A primal hybrid finite element method to solve general compressible, quasi-incompressible and incompressible elasticity using stable H(div)-L2 spaces

Abstract

Hybrid methods are usually derived from an extended variational principle, in which the interelement continuity of the functions subspace is removed and weakly enforced by means of a Lagrange multiplier. In this context, a new primal hybrid finite element formulation is presented, which uses H(div) conforming displacement functions and discontinuos L2 approximation for pressure together with shear traction functions to weakly enforce tangential displacement. This combination allows the simulation of compressible, quasi-incompressible and fully incompressible elastic solids, with convergence rates independent of its bulk modulus. The proposed approach benefits from the property that the divergence of the H(div) displacement functions is De Rham compatible with the (dual) pressure functions. The hybridization of the tangential displacements is weakly enforced through a lower order shear stress space. This leads to a saddle-point problem that is stable over the full range of poisson coefficient (large compressibility up to incompressible). Moreover, a boundary stress (normal and shear) can be recovered that satisfies elementwise equilibrium. Hybridizing the tangent stresses and condensing the internal degrees of freedom, a positive-definite matrix with improved spectral properties can be recovered. The stability, consistency and local conservation features are discussed in details. The formulation is tested and verified for different test cases

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