{1}mix form:

with

${\text{\Delta}}_{I}=\left[{t}_{I}-\left({t}_{I}{x}^{I}\right){b}_{xI}-\left({t}_{I}{y}^{I}\right){b}_{yI}\right]\text{\hspace{0.17em}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=\left(1,1,1,1\right)$

${b}_{xI}=\left({y}_{24}{y}_{31}{y}_{42}{y}_{13}\right)/A\text{\hspace{0.17em}};\text{\hspace{1em}}\left({f}_{ij}=\left({f}_{i}-{f}_{j}\right)/2\right)$

${b}_{yI}=\left({x}_{42}{x}_{13}{x}_{24}{x}_{31}\right)/A$

${\gamma}_{I}=\left[{\Gamma}_{I}-\left({\Gamma}_{J}{x}^{J}\right){b}_{xI}-\left({\Gamma}_{J}{y}^{J}\right){b}_{xI}\right]/4\text{\hspace{0.17em}};\text{\hspace{1em}}\Gamma =\left(1,-1,1,-1\right)$

$A$ is the area of the element.

Where, ${v}_{xI},{v}_{yI},{v}_{zI}$ are the nodal velocities in the x, y, z directions.

Where, ${\omega}_{xI}$ and ${\omega}_{yI}$ are the nodal rotational velocities about the x and y reference axes.

^{1}Belytschko T. and Bachrach W.E., “Efficient implementation of quadrilaterals with high coarse-mesh accuracy”, Computer Methods in Applied Mechanics and Engineering, 54:279-301, 1986.