An improvement for tensile instability in smoothed particle hydrodynamics

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

doi:10.1007/s004660050409

Cited by 149 publications

(95 citation statements)

References 33 publications

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“…where we have made dx ′ → hdx ′ in the integrand and used the scaling relation (5). Using the Gaussian kernel, the integral in Eq.…”

Section: Normalization Condition and C 0 Consistencymentioning

confidence: 99%

“…Therefore, the normalization condition is independent of h provided that the kernel interpolation obeys the scaling relation (5). Alternatively, in the SPH literature the kernels are usually defined in terms of the dimensionless parameter…”

Section: Normalization Condition and C 0 Consistencymentioning

confidence: 99%

“…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%

“…With this choice we obtain the scaling relations n ≈ 2.81N 0.675 and h ≈ 1.29n −0.247 for n and h, respectively. Figure 1 shows the variation of h with n. Based on a balance between the SPH smoothing and discretization errors, Zhu et al [28] derived the parameterizations h ∝ N −1/β and n ∝ N 1−3/β for β ∈ [5,7]. An intermediate value of β ∼ 6 is appropriate when the smoothing is performed with the Wendland kernel (31) on a quasi-ordered particle configuration.…”

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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“…where we have made dx ′ → hdx ′ in the integrand and used the scaling relation (5). Using the Gaussian kernel, the integral in Eq.…”

Section: Normalization Condition and C 0 Consistencymentioning

confidence: 99%

Section: Normalization Condition and C 0 Consistencymentioning

confidence: 99%

Section: Numerical Analysismentioning

confidence: 99%

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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Unsteady fluid–solid interaction by a kernel‐based particle method

Ostad

Mohammadi

2009

Numer Methods Biomed Eng

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SUMMARYA Lagrangian stabilized particle method is adopted for the simulation of unsteady fluid flow and gas dynamics. The velocity field smoothing technique, which was successfully implemented for reducing zero energy modes and tensile instabilities in solid problems, is used for the stabilization of the fluid phase. In this method a smoothing technique has been applied and the field values in particles are prevented from excessive jumps due to instability. This approach can replace traditional stabilization methods such as the artificial viscosity approach. In this paper, the corrective smoothed particle method has been implemented for solving governing partial differential equations of fluid. Also a modification in basic governing equations is performed on boundary interfaces to conserve mass and energy quantities. The method is implemented and verified in one-and two-dimensional compressible flow problems.

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Smoothed particle hydrodynamic modelling of the cerebrospinal fluid for brain biomechanics: Accuracy and stability

Duckworth

Sharp

Ghajari

2021

Numer Methods Biomed Eng

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The Cerebrospinal Fluid (CSF) can undergo shear deformations under head motions. Finite Element (FE) models, which are commonly used to simulate biomechanics of the brain, including traumatic brain injury, employ solid elements to represent the CSF. However, the limited number of elements paired with shear deformations in CSF can decrease the accuracy of their predictions. Large deformation problems can be accurately modelled using the mesh‐free Smoothed Particle Hydrodynamics (SPH) method, but there is limited previous work on using this method for modelling the CSF. Here we explored the stability and accuracy of key modelling parameters of an SPH model of the CSF when predicting relative brain/skull displacements in a simulation of an in vivo mild head impact in human. The Moving Least Squares (MLS) SPH formulation and Ogden rubber material model were found to be the most accurate and stable. The strain and strain rate in the brain differed across the SPH and FE models of CSF. The FE mesh anchored the gyri, preventing them from experiencing the level of strains seen in the in vivo brain experiments and predicted by the SPH model. Additionally, SPH showed higher levels of strains in the sulci compared to the FE model. However, tensile instability was found to be a key challenge of the SPH method, which needs to be addressed in future. Our study provides a detailed investigation of the use of SPH and shows its potential for improving the accuracy of computational models of brain biomechanics.

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An improvement for tensile instability in smoothed particle hydrodynamics

Cited by 149 publications

References 33 publications

On the kernel and particle consistency in smoothed particle hydrodynamics

On the kernel and particle consistency in smoothed particle hydrodynamics

Unsteady fluid–solid interaction by a kernel‐based particle method

Smoothed particle hydrodynamic modelling of the cerebrospinal fluid for brain biomechanics: Accuracy and stability

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