A collocation finite element method with prior matrix condensation

The purpose of the present paper is to demonstrate a method of reducing the final matrix size, so allowing solution for large structures with comparatively small computer storage requirements while retaining the accuracy given by high order displacement functions. The principle used is based on the fact that, of the shell equations to be satisfied at Collocation points, a number are equilibrium conditions which must be satisfied independently of the overall compatibility of forces and deflections for a complete structure. In order to implement this principle the displacement functions can be rewritten with the independent conditions separated from the remainder. Then by writing the equations for external axial and lateral loads and the three possible surface loads, the calculations involving the independent conditions can be performed for each element before final matrix assembly. The final solution for the complete structure then involves a correspondingly smaller number of unknowns. This makes explicit this condition that, of the total number of equations to be satisfied, the majority (six out of eleven for axi-symmetric loading) arise from the requirements of equilibrium conditions which are independent of inter-element effects. It may be noted that when pressure loads are reduced to equivalent forces concentrated at the nodes, as is commonly the case in variational based solutions, four of these degrees of freedom are lost for axial loading and six for unsymmetrical loading. Using the prior condensation method in a typical example the maximum number of equations for simultaneous solution can be reduced from 77 to 28, without any reduction in the effective degrees of freedom of each element