Local error estimates for adaptive mesh refinement

Computational Mechanics, 2019

This paper introduces a local multilevel mesh refinement strategy that automatically stops relating to a user-defined tolerance even in case of local singular solutions. Refinement levels are automatically generated thanks to a criterion based on the direct comparison of the a posteriori error estimate with the local prescribed error. Singular solutions locally increase with the mesh step (e.g. load discontinuities, point load or geometric induced singularities) and are hence characterized by locally large element-wise error whatever the mesh refinement. Then, the refinement criterion may not be self-sufficient to stop the refinement process. Additional stopping criteria are required if no physical-designed estimator wants to be used. Two original geometry-based stopping criteria are proposed that consist in automatically determining the critical region for which the mesh refinement becomes inefficient. Numerical examples show the efficiency of the methodology for stress tensor approximation in L 2-relative or L ∞-absolute norms.

Advances in Engineering Software, 2007

In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper we propose the use of alternative mesh refinement with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria allow to obtain the maximum of accuracy for a prescribed cost. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.

Indian Journal of Engineering and Materials Sciences, 1999

Among the acce pt ab le numerical methods. Finite Element Analysis stands as the most acceptable one for problems characterised by partial differential equations. However. in accuracy in Finite Element An alysis is• unavoidable since a co ntinuum with infinite degrees of freedo m is modelled into finite degrees of freed o m. In addition to the mesh generation tas k being tedious and e rror prone. the accuracy and cost of the analysis de pe nd directly o n size. shape and number of d e ments in the mes h. The procedure of refining the mesh automatic ally based on the error estimate and distribution of the e rror is known as "adap ti ve" mesh refineme nt. A si mplified method called " Divide and Conquer" rule based on "Fuzzy Logic" is used to refi ne th e mes h by using Ir, p and Irp versions. Aut o matic mes h generato r develo ped in thi s paper based on Fuzzy log ic is able to develop well shaped elements. The program for automati c mes h gene ration and subsequent mesh re linement is developed in "C' language and the analys is is carried o ut usi ng " ANSYS " I package. Automatic mesh ge ne ratio n i~ app li cd to problems suc h as dam, square pl ate with a ho le. thi ck sphe rical press ure vessel and a co rbel and e rror less than 5 % is ac hi eved in most of th e cases.

2008

We prove convergence and optimal complexity of an adaptive finite element algorithm on quadrilateral meshes. The local mesh refinement algorithm is based on regular subdivision of marked cells, leading to meshes with hanging nodes. In order to avoid multiple layers of these, a simple rule is defined, which leads to additional refinement. We prove an estimate for the complexity of this refinement technique. As in former work, we use an adaptive marking strategy which only leads to refinement according to an oscillation term, if it is dominant. In comparison to the case of triangular meshes, the a posteriori error estimator contains an additional term which implicitly measure the deviation of a given quadrilateral from a parallelogram. The well-known lower bound of the estimator for the case of conforming P 1 elements does not hold here. We instead prove decrease of the estimator, in order to establish convergence and complexity estimates

International Journal for Numerical Methods in Engineering, 1983

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part I1 of the paper deals with the strategy for adaptive refinement and concentrates on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.

. We present an error estimator based on first- and second-order derivatives recovery for finite element adaptive analysis. At first, we briefly discuss the abstract framework of the adopted error estimation techniques. Some possibilities of derivatives recovery are considered, including the proposal of a directional error estimator. Using the directional error estimator proposed, an adaptive finite element analysis is performed which gives an adapted mesh where the estimated error is uniformly distributed over the domain. The advantages of adapting meshes are well known, but we place particular emphasis on the anisotropic mesh adaptation process generated by the directional error estimator. This mesh adaptation process gives improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori. We apply the above abstract formulation to analyze the behaviour of the recovery technique and the proposed adaptive process for some pa...

1986

We collect in this article a synopsis of methods and results on adaptive finite element methods. We outline methods for constructing a-posteriori error estimates for linear and nonlinear problems in mechanics. Adaptive methods are describeti and a variety of numerical results are given on applications to problems in fluid mechanics.

Numerical Heat Transfer, Part B: Fundamentals, 2014

An a posteriori error estimate suitable for finite-volume adaptive computations is presented. The error estimate combines the least-squares method regressions with the residual computation, which provides information from the grid quality and the governing equations for a better local adaptation of the unstructured grid. The decision algorithm uses the information provided by the error estimate and does not require problem-dependent constants; it also uses a grid interface correction step to provide a smoother and a high-quality adaptive grid. The proposed error estimate and the adaptive refinement algorithm are verified against analytic solution for different two-dimensional problems. In addition, calculations of three-dimensional laminar flows with different types of unstructured grids have demonstrated the applicability of the adaptive method.

Mathematics and Computers in Simulation, 2010

Recently, the adaptive finite element methods have gained a very important position among numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. While the classical a posteriori error estimates are oriented to the use in h-methods the contemporary higher order hp-methods usually require new approaches in a posteriori error estimation. We present a brief review of some error estimation procedures for some particular both linear and nonlinear differential problems with special regards to the needs of the hp-method.

IEEE Transactions on Magnetics, 1990

A sensitivity analysis has been used to determine the energy perturbation of the nodal position in a finite-element mesh. The sensitivity of the nodal position gives the refinement indication and can therefore be used in the adaptive procedure. This method provides an alternative approach to adaptive mesh generation and is illustrated by numerical examples

Journal of Computational Physics, 2007

A new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines. Based on adaptation on an optimal interpolation error estimate in the L p norm (1 < p 6 1) [L. Chen, P. Sun, J. Xu, Multilevel homotopic adaptive finite element methods for convection dominated problems, in: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2004) 459-468], we obtain the optimal anisotropic adaptive meshes in terms of the Hessian matrix of the numerical solution. Such an anisotropic mesh is optimal for anisotropic solutions like the solution of thin film equations on moving contact lines. Thin film flow is described by an important type of nonlinear degenerate fourth order parabolic PDE. In this paper, we address the algorithms and implementation of the new adaptive finite element method for solving such fourth order thin film equations. By means of the resulting algorithm, we are able to capture and resolve the moving contact lines very precisely and efficiently without using any regularization method, even for the extreme degenerate cases, but with fewer grid points and degrees of freedom in contrast to methods on a fixed mesh. As well, we compare the method theoretically and computationally to the positivity-preserving finite difference scheme on a fixed uniform mesh which has proven useful for solving the thin film problem. irregular domains and in higher spatial dimensions with complex boundary conditions, where a posteriori error estimators are available as an essential ingredient of adaptivity. Such estimators are computable quantities depending on the computed solution(s) and data which provide information about the quality of approximation and may thus be used to make judicious mesh modifications. The ultimate purpose is to construct a sequence of meshes which will eventually equidistribute the approximation errors and, as a consequence, the computational effort. To this end, the a posteriori error estimators are split into element indicators which are then employed to make local mesh modifications by refinement (and sometimes coarsening). This naturally leads to loops of the form

Engineering Analysis with Boundary Elements, 2009

In this work, an adaptive technique for application of meshless methods in one-and two-dimensional boundary value problems is described. The proposed method is based on the use of implicit functions for the geometry definition, fixed weighted least squares approximation and an error estimation by means of simple formulas and a robust strategy of refinement based on the own nature of the approximation sub-domains utilised. With all these aspects, the proposed method becomes an attractive alternative for the adaptive solutions to partial differential equations in all scopes of engineering. Numerical results obtained from the computational implementation show the efficiency of the present method.

International Journal for Numerical Methods in Fluids, 2006

New a posteriori error indicators based on edgewise slope-limiting are presented. The L2-norm is employed to measure the error of the solution gradient in both global and element sense. A second-order Newton–Cotes formula is utilized in order to decompose the local gradient error from a 1 finite element solution into a sum of edge contributions. The slope values at edge midpoints are interpolated from the two adjacent vertices. Traditional techniques to recover (superconvergent) nodal gradient values from consistent finite element slopes are reviewed. The deficiencies of standard smoothing procedures—L2-projection and the Zienkiewicz–Zhu patch recovery—as applied to nonsmooth solutions are illustrated for simple academic configurations. The recovered gradient values are corrected by applying a slope limiter edge-by-edge so as to satisfy geometric constraints. The direct computation of slopes at edge midpoints by means of limited averaging of adjacent gradient values is proposed as an inexpensive alternative. Numerical tests for various solution profiles in one and two space dimensions are presented to demonstrate the potential of this postprocessing procedure as an error indicator. Finally, it is used to perform adaptive mesh refinement for compressible inviscid flow simulations. Copyright © 2006 John Wiley & Sons, Ltd.

Applied Sciences, 2021

The adaptive mesh techniques applied to the Finite Element Method have continuously been an active research line. However, these techniques are usually applied to tetrahedra. Here, we use the triangular prismatic element as the discretization shape for a Finite Element Method code with adaptivity. The adaptive process consists of three steps: error estimation, marking, and refinement. We adapt techniques already applied for other shapes to the triangular prisms, showing the differences here in detail. We use five different marking strategies, comparing the results obtained with different parameters. We adapt these strategies to a conformation process necessary to avoid hanging nodes in the resulting mesh. We have also applied two special rules to ensure the quality of the refined mesh. We show the effect of these rules with the Method of Manufactured Solutions and numerical results to validate the implementation introduced.

2017

Error quantification for industrial CFD requires a new paradigm in which a robust flow solver with error quantification capabilities reliably produces solutions with known error bounds. Error quantification hinges on the ability to accurately estimate and efficiently exploit the local truncation error. The goal of this thesis is to develop a reliable truncation error estimator for finite-volume schemes and to use this truncation error estimate to improve flow solutions through defect correction, to correct the output functional, and to adapt the mesh. We use a higher-order flux integral based on lower order solution as an estimation of the truncation error which includes the leading term in the truncation error. Our results show that using this original truncation error estimate is dominated by rough modes and fails to provide the desired convergence for the applications of defect correction, output error estimation and mesh adaptation. So, we tried to obtain an estimate of the trun...