A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress

Gosselet

2016

This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate the algebraic error (due to the use of a DD iterative solver) from the discretization error (due to the FE), which enables the steering of the iterative solver by the discretization error. These lower bounds are also used to improve the goal-oriented error estimation in a substructured context. Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the lower bounds' independence from the substructuring. We also steer the iterative solver by an objective of precision on a quantity of interest. This strategy consists in a sequence of solvings and takes advantage of adaptive remeshing and recycling of search directions.

Section: A Posteriori Error Estimationmentioning

confidence: 99%

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

Gosselet

2016

“…Consequently, once local problems (36) have been solved, tractionsF h are recovered on each edge Γ ∈ J by summing the solutions λ iF…”

Section: By Noticing That Solutions λ If (I)mentioning

confidence: 99%

“…In the case q = p, v * h ∈ V p h and local problems (36) are not solvable. Therefore, the modification introduced in (38) should be used to ensure solvability.…”

Section: By Noticing That Solutions λ If (I)mentioning

confidence: 99%

On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples

Pled

Chamoin

Ladevèze

2011

SUMMARYRobust global/goal-oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation. In the context of Computational Mechanics, the construction of admissible stress fields (i.e. stress tensors which verify the equilibrium equations) is required to set up strict and guaranteed error bounds (using residual based error estimators) and plays an important role in the quality of the error estimates. This work focuses on the different procedures used in the calculation of admissible stress fields, which is a crucial and technically complicated point. The three main techniques that currently exist, called the element equilibration technique (EET), the star-patch equilibration technique (SPET), and the element equilibration + starpatch technique (EESPT), are investigated and compared with respect to three different criteria, namely the quality of associated error estimators, computational cost and easiness of practical implementation into commercial finite element codes.The numerical results which are presented focus on industrial problems; they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators, which have the same convergence rate as the exact global error, is consistent. Two-and three-dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches. The analysis of the results reveals that the SPET is more accurate than EET and EESPT methods, but the corresponding computational cost is higher. Overall, the numerical tests prove the interest of the hybrid method EESPT and show that it is a correct compromise between quality of the error estimate, practical implementation and computational cost. Furthermore, the influence of the cost function involved in the EET and the EESPT is studied in order to optimize the estimators.

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Parret-Fréaud

Gosselet

et al. 2017

This paper investigates the question of the building of admissible stress field in a substructured context. More precisely we analyze the special role played by multiple points. This study leads to (1) an improved recovery of the stress field, (2) an opportunity to minimize the estimator in the case of heterogeneous structures (in the parallel and sequential case), (3) a procedure to build admissible fields for FETI-DP and BDDC methods leading to an error bound which separates the contributions of the solver and of the discretization.