### Abstract

Original language | English |
---|---|

Pages (from-to) | 2485-2507 |

Number of pages | 23 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 198 |

Issue number | 33-36 |

DOIs | |

Publication status | Published - 1 Jul 2009 |

### Fingerprint

### Keywords

- Gauss theorem
- Mesh-less methods
- Simple geometries
- Real computation
- Complicated shape
- Finite elements
- Moving least squares
- Three dimensions
- Discretization
- Parallelization
- Integral terms
- matrix
- Reproducing kernel particle method
- Domain integrals
- MLS approximation
- Shape functions
- Reproducing kernel
- Basis functions
- Meshless
- Reproducing kernel method
- OR fluxes
- Connected domains

### Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*198*(33-36), 2485-2507. https://doi.org/10.1016/j.cma.2009.02.039

**Evaluation of the integral terms in reproducing kernel methods.** / Barbieri, Ettore; Meo, Michele.

Research output: Contribution to journal › Article

*Computer Methods in Applied Mechanics and Engineering*, vol. 198, no. 33-36, pp. 2485-2507. https://doi.org/10.1016/j.cma.2009.02.039

}

TY - JOUR

T1 - Evaluation of the integral terms in reproducing kernel methods

AU - Barbieri, Ettore

AU - Meo, Michele

PY - 2009/7/1

Y1 - 2009/7/1

N2 - Reproducing kernel method (RKM) has its origins in wavelets and it is based on convolution theory. Being their continuous version, RKM is often referred as the general framework for meshless methods. In fact, since in real computation discretization is inevitable, these integrals need to be evaluated numerically, leading to the creation of reproducing kernel particle method RKPM and moving least squares MLS approximation. Nevertheless, in this paper the integrals in RKM are explicitly evaluated for polynomials basis function and simple geometries in one, two and three dimensions even with conforming holes. Moreover, a general formula is provided for complicated shapes also for multiple connected domains. This is possible through a boundary formulation where domain integrals involved in RKM are transformed by Gauss theorem in circular or flux integrals. Parallelization is readily enabled since no preliminary arrangements of nodes is needed for the moments matrix. Furthermore, using symbolic inversion, computation of shape functions in RKM is considerably speeded up.

AB - Reproducing kernel method (RKM) has its origins in wavelets and it is based on convolution theory. Being their continuous version, RKM is often referred as the general framework for meshless methods. In fact, since in real computation discretization is inevitable, these integrals need to be evaluated numerically, leading to the creation of reproducing kernel particle method RKPM and moving least squares MLS approximation. Nevertheless, in this paper the integrals in RKM are explicitly evaluated for polynomials basis function and simple geometries in one, two and three dimensions even with conforming holes. Moreover, a general formula is provided for complicated shapes also for multiple connected domains. This is possible through a boundary formulation where domain integrals involved in RKM are transformed by Gauss theorem in circular or flux integrals. Parallelization is readily enabled since no preliminary arrangements of nodes is needed for the moments matrix. Furthermore, using symbolic inversion, computation of shape functions in RKM is considerably speeded up.

KW - Gauss theorem

KW - Mesh-less methods

KW - Simple geometries

KW - Real computation

KW - Complicated shape

KW - Finite elements

KW - Moving least squares

KW - Three dimensions

KW - Discretization

KW - Parallelization

KW - Integral terms

KW - matrix

KW - Reproducing kernel particle method

KW - Domain integrals

KW - MLS approximation

KW - Shape functions

KW - Reproducing kernel

KW - Basis functions

KW - Meshless

KW - Reproducing kernel method

KW - OR fluxes

KW - Connected domains

UR - http://www.scopus.com/inward/record.url?scp=65849420647&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1016/j.cma.2009.02.039

U2 - 10.1016/j.cma.2009.02.039

DO - 10.1016/j.cma.2009.02.039

M3 - Article

VL - 198

SP - 2485

EP - 2507

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

IS - 33-36

ER -