A meshless collocation method based on the differential reproducing kernel interpolation

Wang, Yung Ming; Chen, Syuan Mu; Wu, Chih Ping

doi:10.1007/s00466-010-0472-6

Cited by 35 publications

(34 citation statements)

References 31 publications

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“…With DC-PSE operators this is possible since the zeroth-order moment is a free parameter that can be used to tune the stability properties of the operators [39]. Setting the zeroth-order moment to zero and evaluating the operators at off-particle locations makes DC-PSE a particle-analog of derivative-reproducing kernel (DRK) Galerkin collocation methods [12,46,44], which are conceptually related to Moving Least-Squares (MLS) schemes [26,6].…”

Section: Approximation Of Derivatives and Particle-particle Interpolamentioning

confidence: 99%

“…(A.2) for b T = p(0), which ensures that the approximation is consistent, as well as the Kronecker delta property ζ p (x q − x p ) = δ pq , which ensures that the approximation is interpolating. Like Wang et al [44], we takeφ to be the quartic spline with cutoff radius 1 and choose a p such that it is smaller than the distance between particle p and its nearest neighbor.…”

Section: Appendix a Interpolating Dc-pse Operatorsmentioning

confidence: 99%

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A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

Reboux

Schrader

Sbalzarini

2012

Journal of Computational Physics

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We present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two-and three-dimensional benchmark problems. These include advection-diffusion, the Burgers equation, the Buckley-Leverett five-spot problem, and curvature-driven level-set surface refinement. AbstractWe present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two-and three-dimensional benchmark problems. These include advection-diffusion, the Burgers equation, the Buckley-Leverett five-spot problem, and curvature-driven level-set surface refinement.

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Section: Approximation Of Derivatives and Particle-particle Interpolamentioning

confidence: 99%

Section: Appendix a Interpolating Dc-pse Operatorsmentioning

confidence: 99%

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

Reboux

Schrader

Sbalzarini

2012

Journal of Computational Physics

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“…A DRK approximation-based collocation method was developed for the quasi-three-dimensional (3D) analysis of multilayered piezoelectric plates and functionally graded magneto-electro-elastic shells by Wu et al [18,20,50]. The novelty of this DRK approximation is in the determination of the shape functions of derivatives of the RK approximation, and these are obtained using a set of differential reproducing conditions without directly differentiating the RK approximation, as is necessary in the conventional RK approximation.…”

Section: Introductionmentioning

confidence: 99%

“…This may cause difficulties and inconvenience when the essential boundary conditions are imposed in the implementation of these methods, especially when solving the weak formulation of physical problems. Wu et al [19], Wang et al [21] and Wu et al [51] thus developed the DRK interpolation and its related meshless collocation and element-free Galerkin methods, in which the shape function at each sampling node was separated into a primitive function possessing the Kronecker delta properties and an enrichment function constituting the reproducing conditions. Subsequently, the shape function of derivatives of the DRK interpolation function at each sampling node was determined by the combinations of the corresponding derivative of the primitive function and the higher-order enrichment function derived using a set of differential reproducing conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Numerous of meshless methods have thus been developed, and can be classified into two categories, which are collocation-and Galerkin-based methods. The former include the smooth particle hydrodynamics [7,8], h − p cloud [9,10], finite point [11,12], point interpolation collocation [13], least square collocation [14][15][16], and reproducing kernel (RK) and differential reproducing kernel (DRK) approximation (or interpolation)-based collocation methods [17][18][19][20][21], while the latter include the reproducing kernel (RK) particle [22], Generalized multiple scale RK particle [23], moving least square [24,25], element free Galerkin [26], meshless local Petrov-Galerkin [27,28], point interpolation [29,30], diffuse element [31], and finite spheres methods [32]. Among these approaches, the unknown approximation or interpolation was constructed using the moving least square, RK, and radial basis function schemes.…”

Section: Introductionmentioning

confidence: 99%

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A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli–Euler beams and Kirchhoff–Love plates

Chen

Wang

2010

Comput Mech

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A Hermite differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving fourth-order differential equations where the field variable and its first-order derivatives are regarded as the primary variables. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the Hermite DRK interpolation, without directly differentiating it. In addition, the shape function of this interpolation at each sampling node is separated into a primitive function possessing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied for the field variable and its first-order derivatives. A weighted least-squares collocation method based on this interpolation is developed for the static analyses of classical beams and plates with fully simple and clamped supports, in which its accuracy and convergence rate are examined, and some guidance for using this method is suggested.

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A robust local polynomial collocation method

Tsay

2012

Numerical Meth Engineering

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In this paper, a robust local polynomial collocation method is presented. Based on collocation, this method is rather simple and straightforward. The present method is developed in a way that the governing equation is satisfied on boundaries as well as boundary conditions. This requirement makes the present method more accurate and robust than conventional collocation methods, especially in estimating the partial derivatives of the solution near the boundary. Studies about the sensitivity of the shape parameter and the local supporting range in the moving least square approach and the convergence of the nodal resolution are carried out by using some benchmark problems. This method is further verified by applying it to a steady-state convectiondiffusion problem. Finally, the present method is applied to calculate the velocity fields of two potential flow problems. More accurate numerical results are obtained. 356N.-J. WU AND T.-K. TSAY methods, in which governing equations are satisfied automatically in the entire domain, are more applicable to problems in which accurate partial derivatives of the solutions around the boundaries are desirable. Typical examples are the application of method of fundamental solutions to water wave problems [18][19][20]. Nevertheless, boundary-type RBF collocation methods are limited to problems governed by some specific equations, such as Laplace or Helmholtz equations. In [21], a modified domain-type RBF collocation method was proposed. With additional satisfaction to the governing equation on boundary collocation points, the gradient of the velocity potential at any free surface node, which represents the velocity vector at that specific node, was accurately estimated. Thus, this method was successfully applied to the simulation of water waves in a swaying tank.Apart from the combination of RBFs, one could use a polynomial to approximate the solution of a PDE and seek the coefficients of all the monomials by applying the collocation technique. However, polynomials have been rarely used globally such as the basis functions being treated in RBF collocation methods. This is because the values of high-order terms could become extremely large at points far from the origin. In most polynomial collocation methods, polynomials are just applied to approximate the solution piecewise around discretized nodes. The moving least square (MLS) approach is always accompanied with the polynomial approximation to transform and combine the local solutions into a global solution form. In each local solution, basis functions are the monomials, and their factors are just the coefficients. In the global solution form, the basis functions are called the shape functions, and their factors are the values of the solution itself. This localizing approach is somewhat like localized RBF collocation methods [22]. Representatives of this family are the hp-meshless cloud method [23] and the finite point method (FPM) [24,25].Being strong-form methods, polynomial collocation methods with MLS approach are reported to ...

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A meshless collocation method based on the differential reproducing kernel interpolation

Cited by 35 publications

References 31 publications

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli–Euler beams and Kirchhoff–Love plates

A robust local polynomial collocation method

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