Remeshing criteria and proper error representations for goal oriented h-adaptivity

Abstract

A key ingredient in h-adaptivity pertains to the transformation of output data from a given error estimator into input data, usually in the form of an element-size distribution, that needs to be supplied to a mesh generator. This paper analyzes the different possibilities of defining remeshing criteria in the framework of goal oriented adaptivity. In standard energy norm driven adaptivity, the optimal mesh is clearly obtained if the local error distribution is uniform. The goal-oriented paradigm introduces new difficulties associated with the different possibilities for the spatial error representation and the signs of the local error contributions.A nodal error representation is introduced in order to improve the communication with the automatic mesh generation tool, precluding the transfer of information from elements to nodes.Numerical experiments demonstrate the ability of the introduced remeshing strategies to drive efficient adaptive procedures and to control the error in quantities of interest. The results of the numerical tests fit the expected properties of the different remeshing strategies.

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

“…whereǫ = ψ − ψH analogously to (32). This bound is less sharp than the parallelogram identity (7) but simpler.…”

“…Here, the adopted strategy is to control the output of the error |l O (ẽ)| by controlling a conservative bound for |l O (ẽ)|. However, other options could be considered as in [32].…”

“…Finally, the new interparticle distance for the particles of the new discretization isH ip k :=H k /α. The goal of this section is to find an expression givingH k as a function of the assessed error in Ω k , and such that the error in the new grid e = u − uH (32) fulfills an accuracy restriction of the type |l O (ẽ)| ≤ ∆ tol . A key point of any adaptive strategy is to properly define the quantity that needs to be controlled.…”

Error Assessment in Structural Transient Dynamics

“…However, the local error contributions provided by the local restrictions of this residual expression are often too pessimistic due to the cancellation effect of the contributions of opposite sign with large absolute values, see [55]. Thus, equation (39) is not used directly to compute error maps for space-time mesh adaptivity.…”

Error control in FORM reliability analysis

“…The control of the quantity e FORM;h can be achieved by using the techniques developped for goal-oriented mesh adaptivity (Diez & Calderon, 2007) applied to the control on the error on the loading effect SðuðyÞÞ. The error estimate is required to decide if the meshes used must be refined in order to obtain good approximation of the failure probability and the choice of the prescribed error (a small prescribed error leads to a very refined mesh) will affect the cost of the reliability analysis.…”