Superconvergent gradient smoothing meshfree collocation method

Wang, Dongdong; Wang, Jiarui; Wu, Jianhua

doi:10.1016/j.cma.2018.06.021

Cited by 73 publications

(18 citation statements)

References 62 publications

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“…That is, when two operators of a certain order are used in conjunction with one another (such as a deformation gradient paired with the force density here), the combined formulation possesses one order of accuracy higher, at least in uniform discretizations, away from the boundary. This observation is consistent with the results of the recursive gradient technique of using two combined operators [37].…”

Section: Patch Testssupporting

confidence: 91%

“…On the other hand, the other cases either slowly converge, or do not converge at all. A slope of two (n) is indicated in the figures, where it can be seen that RKPD obtains rates consistent with nodal collocation of the strong form [17,[36][37][38]. That is, for even orders of accuracy, a rate of n is expected, and is obtained with the current formulation.…”

Section: Rows Of Text In Boldface Indicates the Methods Passes The Patmentioning

confidence: 70%

“…Meanwhile, the present formulation can converge with linear accuracy. This could be attributed to the fact that in essence, the combination of the two linear operators here produce something akin to higher-order differentiation, which is similar to the recursive gradient technique recently introduced [37], where linear basis was shown to be sufficient for convergence in strong form collocation, while also exhibiting the superconvergence phenomenon. The second case considered is a quadratic-order deformation gradient paired with the standard force density, denoted ''PD'' and the high-order force density, denoted ''RKPD'', respectively, in order to test the effect of the different permutations, as well as the proposed RKPD formulation.…”

Section: Rows Of Text In Boldface Indicates the Methods Passes The Patmentioning

confidence: 89%

“…1 for this odd-order approximation indicates superconvergence of the solution. That is, in nodal collocation approaches, the rate of n -1 is observed for odd orders of accuracy [17,36,37,40], with the further implication that linear completeness does not yield convergence. Meanwhile, the present formulation can converge with linear accuracy.…”

Section: Rows Of Text In Boldface Indicates the Methods Passes The Patmentioning

confidence: 99%

“…On the other hand, meshfree collocation approaches exhibit a rate of n ? 1 for a least-squares formulation using more collocation points than source points (approximation functions) [18], while using an equal number of source and collocation points exhibits an odd-even phenomenon where the rate of n is obtained for even orders, and n -1 for odd orders [17,[36][37][38], which has also been observed in isogeometric collocation [39,40]. A recently developed recursive gradient formulation [37] has been developed that exhibits superconvergence, that is, rates of n and n ?…”

Section: Introductionmentioning

confidence: 99%

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Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

2019

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State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree formulation where the integral is discretized by quadrature points, which results in a series of unknowns at the points under the strong-form collocation framework. In this work, the meshfree discretization of state-based peridynamics under the correspondence principle is examined and compared to traditional meshfree methods based on the classical local formulation of solid mechanics. It is first shown that the way in which the peridynamic formulation approximates differentiation can be unified with the implicit gradient approximation, and this is termed the reproducing kernel peridynamic approximation. This allows the construction of non-local deformation gradients with arbitrary-order accuracy, as well as non-local approximations of higher-order derivatives. A high-order accurate non-local divergence of stress is then proposed to replace the force density in the original state-based peridynamics, in order to obtain global arbitrary-order accuracy in the numerical solution. These two operators used in conjunction with one another is termed the reproducing kernel peridynamic method. The strong-form collocation version of the method is tested against benchmark solutions to examine and verify the highorder accuracy and convergence properties of the method. The method is shown to exhibit superconvergent behavior in the nodal collocation setting.

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Section: Patch Testssupporting

confidence: 91%

Section: Rows Of Text In Boldface Indicates the Methods Passes The Patmentioning

confidence: 70%

Section: Rows Of Text In Boldface Indicates the Methods Passes The Patmentioning

confidence: 89%

Section: Rows Of Text In Boldface Indicates the Methods Passes The Patmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

2019

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Strong Form Collocation Meshfree Methods

2023

Meshfree and Particle Methods

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Weak‐form element differential method for solving mechanics and heat conduction problems with abruptly changed boundary conditions

Zheng

Gao

et al. 2020

Numerical Meth Engineering

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SummaryElement differential method (EDM), as a newly proposed numerical method, has been applied to solve many engineering problems because it has higher computational efficiency and it is more stable than other strong‐form methods. However, due to the utilization of strong‐form equations for all nodes, EDM become not so accurate when solving problems with abruptly changed boundary conditions. To overcome this weakness, in this article, the weak‐form formulations are introduced to replace the original formulations of element internal nodes in EDM, which produce a new strong‐weak‐form method, named as weak‐form element differential method (WEDM). WEDM has advantages in both the computational accuracy and the numerical stability when dealing with the abruptly changed boundary conditions. Moreover, it can even achieve higher accuracy than finite element method (FEM) in some cases. In this article, the computational accuracy of EDM, FEM, and WEDM are compared and analyzed. Meanwhile, several examples are performed to verify the robustness and efficiency of the proposed WEDM.

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Superconvergent gradient smoothing meshfree collocation method

Cited by 73 publications

References 62 publications

Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

Strong Form Collocation Meshfree Methods

Weak‐form element differential method for solving mechanics and heat conduction problems with abruptly changed boundary conditions

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