Completeness of corrective smoothed particle method for linear elastodynamics

Chen, J. K.; Beraun, J. E.; Jih, C. J.

doi:10.1007/s004660050516

Cited by 92 publications

(77 citation statements)

References 31 publications

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“…Numerical experiments with suitably chosen test functions in two-space dimensions validate our findings for a number of well-known SPH methods, namely the standard SPH, the CSPM method of Chen et al [5,4], the FPM scheme of Liu and Liu [12], the MSPH method of Zhang and Batra [26], and the recently proposed methodology of Zhu et al [28], where no corrections are required and full consistency is restored by allowing the number of neighbors to increase and the smoothing length to decrease with increasing spatial resolution. In particular, we find that when using the root mean square error (RMSE) as a model evaluation statistics, CSPM and FPM converge to only first-order accuracy, while MSPH, which was previously thought to converge to better than third order, is actually close to second order.…”

Section: Resultssupporting

confidence: 69%

“…We analyze the convergence rate of the SPH approximations for the functions and their derivatives using five different methods: (a) the standard SPH, (b) the CSPM method of Chen et al [5,4], (c) the FPM scheme of Liu and Liu [12], (d) the MSPH method of Zhang and Batra [26], and (e) the methodology recently proposed by Zhu et al [28], which we label SPHn to distinguish it from the standard SPH. Similarly, when CSPM and FPM are run with the Wendland function and varying number of neighbors we shall label them CSPMn and FPMn, respectively.…”

Section: Numerical Analysismentioning

confidence: 99%

“…In general, if up to m derivatives are retained in the series expansions, the resulting kernel and particle approximations will have (m + 1)th-order accuracy or mth-order consistency (i.e., C m consistency). This approach was first developed by Chen et al [5,4] (their corrective smoothed particle method, or CSPM), which solves for the approximation of a function separately from that of its derivatives by neglecting all terms involving derivatives in the former expansion and retaining only first-order terms in the latter expansions. This scheme is equivalent to a Shepard's interpolation for the function [20], and so it should restore C 1 kernel and particle consistency for the interior regions and C 0 consistency at the boundaries.…”

Section: Introductionmentioning

confidence: 99%

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On the kernel and particle consistency in smoothed particle hydrodynamics

Sigalotti

Klapp

Rendón

et al. 2016

Applied Numerical Mathematics

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The problem of consistency of smoothed particle hydrodynamics (SPH) has demanded considerable attention in the past few years due to the ever increasing number of applications of the method in many areas of science and engineering. A loss of consistency leads to an inevitable loss of approximation accuracy. In this paper, we revisit the issue of SPH kernel and particle consistency and demonstrate that SPH has a limiting second-order convergence rate. Numerical experiments with suitably chosen test functions validate this conclusion. In particular, we find that when using the root mean square error as a model evaluation statistics, well-known corrective SPH schemes, which were thought to converge to second, or even higher order, are actually firstorder accurate, or at best close to second order. We also find that observing * Corresponding author Email addresses: leonardo.sigalotti@gmail.com (Leonardo Di G. Sigalotti), jaime.klapp@inin.gob.mx (Jaime Klapp), ottorendon@gmail.com (Otto Rendón), carlosvax@gmail.com (Carlos A. Vargas), franklin.pena@gmail.com (Franklin Peña-Polo) Preprint submitted to Journal of Applied Numerical MathematicsMay 18, 2016 the joint limit when N → ∞, h → 0, and n → ∞, as was recently proposed by Zhu et al., where N is the total number of particles, h is the smoothing length, and n is the number of neighbor particles, standard SPH restores full C 0 particle consistency for both the estimates of the function and its derivatives and becomes insensitive to particle disorder.

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Section: Resultssupporting

confidence: 69%

Section: Numerical Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the kernel and particle consistency in smoothed particle hydrodynamics

Sigalotti

Klapp

Rendón

et al. 2016

Applied Numerical Mathematics

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“…Apart from these forms described above, another discrete Laplacian was formed by Chen et al (1999Chen et al ( , 2001). In the scheme, all the second derivatives are found by solving the set of following equations:…”

Section: Lp-sph07mentioning

confidence: 99%

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

Zhou

Yan

2016

J. Ocean Eng. Mar. Energy

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Three meshless methods, including incompressible smooth particle hydrodynamic (ISPH), moving particle semi-implicit (MPS) and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R) methods, are often employed to model nonlinear or violent water waves and their interaction with marine structures. They are all based on the projection procedure, in which solving Poisson's equation about pressure at each time step is a major task. There are three different approaches to solving Poisson's equation, i.e. (1) discretizing Laplacian directly by approximating the second-order derivatives, (2) transferring Poisson's equation into a weak form containing only gradient of pressure and (3) transferring Poisson's equation into a weak form that does not contain any derivatives of functions to be solved. The first approach is often adopted in ISPH and MPS, while the third one is implemented by the MLPG_R method. This paper attempts to review the most popular, though not all, approaches available in literature for solving the equation.

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“…It has been widely used in different fields as outlined in recent reviews [21][22][23]36]. Details about the standard SPH computation can be found in Liu and Liu's book [37], and the CSP method can be found in [38,39]. To date, this Lagrangian method has shown its advantages in fluid dynamics simulations, and the Lagrangian particle tracking method has been widely used in modelling bubble motions [40], free surface [41], dust [42], and other problems.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Approach for Computational Acoustics with Meshfree Method

Zhang

Smith

Zhang

et al. 2017

Preprint

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Abstract:Although Eulerian approaches are standard in computational acoustics, they are less effective for certain classes of problems like bubble acoustics and combustion noise. A different approach for solving acoustic problems is to compute with individual particles following particle motion. In this paper, a Lagrangian approach to model sound propagation in moving fluid is presented and implemented numerically, using three meshfree methods to solve the Lagrangian acoustic perturbation equations (LAPE) in the time domain. The LAPE split the fluid dynamic equations into a set of hydrodynamic equations for the motion of fluid particles and perturbation equations for the acoustic quantities corresponding to each fluid particle. Then, three meshfree methods, the smoothed particle hydrodynamics (SPH) method, the corrective smoothed particle (CSP) method, and the generalized finite difference (GFD) method, are introduced to solve the LAPE and the linearized LAPE (LLAPE). The SPH and CSP methods are widely used meshfree methods, while the GFD method based on the Taylor series expansion can be easily extended to higher orders. Applications to modeling sound propagation in steady or unsteady fluids in motion are outlined, treating a number of different cases in one and two space dimensions. A comparison of the LAPE and the LLAPE using the three meshfree methods is also presented. The Lagrangian approach shows good agreement with exact solutions. The comparison indicates that the CSP and GFD method exhibit convergence in cases with different background flow. The GFD method is more accurate, while the CSP method can handle higher Courant numbers.

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Completeness of corrective smoothed particle method for linear elastodynamics

Cited by 92 publications

References 31 publications

On the kernel and particle consistency in smoothed particle hydrodynamics

On the kernel and particle consistency in smoothed particle hydrodynamics

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

A Lagrangian Approach for Computational Acoustics with Meshfree Method

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