Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition

Imoto, Yusuke; Tagami, Daisuke

doi:10.14495/jsiaml.9.69

Cited by 6 publications

(7 citation statements)

References 3 publications

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“…In previous works, we established truncation error estimates for an interpolant, approximate gradient operator, and approximate Laplace operator of a generalized particle method in which the particle volumes were given as Voronoi volumes [11,12]. A generalized particle method is a numerical method that typically includes conventional particle methods, such as SPH and MPS.…”

Section: Introductionmentioning

confidence: 99%

Truncation error estimates of approximate operators in a generalized particle method

Imoto

2020

Japan J. Indust. Appl. Math.

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To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method that typically includes other conventional particle methods, such as smoothed particle hydrodynamics or moving particle semi-implicit methods. A new regularity of discrete parameters is proposed via two new indicators based on the Voronoi decomposition of the domain along with two hypotheses of reference weight functions. Then, truncation error estimates are derived for an interpolant, approximate gradient operator, and approximate Laplace operator in the generalized particle method. The convergence rates for these estimates are determined based on the frequency with which they appear in the regularity and hypotheses. Finally, the estimates are computed numerically and the results are shown to be in good agreement with the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Truncation error estimates of approximate operators in a generalized particle method

Imoto

2020

Japan J. Indust. Appl. Math.

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“…Ishijima-Kimura [3] considered the truncation error of the approximate gradient operator under a simple assumption. Imoto-Tagami ( [1], [2]) modified the approximate operators introduced by [4], and derived their truncation error estimates of their approximate operators by using their assumptions on a weight function and the radius of the interaction area. This paper has three purposes.…”

Section: Introductionmentioning

confidence: 99%

“…The first one is to generalize the approximate operators introduce by [4], [1], and [2]. The second one is to derive truncation error estimates for our approximate operators under more general assumptions than them of [1] and [2]. The third one is to give an application example of our main results.…”

Section: Introductionmentioning

confidence: 99%

“…Note also that the four operators Π h , ∇ h , ∆ h , and h are the generalization of the approximate operators introduced by [4], [1], and [2].…”

Section: Introductionmentioning

confidence: 99%

“…Let us state the main difficulty and key idea for deriving our truncation error estimates. Imoto-Tagami ([1], [2]) used the assumption that c * r σ ≤ h m for some c * > 0 and m ∈ N to derive their truncation error estimates for their approximate operators. Since this paper does not use the assumption, it is not easy to derive desirable truncation error estimates for our approximate operators directly.…”

Section: Introductionmentioning

confidence: 99%

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Error Analysis of Approximate Operators for a Particle Method based on Voronoi Diagram

Koba¹,

Sato²

2019

Preprint

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This paper considers several approximate operators used in a particle method based on a Voronoi diagram. Under some assumptions on a weight function, we derive truncation error estimates for our approximate gradient and Laplace operators. Our results show that our approximate gradient and Laplace operators tend to the usual gradient and Laplace operators when the ratio (the radius of the interaction area/the radius of a Voronoi cell) is sufficiently large. The key idea of our approach is to divide the integration region into two ring-shaped areas.2010 Mathematics Subject Classification. 33F05.

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Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier–Stokes equations

2019

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To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier-Stokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible Navier-Stokes equations, and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured. Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the truncation errors of the generalized approximate operators, an optimal weight function is derived by reducing the truncation errors over general particle distributions. The effectiveness of the generalized approximate operators with the optimal weight functions is confirmed using numerical results of truncation errors and driven cavity flow. As an application for flow problems with free surface effects, the explicit particle method is applied to a dam break flow.

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Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition

Cited by 6 publications

References 3 publications

Truncation error estimates of approximate operators in a generalized particle method

Truncation error estimates of approximate operators in a generalized particle method

Error Analysis of Approximate Operators for a Particle Method based on Voronoi Diagram

Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier–Stokes equations

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