Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

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    Abstract

    In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.
    Original languageEnglish
    Pages (from-to)657-675
    Number of pages19
    JournalMathematics of Computation
    Volume72
    Issue number242
    DOIs
    Publication statusPublished - 2003

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    Numerical analysis
    Numerical Analysis
    Theoretical Analysis
    Continuum
    One-dimensional Model
    Approximation
    Interaction
    Engineering
    Ill-posedness
    Simplification
    Nearest Neighbor
    Numerical Methods
    Model
    Finite Element
    Numerical methods
    Atoms
    Estimate
    Community

    Keywords

    • Continuum mechanics
    • Material modelling
    • Finite element method

    Cite this

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    title = "Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model",
    abstract = "In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.",
    keywords = "Continuum mechanics, Material modelling, Finite element method",
    author = "Ping Lin",
    note = "dc.publisher: American Mathematical Society dc.title.alternative: Theoretical and numerical analysis of a material particle model",
    year = "2003",
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    }

    Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. / Lin, Ping.

    In: Mathematics of Computation, Vol. 72, No. 242, 2003, p. 657-675.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

    AU - Lin, Ping

    N1 - dc.publisher: American Mathematical Society dc.title.alternative: Theoretical and numerical analysis of a material particle model

    PY - 2003

    Y1 - 2003

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    AB - In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

    KW - Continuum mechanics

    KW - Material modelling

    KW - Finite element method

    U2 - 10.1090/S0025-5718-02-01456-4

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