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interpolation requirements for implicit gradient-enhanced continuum damage models

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING 2003年第7期19卷 563-572页

作者： Simone, A Askes, H Peerlings, RHJ Sluys, LJ Delft Univ Technol Fac Civil Engn & Geosci NL-2600 GA Delft Netherlands Eindhoven Univ Technol Dept Mech Engn NL-5600 MB Eindhoven Netherlands

Some properties of a gradient-enhanced continuum damage model were discussed by means of a numerical and theoretical study. In particular, it is demonstrated that the discretization of this model does not belong to the class of mixed finite element formulations (although the model has the format of a general coupled problem). Hence, the Babuska-Brezzi condition does not apply to this type of models. Consequently, the use of linear interpolation functions for displacements as well for non-local equivalent strains yields excellent performance in terms of rate of convergence and of convergence of the numerical procedure. Copyright (C) 2003 John Wiley Sons, Ltd.

关键词： gradient-enhanced continuum models Babuska-Brezzi condition stress oscillations interpolation functions mixed methods coupled problems

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interpolation representation of feedforward neural networks

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MATHEMATICAL AND COMPUTER MODELLING 2003年第7-8期37卷 829-847页

作者： Li, HX Li, LX Wang, JY Beijing Normal Univ Dept Math Beijing 100875 Peoples R China Sichuan Normal Univ Dept Math Chengdu 610066 Peoples R China Tsing Hua Univ Dept Automat Beijing 100084 Peoples R China Old Dominion Univ Dept Informat Syst & Decis Sci Norfolk VA 23529 USA Beijing Normal Univ Dept Math Beijing 100875 Peoples R China

Mathematical essence and structures of feedforward neural networks are researched in detail in this paper. First of all, interpolation mechanism of feedforward neural networks is exposed, so we can more clearly understand why a feedforward network is of approximation. For example, a well-known conclusion for arbitrarily a continuous function, there exists a three-layer forward neural network such that the network can approximate the function to within any given precision. It, in fact, is regarded as a natural result of interpolation representation. Then the learning algorithms of feedforward neural networks are discussed by some new ideas. (C) 2003 Elsevier Science Ltd. All rights reserved.

关键词： neural networks fuzzy neural networks interpolation functions mathematical neurons learning algorithms

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Optimum interpolation functions for boundary element method

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2000年第2期24卷 189-200页

作者： Vable, M Ammons, BA Fox, ME Michigan Technol Univ Houghton MI 49931 USA Dateppli Inc Midland MI 48642 USA

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal Values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L-1 norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function. (C) 2000 Elsevier S

关键词： boundary element method interpolation functions numerical error adaptive analysis

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Minimizing the error near discontinuities in boundary element method

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2001年第7期25卷 607-617页

作者： Vable, M Michigan Technol Univ Houghton MI 49931 USA

This paper presents two types of discontinuity modeling elements (DME) that minimize the error near a discontinuity. The DME are elements that are discontinuous at one end, can satisfy continuity requirement up to seventh order at the other end, and may have polynomials of order up to fifteen. The error of approximation in the density function is measured by the L-1 norm, which is minimized with respect to the location of collocation points. Results show that the error for optimum location of collocation points in all cases is smaller than those for uniform location of collocation points and the differences in accuracy grows significantly with the order of polynomials. Two tables report the optimum location of collocation points for the DME for use by other researchers. The DME are used in conjunction with the hr-mesh refinement scheme to study modeling of stress near a stress discontinuity. Results of the study show that the recommendations for modeling density functions near a discontinuity are diametrically opposite to those recommendations for modeling of a smooth density function. (C) 2001 Elsevier Science Ltd. All rights reserved.

关键词： boundary element methods interpolation functions numerical error adaptive analysis

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Automatic derivation of positivity conditions inside boundary elements with the help of the REDLOG computer logic package

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 1999年第10期23卷 847-856页

作者： Ioakimidis, NI Univ Patras Sch Engn Div Appl Math & Mech GR-26110 Patras Greece

The very well-known shape functions are used in classical boundary element analysis for the construction of the polynomial interpolation function p for the approximation to the unknown field quantity u. In this note, we show the usefulness of the recent REDLOG Reduce computer logic package of Dolzmann and Sturm for the construction of positivity and analogous parametric feasibility conditions, based on the nodal values of the polynomial p, inside the whole boundary element. A simple one-dimensional case, based on the classical quadratic element, and a more difficult two-dimensional case, based on a quadratic triangular element, are used for the illustration of the approach, whereas several related conclusions are also drawn. The present results are applicable to cases where the positivity or the negativity or just the boundedness of the unknown quantity is required (such as the case of the pressure distribution in contact problems and the opening displacement in crack problems) and significantly extend the already available recent related computational quantifier elimination research results. (C) 2000 Elsevier Science Ltd. All rights reserved.

关键词： boundary elements computer algebra computer logic feasibility conditions inequality constraints interpolation functions positivity/negativity conditions quadratic elements quantifier elimination triangular elements

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Comparative study of finite element formulations for the semiconductor drift-diffusion equations

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International Journal for Numerical Methods in Engineering 1999年第18期40卷

作者： J. T. Trattles C. M. Johnson Department of Electrical and Electronic Engineering Merz Court University of Newcastle upon Tyne Newcastle upon Tyne NE1 7RU U.K.

A number of transient and steady-state finite element formulations of the semiconductor drift-diffusion equations are studied and compared with respect to their accuracy and efficiency on a simple test structure (the Mock diode). A new formulation, with a consistent interpolation function used to represent the electron and hole carrier densities throughout the set of semiconductor drift-diffusion and Poisson's equations, is introduced. Results highlight the advantages in using consistent interpolation functions showing an increased accuracy in the calculated values and a saving in data storage and execution time. The results also illustrate how the use of different time integration methods affect the number of time steps required during transient simulations. The combination of the fully consistent DFUS with appropriate time integration methods is found to yield a saving of up to 80 per cent of the execution time required for standard spatial/temporal discretization techniques. © 1997 by John Wiley & Sons, Ltd.

关键词： finite element formulations semiconductor equations upwinding schemes time integration methods interpolation functions

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Sequential function approximation for the solution of differential equations

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International Journal for Numerical Methods in Biomedical Engineering 1998年第12期13卷

作者： Andrew J. Meade Jr. Michael Kokkolaras Boris A. Zeldin Department of Mechanical Engineering and Materials Science William Marsh Rice University Houston TX 77251-1892 U.S.A.

A computational method for the solution of differential equations is proposed. With this method an accurate approximation is built by incremental additions of optimal local basis functions. The parallel direct search software package (PDS), that supports parallel objective function evaluations, is used to solve the associated optimization problem efficiently. The advantage of the method is that, although it resembles adaptive methods in computational mechanics, an a priori grid is not necessary. Moreover, the traditional matrix construction and evaluations are avoided. Computational cost is reduced while efficiency is enhanced by the low-dimensional parallel-executed optimization and parallel function evaluations. In addition, the method should be applicable to a broad class of interpolation functions. Results and global convergence rates obtained for one- and two-dimensional boundary value problems are satisfactorily compared to those obtained by the conventional Galerkin finite element method. © 1997 John Wiley & Sons, Ltd.

关键词： sequential function approximation interpolation functions optimization parallel direct search

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An improved element free Galerkin formulation

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International Journal for Numerical Methods in Engineering 1998年第16期40卷

作者： Igor Kaljević Sunil Saigal AYT Corporation 2001 Aerospace Parkway Brook Park OH 44142 U.S.A. Department of Civil and Environmental Engineering Carnegie Mellon University Pittsburgh PA 15213 U.S.A.

An improved formulation of the Element-Free Galerkin Method (EFGM) is presented in this paper. The major shortcoming of the conventional EFGM has been that, due to the use of Moving Least-Squares (MLS) approximation, it does not allow the explicit prescription of boundary conditions. Lagrange multipliers have been employed to circumvent this problem undermining the attractiveness of the method. The proposed EFGM formulation eliminates this shortcoming through the use of a set of MLS interpolants that employ singular weight functions. The validity, accuracy and efficiency of the present formulation are demonstrated through the solution of a variety of example problems. © 1997 John Wiley & Sons, Ltd.

关键词： interpolation functions cell subdivision variable domain of influence

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Sequential function approximation for the solution of differential equations

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING 1997年第12期13卷 977-986页

作者： Meade, AJ Kokkolaras, M Zeldin, BA Rice Univ Dept Mech Engn & Mat Sci Houston TX 77251 USA

A computational method for the solution of differential equations is proposed. With this method an accurate approximation is built by incremental additions of optimal local basis functions. The parallel direct search software package (PDS), that supports parallel objective function evaluations, is used to solve the associated optimization problem efficiently. The advantage of the method is that, although it resembles adaptive methods in computational mechanics, an a priori grid is not necessary. Moreover, the traditional matrix construction and evaluations are avoided. Computational cost is reduced while efficiency is enhanced by the low-dimensional parallel-executed optimization and parallel function evaluations. In addition, the method should be applicable to a broad class of interpolation functions. Results and global convergence rates obtained for one-and two-dimensional boundary value problems are satisfactorily compared to those obtained by the conventional Galerkin finite element method. (C) 1997 John Wiley & Sons, Ltd.

关键词： sequential function approximation interpolation functions optimization parallel direct search

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Comparative study of finite element formulations for the semiconductor drift-diffusion equations

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 1997年第18期40卷 3405-3419页

作者： Trattles, JT Johnson, CM UNIV NEWCASTLE UPON TYNE DEPT ELECT & ELECT ENGNNEWCASTLE TYNE NE1 7RUTYNE & WEARENGLAND

关键词： finite element formulations semiconductor equations upwinding schemes time integration methods interpolation functions

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