Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

Abstract

SUMMARYThis paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non-linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non-linear quantities of interest by means of projection properties in order to take into account higher-order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two-dimensional applications in the context of elastic fatigue problems.

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

“…Since the mid 2000s, attention has been devoted to provide certified bounds for quantities of interest [2,3,4,5,6,7,8,9]. In particular [1] presents a comparison of the performance of two of the main techniques to compute guaranteed bounds for quantities of interest in the context of the advection-reaction-diffusion equation: a standard residual type estimator (hybrid-flux) proposed in [10] and the new flux-free technique proposed in [11].…”

Guaranteed computable bounds on quantities of interest in finite element computations

“…upon inserting (13) and (40). Therefore, letting v D p 2 P m .K/ in (42) allows us to see that (13) and (40) imply (13) and (14) because (39) holds.…”

Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures

“…In Fig. 5, we represent the PGD approximation of ρ 11 1 for different configurations of parameters x 3 and y 3 . The computation of this PGD solution is done once for all, in an offline phase and stored for later use.…”

“…The CRE concept was first introduced as a robust a posteriori error estimator in FE computations [5], enabling to compute both strict and effective discretization error bounds for linear and more generally convex structural mechanics problems, and to lead mesh adaptivity processes. It was primarily used for linear thermal and elasticity problems [6,7] before being extended to nonlinear time dependent problems [8,9] and to goal-oriented error estimation [10][11][12]. The use of CRE for model verification, for which a general overview can be found in [2], requires in particular the computation of admissible dual fields which are fully equilibrated.…”