Stable generalized finite element methods for elasticity crack problems

Cui, Cu

doi:10.1002/nme.6347

Cited by 16 publications

(7 citation statements)

References 48 publications

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“…where D is the main component number [31]. e components of the first D eigenvalues whose cumulative contribution rate reaches a certain value are selected as the principal components.…”

Section: Principal Component Analysismentioning

confidence: 99%

Immovable Cultural Relics Disease Prediction Based on Relevance Vector Machine

Liu

et al. 2020

Mathematical Problems in Engineering

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The preventive cultural relics protection is one of the most concerned contents in archaeology, which includes environmental monitoring and accurate prediction of cultural relics diseases. In view of the deficiency of the analysis of cultural relics data and the prediction of cultural relics diseases, a prediction model of immovable cultural relics diseases based on relevance vector machine (RVM) is proposed. The key factors affecting the disease of immovable cultural relics are found out by the principal component analysis method, and the dimension reduction of data is realized; then, the RVM model under the framework of Bayesian theory is constructed, and the super parameters are estimated by the maximum edge likelihood method; finally, the prediction accuracy of the model is compared with the traditional diseases prediction methods. The experiment results demonstrate that the proposed RVM-based immovable cultural relics disease prediction approach not only has the advantages of more sparse model but also has better prediction accuracy than the traditional radial basis function neural network-based and support vector machine-based methods.

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“…where D is the main component number [31]. e components of the first D eigenvalues whose cumulative contribution rate reaches a certain value are selected as the principal components.…”

Section: Principal Component Analysismentioning

confidence: 99%

Immovable Cultural Relics Disease Prediction Based on Relevance Vector Machine

Liu

et al. 2020

Mathematical Problems in Engineering

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“…3 and that  d-BB 𝛼m ⟨1⟩ =  d-BB 𝛼m ⟨2⟩ =  d-BB 𝛼m ⟨3⟩ for all m = 1, … , 4. The expressions of the functions belonging to the set R d-BB 𝛼 are also presented in Appendix B.…”

mentioning

confidence: 94%

“…Generalized/eXtended finite element method (G/XFEM) formulations able to deliver optimal convergence rates in the energy norm and global matrices with a scaled condition number with the same order as in the finite element method (FEM) have been developed in recent years. [1][2][3][4][5][6][7][8][9][10] This is the case even for linear elastic fracture mechanics (LEFM) problems which have solutions with singularities and discontinuities. It is known, [5][6][7] however, that even optimally convergent first-order G/XFEM for this class of problems is not as computationally efficient as second-order FEMs with quarter-point elements.…”

Section: Introductionmentioning

confidence: 99%

“…The focus is on LEFM problems but the proposed error estimator formulation is quite general and can be adapted to other types of problems such as those involving material interfaces. The second-order G/XFEM formulation for 2-D LEFM problems proposed by Bento et al 7 is used here but other optimally convergent formulations 3,5,6,8 can be used as well. This will be demonstrated in future works, including extensions to fully 3-D LEFM problems.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recovery strategies, a posteriori error estimation, and local error indication for second‐order G/XFEM and FEM

Bento

Proença

Duarte

2023

Numerical Meth Engineering

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This article presents a computationally efficient and straightforward to implement a posteriori error estimator for second-order G/XFEM and FEM approximations. The formulation is based on the recently proposed block-diagonal Zienkiewicz-Zhu (ZZ-BD) a posteriori error estimator. The focus is on linear elastic fracture mechanics (LEFM) problems but the proposed error estimator formulation is general and can be adapted to other types of problems such as those involving material interfaces. The proposed ZZ-BD error estimator is based on a new strategy to recover stress fields from second-order G/XFEM and FEM approximations. This recovery procedure involves locally weighted L 2 projections of raw stresses on an approximation space for discontinuous and singular stress fields. The basis functions for these stress approximations are defined using a low-order partition of unity together with polynomial, discontinuous, and singular recovery enrichment functions. These singular enrichments are provided by the gradient of G/XFEM enrichments for LEFM problems and are, thus, available in most G/XFEM implementations. They also lead to a more computationally efficient stress recovery procedure than the one in the original ZZ-BD error estimator. The adopted G/XFEM are optimally convergent with the error in the energy norm of (h 2 ). Numerical experiments show that the error of the recovered stress field converges at the same rate as the discretization error. They also show that the effectivity index of the error estimator is close to the unity and that it provides good local error indicators for mesh adaptivity algorithms. Second-order FEM for elasticity problems is a special case of the G/XFEM adopted here. This is exploited to define a block-diagonal ZZ error estimator for the FEM as a special case of the proposed estimator for the G/XFEM.

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Mass lumping of stable generalized finite element methods for parabolic interface problems

Gong,

Zhang

2024

Math Methods in App Sciences

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Stable generalized finite element method (SGFEM) for interface problem is based on simple mesh independent of interface curves and possesses merits of optimal convergence rates, conditioning of same order as that of finite element method (FEM), robustness, and freedom from any penalty parameters or stability schemes. This paper investigates mass lumping of SGFEM for parabolic interface problems. The mass lumping is important for computational reduction and maximum principle preserving. The difficulty in the mass lumping of conventional generalized FEM (GFEM) and SGFEM consists in their incorporation of extra enriched functions. The existing mass lumping methods of GFEM cannot work properly due to features of SGFEM that enriched shape functions vanish on mesh nodes, and the nodes are enriched partially rather than totally. In this study, we propose two mass lumping schemes for the SGFEM. The first one is to only lump the FEM part, and enriched parts are kept fixed. This scheme enables us to execute a rigorous optimal convergence analysis. It is noted that a theoretical analysis on the mass lumping of GFEM has not been available yet in the GFEM field. The second scheme is to further compress the enriched part and intersection part of mass matrix separately. The lumped mass matrix is almost diagonal, and there is only a line of nonzero entries in the intersection part of mass matrix. Numerical experiments indicate that two schemes both produce the optimal convergence rates; their errors are almost the same as those of SGFEM without using the mass lumping.

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Stable generalized finite element methods for elasticity crack problems

Cited by 16 publications

References 48 publications

Immovable Cultural Relics Disease Prediction Based on Relevance Vector Machine

Immovable Cultural Relics Disease Prediction Based on Relevance Vector Machine

Recovery strategies, a posteriori error estimation, and local error indication for second‐order G/XFEM and FEM

Mass lumping of stable generalized finite element methods for parabolic interface problems

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