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  • Articles  (19)
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  • error estimation  (19)
  • Wiley-Blackwell  (19)
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  • Institute of Mathematics, Polish Academy of Sciences / Instytut Matematyczny, Polska Akademia Nauk
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  • Articles  (19)
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  • Wiley-Blackwell  (19)
  • American Chemical Society (ACS)
  • Institute of Mathematics, Polish Academy of Sciences / Instytut Matematyczny, Polska Akademia Nauk
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  • 1
    Electronic Resource
    Electronic Resource
    Analytical solutions of mass transport problems for error estimation of finite/infinite element methods (1995)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 11 (1995), S. 13-23 
    Details
    ISSN: 1069-8299
    Keywords: mass transport problems ; error estimation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In order to make an exact estimation of discretization error for finite/infinite element methods, it is necessary to develop analytical solutions for some transient mass transport problems in infinite media. These transient mass transport problems may be viewed as the benchmark problems for the discretization error estimation of a new numerical method so that they generally have the following characteristics: (1) their initial and boundary conditions can be exactly modelled by the finite/infinite element method; (2) their solutions can be rigorously expressed in a closed form. In this paper, several of the aforementioned problems have been constructed and solved mathematically for transient mass transport problems in both 1D and 2D infinite media.
    Additional Material: 7 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640110104
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  • 2
    Electronic Resource
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    The h-, p- and hp-versions of the BEM in elasticity: Numerical results (1995)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 11 (1995), S. 255-265 
    Details
    ISSN: 1069-8299
    Keywords: boundary element method ; error estimation ; hp-version ; Galerkin BEM ; numerical results ; elasticity ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper investigates the convergence of the h-, p- and hp-versions of a variational symmetric boundary element method (BEM) in plane elasticity by numerical experiments. The study discusses mixed boundary value problems on polygonal domains, i.e. problems for which the exact solution is analytic except in a finite number of points. The convergence of the error in energy norm is displayed for all versions of the BEM. All results are also compared with those obtained by the corresponding finite element methods. The theoretically predicted asymptotic convergence rates for all versions of the method can be observed in the numerical experiments. A comparison of computer times is given as well.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640110308
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  • 3
    Electronic Resource
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    Superconvergent patch recovery technique for the finite element method in acoustics (1996)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 12 (1996), S. 581-594 
    Details
    ISSN: 1069-8299
    Keywords: acoustics ; finite element method ; error estimation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A posteriori error estimation has become very popular, mainly in linear elasticity. A robust implementation of the superconvergent patch recovery technique of O. C. Zienkiewicz and J. Z. Zhu is presented for acoustic finite element analyses: the original concepts are extended to complex variables, and both local and global behaviours of the recovery procedure and the error estimation are studied. The numerical tests confirm the improvement of the rates of convergence for the recovered solution and also show the reliability of the error estimator except at frequencies corresponding either to the analytical or to the finite element eigenfrequencies.
    Additional Material: 13 Ill.
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  • 4
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    SUPERCONVERGENT PATCH RECOVERY IN PROBLEMS OF MIXED FORM (1997)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 13 (1997), S. 207-217 
    Details
    ISSN: 1069-8299
    Keywords: plates ; finite elements ; mixed method ; error estimation ; superconvergent patch recovery ; adaptivity ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The superconvergent patch recovery (SPR technique) has been shown to be an effective postprocessing procedure in which an improved solution is obtained based on the original finite element solution. The technique was applied to second order problems with success, but the method has not yet been studied on problems posed in mixed form. This paper demonstrates that the technique can be applied to beam and plate bending problems, characterized by fourth order differential equations. The differential equation is here written as two coupled differential equations of second order leading to a mixed finite element procedure based on approximations of the moment and displacement fields. Two elements of mixed type are handled, namely a triangular plate element with constant moment field and a rectangular element with linearly varying bending moments. Numerical examples are given to show that the postprocessed solution is more accurate and has a higher rate of convergence. © 1997 by John Wiley & Sons, Ltd.
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  • 5
    Electronic Resource
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    CRITERIA TO ACHIEVE NEARLY OPTIMAL MESHES IN THE h-ADAPTIVE FINITE ELEMENT METHOD (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 4039-4061 
    Details
    ISSN: 0029-5981
    Keywords: finite element method ; optimal mesh ; strategy ; refinement ; error estimation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Five Adaptive Modification Strategies (AMSs) are studied as means to obtain nearly optimal meshes using an adaptive finite element modelling system based on the h-version of the FEM. These strategies include the method traditionally seem in the literature as well as four additional methods. The five strategies are tested over five numerical examples, one of them producing convergence oscillations in one example. The principal characteristic of our system is that it permits the user to control certain parameters of the adaptive process, which contributes to an improvement in the results.
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  • 6
    Electronic Resource
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    Error estimation in adaptive BEM by postprocessing interpolation (1998)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 14 (1998), S. 633-645 
    Details
    ISSN: 1069-8299
    Keywords: error estimation ; boundary elements ; adaptive mesh ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A posteriori error estimation for the boundary element method is developed. The nodal error is estimated from the difference of two solutions - one from the original BEM solution, and the other from interpolation of the original solution. Based on nodal errors, a matrix analysis is carried out to calculate the corresponding errors at source points. Two formulae for estimating global percentage errors are proposed in the paper. The first formula uses nodal errors directly to estimate element error, and the second uses an integral form to calculate element error to eliminate extremely high mesh concentration near areas with singular solutions. An h-version adaptive mesh refinement is implemented to study the accuracy of the proposed error estimation. Numerical examples show that the error estimator can correctly guide mesh refinement, and a final mesh can be obtained in a few iterations. © 1998 John Wiley & Sons, Ltd.
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  • 7
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    On understanding and teaching of the finite element method (1995)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 11 (1995), S. 105-115 
    Details
    ISSN: 1069-8299
    Keywords: finite element method ; conceptual diagrams ; error estimation ; postprocessing ; teaching ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The success of the finite element method depends on the highly systematic way the analysis is built up, which means that the structure of the theoretical description and the code is transferable between different applications. It is advisable to teach the subject in such a way that this fact is fully exploited. It is a fact that most FE calculations today do not contain a check of the accuracy/quality of the solution. In the teaching of the FE method nothing is more important than the basic knowledge and understanding of the properties and behaviour of the obtained FE solution. The paper deals with the methodology in presenting the method, by use of conceptual diagrams. It also discusses the interpretation of the results from a classical FE analysis and the extraction of high quality information. Finally, error estimation and postprocessing are discussed. The detailed description of the boundary conditions is lost in the FE solution. In the postprocessing stage they should be taken into account once more.
    Additional Material: 9 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640110204
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  • 8
    Electronic Resource
    Electronic Resource
    Evaluation of some error estimators for the four-noded lagrangian quadrilateral (1995)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 11 (1995), S. 497-506 
    Details
    ISSN: 1069-8299
    Keywords: error estimation ; super convergent patch recovery scheme ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A comparative study of error estimators using a patch recovery scheme with those using simple nodal averaging is made for the four-noded Lagrangian quadrilateral element through two plane stress elasticity problems. It is demonstrated that error estimators using a patch recovery scheme are generally more effective and have an estimated stress field that is closer to the exact one than those using simple nodal averaging.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640110604
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  • 9
    Electronic Resource
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    On solving nearly incompressible 2D problems using an adaptive refinement procedure (1998)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 14 (1998), S. 409-418 
    Details
    ISSN: 1069-8299
    Keywords: adaptive refinement procedure ; nearly incompressible problems ; error estimation ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A numerical study is carried out employing some selected 2D elements and an automatic adaptive refinement procedure for the solution of a problem involving a nearly incompressible material. For a material with Poisson's ratio close to 0·5, it is found that both hybrid elements and high-order Lagrangian triangles can be used to provide a reliable solution. Furthermore, the rate of convergence of the elements is not affected by the high value of Poisson's ratio, and a near-optimal convergence rate is achieved in all the numerical tests run. © 1998 John Wiley & Sons, Ltd.
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  • 10
    Electronic Resource
    Electronic Resource
    A 3-D ADAPTIVE MESH REFINEMENT USING VARIABLE-NODE SOLID TRANSITION ELEMENTS (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 1585-1606 
    Details
    ISSN: 0029-5981
    Keywords: adaptive mesh refinement ; solid transition element ; non-conforming modes ; superconvergent patch recovery ; error estimation ; neighbouring element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An automated three-dimensional adaptive h-refinement strategy using the solid transition elements with variable midside nodes at edges and faces of the element is presented. The basic behaviour of these transition elements were improved by addition of associated non-conforming modes. By introducing these transition elements, some difficulties associated with imposing displacement constraints on irregular nodes to enforce interelement compatibility in the conventional adaptive h-refinement are easily overcome. A superconvergent patch recovery technique is also extended to three-dimensional problem. Numerical examples show the effectiveness of the proposed adaptive mesh refinement scheme using transition elements.
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  • 11
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    CONSISTENT LOCAL ERROR COMPUTATIONS. PART II: FE APPLICATIONS (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 3235-3246 
    Details
    ISSN: 0029-5981
    Keywords: finite elements ; error estimation ; invariants ; mesh refinement ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The notion of invariants associated with differential equations introduced in Part I is used to compute consistent local errors in a finite-element analysis. The procedure is illustrated by two simple examples - a one-dimensional beam problem and a two-dimensional membrane problem. It is also shown how the computed errors may be used for adaptive mesh refinement. However, the primary intent of this paper is to introduce a new concept for calculation of local errors.
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  • 12
    Electronic Resource
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    CONSISTENT LOCAL ERROR COMPUTATIONS PART I: THEORY (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 3219-3234 
    Details
    ISSN: 0029-5981
    Keywords: computational mechanics ; error estimation ; invariants ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In Part I of this paper we examine the notion of invariant quantities in problems encountered in computational mechanics. These invariant quantities are then used to verify results obtained by numerical techniques. Examples in rigid-body dynamics, elastostatics and unsteady-state heat flow are presented. In Part II, we use the invariant quantities to compute consistent local errors in a finite-element analysis.
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  • 13
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    EXTENSION OF THE ZIENKIEWICZ-ZHU ERROR ESTIMATOR TO SHAPE SENSITIVITY ANALYSIS (1997)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 40 (1997), S. 1413-1433 
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    ISSN: 0029-5981
    Keywords: finite element method ; adaptive refinement ; shape sensitivity analysis ; error estimation ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The application of the Zienkiewicz-Zhu estimator was extended to the estimation of the discretization error arising from shape sensitivity analysis using the finite element method. The sensitivity error was quantified from the sensitivity of the energy norm by using an estimator specially developed for this purpose. Sensitivity analyses were carried out using the discrete analytical approach, which introduced no additional errors other than the discretization error. In this work, direct nodal averaging was used for linear triangular elements and the SPR technique for quadratic elements in order to obtain the smoothed stress and sensitivities fields. Two examples with an exact solution are used to analyse the effectivity of the proposed estimator and its convergence with the h-adaptive refinement. © 1997 by John Wiley & Sons, Ltd.
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  • 14
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    On using different finite elements with an automatic adaptive refinement procedure for the solution of 2-D stress analysis problems (1997)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 40 (1997), S. 4547-4576 
    Details
    ISSN: 0029-5981
    Keywords: adaptive refinement procedure ; error estimation ; 2-D stress analysis ; performance of different 2-D elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A series of numerical tests is carried out employing some commonly used finite elements for the solution of 2-D elastostatic stress analysis problems with an automatic adaptive refinement procedure. Different kinds of elements including Lagrangian quadrilateral and triangular elements, serendipity quadrilaterals, incompatible elements and hybrid elements have been tested. It is found that for a general problem involving compressible material and when a moderate accuracy of the final solution is sought, the nine-node Lagrangian (L9) element will be the most effective element, while when an extremely accurate solution is needed, higher order Lagrangian quadrilaterals or triangles will be a suitable choice. However, if only linear elements are available, the well known 5βI linear hybrid element is the best choice. © 1997 John Wiley & Sons, Ltd.
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  • 15
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    Error estimation based on Superconvergent Patch Recovery using statically admissible stress fields (1998)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 42 (1998), S. 443-472 
    Details
    ISSN: 0029-5981
    Keywords: finite element method ; error estimation ; stress recovery ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper we investigate an approach for a posteriori error estimation based on recovery of an improved stress field. The qualitative properties of the recovered stress field necessary to obtain a conservative error estimator, i.e. an upper bound on the true error, are given. A specific procedure for recovery of an improved stress field is then developed. The procedure can be classified as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of the interior equilibrium and the natural boundary conditions. Herein the interior equilibrium is satisfied a priori within each nodal patch. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative estimate. The performance of the developed error estimator is illustrated by investigating two plane strain problems with known closed-form solutions. © 1998 John Wiley & Sons, Ltd.
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  • 16
    Electronic Resource
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    Upper and lower bounds for local electromagnetic quantities (1998)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 42 (1998), S. 499-515 
    Details
    ISSN: 0029-5981
    Keywords: error estimation ; constitutive error formulation ; complementary solutions ; numerical methods ; finite element methods ; virtual work principle ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Most engineering problems are solved by means of numerical methods that are able to provide only approximate solutions, for which it would be extremely useful to have efficient error estimators.Upper and lower bounds for quantities of integral character, like the stored magnetic energy or the ohmic power dissipated in the domain of interest, had been clearly established along with the procedures to obtain them numerically.However, upper and lower bounds for local quantities would be of paramount interest in several fields of applications like Non-Destructive Testing or Nuclear Magnetic Resonance.We present here a procedure for the determination of upper and lower bounds of local field quantities, namely the average value of a field component in an arbitrarily small region. It is based on the introduction of an auxiliary field, and is the natural extension of the method establishing the bounds of global quantities.Our technique can be used for any linear system in stationary conditions for which a virtual work principle can be applied. Its efficiency is demonstrated with the analysis of some stationary 2D and 3D electromagnetic problems. © 1998 John Wiley & Sons, Ltd.
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  • 17
    Electronic Resource
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    An improved REP recovery and the effectivity robustness test (1997)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 40 (1997), S. 3247-3277 
    Details
    ISSN: 0029-5981
    Keywords: recovery ; patch ; adaptivity ; error estimation ; robustness test ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We present in this paper a modified form of the REP gradient recovery process recently published. This new form is not only cheaper but has a much improved performance - which equals and occasionally exceeds the performance of the SPR (Superconvergent Patch Recovery) method.The comparisons are based on the robustness test originally proposed by Babuška et al. This test is described briefly in a manner suitable for those who find some aspects of modern mathematics difficult to follow.The result and comparison of the tests for various repeatable patches with regular and irregular element distribution are made with SPR and the ‘old’ form of REP and are based on the general error estimator introduced by Zienkiewicz and Zhu in 1987. © 1997 John Wiley & Sons, Ltd.
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  • 18
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    A BENCHMARK COMPUTATIONAL STUDY OF FINITE ELEMENT ERROR ESTIMATION (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 3969-3983 
    Details
    ISSN: 0029-5981
    Keywords: benchmark solution ; error estimation ; finite elements ; mesh optimization ; stress smoothing ; adaptive mesh refinement ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Benchmark solutions are presented for a simple linear elastic boundary value problem, as analysed using a range of finite element mesh configurations. For each configuration, various estimates of local (i.e. element) and global discretization error have been computed. These show that the optimal mesh corresponds not only to minimization of global energy (or L2) norms of the error, but also to equalization of element errors as well. Hence, this demonstrates why element error equalization proves successful as a criterion for guiding the process of mesh refinement in mesh adaptivity. The results also demonstrate the effectiveness of the stress projection method for smoothing discontinuous stress fields which, for this investigation, are more extreme as a consequence of the assumption of nearly incompressible material behaviour. In this case, lower order smoothing produces a continuous stress field which is in close agreement with the exact solution.
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  • 19
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    Generalized conservation checks in finite element analysis (1998)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 43 (1998), S. 1295-1307 
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    ISSN: 0029-5981
    Keywords: energy conservation ; mesh refinement ; error estimation ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: By generalizing the conservation of energy statement in classical mechanics, a useful and relatively simple approach has been developed for checking the conservation requirements in finite element analysis. By way of illustration several examples are provided from fields of stress analysis, heat transfer and fluid flow. © 1998 John Wiley & Sons, Ltd.
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