A higher‐order equilibrium finite element method

Abstract

Summary In this paper, a mixed spectral element formulation is presented for planar, linear elasticity. The degrees of freedom for the stress are integrated traction components, ie, surface force components. As a result, the tractions between elements are continuous. The formulation is based on minimization of the complementary energy subject to the constraints that the stress field should satisfy equilibrium of forces and moments. The Lagrange multiplier, which enforces equilibrium of forces, is the displacement field and the Lagrange multiplier, which enforces equilibrium of moments, is the rotation. The formulation satisfies equilibrium of forces pointwise if the body forces are piecewise polynomial. Equilibrium of moments is weakly satisfied. Results of the method are given on orthogonal and curvilinear domains, and an example with a point singularity is given.

Multiscale finite volume method for finite-volume-based simulation of poroelasticity

“…This interpolation allows for convenient integration of the displacement and its derivatives over any interface, specially over finite-volume control volumes. Note that our finite volume formulation provides locally conservative stress field, with much fewer degrees of freedom compared with the mixed-FE formulation [6].…”

“…Classically, finite volume (FV) schemes have been the methods of choice for simulation of flow and transport [2], whereas finite-element (FE) methods have been preferred for mechanical deformation [3]. There exist several examples in the literature where conservative extensions of finite-element (FE) methods (e.g., mixed-FE) have been used for flow and transport simulations (e.g., see [4,5]), as well as mechanical deformation [6]. Similarly, finite-difference [7] and FV methods have been proposed recently for mechanical deformation [8][9][10], motivated by their locallyconservative discrete stress representations with only 1 degree-of-freedom (DOF) per element.…”

Multiscale Finite Volume Method For Finite-Volume-Based Poromechanics Simulations

“…This interpolation allows for convenient integration of the displacement and its derivatives over any interface, specially over finite-volume control volumes. Note that our finite volume formulation provides locally conservative stress field, with much fewer degrees of freedom compared with the mixed-FE formulation [40].…”

“…Classically, finite volume (FV) schemes have been the methods of choice for simulation of flow and transport [3], whereas finite-element (FE) methods have been preferred for mechanical deformation [32]. There exist several examples in the literature where conservative extensions of finite-element (FE) methods (e.g., mixed-FE) have been used for flow and transport simulations (e.g., see [2,36]), as well as mechanical deformation [40]. Similarly, finite-difference [27] and FV methods have been proposed recently for mechanical deformation [19,28,42], motivated by their locally-conservative discrete stress representations with only 1 degree-of-freedom (DOF) per element.…”

Hybrid equilibrium element with high‐order stress fields for accurate elastic dynamic analysis

“…The equilibrium‐based solution of the linear elastic static problem with very high‐order stress field is approached in Reference 16 by the mixed spectral element formulation, in which the symmetry of stress tensor is not assumed a priori, and displacement and rotation are defined as Lagrangian parameters enforcing the forces equilibrium condition in a strong form and the moments equilibrium condition, that is the stress symmetry condition, only weakly. The degrees of freedom for the stress are integrated traction components and stress is co‐diffusive between elements.…”