Generalized interpolation material point method for coupled thermo-mechanical processes

  • Jun Tao
  • Yonggang Zheng
  • Zhen Chen
  • Hongwu Zhang


In this paper, a generalized interpolation material point method (GIMP) for simulating coupled thermo-mechanical processes is developed based on the weak formulations of both conservation of momentum and conservation of energy. The coupling term between the thermal and mechanical field variables is investigated in detail. The convergence behavior of the GIMP is examined with respect to the number of background cells and the number of particles per cell, respectively. A multi-grid approach is proposed to improve the accuracy of the GIMP in dealing with the Dirichlet boundary conditions in thermal analyses. Several representative examples are presented to demonstrate and verify the proposed procedure as compared with analytical or other numerical solutions, including an example on coupled thermo-mechanical failure evolution. It is shown from the obtained results that the proposed procedure might provide a robust spatial-discretization tool for multi-physics simulations.


Generalized interpolation material point method (GIMP) Heat transfer Thermo-mechanical analysis Multi-grid discretization Material failure 



The work was support in part by the National Natural Science Foundation of China (11232003 and 11272003), the U. S. Defense Threat Reduction Agency under grant number HDTRA1-10-1-0022, the Ph.D. Programs Foundation of Ministry of Education of China (20130041110050), and the scholarship from China Scholarship Council (CSC) under the grant number CSC201506060076.


  1. Arani, A.G., Abdollahian, M., Maraghi, Z.K.: Thermo-elastic analysis of a non-axisymmetrically heated FGPM hollow cylinder under multi-physical fields. Int. J. Mech. Mater. Des. 11, 157–171 (2015)CrossRefGoogle Scholar
  2. Atluri, S., Zhu, T.: A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. Balla, M., Hungary, B.: Analytical study of the thermal shock problem of a half-space with various thermoelastic models. Acta Mech. 89, 73–92 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. Bardenhagen, S.G., Kober, E.M.: The generalized interpolation material point method. Comput. Model. Eng. Sci. 5, 477–496 (2004)Google Scholar
  5. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Meth. Eng. 37, 229–256 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. Borouchaki, H., Laug, P., Cherouat, A., Saanouni, K.: Adaptive remeshing in large plastic strain with damage. Int. J. Numer. Meth. Eng. 63, 1–36 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. Brackbill, J.U., Ruppel, H.M.: FLIP: a method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. J. Comput. Phys. 65, 314–343 (1986)MathSciNetCrossRefMATHGoogle Scholar
  8. Cui, X.X., Zhang, X., Sze, K.Y., Zhou, X.: An alternating finite difference material point method for numerical simulation of high explosive explosion problems. Comput. Model. Eng. Sci. 92, 507–538 (2013)MathSciNetGoogle Scholar
  9. Chen, Z., Feng, R., Xin, X., Shen, L.: A computational model for impact failure with shear-induced dilatancy. Int. J. Numer. Meth. Eng. 56, 1979–1997 (2003)CrossRefMATHGoogle Scholar
  10. Chen, Z., Gan, Y., Chen, J.K.: A coupled thermo-mechanical model for simulating the material failure evolution due to localized heating. Comput. Model. Eng. Sci. 26, 123–137 (2008)Google Scholar
  11. Chen, Z., Jiang, S., Gan, Y., Liu, H., Sewell, T.D.: A particle-based multiscale simulation procedure within the material point method framework. Comput. Part. Mech. 1, 147–158 (2014)CrossRefGoogle Scholar
  12. Chen, Z., Shen, L., Mai, Y.W., Shen, Y.G.: A bifurcation-based decohesion model for simulating the transition from localization to decohesion with the MPM. Z. Angew. Math. Phys. 56(5), 908–930 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. Daphalapurkar, N.P., Lu, H., Coker, D., Komanduri, R.: Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method. Int. J. Fract. 143(1), 79–102 (2007)CrossRefMATHGoogle Scholar
  14. Das, R., Cleary, P.W.: Novel application of the mesh-free SPH method for modelling thermo-mechanical responses in arc welding. Int. J. Mech. Mater. Des. 11, 337–355 (2015)CrossRefGoogle Scholar
  15. Davison, de St. Germain J., McCorquodale, J., Johnson, C.R., et al.: Uintah: a massively parallel problem solving environment. In: The 9th IEEE International Symposium on High-Performance Distributed Computing, pp. 33–41 (2000)Google Scholar
  16. Hosseini, S.M., Sladek, J., Sladek, V.: Meshless local Petrov–Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder. Eng. Anal. Bound. Elem. 35, 827–835 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. Jeong, J.H., Jhon, M.S., Halow, J.S., Van Osdol, J.: Smoothed particle hydrodynamics: applications to heat conduction. Comput. Phys. Commun. 153, 71–84 (2003)CrossRefMATHGoogle Scholar
  18. Li, S., Liu, W.K.: Meshfree and particle methods and their applications. Appl. Mech. Rev. 55, 1–34 (2002)CrossRefGoogle Scholar
  19. Lian, Y.P., Liu, Y., Zhang, X.: Coupling of membrane element with material point method for fluid-membrane interaction problems. Int. J. Mech. Mater. Des. 10, 199–211 (2014)CrossRefGoogle Scholar
  20. Liu, Y., Wang, H.K., Zhang, X.: A multiscale framework for high-velocity impact process with combined material point method and molecular dynamics. Int. J. Mech. Mater. Des. 9, 127–139 (2013)MathSciNetCrossRefGoogle Scholar
  21. Liu, Y., Zhang, X., Lu, M.: A meshless method based on least-squares approach for steady-and unsteady-state heat conduction problems. Numer. Heat Transf. Part B Fundam. 47, 257–275 (2005)CrossRefGoogle Scholar
  22. Ma, J., Lu, H., Wang, B., Hornung, R., Wissink, A., Komanduri, R.: Multiscale simulation using generalized interpolation material point (GIMP) method and molecular dynamics (MD). CMC Tech. Sci. Press 4(2), 101 (2006)MathSciNetGoogle Scholar
  23. Nairn, J.A.: Material point method calculations with explicit cracks. Comput. Model. Eng. Sci. 4, 649–664 (2003)MATHGoogle Scholar
  24. Nairn, J. A.: Material Point Method (NairnMPM) and Finite Element Analysis (NairnFEA) Open-Source Software. (2011)
  25. Oterkus, S., Madenci, E., Agwai, A.: Fully coupled peridynamic thermomechanics. J. Mech. Phys. Solids 64, 1–23 (2014a)MathSciNetCrossRefGoogle Scholar
  26. Oterkus, S., Madenci, E., Agwai, A.: Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014b)MathSciNetCrossRefGoogle Scholar
  27. Randles, P.W., Libersky, L.D.: Smoothed particle hydrodynamics: some recent improvements and applications. Comput. Methods Appl. Mech. Eng. 139, 375–408 (1996)MathSciNetCrossRefMATHGoogle Scholar
  28. Singh, I.V.: A numerical solution of composite heat transfer problems using meshless method. Int. J. Heat Mass Transf. 47, 2123–2138 (2004)CrossRefMATHGoogle Scholar
  29. Sladek, J., Sladek, V., Zhang, C.: Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method. Comput. Mater. Sci. 28, 494–504 (2003)CrossRefGoogle Scholar
  30. Sladek, J., Sladek, V., Zhang, C., Tan, C.L.: Meshless local Petrov–Galerkin method for linear coupled thermoelastic analysis. Comput. Model. Eng. Sci. 16, 57–68 (2006)Google Scholar
  31. Sulsky, D., Chen, Z., Schreyer, H.L.: A particle method for history-dependent materials. Comput. Methods Appl. Mech. Eng. 118, 179–196 (1994)MathSciNetCrossRefMATHGoogle Scholar
  32. Sulsky, D., Zhou, S., Schreyer, H.L.: Application of a particle-in-cell method to solid mechanics. Comput. Phys. Commun. 87, 236–252 (1995)CrossRefMATHGoogle Scholar
  33. Tamma, K.K., Namburu, R.R.: Computational approaches with applications to non-classical and classical thermomechanical problems. Appl. Mech. Rev. 50, 514–551 (1997)CrossRefGoogle Scholar
  34. Wang, F., Lin, G., Zheng, B.J., Hu, Z.Q.: An improved local radial point interpolation method for transient heat conduction analysis. Chin. Phys. B 22, 060206 (2013)CrossRefGoogle Scholar
  35. Wieckowski, Z.: The material point method in large strain engineering problems. Comput. Methods Appl. Mech. Eng. 193, 4417–4438 (2004)CrossRefMATHGoogle Scholar
  36. Zhang, H.W., Wang, K.P., Chen, Z.: Material point method for dynamic analysis of saturated porous media under external contact/impact of solid bodies. Comput. Methods Appl. Mech. Eng. 198, 1456–1472 (2009)CrossRefMATHGoogle Scholar
  37. Zhang, W.J., Wang, X.H., Jiang, S.L.: Smooth particle hydrodynamics for transient heat conduction. Chem. Eng. Equip. 10, 18–23 (2010) (in Chinese)Google Scholar
  38. Zhou, S.Z., Zhang, X., Ma, H.L.: Numerical simulation of human head impact using the material point method. Int. J. Comput. Math. 10, 1350014 (2013)MathSciNetCrossRefGoogle Scholar
  39. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals. 6, Oxford (2005)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Jun Tao
    • 1
  • Yonggang Zheng
    • 1
  • Zhen Chen
    • 1
    • 2
  • Hongwu Zhang
    • 1
  1. 1.State Key Laboratory of Structure Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and MechanicsDalian University of TechnologyDalianPeople’s Republic of China
  2. 2.Department of Civil and Environmental EngineeringUniversity of MissouriColumbiaUSA

Personalised recommendations