Proof of linear independence of flat-top PU-based high-order approximation

An, Xinmei; Liu, Xiaoying; Zhao, Zhiye; Liu, He

doi:10.1016/j.enganabound.2014.04.003

Cited by 14 publications

(18 citation statements)

References 21 publications

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“…Actually, as long as the number of flat‐top PU units keeps the same, the relative size of flat‐top elements and PU elements has almost no influence on the accuracy of numerical result. ()…”

Section: Construction Of Locally Refined Flat‐top Pu Meshmentioning

confidence: 99%

“…However, the most commonly used method to avoid the linear dependence is to add flat‐top re gions into the original PU‐based mesh system. () He et al and An et al has proven that it is linearly independent for the regularly patterned flat‐top PU mesh in one‐ and two‐dimensional spaces.…”

Section: Introductionmentioning

confidence: 97%

“…However, the most commonly used method to avoid the linear dependence is to add flat-top re gions into the original PU-based mesh system. [30][31][32][33][34][35][36] He et al 32 and An et al 36 has proven that it is linearly independent for the regularly patterned flat-top PU mesh in one-and two-dimensional spaces. For stress concentration and stress singularity problems, it is necessary to implement local refinement for flat-top PU based mesh, so that, not only the numerical accuracy can be improved with high-order local approximation but also the efficiency can be ensured through local refinement within specific area.…”

Section: Introductionmentioning

confidence: 99%

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Local refinement of flat‐top partition of unity based high‐order approximation

Liu

Zhao

et al. 2018

Numerical Meth Engineering

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Summary The high‐order approximation with regularly patterned flat‐top partition of unity mesh in one‐ and two‐dimensional cases has been proven linearly independent. However, for problems with stress concentration or stress singularity, local refinement within the regular mesh is necessary to improve the accuracy and efficiency. This paper introduces local refinement of flat‐top partition of unity mesh within the framework of high‐order approximation in one‐ and two‐dimensional spaces, respectively. Based on the traditional PU mesh, the construction of locally refined flat‐top PU mesh is straightforward. With the rank deficiency counting approach, linear independence is proven from element level for the locally refined mesh system. Based on the numerical solution procedure presented, two numerical examples are analyzed to verify the proposed approximation method.

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Section: Construction Of Locally Refined Flat‐top Pu Meshmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Local refinement of flat‐top partition of unity based high‐order approximation

Liu

Zhao

et al. 2018

Numerical Meth Engineering

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“…Like other PU-based methods, the linear dependence problem [55] exists in high-order NMM and leads directly to the singularity in a global stiffness matrix. An algorithm for predicting the rank deficiency of the global stiffness matrix was proposed in [56], and this algorithm was extended in [57]. Recently, a new procedure to eliminate the linear dependence was created by [58].…”

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confidence: 99%

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

Fan

Zheng

et al. 2017

Numerical Meth Engineering

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The numerical manifold method (NMM) builds up a unified framework that is used to describe continuous and discontinuous problems; it is an attractive method for simulating a cracking phenomenon. Taking into account the differences between the generalized degrees of freedom of the physical patch and nodal displacement of the element in the NMM, a decomposition technique of generalized degrees of freedom is deduced for mixed mode crack problems. An analytic expression of the energy release rate, which is caused by a virtual crack extension technique, is proposed. The necessity of using a symmetric mesh is demonstrated in detail by analysing an additional error that had previously been overlooked. Because of this necessity, the local mathematical cover refinement is further applied. Finally, four comparison tests are given to illustrate the validity and practicality of the proposed method. The aforementioned aspects are all implemented in the highorder NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h-version high-order NMM.to extract the strain-energy release rates was provided by [18]. Under the context of XFEM, a direct analytical method to extract the mixed-mode strain-energy release rates from Irwin's integral was given by [19], and then this method was extended to high-order XFEM [20,21].Alternatively, the stiffness derivative technique (SDT) and virtual crack extension techniques (VCET) were proposed, respectively [22,23]. Whereafter, the VCET was applied to determine the SIFs of mode-I and mode-II by carrying out virtual crack extension along both the parallel and perpendicular directions to crack surface [24]. A combination of the VCET and field decomposition technique (DFDT) was initially implemented to extract mixed-mode SIFs by carrying out virtual crack extension in only the parallel direction to crack surface [25]. A double VCET for crack growth stability assessment was described by [26]. Based on an energy principle and the VCET, an approach that does not require the use of symmetric crack-tip mesh nor crack-tip singular elements was developed [27]. The VCET was used for simulation of the fatigue crack propagation by [28,29]. In order to avoid using finite difference approximation, which can lead to calculation error, an analytical expression for the energy release rate was derived [30]; another explicit expression for energy changes due to VCET was formulated on the basis of a variation of isoparametric element mappings [31]. A new direct-integration technique for the VCET using variational theory was presented in [32]. A blend of the VCET and DFDT was implemented to decompose three-dimensional mixed-mode energy release rates [33].The concept of shape design sensitivity analysis [34] was applied to calculate the strain-energy release rate [35]. And then, the equivalent domain integral [36] and the interaction integral [37] were used for the sensitivity analysis of cracked bodies. From the view of the shape design s...

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“…5. An, X. M., Liu, X. Y., Zhao, Z. Y. and He, L. (2014), "Proof of linear independence of flat-top PU-based high-order approximation", Engineering…”

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confidence: 99%

Development of flat-top partition of unity based high-order discontinuous deformation analysis

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Proof of linear independence of flat-top PU-based high-order approximation

Cited by 14 publications

References 21 publications

Local refinement of flat‐top partition of unity based high‐order approximation

Local refinement of flat‐top partition of unity based high‐order approximation

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

Development of flat-top partition of unity based high-order discontinuous deformation analysis

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