Rational approach for assumed stress finite elements

Abstract

SUMMARYA new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.

Stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems

“…In the final section, the index of energy compatibility (E-C index) is defined. The higher E-C index of the Wilson displacement mode with respect to the Pian-Sumihara stress mode [9] is verified. The effectiveness of the E-C index on improvement of accuracy and stability is illustrated by two new quadrilateral elements of lower order.…”

“…of E-compatibility of Wilson's quadrilateral with respect to Pian-Sumihara's stress space (proposed in the paper [9] and denoted here by V h P S ) is equal to (1, 1). Proof.…”

“…Proof. Notice that for τ ∈ V h P S , as defined in [9], Let us divide τ ∈ V h P S into τ 1 and τ 2 , τ = τ 1 + τ 2 , τ 1 ∈ V h 0 is a piecewise constant stress. As shown in [10], for τ ∈ V h P S and v ∈ U h W , Let us denote two 4-node combined hybrid elements, constructed by using V h r × U h W and V h P S × U h W , respectively by CH(r) and CH(P S).…”

“…Due to the good performances of several 4-node quadrilateral elements [9]- [14] recently, there is a renewed interest in the construction of finite element schemes that enhance or enrich standard displacement schemes. The two aims of this enhancement are (1) to achieve a greater order of accuracy when using coarse meshes, and (2) to circumvent the locking response in the nearly incompressible regime while preserving the convenience of the displacement finite element methods.…”

“…The two aims of this enhancement are (1) to achieve a greater order of accuracy when using coarse meshes, and (2) to circumvent the locking response in the nearly incompressible regime while preserving the convenience of the displacement finite element methods. The different approaches, such as the assumed stress hybrid method [9,10] and the enhanced strain method [11]- [14] have been considered in the literature. Rational use of accuracy-enhanced displacements of nonconforming bubbles type is fundamental to these methods.…”