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ESPFEM2D: A MATLAB 2D explicit smoothed particle finite element method code for geotechnical large deformation analysis

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Abstract

The Smoothed Particle Finite Element Method (SPFEM) has gained popularity as an effective numerical method for modelling geotechnical problems involving large deformations. To promote the research and application of SPFEM in geotechnical engineering, we present ESPFEM2D, an open-source two-dimensional SPFEM solver developed using MATLAB. ESPFEM2D discretizes the problem domain into computable particle clouds and generates the finite element mesh using Delaunay triangulation and the \( \alpha \)-shape technique to resolve mesh distortion issues. Additionally, it incorporates a nodal integration technique based on strain smoothing, effectively eliminating defects associated with the state variable mapping after remeshing. Furthermore, the solver adopts a simple yet robust approach to prevent the rank-deficiency problem due to under-integration by using only nodes as integration points. The Drucker-Prager model is adopted to describe the soil’s constitutive behavior as a demonstration. Implemented in MATLAB, this open-source solver ensures easy accessibility and readability for researchers interested in utilizing SPFEM. ESPFEM2D can be easily extended and effectively coupled with other existing codes, enabling its application to simulate a wide range of large geomechanical deformation problems. Through rigorous validation using four numerical examples, namely the oscillation of an elastic cantilever beam, non-cohesive soil collapse, cohesive soil collapse, and slope stability analysis, the accuracy, effectiveness and stability of this open-source solver have been thoroughly confirmed.

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References

  1. Benson DJ (1989) An efficient, accurate and simple ALE method for nonlinear finite element programs. Comput Methods Appl Mech Engrg 72:305–350

    Article  ADS  Google Scholar 

  2. Ghosh S, Kikuchi N (1991) An arbitrary Lagrangian-Eulerian finite element method for large deformation analysis of elastic-viscoplastic solids. Comput Methods Appl Mech Engrg 86:27–188

    Article  MathSciNet  Google Scholar 

  3. Bao YD, Sun XH, Zhou X, Zhang YS, Liu YW (2021) Some numerical approaches for landslide river blocking: introduction, simulation, and discussion. Landslides 18(12):3907–3922

    Article  Google Scholar 

  4. Yang ZX, Gao YY, Jardine RJ, Guo WB, Wang D (2020) Large deformation finite-element simulation of displacement-pile installation experiments in sand. J Geotech Geoenviron Eng 146(6):04020044

    Article  Google Scholar 

  5. Ren GF, Wang YX, Tang YQ, Zhao QX, Qiu ZG, Luo WH, Ye ZL (2022) Research on lateral bearing behavior of spliced helical piles with the SPH method. Appl Sci 12(16):8215

    Article  CAS  Google Scholar 

  6. Hu Y, Randolph MF (1998) A practical numerical approach for large deformation problems in soil. Int J Numer Anal Methods Geomech 22:327–350

    Article  Google Scholar 

  7. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181(3):375–389

    Article  ADS  Google Scholar 

  8. Blanc T, Pastor M (2012) A stabilized fractional step, Runge–Kutta Taylor SPH algorithm for coupled problems in geomechanics. Comput Methods Appl Mech Engrg 221:41–53

    Article  MathSciNet  ADS  Google Scholar 

  9. Bui HH, Fukagawa R, Sako K, Ohno S (2008) Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive mode. Int J Numer Anal Methods Geomech 32(12):1537–1570

    Article  Google Scholar 

  10. Peng C, Wu W, Yu HS, Wang C (2015) A SPH approach for large deformation analysis with hypoplastic constitutive model. Acta Geotech 10(6):703–717

    Article  Google Scholar 

  11. Soga K, Alonso E, Yerro A, Kumar K, Bandara S (2016) Trends in large-deformation analysis of landslide mass movements with particular emphasis on the material point method. Géotechnique 66(3):248–273

    Article  Google Scholar 

  12. Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Engrg 118:179–196

    Article  MathSciNet  ADS  Google Scholar 

  13. Yuan WH, Zhen HG, Zheng XC, Wang B, Zhang W (2023) An improved semi-implicit material point method for simulating large deformation problems in saturated geomaterials. Comput Geotech 161:105614

    Article  Google Scholar 

  14. Zhang W, Wu ZZ, Peng C, Li S, Dong YK, Yuan WH (2023) Modelling large-scale landslide using a GPU-accelerated 3D MPM with an efficient terrain contact algorithm. Comput Geotech 158:105411

    Article  Google Scholar 

  15. González Acosta JL, Vardon PJ, Remmerswaal G, Hicks MA (2020) An investigation of stress inaccuracies and proposed solution in the material point method. Comput Mech 65(2):555–581

    Article  MathSciNet  Google Scholar 

  16. Idelsohn SR, Oñate E, Del Pin F (2004) The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int J Numer Methods Eng 61(7):964–989

    Article  MathSciNet  Google Scholar 

  17. Idelsohn SR, Oñate E, Del Pin F, Calvo N (2006) Fluid-structure interaction using the particle finite element method. Comput Methods Appl Mech Engrg 195(17–18):2100–2123

    Article  MathSciNet  ADS  Google Scholar 

  18. Oñate E, Idelsohn SR, Del Pin F, Aubry R (2004) The particle finite element method-an overview. Int J Comput Methods 1(2):267–307

    Article  Google Scholar 

  19. Monforte L, Arroyo M, Carbonell JM, Gens A (2017) Numerical simulation of undrained insertion problems in geotechnical engineering with the particle finite element method (PFEM). Comput Geotech 82:144–156

    Article  Google Scholar 

  20. Rodríguez JM, Carbonell JM, Cante JC, Oliver J (2017) Continuous chip formation in metal cutting processes using the particle finite element method (PFEM). Int J Solids Struct 120:81–102

    Article  Google Scholar 

  21. Zhang X, Krabbenhoft K, Pedroso DM, Lyamin AV, Sheng D, da Silva MV, Wang D (2013) Particle finite element analysis of large deformation and granular flow problems. Comput Geotech 54:133–142

    Article  Google Scholar 

  22. Yuan WH, Zhang W, Dai BB, Wang Y (2019) Application of the particle finite element method for large deformation consolidation analysis. Eng Comput 36(9):3138–3163

    Article  Google Scholar 

  23. Yuan WH, Liu M, Guo N, Dai BB, Zhang W, Wang Y (2023) A temporal stable smoothed particle finite element method for large deformation problems in geomechanics. Comput Geotech 156:105298

    Article  Google Scholar 

  24. Yuan WH, Liu K, Zhang W, Dai BB, Wang Y (2020) Dynamic modeling of large deformation slope failure using smoothed particle finite element method. Landslides 17(7):1591–1603

    Article  Google Scholar 

  25. Yuan WH, Wang B, Zhang W, Jiang J, Feng XT (2019) Development of an explicit smoothed particle finite element method for geotechnical applications. Comput Geotech 106:42–51

    Article  Google Scholar 

  26. Zhang W, Yuan WH, Dai BB (2018) Smoothed particle finite-element method for large-deformation problems in geomechanics. Int J Geomech 18(4):04018010

    Article  Google Scholar 

  27. Zhang W, Zhong ZH, Peng C, Yuan WH, Wu W (2021) GPU-accelerated smoothed particle finite element method for large deformation analysis in geomechanics. Comput Geotech 129:103856

    Article  Google Scholar 

  28. Springel V (2005) The cosmological simulation code GADGET-2. Mon Not R Astron Soc 364(4):1105–1134

    Article  ADS  Google Scholar 

  29. Springel V, Yoshida N, White SDM (2001) GADGET: a code for collisionless and gasdynamical cosmological simulations. New Astron 6:79–117

    Article  CAS  ADS  Google Scholar 

  30. Hopkins PF (2015) A new class of accurate, mesh-free hydrodynamic simulation methods. Mon Not R Astron Soc 450(1):53–110

    Article  CAS  ADS  Google Scholar 

  31. Hopkins PF (2017) A new public release of the GIZMO code. arXiv preprint: arXiv:1712.01294

  32. Gomez-Gesteira M, Crespo AJC, Rogers BD, Dalrymple RA, Dominguez JM, Barreiro A (2012) SPHysics-development of a free-surface fluid solver-part 2: efficiency and test cases. Comput Geosci 48:300–307

    Article  ADS  Google Scholar 

  33. Gomez-Gesteira M, Rogers BD, Crespo AJC, Narayanaswamy M, Dominguez JM (2012) SPHysics-development of a free-surface fluid solver-part 1: theory and formulations. Comput Geosci 48:289–299

    Article  ADS  Google Scholar 

  34. Crespo AJC, Domínguez JM, Rogers BD, Gomez-Gesteira M, Longshaw S, Canelas R, Vacondio R, Barreiro A, Garcia-Feal O (2015) DualSPHysics: open-source parallel CFD solver based on smoothed particle hydrodynamics (SPH). Comput Phys Commun 187:204–216

    Article  CAS  ADS  Google Scholar 

  35. Domínguez JM, Crespo AJC, Valdez-Balderas D, Rogers BD (2013) New multi-GPU implementation for smoothed particle hydrodynamics on heterogeneous clusters. Comput Phys Commun 184(8):1848–1860

    Article  ADS  Google Scholar 

  36. Hérault A, Bilotta G, Vicari A, Rustico E, Negro CD (2011) Numerical simulation of lava flow using a GPU SPH model. Ann Geophys 54(5):600–620

    Google Scholar 

  37. Peng C, Wang S, Wu W, Yu HS, Wang C, Chen JY (2019) LOQUAT: an open-source GPU-accelerated SPH solver for geotechnical modeling. Acta Geotech 14(5):1269–1287

    Article  Google Scholar 

  38. Guo N, Yang ZX (2021) NSPFEM2D: a lightweight 2D node-based smoothed particle finite element method code for modeling large deformation. Comput Geotech 140:104484

    Article  Google Scholar 

  39. de St. Germain JD, McCorquodale J, Parker SG, Johnson CR (2000) Uintah: A massively parallel problem solving environment. In: Proceedings of the 9th IEEE international symposium on high performance distributed computing. IEEE Computer Society, USA, pp 33-41

  40. Dadvand P, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 17(3):253–297

    Article  Google Scholar 

  41. Zhang X, Krabbenhoft K, Sheng DC (2014) Particle finite element analysis of the granular column collapse problem. Granul Matter 16(4):609–619

    Article  Google Scholar 

  42. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50(2):435–466

    Article  Google Scholar 

  43. Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (2009) A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Comput Struct 87(1–2):14–26

    Article  Google Scholar 

  44. Jarušek J (1983) Contact problems with bounded friction coercive case. Czechoslov Math J 33(2):237–261

    Article  MathSciNet  Google Scholar 

  45. Cremonesi M, Frangi A, Perego U (2010) A Lagrangian finite element approach for the analysis of fluid-structure interaction problems. Int J Numer Methods Eng 84(5):610–630

    Article  MathSciNet  Google Scholar 

  46. Field DA (1988) Laplacian smoothing and Delaunay triangulations. Commun Appl Numer Methods 4(6):709–712

    Article  Google Scholar 

  47. Freitag LA, Ollivier-Gooch C (1996) A comparison of tetrahedral mesh improvement techniques. United States

  48. Meduri S, Cremonesi M, Perego U (2019) An efficient runtime mesh smoothing technique for 3D explicit Lagrangian free-surface fluid flow simulations. Int J Numer Methods Eng 117(4):430–452

    Article  MathSciNet  Google Scholar 

  49. Vartziotis D, Wipper J, Schwald B (2009) The geometric element transformation method for tetrahedral mesh smoothing. Comput Methods Appl Mech Engrg 199(1–4):169–182

    Article  MathSciNet  ADS  Google Scholar 

  50. Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Engrg 139:49–74

    Article  MathSciNet  ADS  Google Scholar 

  51. Belytschko T, Guo Y, Kam Liu W, Xiao SP (2000) A unified stability analysis of meshless particle methods. Int J Numer Methods Eng 48:1359–1400

    Article  MathSciNet  Google Scholar 

  52. Hillman M, Chen JS (2016) An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. Int J Numer Methods Eng 107(7):603–630

    Article  MathSciNet  Google Scholar 

  53. Huang TH, Wei HY, Chen JS, Hillman MC (2020) RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations. Comput Part Mech 7(2):393–433

    Article  Google Scholar 

  54. Puso MA, Chen JS, Zywicz E, Elmer W (2008) Meshfree and finite element nodal integration methods. Int J Numer Methods Eng 74(3):416–446

    Article  MathSciNet  Google Scholar 

  55. Silva-Valenzuela R, Ortiz-Bernardin A, Sukumar N, Artioli E, Hitschfeld-Kahler N (2020) A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition. Int J Numer Methods Eng 121(10):2174–2205

    Article  MathSciNet  Google Scholar 

  56. Wei HY, Chen JS, Beckwith F, Baek J (2020) A naturally stabilized semi-Lagrangian meshfree formulation for multiphase porous media with application to landslide modeling. J Eng Mech 146(4):04020012

    Article  Google Scholar 

  57. Ganzenmüller GC (2015) An hourglass control algorithm for lagrangian smooth particle hydrodynamics. Comput Methods Appl Mech Engrg 286:87–106

  58. Yuan WH, Liu M, Dai BB, Wang Y, Chan A, Zhang W (2023) Stabilizing nodal integration in dynamic smoothed particle finite element method: a simple and efficient algorithm. submitted

  59. Yuan WH, Wang HC, Zhang W, Dai BB, Liu K, Wang Y (2021) Particle finite element method implementation for large deformation analysis using Abaqus. Acta Geotech 16(8):2449–2462

    Article  Google Scholar 

  60. Yuan WH, Wang B, Zhang W, Jiang Q, Feng XT (2019) Development of an explicit smoothed particle finite element method for geotechnical applications. Comput Geotech 106:42–51

    Article  Google Scholar 

  61. Chen D, Huang WX, Sloan SW (2019) An alternative updated Lagrangian formulation for finite particle method. Comput Methods Appl Mech Engrg 343:490–505

    Article  MathSciNet  ADS  Google Scholar 

  62. Nguyen CT, Bui HH, Fukagawa R (2015) Failure mechanism of true 2D granular flows. J Chem Eng Japan 48(6):395–402

    Article  CAS  Google Scholar 

  63. Chalk CM, Pastor M, Peakall J, Borman DJ, Sleigh PA, Murphy W, Fuentes R (2020) Stress-particle smoothed particle hydrodynamics: an application to the failure and post-failure behaviour of slopes. Comput Methods Appl Mech Engrg 366:113034

    Article  MathSciNet  ADS  Google Scholar 

  64. Wang L, Zhang X, Lei QH, Panayides S, Tinti S (2022) A three-dimensional particle finite element model for simulating soil flow with elastoplasticity. Acta Geotech 17(12):5639–5653

    Article  PubMed  PubMed Central  Google Scholar 

  65. Bishop AW, Morgenstern N (1960) Stability coefficients for earth slopes. Géotechnique 10(4):129–153

    Article  Google Scholar 

  66. Griffiths DV, Lane PA (1999) Slope stability analysis by finite elements. Géotechnique 49(3):387–403

    Article  Google Scholar 

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Acknowledgements

The research is supported by the National Natural Science Foundation of China (Nos. 52379101 and 41807223), the Key Projects of the National Natural Science Foundation of China (No. 52239008), the Provincial Major Scientific Research Project of General Universities of Guangdong Province (No. 2022KTSCX013), the Youth Innovation Promotion Association CAS (No. 2022379).

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Authors and Affiliations

  1. College of Water Conservancy and Civil Engineering, South China Agricultural University, Guangzhou, 510642, China

    Wei Zhang & Yihui Liu

  2. Department of Civil and Environmental Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China

    Jinhui Li

  3. Department of Engineering Mechanics, Hohai University, Nanjing, 210098, China

    Weihai Yuan

  4. College of Mechanics and Materials, Hohai University, Nanjing, 210098, China

    Weihai Yuan

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  1. Wei Zhang
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Zhang, W., Liu, Y., Li, J. et al. ESPFEM2D: A MATLAB 2D explicit smoothed particle finite element method code for geotechnical large deformation analysis. Comput Mech (2024). https://doi.org/10.1007/s00466-024-02441-z

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