On the equivalence of various hybrid finite elements and a new orthogonalization method for explicit element stiffness formulation

Hybrid finite element methods are superior to conventional displacement-based finite element formulations in many aspects and to date many types of hybrid elements have been proposed based on different assumed stress fields. However, some of them are essentially equivalent to each other and there is no general approach available yet to determine their corresponding relationships. In addition, even though the hybrid techniques have many advantages compared to the displacement-based elements, there is an obvious disadvantage, the additional computational cost of the matrix inversions required in constructing element stiffness matrix. In this paper, we show that a linearly independent transformation of the assumed stress field for a given element leads to an equivalent hybrid finite element. This is verified through the proof of equivalence between various existing hybrid elements proposed by different researchers. To improve the computational efficiency, a new inner product including the compliant matrix of material is introduced. With this new definition of inner product, a simplified orthogonalization procedure for the assumed stress modes is then proposed. Based on the orthonorm stress modes obtained using the proposed orthogonalization, an explicit formulation of the hybrid element stiffness matrix is presented. The effectiveness of the present method is validated by numerical examples.