Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization

Abstract

This paper focuses on the construction of statically admissible stress fields (SA-fields) for a posteriori error estimation. In the context of verification, the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution. The reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the SA-fields. This paper examines the strong prolongation hypothesis, which is the starting point of the Element Equilibration Technique (EET). We manage to characterize the full space of SA-fields satisfying the prolongation equation so that optimizing inside this space is possible. The computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled. In this paper, we describe the new technique in details and compare it to the classical EET and to the flux-free technique for different 2D mechanical problems. The method is explained on first degree triangular elements, but we show how extensions to different elements and to 3D are straightforward.

Improved recovery of admissible stress in domain decomposition methods — application to heterogeneous structures and new error bounds for FETI‐DP

“…Besides, it should be noted that the improvement presented here may also be applied to newer EESPT algorithm [17], or the STARFLEET method [29], since the common basis is shared by these approaches.…”

“…Among these post-processors, we can cite the Element Equilibration Technique (EET) [18,21] and its recent variant [17,28,29], and flux-free techniques based on partition of unity [4,25,26,9,23,24]. Note that whatever the approach, the numerical cost is never negligible.…”

Strict lower bounds with separation of sources of error in non‐overlapping domain decomposition methods

“…The construction ofσ P SApΩq is a complex problem solved by various approaches [14,20,23,26]. The techniques [14,23,26] are two-steps procedures.…”

“…The techniques [14,23,26] are two-steps procedures. The first step consist in building a set of equilibrated tractions or works along the edges of the elements of the mesh.…”

“…• pu psq N q s P KAp Ť Ω psq q: displacement field which results from a Neumann problem and which is not globally admissible q s P SApΩq using dedicated methods such as [14,20,23,26].…”