DOI: 10.1016/s0045-7825(96)01132-2. Moving least-square reproducing kernel methods (I) Methodology and convergence

Moving least-square reproducing kernel methods (I) Methodology and convergence

10.1016/s0045-7825(96)01132-2

Crossref journal-article

Elsevier BV

Computer Methods in Applied Mechanics and Engineering (78)

Bibliography

Liu, W.-K., Li, S., & Belytschko, T. (1997). Moving least-square reproducing kernel methods (I) Methodology and convergence. Computer Methods in Applied Mechanics and Engineering, 143(1â2), 113â154.

Authors 3

Wing-Kam Liu (first)
Shaofan Li (additional)
Ted Belytschko (additional)

References 27 Referenced 437

10.1016/0021-9991(82)90025-0 / J. Comput. Phys. / Kernel estimates as a basis for general particle methods in hydrodynamics by Gingold (1982)
10.1137/0903027 / SIAM J. Scient. Stat. Comput. / Why particle methods work by Monaghan (1982)
10.1007/BF00364252 / Comput. Mech. / Generalizing the finite element method: Diffuse approximation and diffuse elements by Nayroles (1992)
10.1002/nme.1620370205 / Int. J. Numer. Methods Engrg. / Element free Galerkin methods by Belytschko (1994)
10.1088/0965-0393/2/3A/007 / Modelling Simul. Mater. Sci. Engrg. / Fracture and crack growth by element free Galerkin methods by Belytschko (1994)
10.1002/fld.1650200824 / Int. J. Numer. Methods Fluids / Reproducing kernel particle methods by Liu (1995)
10.1002/fld.1650211010 / Int. J. Numer. Methods Fluids / Wavelet and multiple scale reproducing kernel methods by Liu (1995)
10.1002/nme.1620381005 / Int. J. Numer. Methods Engrg. / Reproducing kernel particle methods for structural dynamics by Liu (1995)
{'issue': '1', 'key': '10.1016/S0045-7825(96)01132-2_BIB9', 'first-page': '3', 'article-title': 'An introduction to wavelet reproducing kernel particle methods', 'volume': '8', 'author': 'Liu', 'year': '1995', 'journal-title': 'USACM Bulletin'} / USACM Bulletin / An introduction to wavelet reproducing kernel particle methods by Liu (1995)
10.1090/S0025-5718-1981-0616367-1 / Math. Comput. / Surfaces generated by moving least square methods by Lancaster (1980)
10.1016/S0045-7825(96)01082-1 / Comput. Methods Appl. Mech. Engrg. / Moving least square reproducing kernel method. Part II: Fourier analysis by Li (1996)
{'key': '10.1016/S0045-7825(96)01132-2_BIB12', 'series-title': 'Wavelet interpolation and approximate solutions of elliptic partial differential equations', 'author': 'Wells', 'year': '1993'} / Wavelet interpolation and approximate solutions of elliptic partial differential equations by Wells (1993)
{'key': '10.1016/S0045-7825(96)01132-2_BIB13', 'series-title': 'Hp Clouds—a meshless method to solve boundary-value problems', 'author': 'Duarte', 'year': '1995'} / Hp Clouds—a meshless method to solve boundary-value problems by Duarte (1995)
{'key': '10.1016/S0045-7825(96)01132-2_BIB14', 'series-title': 'Principles of Mathematical Analysis', 'author': 'Rudin', 'year': '1976'} / Principles of Mathematical Analysis by Rudin (1976)
{'key': '10.1016/S0045-7825(96)01132-2_BIB15', 'series-title': 'Analytic Inequalities', 'author': 'Mitrinovic̀', 'year': '1970'} / Analytic Inequalities by Mitrinovic̀ (1970)
{'key': '10.1016/S0045-7825(96)01132-2_BIB16', 'series-title': 'Interpolation and Approximation', 'author': 'Davis', 'year': '1975'} / Interpolation and Approximation by Davis (1975)
{'key': '10.1016/S0045-7825(96)01132-2_BIB17', 'series-title': 'An Introduction to Wavelet', 'author': 'Chui', 'year': '1992'} / An Introduction to Wavelet by Chui (1992)
10.21236/ADA301760 / The partition of unity finite element method by Babuška (1995)
10.1016/0021-9045(73)90029-4 / J. Approx. Theory / Spline approximation by quasi-interpolants by de Boor (1975)
{'key': '10.1016/S0045-7825(96)01132-2_BIB20', 'series-title': 'Matrix Analysis', 'author': 'Horn', 'year': '1985'} / Matrix Analysis by Horn (1985)
{'key': '10.1016/S0045-7825(96)01132-2_BIB21', 'series-title': 'Matrix Computations', 'author': 'Golub', 'year': '1989'} / Matrix Computations by Golub (1989)
{'key': '10.1016/S0045-7825(96)01132-2_BIB22', 'series-title': 'The Finite Element Method For Elliptic Problems', 'author': 'Ciarlet', 'year': '1978'} / The Finite Element Method For Elliptic Problems by Ciarlet (1978)
10.1007/BF01436561 / Numer. Math. / The finite element method with Lagrangian multipliers by Babuška (1973)
{'key': '10.1016/S0045-7825(96)01132-2_BIB24', 'series-title': 'The Finite Element Method: Linear Static and Dynamic Finite Element Analysis', 'author': 'Hughes', 'year': '1987'} / The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by Hughes (1987)
{'key': '10.1016/S0045-7825(96)01132-2_BIB25', 'series-title': 'Mathematical Aspects of Finite Elements in Partial Differential Equations', 'article-title': 'An H−1-Galerkin procedure for the two-point boundary value', 'author': 'Rachford', 'year': '1974'} / Mathematical Aspects of Finite Elements in Partial Differential Equations / An H−1-Galerkin procedure for the two-point boundary value by Rachford (1974)
10.1016/0045-7825(77)90058-5 / Comput. Methods Appl. Mech. Engrg. / Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems by Babuška (1977)
{'key': '10.1016/S0045-7825(96)01132-2_BIB27', 'series-title': 'Wavelets, CBMS-NSF Series in Applied Mathematics', 'author': 'Daubechies', 'year': '1992'} / Wavelets, CBMS-NSF Series in Applied Mathematics by Daubechies (1992)

Dates

Type	When
Created	23 years, 1 month ago (July 25, 2002, 6:53 a.m.)
Deposited	6 years, 4 months ago (April 27, 2019, 12:57 p.m.)
Indexed	2 days, 1 hour ago (Sept. 4, 2025, 9:14 a.m.)
Issued	28 years, 5 months ago (April 1, 1997)
Published	28 years, 5 months ago (April 1, 1997)
Published Print	28 years, 5 months ago (April 1, 1997)

Funders 0

None

BibTeX

@article{Liu_1997, title={Moving least-square reproducing kernel methods (I) Methodology and convergence}, volume={143}, ISSN={0045-7825}, url={http://dx.doi.org/10.1016/s0045-7825(96)01132-2}, DOI={10.1016/s0045-7825(96)01132-2}, number={1–2}, journal={Computer Methods in Applied Mechanics and Engineering}, publisher={Elsevier BV}, author={Liu, Wing-Kam and Li, Shaofan and Belytschko, Ted}, year={1997}, month=apr, pages={113–154} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        9,
        4
      ]
    ],
    "date-time": "2025-09-04T13:14:21Z",
    "timestamp": 1756991661060
  },
  "reference-count": 27,
  "publisher": "Elsevier BV",
  "issue": "1-2",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            1997,
            4,
            1
          ]
        ],
        "date-time": "1997-04-01T00:00:00Z",
        "timestamp": 859852800000
      },
      "content-version": "tdm",
      "delay-in-days": 0,
      "URL": "https://www.elsevier.com/tdm/userlicense/1.0/"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1997,
        4
      ]
    ]
  },
  "DOI": "10.1016/s0045-7825(96)01132-2",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2002,
        7,
        25
      ]
    ],
    "date-time": "2002-07-25T10:53:54Z",
    "timestamp": 1027594434000
  },
  "page": "113-154",
  "source": "Crossref",
  "is-referenced-by-count": 437,
  "title": "Moving least-square reproducing kernel methods (I) Methodology and convergence",
  "prefix": "10.1016",
  "volume": "143",
  "author": [
    {
      "given": "Wing-Kam",
      "family": "Liu",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "Shaofan",
      "family": "Li",
      "sequence": "additional",
      "affiliation": []
    },
    {
      "given": "Ted",
      "family": "Belytschko",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "78",
  "reference": [
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB1",
      "doi-asserted-by": "crossref",
      "first-page": "429",
      "DOI": "10.1016/0021-9991(82)90025-0",
      "article-title": "Kernel estimates as a basis for general particle methods in hydrodynamics",
      "volume": "46",
      "author": "Gingold",
      "year": "1982",
      "journal-title": "J. Comput. Phys."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB2",
      "doi-asserted-by": "crossref",
      "first-page": "422",
      "DOI": "10.1137/0903027",
      "article-title": "Why particle methods work",
      "volume": "3",
      "author": "Monaghan",
      "year": "1982",
      "journal-title": "SIAM J. Scient. Stat. Comput."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB3",
      "doi-asserted-by": "crossref",
      "first-page": "307",
      "DOI": "10.1007/BF00364252",
      "article-title": "Generalizing the finite element method: Diffuse approximation and diffuse elements",
      "volume": "10",
      "author": "Nayroles",
      "year": "1992",
      "journal-title": "Comput. Mech."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB4",
      "doi-asserted-by": "crossref",
      "first-page": "229",
      "DOI": "10.1002/nme.1620370205",
      "article-title": "Element free Galerkin methods",
      "volume": "37",
      "author": "Belytschko",
      "year": "1994",
      "journal-title": "Int. J. Numer. Methods Engrg."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB5",
      "doi-asserted-by": "crossref",
      "first-page": "519",
      "DOI": "10.1088/0965-0393/2/3A/007",
      "article-title": "Fracture and crack growth by element free Galerkin methods",
      "volume": "2",
      "author": "Belytschko",
      "year": "1994",
      "journal-title": "Modelling Simul. Mater. Sci. Engrg."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB6",
      "doi-asserted-by": "crossref",
      "first-page": "1081",
      "DOI": "10.1002/fld.1650200824",
      "article-title": "Reproducing kernel particle methods",
      "volume": "20",
      "author": "Liu",
      "year": "1995",
      "journal-title": "Int. J. Numer. Methods Fluids"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB7",
      "doi-asserted-by": "crossref",
      "first-page": "901",
      "DOI": "10.1002/fld.1650211010",
      "article-title": "Wavelet and multiple scale reproducing kernel methods",
      "volume": "21",
      "author": "Liu",
      "year": "1995",
      "journal-title": "Int. J. Numer. Methods Fluids"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB8",
      "doi-asserted-by": "crossref",
      "first-page": "1655",
      "DOI": "10.1002/nme.1620381005",
      "article-title": "Reproducing kernel particle methods for structural dynamics",
      "volume": "38",
      "author": "Liu",
      "year": "1995",
      "journal-title": "Int. J. Numer. Methods Engrg."
    },
    {
      "issue": "1",
      "key": "10.1016/S0045-7825(96)01132-2_BIB9",
      "first-page": "3",
      "article-title": "An introduction to wavelet reproducing kernel particle methods",
      "volume": "8",
      "author": "Liu",
      "year": "1995",
      "journal-title": "USACM Bulletin"
    },
    {
      "issue": "155",
      "key": "10.1016/S0045-7825(96)01132-2_BIB10",
      "doi-asserted-by": "crossref",
      "first-page": "141",
      "DOI": "10.1090/S0025-5718-1981-0616367-1",
      "article-title": "Surfaces generated by moving least square methods",
      "volume": "37",
      "author": "Lancaster",
      "year": "1980",
      "journal-title": "Math. Comput."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB11",
      "doi-asserted-by": "crossref",
      "first-page": "159",
      "DOI": "10.1016/S0045-7825(96)01082-1",
      "article-title": "Moving least square reproducing kernel method. Part II: Fourier analysis",
      "volume": "139",
      "author": "Li",
      "year": "1996",
      "journal-title": "Comput. Methods Appl. Mech. Engrg."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB12",
      "series-title": "Wavelet interpolation and approximate solutions of elliptic partial differential equations",
      "author": "Wells",
      "year": "1993"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB13",
      "series-title": "Hp Clouds\u2014a meshless method to solve boundary-value problems",
      "author": "Duarte",
      "year": "1995"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB14",
      "series-title": "Principles of Mathematical Analysis",
      "author": "Rudin",
      "year": "1976"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB15",
      "series-title": "Analytic Inequalities",
      "author": "Mitrinovic\u0300",
      "year": "1970"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB16",
      "series-title": "Interpolation and Approximation",
      "author": "Davis",
      "year": "1975"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB17",
      "series-title": "An Introduction to Wavelet",
      "author": "Chui",
      "year": "1992"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB18",
      "doi-asserted-by": "crossref",
      "DOI": "10.21236/ADA301760",
      "article-title": "The partition of unity finite element method",
      "author": "Babu\u0161ka",
      "year": "1995"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB19",
      "doi-asserted-by": "crossref",
      "first-page": "19",
      "DOI": "10.1016/0021-9045(73)90029-4",
      "article-title": "Spline approximation by quasi-interpolants",
      "volume": "8",
      "author": "de Boor",
      "year": "1975",
      "journal-title": "J. Approx. Theory"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB20",
      "series-title": "Matrix Analysis",
      "author": "Horn",
      "year": "1985"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB21",
      "series-title": "Matrix Computations",
      "author": "Golub",
      "year": "1989"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB22",
      "series-title": "The Finite Element Method For Elliptic Problems",
      "author": "Ciarlet",
      "year": "1978"
    },
    {
      "issue": "20",
      "key": "10.1016/S0045-7825(96)01132-2_BIB23",
      "doi-asserted-by": "crossref",
      "first-page": "179",
      "DOI": "10.1007/BF01436561",
      "article-title": "The finite element method with Lagrangian multipliers",
      "author": "Babu\u0161ka",
      "year": "1973",
      "journal-title": "Numer. Math."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB24",
      "series-title": "The Finite Element Method: Linear Static and Dynamic Finite Element Analysis",
      "author": "Hughes",
      "year": "1987"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB25",
      "series-title": "Mathematical Aspects of Finite Elements in Partial Differential Equations",
      "article-title": "An H\u22121-Galerkin procedure for the two-point boundary value",
      "author": "Rachford",
      "year": "1974"
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB26",
      "doi-asserted-by": "crossref",
      "first-page": "175",
      "DOI": "10.1016/0045-7825(77)90058-5",
      "article-title": "Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems",
      "volume": "11",
      "author": "Babu\u0161ka",
      "year": "1977",
      "journal-title": "Comput. Methods Appl. Mech. Engrg."
    },
    {
      "key": "10.1016/S0045-7825(96)01132-2_BIB27",
      "series-title": "Wavelets, CBMS-NSF Series in Applied Mathematics",
      "author": "Daubechies",
      "year": "1992"
    }
  ],
  "container-title": "Computer Methods in Applied Mechanics and Engineering",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://api.elsevier.com/content/article/PII:S0045782596011322?httpAccept=text/xml",
      "content-type": "text/xml",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "https://api.elsevier.com/content/article/PII:S0045782596011322?httpAccept=text/plain",
      "content-type": "text/plain",
      "content-version": "vor",
      "intended-application": "text-mining"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2019,
        4,
        27
      ]
    ],
    "date-time": "2019-04-27T16:57:08Z",
    "timestamp": 1556384228000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://linkinghub.elsevier.com/retrieve/pii/S0045782596011322"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1997,
        4
      ]
    ]
  },
  "references-count": 27,
  "journal-issue": {
    "issue": "1-2",
    "published-print": {
      "date-parts": [
        [
          1997,
          4
        ]
      ]
    }
  },
  "alternative-id": [
    "S0045782596011322"
  ],
  "URL": "http://dx.doi.org/10.1016/S0045-7825(96)01132-2",
  "relation": {},
  "ISSN": [
    "0045-7825"
  ],
  "subject": [],
  "container-title-short": "Computer Methods in Applied Mechanics and Engineering",
  "published": {
    "date-parts": [
      [
        1997,
        4
      ]
    ]
  }
}