J.J. Ródenas, J.P. Moitinho de Almeida, P. Díez, O.A. González-Estrada, O.J.B. Almeida Pereira
ABSTRACT
This paper presents a comparison of two bounding techniques for the global
(energy) discretization error of plane elasticity solutions obtained using the
Finite Element Method. Benchmark problems have been used to evaluate the level
of accuracy provided by each technique. Pros and cons of each technique are
also exposed.
The first method, developed by Díez et al, is based on a recovery-type error
estimator. The upper bound property requires the recovered solution to be both
statically equilibrated and continuous. The equilibrium is obtained locally
(patch-by-patch) using an enhanced version of the Superconvergent Patch
Recovery technique that enforces the satisfaction of the equilibrium equations
introducing constraint equations using the Lagrange multipliers technique. The
continuity is enforced by a postprocessing based on the partition of the unity
concept that slightly modifies the equilibrium, providing a nearly statically
admissible stress field. Correction terms are then used to account for the
equilibrium defaults introduced by the postprocessing technique in order to
provide an accurate upper bound of the error in energy norm.
The second method, directly based on the concept of dual analysis, as proposed by Fraeijs de Veubeke, solves two finite element models of complementary nature, one compatible, the other equilibrated, from which the upper bound of the energy error is directly obtained. As the solution process optimizes the energy within each approximation space, optimal bounds are obtained, at the cost of solving a second global problem.