Skip to main content
SpringerLink
Log in

Analysis of Navier–Stokes equations by a BC/GE embedded local meshless method

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A meshless numerical model is developed for the simulation of two-dimensional incompressible viscous flows. We directly deal with the pressure–velocity coupling system of the Navier–Stokes equations. With an efficient time marching scheme, the flow problem is separated into a series of time-independent boundary value problems (BVPs) in which we seek the pressure distribution at discretized time instants. Unlike in conventional works that need iterative time marching processes, numerical results of the present model are obtained straightforwardly. Iteration is implemented only when dealing with the linear simultaneous equations while solving the BVPs. The method for solving these BVPs is a strong form meshless method which employs the local polynomial collocation with the weighted-least-squares (WLS) approach. By embedding all the constraints into the local approximation, i.e. ensuring the satisfaction of governing equation at both the internal and boundary nodes and the satisfaction of the boundary conditions (BCs) at boundary points, this strong form method is more stable and robust than those just collocate one boundary condition at one boundary node. We innovatively use this concept to embed the satisfaction of the continuity equation into the local approximation of the velocity components. Consequently, their spatial derivatives can be accurately calculated. The nodal arrangement is quite flexible in this method. One can set the nodal resolution finer in areas where the flow pattern is complicated and coarser in other regions. Three benchmark problems are chosen to test the performance of the present novel model. Numerical results are well compared with data found in reference papers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Armfield, S.W.: Finite difference solutions of the Navier–Stokes equations on staggered and non-staggered grids. Comput. Fluids 20, 1–17 (1991). https://doi.org/10.1016/0045-7930(91)90023-B

    Article  MATH  Google Scholar 

  2. Nikitin, N.: Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates. J. Comput. Phys. 217, 759–781 (2006). https://doi.org/10.1016/j.jcp.2006.01.036

    Article  MathSciNet  MATH  Google Scholar 

  3. Reis, G.A., Tasso, I.V.M., Souza, L.F., Cuminato, J.A.: A compact finite differences exact projection method for the Navier–Stokes equations on a staggered grid with fourth-order spatial precision. Comput. Fluids 118, 19–31 (2015). https://doi.org/10.1016/j.compfluid.2015.06.015

    Article  MathSciNet  MATH  Google Scholar 

  4. Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes equations. Springer, London (1986). https://doi.org/10.1007/978-3-642-61623-5

    Book  MATH  Google Scholar 

  5. Glowinski, R., Pironneau, O.: Finite element methods for Navier–Stokes equations. Annu. Rev. Fluid Mech. 24, 167–204 (1992). https://doi.org/10.1146/annurev.fl.24.010192.001123

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, B.Q.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer, London (2006). https://doi.org/10.1007/1-84628-205-5

    Book  MATH  Google Scholar 

  7. Li, L.: A split-step finite-element method for incompressible Navier–Stokes equations with high-order accuracy up-to the boundary. J. Comput. Phys. 408, 109274 (2020). https://doi.org/10.1016/j.jcp.2020.109274

    Article  MathSciNet  MATH  Google Scholar 

  8. Dalal, A., Eswaran, V., Biswas, G.: A finite-volume method for Navier–Stokes equations on unstructured meshes. Numer. Heat Tranf. B-Fundam. 54, 238–259 (2008). https://doi.org/10.1080/10407790802182653

    Article  Google Scholar 

  9. Trebotich, D., Graves, D.T.: An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries. Commun. Appl. Math. Comput. Sci. 10, 43–82 (2015). https://doi.org/10.2140/camcos.2015.10.43

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, J., Lin, X., Chen, Z.: Finite Volume Methods for the Incompressible Navier–Stokes Equations. Springer, London (2022)

    Book  MATH  Google Scholar 

  11. Chorin, A.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967). https://doi.org/10.1016/0021-9991(67)90037-X

    Article  MathSciNet  MATH  Google Scholar 

  12. Issa, R.I.: Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65 (1986). https://doi.org/10.1016/0021-9991(86)90099-9

    Article  MathSciNet  MATH  Google Scholar 

  13. Monaghan, J.J.: Smoothed particle hydrodynamics and its diverse applications. Annu. Rev. Fluid Mech. 44, 323–346 (2012). https://doi.org/10.1146/annurev-fluid-120710-101220

    Article  MathSciNet  MATH  Google Scholar 

  14. Lind, S.J., Rogers, B.D., Stansby, P.K.: Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling. Proc. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 476, 20190801 (2020). https://doi.org/10.1098/rspa.2019.0801

    Article  MathSciNet  MATH  Google Scholar 

  15. Khayyer, A., Gotoh, H.: Modified moving particle semi-implicit methods for the prediction of 2D wave impact pressure. Coast. Eng. 56, 419–440 (2009). https://doi.org/10.1016/j.coastaleng.2008.10.004

    Article  Google Scholar 

  16. Kondo, M., Koshizuka, S.: Improvement of stability in moving particle semi-implicit method. Int. J. Numer. Methods Fluids 65, 638–654 (2011). https://doi.org/10.1002/fld.2207

    Article  MathSciNet  MATH  Google Scholar 

  17. Khayyer, A., Gotoh, H.: Enhancement of stability and accuracy of the moving particle semi-implicit method. J. Comput. Phys. 230, 3093–3118 (2011). https://doi.org/10.1016/j.jcp.2011.01.009

    Article  MathSciNet  MATH  Google Scholar 

  18. Koshizuka, S., Shibata, K., Kondo, M., Matsunaga, T.: Moving Particle Semi-implicit Method, A Meshfree Particle Method for Fluid Dynamics. Academic Press, Cambridge (2018). https://doi.org/10.1016/C2016-0-03952-9

    Book  Google Scholar 

  19. Luo, M., Khayyer, A., Lin, P.: Particle methods in ocean and coastal engineering. Appl. Ocean Res. 114, 102734 (2021). https://doi.org/10.1016/j.apor.2021.102734

    Article  Google Scholar 

  20. Ma, Q.W.: Meshless local Petrov–Galerkin method for two-dimensional nonlinear water wave problems. J. Comput. Phys. 205, 611–625 (2005). https://doi.org/10.1016/j.jcp.2004.11.010

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, Q.W.: MLPG based on Rankine source solution for simulating nonlinear water waves. CMES-Comput. Model. Eng. Sci. 9, 193–209 (2005). https://doi.org/10.3970/CMES.2005.009.193

    Article  MATH  Google Scholar 

  22. Ma, Q.W., Zhou, J.T.: MLPG_R method for numerical solution of 2D breaking waves. CMES-Comput. Model. Eng. Sci. 43, 277–303 (2009). https://doi.org/10.3970/cmes.2009.043.277

    Article  MathSciNet  Google Scholar 

  23. Zhou, J.T., Ma, Q.W.: MLPG method based on Rankine source solution for modelling 3D breaking waves. CMES-Comput. Model. Eng. Sci. 56, 179–210 (2010). https://doi.org/10.3970/cmes.2010.056.179

    Article  MathSciNet  MATH  Google Scholar 

  24. Ma, Q.W., Zhou, Y., Yan, S.: A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves. J Ocean Eng Mar Energy 2, 279–299 (2016). https://doi.org/10.1007/s40722-016-0063-5

    Article  Google Scholar 

  25. Sriram, V., Ma, Q.W.: Review on the local weak form-based meshless method (MLPG): developments and applications in ocean engineering. Appl. Ocean Res. 116, 102883 (2021). https://doi.org/10.1016/j.apor.2021.102883

    Article  Google Scholar 

  26. Hannani, S.K., Sadeghi, M.M.: Navier–Stokes calculations using a finite point meshless method. Sci. Iran 12, 151–166 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Fang, J., Parriaux, A.: A regularized Lagrangian finite point method for the simulation of incompressible viscous flows. J. Comput. Phys. 227, 8894–8908 (2008). https://doi.org/10.1016/j.jcp.2008.06.031

    Article  MathSciNet  MATH  Google Scholar 

  28. Suchde, P., Kuhnert, J., Tiwari, S.: On meshfree GFDM solvers for the incompressible Navier–Stokes equations. Comput. Fluids 165, 1–12 (2018). https://doi.org/10.1016/j.compfluid.2018.01.008

    Article  MathSciNet  MATH  Google Scholar 

  29. Young, D.L., Lin, M.C.H., Tsai, C.C.: Analysis of high Reynolds free surface flows. J. Mech. 38, 454–472 (2022)

    Article  Google Scholar 

  30. Liu, G.R., Gu, Y.T.: An Introduction to Meshfree Methods and their Programming. Springer, Dordrecht (2005). https://doi.org/10.1007/1-4020-3468-7

    Book  Google Scholar 

  31. Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008). https://doi.org/10.1016/j.matcom.2008.01.003

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, J.-S., Hillman, M., Chi, S.-W.: Meshfree methods: progress made after 20 Years. J. Eng. Mech. ASCE 143, 04017001 (2017). https://doi.org/10.1061/(ASCE)EM.1943-7889.0001176

    Article  Google Scholar 

  33. Jin, X.Z., Li, G., Aluru, N.R.: New approximations and collocation schemes in the finite cloud method. Comput. Struct. 83, 1366–1385 (2005). https://doi.org/10.1016/j.compstruc.2004.08.030

    Article  MathSciNet  Google Scholar 

  34. Wu, N.-J., Chang, K.-A.: Simulation of free­surface waves in liquid sloshing using a domain type meshless method. Int. J. Numer. Methods Fluids 67, 269–288 (2011). https://doi.org/10.1002/fld.2346

    Article  MathSciNet  Google Scholar 

  35. Wu, N.-J., Tsay, T.-K.: A robust local polynomial collocation method. Int. J. Numer. Methods Eng. 93, 355–375 (2013). https://doi.org/10.1002/nme.4380

    Article  MathSciNet  MATH  Google Scholar 

  36. Wu, N.-J., Tsay, T.-K., Chen, Y.-Y.: Generation of stable solitary waves by a piston-type wave maker. Wave Motion 51, 240–255 (2014). https://doi.org/10.1016/j.wavemoti.2013.07.005

    Article  MathSciNet  MATH  Google Scholar 

  37. Wu, N.-J., Hsiao, S.-C., Wu, H.-L.: Mesh-free simulation of liquid sloshing subjected to harmonic excitations. Eng. Anal. Bound. Elem. 64, 90–100 (2016). https://doi.org/10.1016/j.enganabound.2015.12.001

    Article  MathSciNet  MATH  Google Scholar 

  38. Hsiao, S.-C., Shih, M.-Y., Wu, N.-J.: Simulation of propagation and run-up of three dimensional landslide-induced waves using a meshless method. Water 10, 552 (2018). https://doi.org/10.3390/w10050552

    Article  Google Scholar 

  39. Kupradze, V.D., Aleksidze, M.A.: The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput. Math. Math. Phys. 4, 82–126 (1964). https://doi.org/10.1016/0041-5553(64)90006-0

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, N.-J., Tsay, T.-K.: Applicability of the method of fundamental solutions to 3-D wave–body interaction with fully nonlinear free surface. J. Eng. Math. 63, 61–78 (2009). https://doi.org/10.1007/s10665-008-9250-2

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun, Y.: A meshless method based on the method of fundamental solution for solving the steady-state heat conduction problems. Int. J. Heat Mass Transf. 97, 891–907 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.03.002

    Article  Google Scholar 

  42. Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120, 118–152 (2020). https://doi.org/10.1016/j.enganabound.2020.08.013

    Article  MathSciNet  MATH  Google Scholar 

  43. Javed, A., Djijdeli, K., Xing, J.T.: Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier-Strokes equations. Comput. Fluids 89, 38–52 (2014). https://doi.org/10.1016/j.compfluid.2013.10.028

    Article  MathSciNet  MATH  Google Scholar 

  44. Lin, C.Y., Gu, M.H., Young, D.L., Sladek, J., Sladek, V.: The localized method of approximated particular solutions for solving two-dimensional incompressible viscous flow field. Eng. Anal. Bound. Elem. 57, 23–36 (2015). https://doi.org/10.1016/j.enganabound.2014.11.035

    Article  MathSciNet  MATH  Google Scholar 

  45. Xie, Y., Zhao, X., Rubinato, M., Yu, Y.: An improved meshfree scheme based on radial basis functions for solving incompressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids. 93, 2842–2862 (2021). https://doi.org/10.1002/fld.5012

    Article  MathSciNet  Google Scholar 

  46. Tsai, C.C., Young, D.L., Cheng, A.H.-D.: Meshless BEM for three-dimensional Stokes flows, CMES-Comp. Model. Eng. Sci. 3, 117–128 (2002). https://doi.org/10.3970/cmes.2002.003.117

    Article  MATH  Google Scholar 

  47. Bourantas, G.C., Zwick, B.F., Joldes, G.R., Loukopoulos, V.C., Tavner, A.C.R., Wittek, A., Miller, K.: An explicit meshless point collocation solver for incompressible Navier–Stokes equations. Fluids 4, 164 (2019). https://doi.org/10.3390/fluids4030164

    Article  Google Scholar 

  48. Ebrahimijahan, A., Dehghan, M., Abbaszadeh, M.: Simulation of the incompressible Navier–Stokes via integrated radial basis function based on finite difference scheme. Eng. Comput. (2022). https://doi.org/10.1007/s00366-021-01543-z

    Article  MATH  Google Scholar 

  49. Chinchapatnam, P.P., Djidjeli, K., Nair, P.B.: Radial basis function meshless method for the steady incompressible Navier–Stokes equations. Int. J. Comput. Math. 84, 1509–1521 (2007). https://doi.org/10.1080/00207160701308309

    Article  MathSciNet  MATH  Google Scholar 

  50. Bustamantea, C.A., Powerb, H., Florez, W.F.: A global meshless collocation particular solution method for solving the two-dimensional Navier–Stokes system of equations. Comput. Math. Appl. 65, 1939–1955 (2013). https://doi.org/10.1016/j.camwa.2013.04.014

    Article  MathSciNet  Google Scholar 

  51. Wang, Z.H., Huang, Z., Zhang, W., Xi, G.: A Meshless local radial basis function method for two-dimensional incompressible Navier–Stokes equations. Numer. Heat Tranf. B-Fundam. 67, 320–337 (2015). https://doi.org/10.1080/10407790.2014.955779

    Article  Google Scholar 

  52. Rammane, M., Mesmoudi, S., Tri, A., Braikat, B., Damil, N.: Solving the incompressible fluid flows by a high-order mesh-free approach. Int. J. Numer. Methods Fluids 92, 422–435 (2020). https://doi.org/10.1002/fld.4789

    Article  MathSciNet  MATH  Google Scholar 

  53. Dehghan, M., Abbaszadeh, M.: Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation. Comput. Methods Appl. Mech. Eng. 311, 856–888 (2016). https://doi.org/10.1016/j.cma.2016.09.008

    Article  MathSciNet  MATH  Google Scholar 

  54. Abbaszadeh, M., Dehghan, M.: Investigation of the Oldroyd model as a generalized incompressible Navier–Stokes equation via the interpolating stabilized element free Galerkin technique. Appl. Numer. Math. 150, 274–294 (2020). https://doi.org/10.1016/j.apnum.2019.08.025

    Article  MathSciNet  MATH  Google Scholar 

  55. Hu, G., Li, R., Zhang, X.: A novel stabilized Galerkin meshless method for steady incompressible Navier–Stokes equations. Eng. Anal. Bound. Elem. 133, 95–106 (2021). https://doi.org/10.1016/j.enganabound.2021.08.017

    Article  MathSciNet  MATH  Google Scholar 

  56. Abbaszadeh, M., Dehghan, M.: Reduced order modeling of time-dependent incompressible Navier–Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD/DEIM method. Comput. Methods Appl. Mech. Eng. 364, 112914 (2020). https://doi.org/10.1016/j.cma.2020.112914

    Article  MathSciNet  MATH  Google Scholar 

  57. Dehghan, M., Narimani, N.: An element-free Galerkin meshless method for simulating the behavior of cancer cell invasion of surrounding tissue. Appl. Math. Model. 59, 500–513 (2018). https://doi.org/10.1016/j.apm.2018.01.034

    Article  MathSciNet  MATH  Google Scholar 

  58. Narimani, N., Dehghan, M.: A direct RBF-PU method for simulating the infiltration of cytotoxic T-lymphocytes into the tumor microenvironment. Commun. Nonlinear Sci. Numer. Simul. 114, 106616 (2022). https://doi.org/10.1016/j.cnsns.2022.106616

    Article  MathSciNet  MATH  Google Scholar 

  59. Dehghan, M., Narimani, N.: Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction–diffusion equation. Comput. Appl. Math. 37, 6955–6971 (2018). https://doi.org/10.1007/s40314-018-0716-1

    Article  MathSciNet  MATH  Google Scholar 

  60. Limache, A.S.I., Rossi, R., Oñate, E.: The violation of objectivity in Laplace formulation of the Navier–Stokes equations. Int. J. Numer. Methods Fluids 54(6–8), 639–664 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  61. Oñate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L.: A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996). https://doi.org/10.1002/(SICI)1097-0207(19961130)39:22%3c3839::AID-NME27%3e3.0.CO;2-R

    Article  MathSciNet  MATH  Google Scholar 

  62. Oñate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L., Sacco, C.: A stabilized finite point method for analysis of fluid mechanics problems. Comput. Meth. Appl. Mech. Eng. 139, 315–346 (1996). https://doi.org/10.1016/S0045-7825(96)01088-2

    Article  MathSciNet  MATH  Google Scholar 

  63. Liszka, T., Orkisz, J.: The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Struct. 11, 83–95 (1980). https://doi.org/10.1016/0045-7949(80)90149-2

    Article  MathSciNet  MATH  Google Scholar 

  64. Benito, J.J., Ureña, F., Gavete, L.: Influence of several factors in the generalized finite difference method. Appl. Math. Model. 25, 1039–1053 (2001). https://doi.org/10.1016/S0307-904X(01)00029-4

    Article  MATH  Google Scholar 

  65. Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982). https://doi.org/10.1016/0021-9991(82)90058-4

    Article  MATH  Google Scholar 

  66. Botella, O., Peyret, R.: Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27, 421–433 (1998). https://doi.org/10.1016/S0045-7930(98)00002-4

    Article  MATH  Google Scholar 

  67. Bruneau, C.-H., Saad, M.: The 2D lid-driven cavity problem revisited. Comput. Fluids 35, 326–348 (2006). https://doi.org/10.1016/j.compfluid.2004.12.004

    Article  MATH  Google Scholar 

  68. Li, M., Tang, T.: Steady viscous flow in a triangular cavity by efficient numerical techniques. Comput. Math. Appl. 31, 55–65 (1996). https://doi.org/10.1016/0898-1221(96)00052-1

    Article  MathSciNet  MATH  Google Scholar 

  69. Roshko, A.: On the drag and shedding frequency of two-dimensional bluff bodies. NACA Tech. Note 3169, pp. 1–29 (1954). https://digital.library.unt.edu/ark:/67531/metadc57034/m2/1/high_res_d/19930083869.pdf

  70. Williamson, C.H.K.: Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579–627 (1989). https://doi.org/10.1017/S0022112089002429

    Article  Google Scholar 

  71. Fey, U., König, M., Eckelmann, H.: A new Strouhal–Reynolds-number relationship for the circular cylinder in the range 47<Re<2×105. Phys. Fluids 10, 1547 (1998). https://doi.org/10.1063/1.869675

    Article  Google Scholar 

  72. Liu, C., Zheng, X., Sung, C.H.: Preconditioned multigrid methods for unsteady incompressible flows. J. Comput. Phys. 139, 35–57 (1998). https://doi.org/10.1006/jcph.1997.5859

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the funding supported by the National Science and Technology Council, Taiwan (Grant No. 110-2223-E-019 -001 -MY3).

Author information

Authors and Affiliations

  1. Department of Marine Environmental Informatics, National Taiwan Ocean University, Keelung, 202301, Taiwan

    Nan-Jing Wu

  2. Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 202301, Taiwan

    Nan-Jing Wu

  3. Department of Civil Engineering, and Hydrotech Research Institute, National Taiwan University, Taipei, 10617, Taiwan

    Der-Liang Young

  4. Core Tech (Moldex3D), Chubei, 30265, Taiwan

    Der-Liang Young

Authors
  1. Nan-Jing Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Der-Liang Young
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nan-Jing Wu.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, NJ., Young, DL. Analysis of Navier–Stokes equations by a BC/GE embedded local meshless method. Acta Mech 234, 3843–3867 (2023). https://doi.org/10.1007/s00707-023-03589-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03589-0

Search

Navigation