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Solution Errors (solution + error)

Distribution by Scientific Domains

Engineering	80%

Selected Abstracts

On the spectrum of the electric field integral equation and the convergence of the moment method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2001
Karl F. Warnick
Abstract Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self-coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low-order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2001
Owe Axelsson
Abstract Some techniques suitable for the control of the solution error in the preconditioned conjugate gradient method are considered and compared. The estimation can be performed both in the course of the iterations and after their termination. The importance of such techniques follows from the non-existence of some reasonable a priori error estimate for very ill-conditioned linear systems when sufficient information about the right-hand side vector is lacking. Hence, some a posteriori estimates are required, which make it possible to verify the quality of the solution obtained for a prescribed right-hand side. The performance of the considered error control procedures is demonstrated using real-world large-scale linear systems arising in computational mechanics. Copyright © 2001 John Wiley & Sons, Ltd. [source]

A modified node-to-segment algorithm passing the contact patch test

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Giorgio Zavarise
Abstract Several investigations have shown that the classical one-pass node-to-segment (NTS) algorithms for the enforcement of contact constraints fail the contact patch test. This implies that the algorithms may introduce solution errors at the contacting surfaces, and these errors do not necessarily decrease with mesh refinement. The previous research has mainly focused on the Lagrange multiplier method to exactly enforce the contact geometry conditions. The situation is even worse with the penalty method, due to its inherent approximation that yields a solution affected by a non-zero penetration. The aim of this study is to analyze and improve the contact patch test behavior of the one-pass NTS algorithm used in conjunction with the penalty method for 2D frictionless contact. The paper deals with the case of linear elements. For this purpose, several sequential modifications of the basic formulation have been considered, which yield incremental improvements in results of the contact patch test. The final proposed formulation is a modified one-pass NTS algorithm which is able to pass the contact patch test also if used in conjunction with the penalty method. In other words, this algorithm is able to correctly reproduce the transfer of a constant contact pressure with a constant proportional penetration. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A study on the convergence of least-squares meshfree method under inaccurate integration

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2003
Sang-Hoon Park
Abstract In the authors' previous work, it has been shown through numerical examples that the least-squares meshfree method (LSMFM) is highly robust to the integration errors while the Galerkin meshfree method is very sensitive to them. A mathematical study on the convergence of the solution of LSMFM under inaccurate integration is presented. New measures are introduced to take into account the integration errors in the error estimates. It is shown that, in LSMFM, solution errors are bounded by approximation errors even when integration is not accurate. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Positive-definite q -families of continuous subcell Darcy-flux CVD(MPFA) finite-volume schemes and the mixed finite element method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008
Michael G. Edwards
Abstract A new family of locally conservative cell-centred flux-continuous schemes is presented for solving the porous media general-tensor pressure equation. A general geometry-permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control-volume distributed subcell flux-continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2:259,290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive-definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical-space schemes are shown to be non-symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non-symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical-space and subcell-space q -families of schemes. M -matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical-space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell-wise constant tensor schemes and that subcell tensor approximation using the control-volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd. [source]