An efficient methodology for stress‐based finite element approximations in two‐dimensional elasticity

Abstract

Summary We present an efficient approach to compute the matrices used in finite element formulations where self‐equilibrated (zero divergence) approximations of the stress field are used. The fundamental aspect of this approach is that it is applicable to polynomial approximations of high degree (it was tested up to degree 10), providing closed‐form expressions for the elementary matrices involved. The components of the stress field are defined as a function of the coordinates in the master element, facilitating the postprocessing of the finite element results and allowing for the consideration of geometric parameters in reduced order models.

Error estimation for proper generalized decomposition solutions: A dual approach

“…This is a relevant advance for the PGD technique, which is an extremely powerful tool for the simulation of parametric problems, where an accurate assessment of the error of the solutions is required. We also applied a novel way of determining equilibrated stress approximations, which is described in detail in another paper, 13 that leads to analytic expressions for the integrals involved and isolates the effects of the geometry in these computations.…”

“…The approach used for the equilibrium formulation differs from that used in Reference 12 because it works with the stress approximation defined in the frame of the master element, allowing for analytical expressions of the integrals involved, which can potentially account for variations in the geometry of the mesh. The details of this implementation are presented in Reference 13 and briefly described next.…”

An efficient methodology for complementary finite element approximations in three‐dimensional elasticity

“…It is our understanding that the hybrid equilibrium formulation 2 is the key to solve most of the problems implied by working with a tensor field. The methodology in Reference 3, which we now complete for 3D problems, bridges the gap in terms of efficiency, by providing a set of procedures for dealing with stress based approximations of a general degree that greatly simplifies the computations involved. For completeness most definitions used therein are repeated here.…”

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

“…The equilibrium-based formulation was also developed in the hybrid form and several contributions [3][4][5][6][7][8] are available in the literature. In the hybrid equilibrium element (HEE) formulation, stress fields implicitly satisfy domain equilibrium equations and are independently defined for each finite element by polynomial stress functions of arbitrary order, with a set of so-called generalized stresses as degrees of freedom, which do not represent nodal values.…”