A systematic and quantitative method to determine the optimal assumed stress fields for hybrid stress finite elements

• Define a new inner product with material weighting matrix where the modes have specific physical meanings of flexibility.
• Point out that the modes in conventional energy inner product are only mathematical vectors without any physical meaning.
• Calculate the exact similarity degrees between different stress modes using our new inner product.
• Derive the basic stress modes from displacement field and broken them into a set of sub-modes.
• Select sub-modes with largest similarity degrees with their basic modes as the optimal assumed stress modes for hybrid element.

A novel method is developed to determine the optimal stress fields for the hybrid stress element. It provides a straightforward way as to how and why the resulting element can improve its displacement counterpart. A new inner product with material weighting matrix is defined to derive this quantitative method. It reveals the relationship in quantity of exact similarity degrees between different stress modes. It is different from the methods based on the conventional energy product which can only tell whether or not the stress and strain are orthogonal to each other because they are considered as mathematical vectors without any physical meaning. The strategy including two steps is proposed to determine the desired stress field. Firstly, the basic stress modes are broken into a set of sub-modes, where the necessary and unnecessary sub-modes are independent from each other because all of them are uniaxial. Secondly, all sub-modes are compared with their basic mode. The sub-mode with largest similarity degree with the basic mode implies that it represents the most important features inside the basic mode so it is selected as the optimal assumed stress mode for hybrid element. The 2D 4-node and 3D 8-node hybrid elements are illustrated by the present approach. Numerical examples are provided to compare the performances of element derived from different assumed stress fields.