An alternate stable midpoint quadrature to improve the element stiffness matrix of quadrilaterals for application of functionally graded materials (FGM)

An efficient, stable and accurate quadrilateral element and its improved stiffness matrix on the midpoint quadrature concept is proposed in this research study. As a first approximation, the integrating point is considered as midpoint of the element of the mapped 2-square in the (ξ, η) plane (same as one-point Gauss-Quadrature). As a second approximation or stabilizing function, integrating points are assumed to be either at the midpoint of the four quadrants or four element edges of the mapped 2-square element in the (ξ, η) plane and these interpolated data are assembled. An appropriate weighted addition of the two approximations is found to result in a better and stable stiffness matrix than equivalent time delayed value of Gauss Quadrature.