New shape functions for arbitrary discontinuities without additional unknowns

Thomas Peter Fries, Ted Belytschko

Research output: Chapter in Book/Report/Conference proceedingChapterResearchpeer-review

Abstract

A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.

Original languageEnglish
Title of host publicationMeshfree Methods for Partial Differential Equations III
Pages87-103
Number of pages17
Volume57
DOIs
Publication statusPublished - 2007

Publication series

NameLecture Notes in Computational Science and Engineering
Volume57
ISSN (Print)14397358

Fingerprint

Shape Function
Discontinuity
Unknown
Arbitrary
Mesh
Moving Least Squares
Extended Finite Element Method
Partition of Unity
Approximation Space
Level Set Method
Special Functions
Weight Function
Finite Element
Finite element method

Keywords

  • Discontinuity
  • Moving least-squares
  • Partition of unity

ASJC Scopus subject areas

  • Engineering(all)
  • Computational Mathematics
  • Modelling and Simulation
  • Control and Optimization
  • Discrete Mathematics and Combinatorics

Cite this

Fries, T. P., & Belytschko, T. (2007). New shape functions for arbitrary discontinuities without additional unknowns. In Meshfree Methods for Partial Differential Equations III (Vol. 57, pp. 87-103). (Lecture Notes in Computational Science and Engineering; Vol. 57). https://doi.org/10.1007/978-3-540-46222-4_6

New shape functions for arbitrary discontinuities without additional unknowns. / Fries, Thomas Peter; Belytschko, Ted.

Meshfree Methods for Partial Differential Equations III. Vol. 57 2007. p. 87-103 (Lecture Notes in Computational Science and Engineering; Vol. 57).

Research output: Chapter in Book/Report/Conference proceedingChapterResearchpeer-review

Fries, TP & Belytschko, T 2007, New shape functions for arbitrary discontinuities without additional unknowns. in Meshfree Methods for Partial Differential Equations III. vol. 57, Lecture Notes in Computational Science and Engineering, vol. 57, pp. 87-103. https://doi.org/10.1007/978-3-540-46222-4_6
Fries TP, Belytschko T. New shape functions for arbitrary discontinuities without additional unknowns. In Meshfree Methods for Partial Differential Equations III. Vol. 57. 2007. p. 87-103. (Lecture Notes in Computational Science and Engineering). https://doi.org/10.1007/978-3-540-46222-4_6
Fries, Thomas Peter ; Belytschko, Ted. / New shape functions for arbitrary discontinuities without additional unknowns. Meshfree Methods for Partial Differential Equations III. Vol. 57 2007. pp. 87-103 (Lecture Notes in Computational Science and Engineering).
@inbook{5169cdbb4a0245198b6334d6ca867a03,
title = "New shape functions for arbitrary discontinuities without additional unknowns",
abstract = "A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.",
keywords = "Discontinuity, Moving least-squares, Partition of unity",
author = "Fries, {Thomas Peter} and Ted Belytschko",
year = "2007",
doi = "10.1007/978-3-540-46222-4_6",
language = "English",
isbn = "9783540462149",
volume = "57",
series = "Lecture Notes in Computational Science and Engineering",
pages = "87--103",
booktitle = "Meshfree Methods for Partial Differential Equations III",

}

TY - CHAP

T1 - New shape functions for arbitrary discontinuities without additional unknowns

AU - Fries, Thomas Peter

AU - Belytschko, Ted

PY - 2007

Y1 - 2007

N2 - A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.

AB - A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.

KW - Discontinuity

KW - Moving least-squares

KW - Partition of unity

UR - http://www.scopus.com/inward/record.url?scp=84880484320&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-46222-4_6

DO - 10.1007/978-3-540-46222-4_6

M3 - Chapter

SN - 9783540462149

VL - 57

T3 - Lecture Notes in Computational Science and Engineering

SP - 87

EP - 103

BT - Meshfree Methods for Partial Differential Equations III

ER -