Boundary element-free method for elastodynamics

Cheng, Yumin; Peng, Miaojuan

doi:10.1360/142004-25

Cited by 121 publications

(59 citation statements)

References 10 publications

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“…They can solve many problems in physics and engineering that are not suited to conventional numerical methods, with a minimum of meshing or no meshing at all [2]. For more study about these methods, for example, see [3,9,14,16,[19][20][21]40].…”

Section: Introductionmentioning

confidence: 99%

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

Sepehrian

Shamohammadi

2018

Arab. J. Math.

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A radial basis function method for solving time-fractional KdV equation is presented. The Caputo derivative is approximated by the high order formulas introduced in Buhman (Proc. Edinb. Math. Soc. 36:319-333, 1993). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point predictor-corrector method for solving the system is introduced. The efficiency and accuracy of our method are demonstrated through several illustrative examples. By the examples, the experimental convergence order is approximately 4 − α, where α is the order of time derivative.Mathematics Subject Classification 65D05 · 65M06 · 65N06 · 65N35

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Section: Introductionmentioning

confidence: 99%

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

Sepehrian

Shamohammadi

2018

Arab. J. Math.

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“…Combining different approximation functions in the meshless method with the boundary integral equation method, researchers have developed meshless boundary integral equation methods, such as the boundary node method (BNM) [3][4][5], the local boundary integral equation (LBIE) method [6][7][8], and the boundary element-free method (BEFM) [9][10][11][12][13][14][15][16][17][18][19].…”

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confidence: 99%

“…The meshless method based on the MLS approximation can generate a solution possessing great precision. The meshless boundary integral equation methods are developed by combining the MLS approximation with boundary integral equation methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The boundary node method (BNM) is one of the meshless boundary integral equation methods, and Mukherjee et al [3][4][5] used it to solve potential problems and linear elasticity problems.…”

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confidence: 99%

“…A disadvantage of the conventional least-squares method is that the final algebra equations system is sometimes ill-conditioned or singular. Then Cheng and Liew et al [9,10] improved the moving least-squares approximation. In the improved moving least-squares approximation, the algebra equation system is not ill-conditioned or singular, and the solution of the algebra equation system can be obtained directly without obtaining the inverse matrix.…”

mentioning

confidence: 99%

“…There are also fewer coefficients in the improved moving least-squares approximation than there are in the MLS approximation, and hence the computing speed and efficiency have increased. Combining the boundary integral equation method with the improved moving least-squares approximation, Cheng and Liew et al [9][10][11][12][13][14][15][16][17][18][19] come up with a direct meshless boundary integral equation method, called boundary element-free method (BEFM), to solve the problems, such as potential problems, elasticity, elastodynamics, and fracture. And the improved element-free Galerkin method based on the improved moving least-squares approximation was discussed by Zhang, Liew and Cheng [21][22][23].…”

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confidence: 99%

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An interpolating boundary element-free method (IBEFM) for elasticity problems

Ren

Cheng

2010

Sci. China Phys. Mech. Astron.

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The paper begins by discussing the interpolating moving least-squares (IMLS) method. Then the formulae of the IMLS method obtained by Lancaster are revised. On the basis of the boundary element-free method (BEFM), combining the boundary integral equation method with the IMLS method improved in this paper, the interpolating boundary element-free method (IBEFM) for two-dimensional elasticity problems is presented, and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained. In the IMLS method in this paper, the shape function satisfies the property of Kronecker δ function, and then in the IBEFM the boundary conditions can be applied directly and easily. The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables. Thus it gives a greater computational precision. Numerical examples are presented to demonstrate the method. moving least-squares (MLS) approximation, interpolating moving least-squares (IMLS) method, boundary integral equation, meshless method, boundary element-free method (BEFM), interpolating boundary element-free method (IBEFM), elasticity problem PACS: 02.60.Ed, 02.70.-c, 02.70.Pt, 46.15.-x In recent years, more and more attention has been paid to research on the meshless (or meshfree) method [1,2]. The meshless method has some advantages over the traditional computational methods, such as finite element method (FEM) and boundary element method (BEM). The meshless boundary integral equation method points to one of the important research directions of the meshless method. Combining different approximation functions in the meshless method with the boundary integral equation method, researchers have developed meshless boundary integral equation methods, such as the boundary node method (BNM) [3-5], the local boundary integral equation (LBIE) method [6-8], and the boundary element-free method (BEFM) [9-19]. The moving least-squares (MLS) approximation is one of the bases of the meshless method [1,2,20]. The meshless method based on the MLS approximation can generate a solution possessing great precision. The meshless boundary integral equation methods are developed by combining the MLS approximation with boundary integral equation methods [3-19]. The boundary node method (BNM) is one of the meshless boundary integral equation methods, and Mukherjee et al. [3-5] used it to solve potential problems and linear elasticity problems. Another equally important method of the meshless boundary integral equation methods is the local boundary integral equation (LBIE) method, which Atluri et al. [6-8] present to solve linear and nonlinear boundary value problems.In the BNM, the basic unknown quantities are approximations of the nodal variables. However, as they are not the real nodal variables, the boundary conditions cannot be directly applied. In the LBIE method, the traction term is not included in the local boundary integral equations.

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The dimension splitting reproducing kernel particle method for three‐dimensional potential problems

Peng

Cheng

2019

Numerical Meth Engineering

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Summary In this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three‐dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two‐dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method is used in the dimension splitting direction. Then, by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element‐free Galerkin method for 3D potential problems.

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Boundary element-free method for elastodynamics

Cited by 121 publications

References 10 publications

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

An interpolating boundary element-free method (IBEFM) for elasticity problems

The dimension splitting reproducing kernel particle method for three‐dimensional potential problems

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