A transport point method for complex flow problems with free surface

  • Yan Song
  • Yan Liu
  • Xiong ZhangEmail author


The material point method (MPM) has been widely used in a broad area of engineering. However, it still suffers from the accuracy problem. One of the most important sources of the accuracy problem is the error of particle quadrature. The discontinuity of the gradient of the shape function causes severe crossing-mesh error. This paper proposes a new transport point method (TPM) which employs a mixed quadrature scheme combining Gaussian quadrature in the internal cells and particle quadrature in the boundary cells. The main distinction between the TPM and MPM is the quantities carried by particles. The transport points in TPM do not have volume and only carry intensive quantities and have dual properties of both Euler and Lagrange. The MLS method is used to reconstruct quantities at cell centers and nodes, and the integral weight is the cell volume. A point rearrangement algorithm is proposed so that the transport points can be added, moved or deleted arbitrarily as long as the accuracy of the reconstructed flow field is maintained, which can be used to eliminate the numerical fracture and impose inlet condition.


Transport point method Material point method Mixed quadrature scheme Inlet condition 


Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Zhang X, Chen Z, Lian YP, Liu Y (2017) Material point method. Elsevier, AmsterdamCrossRefGoogle Scholar
  2. 2.
    Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118(1–2):179–196MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang DZ, Ma X, Giguere PT (2011) Material point method enhanced by modified gradient of shape function. J Comput Phys 230(16):6379–6398MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zhang DZ, Zou Q, VanderHeyden WB, Ma X (2008) Material point method applied to multiphase flows. J Comput Phys 227(6):3159–3173MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Huang P, Zhang X, Ma S, Huang X (2011) Contact algorithms for the material point method in impact and penetration simulation. Int J Numer Methods Eng 85(4):498–517CrossRefzbMATHGoogle Scholar
  6. 6.
    Zhang X, Huang P (2010) An object-oriented mpm framework for simulation of large deformation and contact of numerous grains. CMES Comput Model Eng Sci 55(1):61–88MathSciNetGoogle Scholar
  7. 7.
    Huang P, Zhang X, Ma S, Wang H (2008) Shared memory openmp parallelization of explicit mpm and its application to hypervelocity impact. CMES Comput Model Eng Sci 38(2):119–148zbMATHGoogle Scholar
  8. 8.
    Lian Y, Zhang X, Liu Y (2011) Coupling of finite element method with material point method by local multi-mesh contact method. Comput Methods Appl Mech Eng 200(47–48):3482–3494MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bardenhagen SG, Nairn JA, Lu H (2011) Simulation of dynamic fracture with the material point method using a mixed j-integral and cohesive law approach. Int J Fract 170(1):49–66CrossRefzbMATHGoogle Scholar
  10. 10.
    Daphalapurkar NP, Lu H, Coker D, Komanduri R (2007) Simulation of dynamic crack growth using the generalized interpolation material point (gimp) method. Int J Fract 143(1):79–102CrossRefzbMATHGoogle Scholar
  11. 11.
    Chen Z, Hu W, Shen L, Xin X, Brannon R (2002) An evaluation of the mpm for simulating dynamic failure with damage diffusion. Eng Fract Mech 69(17):1873–1890CrossRefGoogle Scholar
  12. 12.
    Chen Z, Feng R, Xin X, Shen L (2003) A computational model for impact failure with shear-induced dilatancy. Int J Numer Methods Eng 56(14):1979–1997CrossRefzbMATHGoogle Scholar
  13. 13.
    Sulsky D, Schreyer L (2004) Mpm simulation of dynamic material failure with a decohesion constitutive model. Eur J Mech A Solids 23(3):423–445CrossRefzbMATHGoogle Scholar
  14. 14.
    Sulsky D, Peterson K (2011) Toward a new elastic-decohesive model of arctic sea ice. Physica D Nonlinear Phenom 240(20):1674–1683CrossRefGoogle Scholar
  15. 15.
    Liang Y, Benedek T, Zhang X, Liu Y (2017) Material point method with enriched shape function for crack problems. Comput Methods Appl Mech Eng 322:541–562MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lian Y, Zhang X, Zhou X, Ma Z (2011) A femp method and its application in modeling dynamic response of reinforced concrete subjected to impact loading. Comput Methods Appl Mech Eng 200(17–20):1659–1670CrossRefzbMATHGoogle Scholar
  17. 17.
    Lian Y, Zhang X, Liu Y (2012) An adaptive finite element material point method and its application in extreme deformation problems. Comput Methods Appl Mech Eng 241:275–285CrossRefzbMATHGoogle Scholar
  18. 18.
    Bardenhagen SG, Kober EM (2004) The generalized interpolation material point method. Comput Model Eng Sci 5(6):477–496Google Scholar
  19. 19.
    Sadeghirad A, Brannon RM, Burghardt J (2011) A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. Int J Numer Methods Eng 86(12):1435–1456MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jiang C, Schroeder C, Selle A, Teran J, Stomakhin A (2015) The affine particle-in-cell method. ACM Trans Gr 34(4):51zbMATHGoogle Scholar
  21. 21.
    Steffen M, Kirby RM, Berzins M (2008) Analysis and reduction of quadrature errors in the material point method (mpm). Int J Numer Methods Eng 76(6):922–948MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sulsky D, Gong M (2016) Improving the material-point method. In: Weinberg K, Pandolfi A (eds) Innovative mumerical approaches for multi-field and multi-scale problems, lecture notes in applied and computational mechanics, vol 81. Springer International Publishing, Switzerland, pp 217–240 CrossRefGoogle Scholar
  23. 23.
    Ma S, Zhang X, Lian Y, Zhou X (2009) Simulation of high explosive explosion using adaptive material point method. Comput Model Eng Sci 39(2):101MathSciNetzbMATHGoogle Scholar
  24. 24.
    Monteleone A, Monteforte M, Napoli E (2017) Inflow/outflow pressure boundary conditions for smoothed particle hydrodynamics simulations of incompressible flows. Comput Fluids 159:9–22MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhao X, Bolognin M, Liang D, Rohe A, Vardon PJ (2019) Development of in/outflow boundary conditions for mpm simulation of uniform and non-uniform open channel flows. Comput Fluids 179:27–33MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kafaji I (2013) Formulation of a dynamic material point method (MPM) for geomechanical problems. PhD thesis, University of Stuttgart, Stuttgart, GermanyGoogle Scholar
  27. 27.
    Beuth, L (2012) Formulation and application of a quasi-static material point method. PhD thesis, University of Stuttgart, Stuttgart, GermanyGoogle Scholar
  28. 28.
    Chen ZP, Zhang X, Sze KY, Kan L, Qiu XM (2018) vp material point method for weakly compressible problems. Comput Fluids 176:170–181MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhou Z, De Kat J, Buchner B (1999) A nonlinear 3d approach to simulate green water dynamics on deck. In: Proceedings of the seventh international conference on numerical ship hydrodynamics, Nantes, France, pp 1–15Google Scholar
  30. 30.
    Lobovskỳ L, Botia-Vera E, Castellana F, Mas-Soler J, Souto-Iglesias A (2014) Experimental investigation of dynamic pressure loads during dam break. J Fluids Struct 48:407–434CrossRefGoogle Scholar

Copyright information

© OWZ 2019

Authors and Affiliations

  1. 1.School of Aerospace EngineeringTsinghua UniversityBeijingPeople’s Republic of China

Personalised recommendations