Reproducing kernel element interpolation: Globally conforming Im/Cn/Pk Hierarchies

Shaofan Li; Daniel C. Simkins; Hongsheng Lu; Wing K Liu

Reproducing kernel element interpolation: Globally conforming I^m/Cⁿ/P^k Hierarchies

Shaofan Li^*, Daniel C. Simkins, Hongsheng Lu, Wing K Liu

^*Corresponding author for this work

Mechanical Engineering

Research output: Chapter in Book/Report/Conference proceeding › Chapter

2 Scopus citations

Abstract

In this work, arbitrarily smooth, globally compatible, I^m/Cⁿ/P^k interpolation hierarchies are constructed in the framework of reproducing kernel element method (RKEM) for multi-dimensional domains. This is the first interpolation hierarchical structure that has been ever constructed with both minimal degrees of freedom and higher order continuity and reproducing conditions over multi-dimensional domains. The proposed hierarchical structure possesses the generalized Kronecker property, i.e., ∂^αΨ_I ^(β)/∂x^α(x_J) = δ_IJδ_αβ, {pipe}α{pipe}, {pipe}β{pipe} ≤ m. The newly constructed globally conforming interpolant is a hybrid of global partition polynomials (C^∞) and a smooth (Cⁿ) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.

Original language	English (US)
Title of host publication	Meshfree Methods for Partial Differential Equations II
Pages	109-132
Number of pages	24
State	Published - Dec 1 2005

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	43
ISSN (Print)	1439-7358

ASJC Scopus subject areas

Modeling and Simulation
General Engineering
Discrete Mathematics and Combinatorics
Control and Optimization
Computational Mathematics

Cite this

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title = "Reproducing kernel element interpolation: Globally conforming Im/Cn/Pk Hierarchies",

abstract = "In this work, arbitrarily smooth, globally compatible, Im/Cn/Pk interpolation hierarchies are constructed in the framework of reproducing kernel element method (RKEM) for multi-dimensional domains. This is the first interpolation hierarchical structure that has been ever constructed with both minimal degrees of freedom and higher order continuity and reproducing conditions over multi-dimensional domains. The proposed hierarchical structure possesses the generalized Kronecker property, i.e., ∂αΨI (β)/∂xα(xJ) = δIJδαβ, {pipe}α{pipe}, {pipe}β{pipe} ≤ m. The newly constructed globally conforming interpolant is a hybrid of global partition polynomials (C∞) and a smooth (Cn) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.",

author = "Shaofan Li and Simkins, {Daniel C.} and Hongsheng Lu and Liu, {Wing K}",

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series = "Lecture Notes in Computational Science and Engineering",

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T1 - Reproducing kernel element interpolation

T2 - Globally conforming Im/Cn/Pk Hierarchies

AU - Li, Shaofan

AU - Simkins, Daniel C.

AU - Lu, Hongsheng

AU - Liu, Wing K

PY - 2005/12/1

Y1 - 2005/12/1

N2 - In this work, arbitrarily smooth, globally compatible, Im/Cn/Pk interpolation hierarchies are constructed in the framework of reproducing kernel element method (RKEM) for multi-dimensional domains. This is the first interpolation hierarchical structure that has been ever constructed with both minimal degrees of freedom and higher order continuity and reproducing conditions over multi-dimensional domains. The proposed hierarchical structure possesses the generalized Kronecker property, i.e., ∂αΨI (β)/∂xα(xJ) = δIJδαβ, {pipe}α{pipe}, {pipe}β{pipe} ≤ m. The newly constructed globally conforming interpolant is a hybrid of global partition polynomials (C∞) and a smooth (Cn) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.

AB - In this work, arbitrarily smooth, globally compatible, Im/Cn/Pk interpolation hierarchies are constructed in the framework of reproducing kernel element method (RKEM) for multi-dimensional domains. This is the first interpolation hierarchical structure that has been ever constructed with both minimal degrees of freedom and higher order continuity and reproducing conditions over multi-dimensional domains. The proposed hierarchical structure possesses the generalized Kronecker property, i.e., ∂αΨI (β)/∂xα(xJ) = δIJδαβ, {pipe}α{pipe}, {pipe}β{pipe} ≤ m. The newly constructed globally conforming interpolant is a hybrid of global partition polynomials (C∞) and a smooth (Cn) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.

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M3 - Chapter

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Reproducing kernel element interpolation: Globally conforming I^m/Cⁿ/P^k Hierarchies

Abstract

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