Elimination of spurious pressure and kinematic modes in biquadratic nine‐node plane element

Abstract

AbstractIn two‐dimensional penalty finite element analysis of incompressible materials, Q9/3P, the 9‐node quadrilateral with three assumed quasi‐pressure modes, is the most popular element not bothered by spurious pressure. In constructing the penalty matrix of Q9/3P, it is necessary to form a 3×18 matrix and a 3×3 symmetric matrix. The inverse of the symmetric matrix is then post‐ and pre‐multiplied by the 3×18 matrix and its transpose, respectively. By employing a rank subtraction technique, a new and more efficient implementation scheme is devised for the penalty matrix. Same as the conventional Q9 element, when Q9/3P is fully integrated, it becomes expensive and too stiff. On the other hand, there are two spurious kinematic modes should the element be sub‐integrated. In the proposed Q9/3P element, the two mechanisms will be annihilated by judiciously chosen higher‐order assumed stress modes in conjunction with a modified Hellinger‐Reissner functional. It will be demonstrated that the element is of good accuracy and high efficiency.