Evaluation of the stiffness matrix in static and dynamic elasticity problems

Abstract

A general evaluation technique (GET) for the stiffness matrix in the finite element methods (FEM) using a modified integration rule with alternate integration points $$r\in [0, 1]$$ rather than the standard Gauss points is proposed. The GET is examined using quadrilateral elements for elasticity problems. For the first time, we have found that the desired softening and stiffening effect can be achieved with adjustments of the integration point r. This allows the FEM model to achieve better accuracy and handle special problems, such as hourglass instability and volumetric locking. Ideal regions for the integration point r are found to overcome the hourglass and volumetric locking issues for the overestimation problems. In addition, the exact solution of the FEM model with optimal r value in terms of strain energy can be always obtained for general overestimation problems of elasticity. More importantly, the optimal integration point r obtained from the static case can be directly applied to dynamic problems to improve the transient displacement significantly. The intensive numerical examples including the static, dynamic, compressible and nearly incompressible material problems are analyzed to verify the accuracy and properties of the GET. Furthermore, the implementation of the GET is extremely simple without increasing computational cost.