Two-Dimensional Fixed Grid Based Finite Element Structural Analysis

This paper introduces a new fixed mesh structural analysis technique based on isoparametric formulations from classic finite element analysis. Fixed mesh methods are popular in boundary based optimisation as they avoid mesh distortion problems and reanalysis is simple and efficient. The area ratio based fixed grid method is often employed due to its favourable simplicity in implementation. However, maximum errors occur along the boundary due to homogenization of stiffness. Also errors are area ratio dependant, producing an undesirable variation of errors along the boundary. The aim of the isoparametric fixed grid method was to reduce the error dependency on area ratio without significantly reducing the efficiency of the area ratio formulation. The two dimensional isoparametric method divides boundary elements into three types depending on their inside area shape. Quadrilateral elements were formulated using a bilinear isoparametric formulation and an algebraic expression for the stiffness matrix was derived using a priori information about the element shape. Triangular elements were formulated as constant strain triangles and pentagonal element stiffness matrices were approximated by linear interpolation of quadrilateral and inside element matrices. Numerical examples were used to compare the isoparametric to the area ratio fixed grid method using a fitted mesh as a baseline for displacement error calculation. The examples showed some promising benefits in accuracy of the isoparametric fixed grid method, but also revealed areas for improvement.