A study of optimal membrane triangles with drilling freedoms

Abstract

This article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms. The term “optimal” is used in the sense of exact inplane pure-bending response of rectangular mesh units of arbitrary aspect ratio. Following a comparative summary of element formulation approaches, the construction of an optimal 3-node triangle using the ANDES template is shown to be unique if energy orthogonality constraints are enforced a priori. Two other formulation are examined and compared with the optimal model. Retrofitting the conventional LST (Linear Strain Triangle) element by midpoint-migrating by congruential transformations is shown to be unable to produce an optimal element while rank deficiency is inevitable. Use of the quadratic strain field of the 1988 Allman triangle, or linear filtered versions thereof, is also unable to reproduce the optimal element. Moreover, these elements exhibit aspect ratio lock. These predictions are verified on benchmark examples.