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1 – 10 of over 8000The analysis of error estimation is addressed in the framework ofviscoplasticity problems, this is to say, of incompressible andnon‐linear materials. Firstly, Zienkiewicz—Zhu(Z2…
Abstract
The analysis of error estimation is addressed in the framework of viscoplasticity problems, this is to say, of incompressible and non‐linear materials. Firstly, Zienkiewicz—Zhu (Z2) type error estimators are studied. They are based on the comparison between the finite element solution and a continuous solution which is computed by smoothing technique. From numerical examples, it is shown that the choice of a finite difference smoothing method (Orkisz’ method) improves the precision and the efficiency of this type of estimator. Then a Δ estimator is introduced. It makes it possible to take into account the fact that the smoothed solution does not verify the balance equations. On the other hand, it leads us to introduce estimators for the velocity error according to the L2 and L∞norms, since in metal forming this error is as important as the energy error. These estimators are applied to an industrial problem of extrusion, demonstrating all the potential of the adaptive remeshing method for forming processes.
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Michael Bleaney and Zhiyong Li
This paper aims to investigate the performance of estimators of the bid-ask spread in a wide range of circumstances and sampling frequencies. The bid-ask spread is important for…
Abstract
Purpose
This paper aims to investigate the performance of estimators of the bid-ask spread in a wide range of circumstances and sampling frequencies. The bid-ask spread is important for many reasons. Because spread data are not always available, many methods have been suggested for estimating the spread. Existing papers focus on the performance of the estimators either under ideal conditions or in real data. The gap between ideal conditions and the properties of real data are usually ignored. The consistency of the estimates across various sampling frequencies is also ignored.
Design/methodology/approach
The estimators and the possible errors are analysed theoretically. Then we perform simulation experiments, reporting the bias, standard deviation and root mean square estimation error of each estimator. More specifically, we assess the effects of the following factors on the performance of the estimators: the magnitude of the spread relative to returns volatility, randomly varying of spreads, the autocorrelation of mid-price returns and mid-price changes caused by trade directions and feedback trading.
Findings
The best estimates come from using the highest frequency of data available. The relative performance of estimators can vary quite markedly with the sampling frequency. In small samples, the standard deviation can be more important to the estimation error than bias; in large samples, the opposite tends to be true.
Originality/value
There is a conspicuous lack of simulation evidence on the comparative performance of different estimators of the spread under the less than ideal conditions that are typical of real-world data. This paper aims to fill this gap.
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Ch. Pinto, R. Avilés, J. Albizuri and A. Hernández
In this second part of the paper, some properties of the discretization error estimators are presented, although their theoretical background was already developed in the first…
Abstract
In this second part of the paper, some properties of the discretization error estimators are presented, although their theoretical background was already developed in the first part. Two numerical examples have been selected and will be used to check some properties of these error estimators. In addition to this, some practical conclusions will be addressed from the results and graphical output of the implemented procedure.
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Zuqi Tang, Yvonnick Le Menach, Emmanuel Creuse, Serge Nicaise, Francis Piriou and Nicolas Nemitz
In this paper, the aim is to propose a residual‐based error estimator to evaluate the numerical error induced by the computation of the electromagnetic systems using a finite…
Abstract
Purpose
In this paper, the aim is to propose a residual‐based error estimator to evaluate the numerical error induced by the computation of the electromagnetic systems using a finite element method in the case of the harmonic A‐φ formulation.
Design/methodology/approach
The residual based error estimator used in this paper verifies the mathematical property of global and local error estimation (reliability and efficiency).
Findings
This estimator used is based on the evaluation of quantities weakly verified in the case of harmonic A‐φ formulation.
Originality/value
In this paper, it is shown that the proposed estimator, based on the mathematical developments, is hardness in the case of the typical applications.
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A. Rieger and P. Wriggers
Several a posteriori error indicators and error estimators for frictionless contact problems are compared. In detail, residual based error estimators, error indicators relying on…
Abstract
Several a posteriori error indicators and error estimators for frictionless contact problems are compared. In detail, residual based error estimators, error indicators relying on superconvergence properties and error estimators based on duality principles are investigated. Applications are to 2D solids under the hypothesis of nonlinear elastic material behaviour associated with finite deformations. A penalization technique is applied to enforce multilateral boundary conditions due to contact. The approximate solution of the problem is obtained by using the finite element method. Several numerical results are reported to show the applicability of the adaptive algorithm to the considered problems.
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D.H. Wu and I.G. Currie
An à‐posteriori error indicator for solving viscous incompressible flow problems is analyzed in this paper. The indicator named “velocity angle error estimator” is based on the…
Abstract
An à‐posteriori error indicator for solving viscous incompressible flow problems is analyzed in this paper. The indicator named “velocity angle error estimator” is based on the spatial derivative of velocity direction fields and it can detect local flow features, such as vortices and separation, and resolve flow details precisely. The refinement indicator corresponds to the antisymmetric part of the deformation‐rate‐tensor, and it is sensitive to the second derivative of the velocity angle field. Rationality discussions reveal that the à‐posteriori error indicator is a curvature error indicator, and its value reflects the accuracy of streamline curves. It is also found that the velocity angle error indicator contains the nonlinear convective term of the Navier–Stokes equations, and it identifies and computes the direction difference when the convective acceleration direction and the flow velocity direction have a disparity. Numerical simulation is presented to illustrate the use of the velocity angle error indicator.
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Octavio Andrés González‐Estrada, Juan José Ródenas, Stéphane Pierre Alain Bordas, Marc Duflot, Pierre Kerfriden and Eugenio Giner
The purpose of this paper is to assess the effect of the statical admissibility of the recovered solution and the ability of the recovered solution to represent the singular…
Abstract
Purpose
The purpose of this paper is to assess the effect of the statical admissibility of the recovered solution and the ability of the recovered solution to represent the singular solution; also the accuracy, local and global effectivity of recovery‐based error estimators for enriched finite element methods (e.g. the extended finite element method, XFEM).
Design/methodology/approach
The authors study the performance of two recovery techniques. The first is a recently developed superconvergent patch recovery procedure with equilibration and enrichment (SPR‐CX). The second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is enriched with singular terms for a better description of the singular nature of the solution.
Findings
Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the enrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz‐Zhu type error estimators for this class of problems. The results also reveal significant improvements in the effectivities yielded by statically admissible recovered solutions.
Originality/value
The paper shows that both extended recovery procedures and statical admissibility are key to an accurate assessment of the quality of enriched finite element approximations.
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Zuqi Tang, Yvonnick Le-menach, E. Creusé, S. Nicaise, F. Piriou and N. Némitz
The purpose of this paper is to propose some a posteriori residual error estimators (REEs)to evaluate the accuracy of the finite element method for quasi-static electromagnetic…
Abstract
Purpose
The purpose of this paper is to propose some a posteriori residual error estimators (REEs)to evaluate the accuracy of the finite element method for quasi-static electromagnetic problems with mixed boundary conditions. Both classical magnetodynamic A-ϕ and T-Ω formulations in harmonic case are analysed. As an example of application the estimated error maps of an electromagnetic system are studied. At last, a remeshing process is done according to the estimated error maps.
Design/methodology/approach
The paper proposes to analyze the efficiency of numerical REEs in the case of magnetodynamic harmonic formulations. The deal is to determine the areas where it is necessary to improve the mesh. Moreover the error estimators are applied for structures with mixed boundary conditions.
Findings
The studied application shows the possibilities of the residual error estimators in the case of electromagnetic structures. The comparison of the remeshed show the improvement of the obtained solution when the authors compare with a reference one.
Research limitations/implications
The paper provides some interesting results in the case of magnetodynamic harmonic formulations in terms of potentials. Both classical formulations are studied.
Practical implications
The paper provides some informations to develop the proposed formulations in the software using finite element method.
Social implications
The paper deals with the possibility to improve the determination of the meshes in the analysis of electromagnetic structure with the finite element method. The proposed method can be a good solution to obtain an optimal mesh for a given numerical error.
Originality/value
The paper proposes some elements of solution for the numerical analysis of electromagnetic structures. More particularly the results can be used to determine the good meshes of the finite element method.
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A. Hernández, Ch. Pinto, J. Albizuri and O. Altuzarra
This paper focuses on the study of finite element (FE) analysis reliability of non‐linear planar structural problems (large strains and plasticity). In this first part, some error…
Abstract
This paper focuses on the study of finite element (FE) analysis reliability of non‐linear planar structural problems (large strains and plasticity). In this first part, some error estimators of the flux projection type have been developed over the strain power density concept. Spatial, temporal and global error estimators are proposed. From this point, the authors analyse the behaviour of different discretization error components as a function of parameters such as load step or the number of degrees of freedom of the FE model. In the second part of this work, several properties of these estimators are checked with the application to some numerical examples.
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K.C. CHELLAMUTHU and NATHAN IDA
Two different ‘a posteriori’ error estimation techniques are proposed in this paper. The effectiveness of the error estimates in adaptive mesh refinement for 2D and 3D…
Abstract
Two different ‘a posteriori’ error estimation techniques are proposed in this paper. The effectiveness of the error estimates in adaptive mesh refinement for 2D and 3D electrostatic problems are also analyzed with numerical test results. The post‐processing method employs an improved solution to estimate the error, whereas the gradient of field method utilizes the gradient of the field solution for estimating the ‘a posterior’ error. The gradient of field method is computationally inexpensive, since it solves a local problem on a patch of elements. The error estimates are tested by solving a set of self‐adjoint boundary value problems in 2D and 3D using a hierarchical minimal tree based mesh refinement algorithm. The numerical test results and the performance evaluation establish the effectiveness of the proposed error estimates for adaptive mesh refinement.