Optimising the stiffness matrix integration of n-noded 3D finite elements

The integration of the stiffness and mass matrices in finite element analysis is a time consuming task. When dealing with large problems having very fine discretisations, the finite element mesh becomes considerably large and several thousands of elements are usually needed. Moreover, when dealing with nonlinear dynamic problems, the CPU time required to obtain the solution increases dramatically because of the large number of times the global matrix should be computed and assembled. This is the reason why any reduction in computer time (even being small) when evaluating the problem matrices is of the most concern for engineers and analysts. The integration of the stiffness matrix of n-noded high-order hexahedral finite elements is carried out by taking advantage of some mathematical relations found among the nine terms of the nodal stiffness matrix, previously found for the more simple brick element. Significant time savings were obtained in the 20-noded finite element example case.