The piecewise polynomial partition of unity functions for the generalized finite element methods

Oh, Hae‐Soo; Kim, June G.; Hong, Won-Tak

doi:10.1016/j.cma.2008.02.035

Cited by 42 publications

(47 citation statements)

References 14 publications

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“…Fortran code for r (x, y) can be found in our previous paper [22]. In [20], we prove the decomposition of suppψ δ k into subsets on which the convolution PU function is a polynomial.…”

Section: Definition 31mentioning

confidence: 99%

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The reproducing singularity particle shape functions for problems containing singularities

Jeong

Kim

2007

Comput Mech

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In this paper, we construct particle shape functions that reproduce singular functions as well as polynomial functions. We also construct piecewise polynomial convolution partition of unity functions by taking the convolution of the scaled conical window function with the characteristic functions of quadrangular patches (we provide the computer code for this construction). We demonstrate that the reproducing singular particle shape functions yield highly accurate numerical solutions for the singularity problems with crack singularity or a jump boundary data singularity.

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“…Fortran code for r (x, y) can be found in our previous paper [22]. In [20], we prove the decomposition of suppψ δ k into subsets on which the convolution PU function is a polynomial.…”

Section: Definition 31mentioning

confidence: 99%

“…5), we use the conformal mapping T α , α = 1/2. Thus, the components J * i j of the inverse matrix and the determinant |J (T s )| are those in (19) and (20).…”

Section: Numerical Examplementioning

confidence: 99%

The reproducing singularity particle shape functions for problems containing singularities

Jeong

Kim

2007

Comput Mech

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“…The technique we used is the so-called partition of unity finite element method (PUFEM) [15,20]. We refer the readers to [16][17][18][19] and references therein for recent developments and applications of the PUFEM. Although pointed out in the fundamental paper of Melenk and Babuška [20], the ability of the PUFEM to construct finite element spaces of high regularity has not been fully…”

Section: Introductionmentioning

confidence: 99%

A new family of high regularity elements

Sun

2011

Numerical Methods Partial

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In this article, we propose a new family of high regularity finite element spaces. The global approximation spaces are obtained in two steps. We first build an open cover of the computational domain and local approximation spaces on each patch of the cover. Then we construct partition of unity functions subordinate to the open cover depending on the regularity requirement. The basis functions of the global space is given by the products of the local basis functions and the corresponding partition of unity functions. The method can be used to construct finite element spaces of any desired regularity. Approximation properties and implementation details are discussed. Numerical examples for the biharmonic equation are presented to show the effectiveness of the proposed method.

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“…Over the past two decades, the concept of partition of unity (PU) approximations has been established and developed into different types of PU-based methods for solid mechanics including the Partition of Unity method [1][2], hp clouds [3], the generalized finite element method [4], the octree partition of unity method (OctPUM) [5] and others [6][7][8][9]. PUFEMs have attracted much interest from researchers in computational solid mechanics as they offer several advantages over the conventional finite element method (FEM), such as a free choice of local approximation functions, which allows flexibility for modelling complicated problems, and the construction of high order approximations without the addition of extra nodes.…”

Section: Introductionmentioning

confidence: 99%

A new partition of unity finite element free from the linear dependence problem and possessing the delta property

Cai

Zhuang

Augarde

2010

Computer Methods in Applied Mechanics and Engineering

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Charles (2010) 'A new partition of unity nite element free from the linear dependence problem and possessing the delta property.', Computer methods in applied mechanics and engineering., 199 (17-20). pp. 1036-1043. Further information on publisher's website:http://dx.doi.org/10.1016/j.cma. 2009.11.019 Publisher's copyright statement:Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT AbstractPartition-of-unity based finite element methods (PUFEMs) have appealing capabilities for p-adaptivity and local refinement with minimal or even no remeshing of the problem domain. However, PUFEMs suffer from a number of problems that practically limit their application, namely the linear dependence (LD) problem, which leads to a singular global stiffness matrix, and the difficulty with which essential boundary conditions can be imposed due to the lack of the Kronecker delta property. In this paper we develop a new PU-based triangular element using a dual local approximation scheme by treating boundary and interior nodes separately. The present method is free from the LD problem and essential boundary conditions can be applied directly as in the FEM. The formulation uses triangular elements, however the essential idea is readily extendable to other types of meshed or meshless formulation based on a PU approximation. The computational cost of the present method is comparable to other PUFEM elements described in the literature. The proposed method can be simply understood as a PUFEM with composite shape functions possessing the delta property and appropriate compatibility.

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The piecewise polynomial partition of unity functions for the generalized finite element methods

Cited by 42 publications

References 14 publications

The reproducing singularity particle shape functions for problems containing singularities

The reproducing singularity particle shape functions for problems containing singularities

A new family of high regularity elements

A new partition of unity finite element free from the linear dependence problem and possessing the delta property

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