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An upwind moving least squares approximation to solve convection-dominated problems: : An application in mixed discrete least squares meshfree method

Published: 17 July 2024 Publication History

Highlights

The presented novel upwind moving least squares approximation can satisfy positive operator condition.
Suggested upwind method can successfully overcome the problem of nonphysical oscillation encountering convection-dominated partial differential equations.
The proposed upwind method is more accurate than the existing version of mixed discrete least squares meshfree method for solving convection-dominated equations.
The suggested method can be utilized in any meshfree method.

Abstract

Moving Least Squares (MLS), as a series representation type of approximation, is broadly used in a wide array of meshfree methods. However, using the existing standard form of the MLS causes an unphysical oscillation for the meshfree methods encountering the convection-dominated partial differential equations (PDEs). In this study, several approaches are investigated to enhance the MLS approximation for solving convection-dominated problems. A novel upwind version of MLS approximation called shifted upward MLS (SU-MLS), is presented, which is based on strengthening the effect of upwind nodal points in approximation by wisely adjusting the weight function. Regarding the presented theoretical/numerical investigation, the proposed SU-MLS approximation can yield monotone solutions encountering the convection-dominated PDEs, unlike the existing standard form of MLS. The suggested SU-MLS approximation is, then, utilized in the mixed discrete least squares meshfree (MDLSM) method, rather than the standard MLS which is conventionally used in existing MDLSM. The novel method, which uses SU-MLS, is named upwind MDLSM (UMDLSM). Several numerical examples are investigated, and the results are compared to existing MDLSM. The obtained results indicate that the suggested UMDLSM is remarkably more accurate than the existing MDLSM in convection-dominated PDEs. Furthermore, while the existing MDLSM dramatically suffers from spurious oscillations (wiggling) when the Peclet number is high, the presented UMDLSM can yield monotone and accurate solutions.

References

[1]
G.R. Liu, Meshfree methods: Moving Beyond the Finite Element Method, CRC press, 2009.
[2]
G.R. Liu, Y.T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer Science & Business Media, 2005.
[3]
G.R. Liu, M.B. Liu, Smoothed Particle hydrodynamics: a Meshfree Particle Method, World scientific, 2003.
[4]
J.J. Monaghan, Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys. 30 (1992) 543–574.
[5]
J.J. Monaghan, Simulating free surface flows with SPH, J. Comput. Phys. 110 (1994) 399–406.
[6]
R. Xu, P. Stansby, D. Laurence, Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, J. Comput. Phys. 228 (2009) 6703–6725.
[7]
S. Koshizuka, A particle method for incompressible viscous flow with fluid fragmentation, Comput. Fluid Dyn. J. (1996).
[8]
S. Koshizuka, A. Nobe, Y. Oka, Numerical analysis of breaking waves using the moving particle semi-implicit method, Int. J. Numer. Methods Fluids. 26 (1998) 751–769.
[9]
A. Khayyer, H. Gotoh, Enhancement of stability and accuracy of the moving particle semi-implicit method, J. Comput. Phys. 230 (2011) 3093–3118.
[10]
Z. Chen, Z. Zong, M.B. Liu, L. Zou, H.T. Li, C. Shu, An SPH model for multiphase flows with complex interfaces and large density differences, J. Comput. Phys. 283 (2015) 169–188.
[11]
Z.B. Wang, R. Chen, H. Wang, Q. Liao, X. Zhu, S.Z. Li, An overview of smoothed particle hydrodynamics for simulating multiphase flow, Appl. Math. Model. 40 (2016) 9625–9655.
[12]
X. Liu, K. Morita, S. Zhang, Direct numerical simulation of incompressible multiphase flow with vaporization using moving particle semi-implicit method, J. Comput. Phys. 425 (2021).
[13]
J. Wang, X. Zhang, Improved Moving Particle Semi-implicit method for multiphase flow with discontinuity, Comput. Methods Appl. Mech. Eng. 346 (2019) 312–331.
[14]
C.H. Lee, A.J. Gil, A. Ghavamian, J. Bonet, A Total Lagrangian upwind Smooth Particle Hydrodynamics algorithm for large strain explicit solid dynamics, Comput. Methods Appl. Mech. Eng. 344 (2019) 209–250.
[15]
Y. Amini, H. Emdad, M. Farid, A new model to solve fluid–hypo-elastic solid interaction using the smoothed particle hydrodynamics (SPH) method, Eur. J. Mech. 30 (2011) 184–194.
[16]
A. Khayyer, N. Tsuruta, Y. Shimizu, H. Gotoh, Multi-resolution MPS for incompressible fluid-elastic structure interactions in ocean engineering, Appl. Ocean Res. 82 (2019) 397–414.
[17]
M. Rakhsha, A. Pazouki, R. Serban, D. Negrut, Using a half-implicit integration scheme for the SPH-based solution of fluid–solid interaction problems, Comput. Methods Appl. Mech. Eng. 345 (2019) 100–122.
[18]
Y. Tang, Q. Jiang, C. Zhou, A Lagrangian-based SPH-DEM model for fluid–solid interaction with free surface flow in two dimensions, Appl. Math. Model. 62 (2018) 436–460.
[19]
M. Soleimani, S. Sahraee, P. Wriggers, Red blood cell simulation using a coupled shell–fluid analysis purely based on the SPH method, Biomech. Model. Mechanobiol. 18 (2019) 347–359.
[20]
T. Belytschko, Y.Y. Lu, L. Gu, Element-free Galerkin methods, Int. J. Numer. Methods Eng. 37 (1994) 229–256.
[21]
S.N. Atluri, T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech 22 (1998) 117–127.
[22]
H. Arzani, M.H. Afshar, Solving Poisson's equations by the discrete least square meshless method, WIT Trans. Modelling Simul. 42 (2006) 23–31.
[23]
T. Belytschko, Y.Y. Lu, L. Gu, Crack propagation by element-free Galerkin methods, Eng. Fract. Mech 51 (1995) 295–315.
[24]
Y. Shao, Q. Duan, S. Qiu, Consistent element-free Galerkin method for three-dimensional crack propagation based on a phase-field model, Comput. Mater. Sci. 179 (2020).
[25]
M.J. Schulte, M. Robinett, N. Weidle, C.J. Duran, M.C. Flickinger, Experiments and finite element modeling of hydrodynamics and mass transfer for continuous gas-to-liquid biocatalysis using a biocomposite falling film reactor, Chem. Eng. Sci. 209 (2019).
[26]
I.V. Singh, P.K. Jain, Parallel EFG algorithm for heat transfer problems, Adv. Eng. Softw. 36 (2005) 554–560.
[27]
A. Singh, I.V. Singh, R. Prakash, Meshless element free Galerkin method for unsteady nonlinear heat transfer problems, Int. J. Heat Mass Transf. 50 (2007) 1212–1219.
[28]
M. Abbaszadeh, M. Dehghan, A. Khodadadian, N. Noii, C. Heitzinger, T. Wick, A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier–Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations, J. Comput. Phys. 426 (2021).
[29]
X.H. Wu, W.Q. Tao, S.P. Shen, X.W. Zhu, A stabilized MLPG method for steady state incompressible fluid flow simulation, J. Comput. Phys. 229 (2010) 8564–8577.
[30]
H. Lin, S.N. Atluri, The meshless local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations, C. Comput. Model. Eng. Sci. 2 (2001) 117–142.
[31]
A.R. Mojdehi, A. Darvizeh, A. Basti, Nonlinear Dynamic Analysis of Three-Dimensional Elasto-Plastic Solids by the Meshless Local Petrov-Galerkin(MLPG) Method, Comput. Mater. Contin. 29 (2012) 15–39.
[32]
J. Sladek, V. Sladek, M. Jus, The MLPG for crack analyses in composites with flexoelectricity effects, Compos. Struct. 204 (2018) 105–113.
[33]
S. Faraji, M.H. Afshar, J. Amani, Mixed discrete least square meshless method for solution of quadratic partial differential equations, Sci. Iran. 21 (2014) 492–504.
[34]
S. Faraji Gargari, M. Kolahdoozan, M.H. Afshar, Mixed Discrete Least Squares Meshfree method for solving the incompressible Navier–Stokes equations, Eng. Anal. Bound. Elem. 88 (2018),.
[35]
S. Faraji Gargari, M. Kolahdoozan, M.H. Afshar, Collocated mixed discrete least squares meshless (CMDLSM) method for solving quadratic partial differential equations, Sci. Iran. (2018) 25,.
[36]
J. Amani, M.H. Afshar, M. Naisipour, Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions, Eng. Anal. Bound. Elem. 36 (2012) 894–902.
[37]
S.F. Gargari, M. Kolahdoozan, M.H. Afshar, Mixed Discrete Least Squares Meshfree method for solving the incompressible Navier–Stokes equations, Eng. Anal. Bound. Elem. 88 (2018) 64–79.
[38]
S.F. Gargari, M. Kolahdoozan, M.H. Afshar, S. Dabiri, An Eulerian–Lagrangian mixed discrete least squares meshfree method for incompressible multiphase flow problems, Appl. Math. Model. 76 (2019) 193–224.
[39]
N. Eini, M.H. Afshar, S. Faraji Gargari, G. Shobeyri, A. Afshar, A fully Lagrangian mixed discrete least squares meshfree method for simulating the free surface flow problems, Eng. Comput. (2020) 1–21.
[40]
M. Meenal, T.I. Eldho, Two-dimensional contaminant transport modeling using meshfree point collocation method (PCM), Eng. Anal. Bound. Elem. 36 (2012) 551–561,.
[41]
S. Boddula, T.I. Eldho, A moving least squares based meshless local petrov-galerkin method for the simulation of contaminant transport in porous media, Eng. Anal. Bound. Elem. 78 (2017) 8–19,.
[42]
P. Majumder, T.I. Eldho, Reactive contaminant transport simulation using the analytic element method, random walk particle tracking and kernel density estimator, J. Contam. Hydrol. 222 (2019) 76–88,.
[43]
G. Fourtakas, P.K. Stansby, B.D. Rogers, S.J. Lind, An Eulerian–Lagrangian incompressible SPH formulation (ELI-SPH) connected with a sharp interface, Comput. Methods Appl. Mech. Eng. 329 (2018) 532–552.
[44]
M. Antuono, P.N. Sun, S. Marrone, A. Colagrossi, The δ-ALE-SPH model: An arbitrary Lagrangian-Eulerian framework for the δ-SPH model with particle shifting technique, Comput. Fluids. 216 (2021).
[45]
S. Tiwari, A. Klar, G. Russo, A meshfree arbitrary Lagrangian-Eulerian method for the BGK model of the Boltzmann equation with moving boundaries, J. Comput. Phys. 458 (2022).
[46]
A. Golbabai, N. Kalarestaghi, Improved localized radial basis functions with fitting factor for dominated convection-diffusion differential equations, Eng. Anal. Bound. Elem. 92 (2018) 124–135.
[47]
Y. Gu, G.R. Liu, Meshless techniques for convection dominated problems, Comput. Mech. 38 (2006) 171–182.
[48]
M. Cheng, G.R. Liu, A novel finite point method for flow simulation, Int. J. Numer. Methods Fluids. 39 (2002) 1161–1178.
[49]
H. Lin, S.N. Atluri, Meshless local Petrov-Galerkin(MLPG) method for convection diffusion problems, C. Model. Eng. Sci. 1 (2000) 45–60.
[50]
X.H. Wu, Y.J. Dai, W.Q. Tao, MLPG/SUPG method for convection-dominated problems, Numer. Heat Transf. Part B Fundam. 61 (2012) 36–51.
[51]
C.H. Lee, A.J. Gil, O.I. Hassan, J. Bonet, S. Kulasegaram, A variationally consistent Streamline Upwind Petrov–Galerkin Smooth Particle Hydrodynamics algorithm for large strain solid dynamics, Comput. Methods Appl. Mech. Eng. 318 (2017) 514–536.
[52]
T. Gao, T. Liang, L. Fu, A new smoothed particle hydrodynamics method based on high-order moving-least-square targeted essentially non-oscillatory scheme for compressible flows, J. Comput. Phys. (2023).
[53]
D. Avesani, M. Dumbser, A. Bellin, A new class of Moving-Least-Squares WENO–SPH schemes, J. Comput. Phys. 270 (2014) 278–299.
[54]
S. Peddavarapu, R. Srinivasan, Local maximum-entropy approximation based stabilization methods for the convection diffusion problems, Eng. Anal. Bound. Elem. 146 (2023) 531–554.
[55]
T.H. Huang, Stabilized and variationally consistent integrated meshfree formulation for advection-dominated problems, Comput. Methods Appl. Mech. Eng. 403 (2023).
[56]
M. Dehghan, M. Abbaszadeh, An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation, Eng. Anal. Bound. Elem. 92 (2018) 244–256.
[57]
M. Abbaszadeh, M. Dehghan, Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection–diffusion equation, Eng. Comput. 37 (2021) 873–889.
[58]
H.G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods For Singularly Perturbed Differential equations: Convection-Diffusion-Reaction and Flow Problems, Springer Science & Business Media, 2008.
[59]
A. Owen, Artificial diffusion in the numerical modelling of the advective transport of salinity, Appl. Math. Model. 8 (1984) 116–120.
[60]
A.W. Vreman, Stabilization of the Eulerian model for incompressible multiphase flow by artificial diffusion, J. Comput. Phys. 230 (2011) 1639–1651.
[61]
Q. Cai, S. Kollmannsberger, E. Sala-Lardies, A. Huerta, E. Rank, On the natural stabilization of convection dominated problems using high order Bubnov–Galerkin finite elements, Comput. Math. with Appl. 66 (2014) 2545–2558.
[62]
D.W. Kelly, S. Nakazawa, O.C. Zienkiewicz, J. Heinrich, A note on upwinding and anisotropic balancing dissipation in finite element approximations to convective diffusion problems, Int. J. Numer. Methods Eng. 15 (1980) 1705–1711.
[63]
R. Courant, E. Isaacson, M. Rees, On the solution of nonlinear hyperbolic differential equations by finite differences, Commun. Pure Appl. Math. 5 (1952) 243–255.
[64]
J.J.H. Miller, E. O'riordan, G.I. Shishkin, On piecewise-uniform meshes for upwind-and central-difference operators for solving singularly perturbed problems, IMA J. Numer. Anal. 15 (1995) 89–99.
[65]
P.M. Gresho, R.L. Lee, Don't suppress the wiggles—They're telling you something!, Comput. Fluids. 9 (1981) 223–253.
[66]
A.E.P. Veldman, Computational fluid dynamics, Lect. Notes, Univ. Groningen, Netherlands. (2001).
[67]
B.P. Leonard, A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Eng. 19 (1979) 59–98.
[68]
C. Hirsch, Numerical Computation of Internal and External Flows, Computational Methods for Inviscid and Viscous Flows, Chichester, 1990, Vol. 2.
[69]
F. Moukalled, L. Mangani, M. Darwish, F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method, Springer, 2016.
[70]
J.C. Heinrich, P.S. Huyakorn, O.C. Zienkiewicz, A. Mitchell, An'upwind'finite element scheme for two-dimensional convective transport equation, Int. J. Numer. Methods Eng. 11 (1977) 131–143.
[71]
T.P. Fries, H. Matthies, A review of Petrov-Galerkin stabilization approaches and an extension to meshfree methods, Inst. für wiss, Rechnen (2004).
[72]
T.P. Fries, H.G. Matthies, A stabilized and coupled meshfree/meshbased method for the incompressible Navier–Stokes equations—Part I: stabilization, Comput. Methods Appl. Mech. Eng. 195 (2006) 6205–6224.
[73]
A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng. 32 (1982) 199–259.
[74]
C.R. Swaminathan, V.R. Voller, Streamline upwind scheme for control-volume finite elements, part II. Implementation and comparison with the SUPG finite-element scheme, Numer. Heat Transf, Part B Fundam 22 (1992) 109–124.
[75]
C.R. Swaminathan, V.R. Voller, S.V. Patankar, A streamline upwind control volume finite element method for modeling fluid flow and heat transfer problems, Finite Elem. Anal. Des. 13 (1993) 169–184.
[76]
C.R. Swaminathan, V.R. Voller, Streamline upwind scheme for control-volume finite elements, part I. Formulations, Numer. Heat Transf, Part B Fundam 22 (1992) 95–107.
[77]
D. Kuzmin, S. Turek, High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter, J. Comput. Phys. 198 (2004) 131–158.
[78]
D. Kuzmin, Explicit and implicit FEM-FCT algorithms with flux linearization, J. Comput. Phys. 228 (2009) 2517–2534.
[79]
Y.T. Zhang, C.W. Shu, ENO and WENO schemes, Handb. Numer. Anal., Elsevier, 2016, pp. 103–122.
[80]
M. Hajipour, A. Malek, High accurate NRK and MWENO scheme for nonlinear degenerate parabolic PDEs, Appl. Math. Model. 36 (2012) 4439–4451.
[81]
T. Arbogast, C.S. Huang, X. Zhao, D.N. King, A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations, Comput. Methods Appl. Mech. Eng. 368 (2020).
[82]
Z. Huang, G. Lin, A.M. Ardekani, A mixed upwind/central WENO scheme for incompressible two-phase flows, J. Comput. Phys. 387 (2019) 455–480.
[83]
N. Zhan, R. Chen, Y. You, Three-dimensional high-order finite-volume method based on compact WENO reconstruction with hybrid unstructured grids, J. Comput. Phys. (2023).
[84]
A.R. Firoozjaee, M.H. Afshar, Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations, Eng. Anal. Bound. Elem. 33 (2009) 83–92.
[85]
F.E. Erami, A.R. Firoozjaee, Numerical solution of bed load transport equations using discrete least squares meshless (DLSM) method, Appl. Math. Model. 77 (2020) 1095–1109.
[86]
X. Guo, A meshless regularized local boundary integral equation method and the selection of weight function and geometrical parameters, Eng. Anal. Bound. Elem. 117 (2020) 221–231,.
[87]
V. John, Numerical Methods for Scalar Convection-Dominated Problems, (2013).
[88]
Y.M. Cheng, F.N. Bai, M.J. Peng, A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity, Appl. Math. Model. 38 (2014) 5187–5197,.
[89]
M. Dehghan, M. Abbaszadeh, A. Mohebbi, Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: Meshless interpolating element free Galerkin (IEFG) and finite element methods, Eng. Anal. Bound. Elem 64 (2016) 205–221,.
[90]
J.F. Wang, F.X. Sun, Y.M. Cheng, A.X. Huang, Error estimates for the interpolating moving least-squares method, Appl. Math. Comput. 245 (2014) 321–342,.
[91]
F.X. Sun, J.F. Wang, Y.M. Cheng, A.X. Huang, Error estimates for the interpolating moving least-squares method in n-dimensional space, Appl. Numer. Math. 98 (2015) 79–105,.
[92]
Q. Wang, W. Zhou, Y. Cheng, G. Ma, X. Chang, Y. Miao, E. Chen, Regularized moving least-square method and regularized improved interpolating moving least-square method with nonsingular moment matrices, Appl. Math. Comput. 325 (2018) 120–145,.
[93]
S.F. gargari, M. Kolahdoozan, M.H. Afshar, Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems, Sci. Iran. 25 (2018) 565–578.

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Published In

cover image Journal of Computational Physics
Journal of Computational Physics  Volume 506, Issue C
Jun 2024
654 pages

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Academic Press Professional, Inc.

United States

Publication History

Published: 17 July 2024

Author Tags

  1. Moving least squares (MLS)
  2. Meshfree methods
  3. Convection-dominated PDEs
  4. DLSM
  5. Upwind
  6. Positivity analysis

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