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Error Indicator (error + indicator)

Distribution by Scientific Domains

Engineering	86%

Selected Abstracts

An adaptive multigrid iterative approach for frictional contact problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2006
S. A. Mohamed
Abstract The objective of this paper is the construction of a robust strategy towards adaptively solving Signorini's frictional contact problems. The frictional contact problem between a linearly elastic body and rigid foundation is formulated as a classical boundary value problem of the elastic body but associated with special inequality conditions on the contact surface. A new iterative approach is presented to solve the problem on a given mesh. In the first iteration the candidate nodes are assumed to be in micro-slip contact and then proceeding to update the contact status according to the actual displacements and stresses obtained at the end of each increment. An efficient multigrid method is developed to solve the discrete problems of different iterations. The proposed iterative procedure is integrated with an error indicator and automatic grid generator to construct an adaptive multigrid method. Numerical results of the convergence rates, automatically generated grid sequence, contact stresses and strains as well as two parametric studies are presented to prove the efficiency of the proposal. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz,Zhu error estimator

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2003
M. Picasso
Abstract The framework of Formaggia and Perotto (Numerische Mathematik 2001; 89: 641,667) is considered to derive a new anisotropic error indicator for a Laplace problem in the energy norm. The matrix containing the error gradient is approached using a Zienkiewicz,Zhu error estimator. A numerical study of the effectivity index is proposed for anisotropic unstructured meshes, showing that our indicator is sharp. An anisotropic adaptive algorithm is implemented, aiming at controlling the estimated relative error. Copyright © 2003 John Wiley & Sons, Ltd. [source]

An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2008
Lourenço Beirão da Veiga
Abstract We present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form. We show that this estimator is reliable and efficient with respect to an energy-type error comprising both flux and pressure. Its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes, different permeability tensors, and different types of computational meshes. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A posteriori error estimation for extended finite elements by an extended global recovery

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2008
Marc Duflot
Abstract This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity, the proposed scheme is ideally suited to industrial applications. The procedure is based on global minimization of the L2 norm of the difference between the raw strain field (C,1) and the recovered (C0) strain field. The methodology engineered in this paper extends the ideas of Oden and Brauchli (Int. J. Numer. Meth. Engng 1971; 3) and Hinton and Campbell (Int. J. Numer. Meth. Engng 1974; 8) by enriching the approximation used for the construction of the recovered derivatives (strains) with the gradients of the functions employed to enrich the approximation employed for the primal unknown (displacements). We show linear elastic fracture mechanics examples, both in simple two-dimensional settings, and for a three-dimensional structure. Numerically, we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement. Consequently, the approximate error converges to the exact error, indicating that the error indicator is valid. Additionally, the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased, before standard h - and p -adaptivities are applied; we suggest to coin this methodology e-adaptivity. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Robust adaptive remeshing strategy for large deformation, transient impact simulations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2006
Tobias Erhart
Abstract In this paper, an adaptive approach, with remeshing as essential ingredient, towards robust and efficient simulation techniques for fast transient, highly non-linear processes including contact is discussed. The necessity for remeshing stems from two sources: the capability to deal with large deformations that might even require topological changes of the mesh and the desire for an error driven distribution of computational resources. The overall computational approach is sketched, the adaptive remeshing strategy is presented and the crucial aspect, the choice of suitable error indicator(s), is discussed in more detail. Several numerical examples demonstrate the performance of the approach. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Automatic energy conserving space,time refinement for linear dynamic structural problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005
P. Cavin
Abstract In this paper a local space,time automatic refinement method (STAR method) is developed to efficiently solve time-dependent problems using FEM techniques. The automatic process is driven by an energy or a displacement error indicator which controls the precision of the result. The STAR method solves the numerical problem on grids with different mesh size. For the Newmark schemes, a general demonstration, using the energy method, gives the interface conditions between two successive grids which is compatible with the stability of the scheme. Finally, using a linear one-dimensional example, the convergence of the method and the precision of the results are discussed. Copyright © 2005 John Wiley & Sons, Ltd. [source]

An a posteriori error estimator for the p - and hp -versions of the finite element method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005
J. E. Tarancón
Abstract An a posteriori error estimator is proposed in this paper for the p - and hp -versions of the finite element method in two-dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42:561,587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non-uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p - and hp -adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2002
K. H. Muci-Küchler
Abstract In this work, a new global reanalysis technique for the efficient computation of stresses and error indicators in two-dimensional elastostatic problems is presented. In the context of the boundary element method, the global reanalysis technique can be viewed as a post-processing activity that is carried out once an analysis using Lagrangian elements has been performed. To do the reanalysis, the functional representation for the displacements is changed from Lagrangian to Hermite, introducing the nodal values of the tangential derivatives of those quantities as additional degrees of freedom. Next, assuming that the nodal values of the displacements and the tractions remain practically unchanged from the ones obtained in the analysis using Lagrangian elements, the tangent derivative boundary integral equations are collocated at each functional node in order to determine the additional degrees of freedom that were introduced. Under this scheme, a second system of equations is generated and, once it is solved, the nodal values of the tangential derivatives of the displacements are obtained. This approach gives more accurate results for the stresses at the nodes since it avoids the need to differentiate the shape functions in order to obtain the normal strain in the tangential direction. When compared with the use of Hermite elements, the global reanalysis technique has the attraction that the user does not have to give as input data the additional information required by this type of elements. Another important feature of the proposed approach is that an efficient error indicator for the values of the stresses can also be obtained comparing the values for the stresses obtained through the use of Lagrangian elements and the global reanalysis technique. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Adaptive strategy of the supersonic turbulent flow over a backward-facing step

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2004
Shih-Ying YangArticle first published online: 16 MAR 200
Abstract An adaptive strategy incorporating mesh remeshing and refining is developed to study the supersonic turbulent flow over a backward-facing step on a mixed quadrilateral,triangular mesh. In the Cartesian co-ordinate system, the unsteady Favre-averaged Navier,Stokes equations with a low-Reynolds-number k,,turbulence model are solved using a locally implicit scheme with an anisotropic dissipation model. In the present adaptive strategy, two error indicators for both mesh remeshing and refining, respectively, are presented. The remeshing error indicator incorporates unified magnitude of substantial derivative of pressure and that of vorticity magnitude, whereas the refining error indicator incorporates unified magnitude of substantial derivative of pressure and that of weighted vorticity magnitude. To assess the present approach, the transonic turbulent flow around an NACA 0012 airfoil is performed. Based on the comparison with the experimental data, the accuracy of the present approach is confirmed. According to the high-resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Anisotropic a posteriori error estimate for an optimal control problem governed by the heat equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006
Marco Picasso
Abstract The abstract framework of Becker et al. is considered to solve an optimal control problem governed by a parabolic equation. Existence and uniqueness of a solution are proved using the inf-sup framework and space-time functional spaces. A Crank-Nicolson time discretization is proposed, together with continuous, piecewise linear finite elements in space. Existence and uniqueness of a solution to the discretized problem is also proved using the inf-sup framework. An a posteriori error estimate is proposed, the goal being to control the error between the true and computed cost functional. The error estimate remains valid on strongly anisotropic meshes and an anisotropic error indicator is proposed when the time step is small. Finally, the quality of this error indicator is studied numerically on isotropic and anisotropic meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source]

Conservative semi-Lagrangian advection on adaptive unstructured meshes

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2004
Armin Iske
Abstract A conservative semi-Lagrangian method is designed in order to solve linear advection equations in two space variables. The advection scheme works with finite volumes on an unstructured mesh, which is given by a Voronoi diagram. Moreover, the mesh is subject to adaptive modifications during the simulation, which serves to effectively combine good approximation quality with small computational costs. The required adaption rules for the refinement and the coarsening of the mesh rely on a customized error indicator. The implementation of boundary conditions is addressed. Numerical results finally confirm the good performance of the proposed conservative and adaptive advection scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 388,411, 2004 [source]

Galerkin-type space-time finite elements for volumetrically coupled problems

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Holger Steeb Dipl.-Ing.
The study focuses on error estimation techniques for a coupled problem with two constituents based on the Theory of Porous Media. After developing space-time finite elements for this mixed problem, we extend the numerical scheme to a coupled space-time adaptive strategy. Therefore, an adjoint or dual problem is formulated and discussed, which is solved lateron numerically. One advantage of the presented technique is the high flexibility of the error indicator with respect to the error measure. [source]

Estimating spatial and parameter error in parameterized nonlinear reaction,diffusion equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2007
B. R. Carnes
Abstract A new approach is proposed for the a posteriori error estimation of both global spatial and parameter error in parameterized nonlinear reaction,diffusion problems. The technique is based on linear equations relating the linearized spatial and parameter error to the weak residual. Computable local element error indicators are derived for local contributions to the global spatial and parameter error, along with corresponding global error indicators. The effectiveness of the error indicators is demonstrated using model problems for the case of regular points and simple turning points. In addition, a new turning point predictor and adaptive algorithm for accurately computing turning points are introduced. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Hamiltonian-based error computations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2006
Y. L. Kuo
Abstract This paper presents two sets of the Hamiltonian for checking errors of approximated solutions. The first set can be applied to those problems having any number of independent and dependent variables. This set of the Hamiltonian can effectively indicate the errors of approximated solutions when requiring a high accuracy. The second set of the Hamiltonian has the invariant property when the Lagrangian is not an explicit function of time, even for non-conservative systems. Both sets can be formulated as error indicators to check errors of approximated solutions. Three illustrative examples demonstrate the error analyses of finite element solutions. Copyright © 2005 John Wiley & Sons, Ltd. [source]

An adaptive spacetime discontinuous Galerkin method for cohesive models of elastodynamic fracture

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
Reza Abedi
Abstract This paper describes an adaptive numerical framework for cohesive fracture models based on a spacetime discontinuous Galerkin (SDG) method for elastodynamics with elementwise momentum balance. Discontinuous basis functions and jump conditions written with respect to target traction values simplify the implementation of cohesive traction,separation laws in the SDG framework; no special cohesive elements or other algorithmic devices are required. We use unstructured spacetime grids in a h -adaptive implementation to adjust simultaneously the spatial and temporal resolutions. Two independent error indicators drive the adaptive refinement. One is a dissipation-based indicator that controls the accuracy of the solution in the bulk material; the second ensures the accuracy of the discrete rendering of the cohesive law. Applications of the SDG cohesive model to elastodynamic fracture demonstrate the effectiveness of the proposed method and reveal a new solution feature: an unexpected quasi-singular structure in the velocity response. Numerical examples demonstrate the use of adaptive analysis methods in resolving this structure, as well as its importance in reliable predictions of fracture kinetics. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators

Adaptive strategy of the supersonic turbulent flow over a backward-facing step

Adaptive finite element procedures for elastoplastic problems at finite strains

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
A. Koch Dipl.-Ing.
A major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step,size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step,size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density,function,based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples. [source]