### 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 language | English |
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Title of host publication | Meshfree Methods for Partial Differential Equations III |

Pages | 87-103 |

Number of pages | 17 |

Volume | 57 |

DOIs | |

Publication status | Published - 2007 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 57 |

ISSN (Print) | 14397358 |

### Fingerprint

### 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

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Research › peer-review

*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

}

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 -