skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: A MUSCL-SCNI approach for meshfree modeling of shock waves in fluids

Abstract

Here, a stable and nodally integrated meshfree formulation for modeling shock waves in fluids is developed. The reproducing kernel approximation is employed to discretize the conservation equations for compressible flow, and a flux vector splitting approach is applied to allow proper numerical treatments for the advection and pressure parts, respectively, based on the characteristics of each flux term. To capture the essential shock physics in fluids, including the Rankine–Hugoniot jump conditions and the entropy condition, local Riemann enrichment is introduced under the stabilized conforming nodal integration (SCNI) framework. Meanwhile, numerical instabilities associated with the advection flux are eliminated by adopting a modified upwind scheme. To further enhance accuracy, a MUSCL-type method is introduced in conjunction with an oscillation limiter to avoid Gibbs phenomenon and ensure monotonic piecewise linear reconstruction in the smooth region. The present meshfree formulation is free from tunable artificial parameters and is capable of capturing shock and rarefaction waves without over/undershoots. Finally, several numerical examples are analyzed to demonstrate the effectiveness of the proposed MUSCL-SCNI approach in meshfree modeling of complex shock phenomena, including shock diffraction, shock–vortex interaction, and high energy explosion processes.

Authors:
 [1]; ORCiD logo [1];  [1];  [2];  [2];  [3];  [3];  [3]
  1. University of California, San Diego, La Jolla, CA (United States)
  2. U.S. Army Engineer Research and Development Center, Vicksburg, MS (United States)
  3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1526226
Report Number(s):
SAND-2019-5754J
Journal ID: ISSN 2196-4378; 675750
Grant/Contract Number:  
AC04-94AL85000; NA0003525
Resource Type:
Accepted Manuscript
Journal Name:
Computational Particle Mechanics
Additional Journal Information:
Journal Volume: 7; Journal Issue: 2; Journal ID: ISSN 2196-4378
Publisher:
Springer Nature
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; Shock modeling; Compressible flow; RKPM; Riemann-SCNI; MUSCL-SCNI; Oscillation limiter

Citation Formats

Huang, Tsung-Hui, Chen, Jiun-Shyan, Wei, Haoyan, Roth, Michael J., Sherburn, Jesse A., Bishop, Joseph E., Tupek, Michael R., and Fang, Eliot H. A MUSCL-SCNI approach for meshfree modeling of shock waves in fluids. United States: N. p., 2019. Web. https://doi.org/10.1007/s40571-019-00248-x.
Huang, Tsung-Hui, Chen, Jiun-Shyan, Wei, Haoyan, Roth, Michael J., Sherburn, Jesse A., Bishop, Joseph E., Tupek, Michael R., & Fang, Eliot H. A MUSCL-SCNI approach for meshfree modeling of shock waves in fluids. United States. https://doi.org/10.1007/s40571-019-00248-x
Huang, Tsung-Hui, Chen, Jiun-Shyan, Wei, Haoyan, Roth, Michael J., Sherburn, Jesse A., Bishop, Joseph E., Tupek, Michael R., and Fang, Eliot H. Fri . "A MUSCL-SCNI approach for meshfree modeling of shock waves in fluids". United States. https://doi.org/10.1007/s40571-019-00248-x. https://www.osti.gov/servlets/purl/1526226.
@article{osti_1526226,
title = {A MUSCL-SCNI approach for meshfree modeling of shock waves in fluids},
author = {Huang, Tsung-Hui and Chen, Jiun-Shyan and Wei, Haoyan and Roth, Michael J. and Sherburn, Jesse A. and Bishop, Joseph E. and Tupek, Michael R. and Fang, Eliot H.},
abstractNote = {Here, a stable and nodally integrated meshfree formulation for modeling shock waves in fluids is developed. The reproducing kernel approximation is employed to discretize the conservation equations for compressible flow, and a flux vector splitting approach is applied to allow proper numerical treatments for the advection and pressure parts, respectively, based on the characteristics of each flux term. To capture the essential shock physics in fluids, including the Rankine–Hugoniot jump conditions and the entropy condition, local Riemann enrichment is introduced under the stabilized conforming nodal integration (SCNI) framework. Meanwhile, numerical instabilities associated with the advection flux are eliminated by adopting a modified upwind scheme. To further enhance accuracy, a MUSCL-type method is introduced in conjunction with an oscillation limiter to avoid Gibbs phenomenon and ensure monotonic piecewise linear reconstruction in the smooth region. The present meshfree formulation is free from tunable artificial parameters and is capable of capturing shock and rarefaction waves without over/undershoots. Finally, several numerical examples are analyzed to demonstrate the effectiveness of the proposed MUSCL-SCNI approach in meshfree modeling of complex shock phenomena, including shock diffraction, shock–vortex interaction, and high energy explosion processes.},
doi = {10.1007/s40571-019-00248-x},
journal = {Computational Particle Mechanics},
number = 2,
volume = 7,
place = {United States},
year = {2019},
month = {5}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 2 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Hydrodynamic meshfree method for high-rate solid dynamics using a Rankine-Hugoniot enhancement in a Riemann-SCNI framework: HYDRODYNAMIC MESHFREE METHOD
journal, May 2016

  • Roth, Michael J.; Chen, Jiun-Shyan; Danielson, Kent T.
  • International Journal for Numerical Methods in Engineering, Vol. 108, Issue 12
  • DOI: 10.1002/nme.5266

Uniformly high order accurate essentially non-oscillatory schemes, III
journal, August 1987


On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws
journal, January 1983

  • Harten, Amiram; Lax, Peter D.; Leer, Bram van
  • SIAM Review, Vol. 25, Issue 1
  • DOI: 10.1137/1025002

Linear Elastic Response of Tubes to Internal Detonation Loading
journal, May 2002

  • Beltman, W. M.; Shepherd, J. E.
  • Journal of Sound and Vibration, Vol. 252, Issue 4
  • DOI: 10.1006/jsvi.2001.4039

Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods
journal, April 1981


Numerical Instabilities in Upwind Methods: Analysis and Cures for the “Carbuncle” Phenomenon
journal, January 2001

  • Pandolfi, Maurizio; D'Ambrosio, Domenic
  • Journal of Computational Physics, Vol. 166, Issue 2
  • DOI: 10.1006/jcph.2000.6652

Stable and flux-conserved meshfree formulation to model shocks
journal, January 2016

  • Roth, Michael J.; Chen, Jiun-Shyan; Slawson, Thomas R.
  • Computational Mechanics, Vol. 57, Issue 5
  • DOI: 10.1007/s00466-016-1260-8

A Damage Particle Method for Smeared Modeling of Brittle Fracture
journal, January 2018


Multiresolution reproducing kernel particle method for computational fluid dynamics
journal, June 1997


Approximate Riemann solvers, parameter vectors, and difference schemes
journal, October 1981


Efficient implementation of essentially non-oscillatory shock-capturing schemes
journal, August 1988


A finite point method for adaptive three-dimensional compressible flow calculations
journal, July 2009

  • Ortega, Enrique; Oñate, Eugenio; Idelsohn, Sergio
  • International Journal for Numerical Methods in Fluids, Vol. 60, Issue 9
  • DOI: 10.1002/fld.1892

Meshfree Methods: Progress Made after 20 Years
journal, April 2017


On the shock-vortex interaction in Schardin's problem
journal, November 2000


Use of artificial viscosity in multidimensional fluid dynamic calculations
journal, July 1980


A stabilized conforming nodal integration for Galerkin mesh-free methods
journal, January 2000


A general, non-iterative Riemann solver for Godunov's method
journal, October 1985


Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method
journal, July 1979


Robust HLL-type Riemann solver capable of resolving contact discontinuity
journal, June 2012


VI. On the rate of explosion in gases
journal, January 1899

  • Chapman, D. L.
  • The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 47, Issue 284
  • DOI: 10.1080/14786449908621243

Numerical solutions of Euler equations by using a new flux vector splitting scheme
journal, July 1993

  • Zha, G-C.; Bilgen, E.
  • International Journal for Numerical Methods in Fluids, Vol. 17, Issue 2
  • DOI: 10.1002/fld.1650170203

Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures
journal, December 1996

  • Chen, Jiun-Shyan; Pan, Chunhui; Wu, Cheng-Tang
  • Computer Methods in Applied Mechanics and Engineering, Vol. 139, Issue 1-4
  • DOI: 10.1016/S0045-7825(96)01083-3

The weighted reconstruction of reproducing kernel particle method for one-dimensional shock wave problems
journal, February 2018


The design and application of upwind schemes on unstructured meshes
conference, February 2013

  • Barth, Timothy; Jespersen, Dennis
  • 27th Aerospace Sciences Meeting
  • DOI: 10.2514/6.1989-366

Flux splitting schemes for the Euler equations
journal, November 2012


Total variation diminishing Runge-Kutta schemes
journal, January 1998

  • Gottlieb, Sigal; Shu, Chi-Wang
  • Mathematics of Computation of the American Mathematical Society, Vol. 67, Issue 221
  • DOI: 10.1090/S0025-5718-98-00913-2

High Frequency Cinematography in the Shock Tube
journal, January 1957


A finite point method for compressible flow: A FINITE POINT METHOD FOR COMPRESSIBLE FLOW
journal, December 2001

  • Löhner, Rainald; Sacco, Carlos; Oñate, Eugenio
  • International Journal for Numerical Methods in Engineering, Vol. 53, Issue 8
  • DOI: 10.1002/nme.334

A New Flux Splitting Scheme
journal, July 1993

  • Liou, Meng-Sing; Steffen, Christopher J.
  • Journal of Computational Physics, Vol. 107, Issue 1
  • DOI: 10.1006/jcph.1993.1122

A new class of Moving-Least-Squares WENO–SPH schemes
journal, August 2014

  • Avesani, Diego; Dumbser, Michael; Bellin, Alberto
  • Journal of Computational Physics, Vol. 270
  • DOI: 10.1016/j.jcp.2014.03.041

Simplified Second-Order Godunov-Type Methods
journal, May 1988

  • Davis, S. F.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 3
  • DOI: 10.1137/0909030

Uniformly High-Order Accurate Nonoscillatory Schemes. I
journal, April 1987

  • Harten, Ami; Osher, Stanley
  • SIAM Journal on Numerical Analysis, Vol. 24, Issue 2
  • DOI: 10.1137/0724022

Shock simulation by the particle method SPH
journal, November 1983


Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection
journal, March 1977


Efficient Implementation of Weighted ENO Schemes
journal, June 1996

  • Jiang, Guang-Shan; Shu, Chi-Wang
  • Journal of Computational Physics, Vol. 126, Issue 1
  • DOI: 10.1006/jcph.1996.0130

A Finite-Volume Particle Method for Compressible Flows
journal, December 2000

  • Hietel, Dietmar; Steiner, Konrad; Struckmeier, Jens
  • Mathematical Models and Methods in Applied Sciences, Vol. 10, Issue 09
  • DOI: 10.1142/S0218202500000604

A Method for the Numerical Calculation of Hydrodynamic Shocks
journal, March 1950

  • VonNeumann, J.; Richtmyer, R. D.
  • Journal of Applied Physics, Vol. 21, Issue 3
  • DOI: 10.1063/1.1699639

SPH and Riemann Solvers
journal, September 1997


A stabilized nodally integrated meshfree formulation for fully coupled hydro-mechanical analysis of fluid-saturated porous media
journal, December 2016


Reformulation of Smoothed Particle Hydrodynamics with Riemann Solver
journal, June 2002


Filters, reproducing kernel, and adaptive meshfree method
journal, July 2003


Reproducing kernel particle methods
journal, April 1995

  • Liu, Wing Kam; Jun, Sukky; Zhang, Yi Fei
  • International Journal for Numerical Methods in Fluids, Vol. 20, Issue 8-9
  • DOI: 10.1002/fld.1650200824

Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III
journal, February 1997

  • Harten, Ami; Engquist, Bjorn; Osher, Stanley
  • Journal of Computational Physics, Vol. 131, Issue 1
  • DOI: 10.1006/jcph.1996.5632

Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
journal, August 1997


    Works referencing / citing this record:

    RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations
    journal, August 2019

    • Huang, Tsung-Hui; Wei, Haoyan; Chen, Jiun-Shyan
    • Computational Particle Mechanics, Vol. 7, Issue 2
    • DOI: 10.1007/s40571-019-00272-x