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 . nodal displacement vector of element basic deformation modes of nodal displacement vector rigid body motion modes of nodal displacement vector nodal displacement vector for node for a nodal displacement vector for node for matrix for matrix for coefficients of polynomials for displacement field applied body force per unit volume geometry matrix DN matrix for material s elastic stiffness matrix material constant for 2D in Eqs . and for 3D in Eq . material s elastic stiffness coefficients similarity degree between and similarity degree between and matrix of differential operator Young s modulus of material nodal force as defined in Eq . leverage matrix as defined in Eq . leverage matrix for leverage matrix for shear modulus of material flexibility matrix as defined in Eq . flexibility matrix for flexibility matrix for identity matrix of energy product as defined in Eq . stiffness matrix for hybrid stress element stiffness matrix for displacement element number of basic deformation modes matrix as defined in Eq . sub matrix for in Eq . number of stress parameters for number of stress parameters for number of nodal displacements for an element dimension of subspace for 2D and for 3D number of nodes for an element shape function matrix matrix for shape functions stress matrix for stress matrix for stress matrix for number of rigid body modes transformation matrix in Eq . material s elastic compliance matrix material s elastic compliance coefficients element boundary of element traction or natural boundary applied boundary traction transformation matrix of equivalent stress modes constants displacement field displacement terms of first order basic deformation mode of element. 


Highlights:
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.