A General Lagrangian Formulation for the Computation of A Posteriori Finite Element Bounds

2003 ◽  
pp. 159-206 ◽  
Author(s):  
Anthony T. Patera ◽  
Jaume Peraire
Keyword(s):  
Finite Element ◽  
Lagrangian Formulation ◽  
A Posteriori

Effective Convergence Checks for Verifying Finite Element Stresses at Three-Dimensional Stress Concentrations

2018 ◽  
Vol 3 (3) ◽  
Author(s):  
J. R. Beisheim ◽  
G. B. Sinclair ◽  
P. J. Roache
Keyword(s):  
Finite Element ◽  
Error Estimation ◽  
A Posteriori Error Estimates ◽  
Three Dimensional ◽  
Posteriori Error ◽  
Test Problems ◽  
A Posteriori ◽  
Stress Concentrations ◽  
Element Analysis ◽  
A Posteriori Error

Current computational capabilities facilitate the application of finite element analysis (FEA) to three-dimensional geometries to determine peak stresses. The three-dimensional stress concentrations so quantified are useful in practice provided the discretization error attending their determination with finite elements has been sufficiently controlled. Here, we provide some convergence checks and companion a posteriori error estimates that can be used to verify such three-dimensional FEA, and thus enable engineers to control discretization errors. These checks are designed to promote conservative error estimation. They are applied to twelve three-dimensional test problems that have exact solutions for their peak stresses. Error levels in the FEA of these peak stresses are classified in accordance with: 1–5%, satisfactory; 1/5–1%, good; and <1/5%, excellent. The present convergence checks result in 111 error assessments for the test problems. For these 111, errors are assessed as being at the same level as true exact errors on 99 occasions, one level worse for the other 12. Hence, stress error estimation that is largely reasonably accurate (89%), and otherwise modestly conservative (11%).

scholarly journals A posteriori error estimates for a multi-scale finite-element method

2021 ◽  
Vol 40 (4) ◽  
Author(s):  
Khallih Ahmed Blal ◽  
Brahim Allam ◽  
Zoubida Mghazli
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
A Posteriori Error Estimates ◽  
Numerical Test ◽  
Posteriori Error ◽  
Diffusion Problem ◽  
A Posteriori ◽  
Essential Information ◽  
Multi Scale ◽  
Element Method

AbstractWe are interested in the discretization of a diffusion problem with highly oscillating coefficient, by a multi-scale finite-element method (MsFEM). The objective of this method is to capture the multi-scale structure of the solution via local basis functions which contain the essential information on small scales. In this paper, we perform an a posteriori analysis of this discretization. The main result consists of building error indicators with respect to both small and large meshes used in this method. We present a numerical test in which the experiments are in good coherency with the results of analysis.

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