Averaging Techniques for a Posteriori Error Control in Finite Element and Boundary Element Analysis

2007 ◽  
pp. 29-59 ◽  
Author(s):  
Carsten Carstensen ◽  
Dirk Praetorius
Keyword(s):  
Finite Element ◽  
Boundary Element ◽  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
Element Analysis ◽  
A Posteriori Error ◽  
Boundary Element Analysis ◽  
Averaging Techniques

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%).

Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity

2020 ◽  
Author(s):  
Carsten Carstensen ◽  
Gouranga Mallik ◽  
Neela Nataraj
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Best Approximation ◽  
Error Control ◽  
A Priori ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Semilinear Problems ◽  
Abstract Framework

Abstract The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier–Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton–Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix–Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with $L^\infty $ coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.

Adaptive Refinement for 3-D Finite Element Analysis

1993 ◽  
Author(s):  
S. N. Muthukrishnan ◽  
R. V. Namblar ◽  
K. L. Lawrence
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Posteriori Error ◽  
Adaptive Refinement ◽  
A Posteriori ◽  
Element Analysis ◽  
A Posteriori Error ◽  
Simple Strategy ◽  
Element Size ◽  
Adaptive Nature

Abstract A simple strategy to adaptively refine three dimensional tetrahedral meshes has been implemented. A method of refinement for both plane faced and curve faced elements is presented in this paper. Examples are presented in which the rcmeshing is based on a refinement ratio determined from an a-posteriori error indicator obtained from the finite element solution of the problem. The resulting finite element meshes have a smooth gradient in element size. Example meshes are included to show the adaptive nature of the remesher when applied over several solution cycles.

Sign in / Sign up

Export Citation Format

Share Document