The Method of Fundamental Solutions for Two- and Three-Dimensional Transient Heat Conduction Problems Involving Point and Curved Line Heat Sources

  • Mehrdad MohammadiEmail author
Research Paper


The objective of this paper is to formulate the method of fundamental solutions (MFS) for solving transient heat conduction problems involving concentrated heat sources with time- and space-dependent intensities. The heat sources can be concentrated on points or arbitrary curved paths. In the proposed MFS formulation, a linear combination of time-dependent fundamental solutions and particular solutions is considered for approximating the solution. The particular solutions are expressed as time- and space-dependent integrals, which are computed effectively without considering any internal points or internal cells and without using any kind of time transformation. Several 2D and 3D numerical examples are analyzed. The finite element method (FEM) with a fine mesh and with a small size time step is used to find an accurate reference solution. In comparison to the FEM solution, the proposed MFS formulation achieves accurate results with a small number of source points and a large size time step.


Method of fundamental solutions Transient heat conduction Curved line heat source Point heat source Meshfree 


  1. Alves CJS, Martins NFM, Valtchev SS (2018) Trefftz methods with cracklets and their relation to BEM and MFS. Eng Anal Bound Elem 95:93–104MathSciNetCrossRefGoogle Scholar
  2. Balakrishnan K, Sureshkumar R, Ramachandran PA (2000) An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems. Comput Math Appl 43:289–304MathSciNetCrossRefGoogle Scholar
  3. Burgess G, Mahajerin E (1984) A comparison of the boundary element and superposition methods. Comput Struct 19:697–705CrossRefGoogle Scholar
  4. Chantasiriwan S (2006) Methods of fundamental solutions for time dependent heat conduction problems. Int J Numer Methods Eng 66:147–165CrossRefGoogle Scholar
  5. Chen CS, Golberg MA, Hon YC (1998) The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations. Int J Numer Methods Eng 43:1421–1435CrossRefGoogle Scholar
  6. Cui M, Xu BB, Feng WZ, Zhang Y, Gao XW, Peng HF (2018) A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity. Numer Heat Transf B Fundam 73:1–18CrossRefGoogle Scholar
  7. Curran DAS, Cross M, Lewis BA (1980) Solution of parabolic differential equations by the boundary element method using discretisation in time. Appl Math Model 4:398–400MathSciNetCrossRefGoogle Scholar
  8. DeMey G (1985) The auxiliary boundary element method for time dependent problems. J Comput Appl Math 12–13:239–245MathSciNetCrossRefGoogle Scholar
  9. Dong CF, Sun FY, Meng BQ (2007) A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Eng Anal Bound Elem 31:75–82CrossRefGoogle Scholar
  10. Dowden JM, Ducharme R, Kapadia PD (1998) Time-dependent line and point sources: a simple model for time-dependent welding processes. Lasers Eng 7:215–228Google Scholar
  11. Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9:69–95MathSciNetCrossRefGoogle Scholar
  12. Grabski JK (2019) On the sources placement in the method of fundamental solutions for time-dependent heat conduction problems. Comput Math Appl. CrossRefGoogle Scholar
  13. Hansen P (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34:561–580MathSciNetCrossRefGoogle Scholar
  14. Hematiyan MR, Karami G (2008) A meshless boundary element method formulation for transient heat conduction problems with heat sources. Sci Iran 15:348–359zbMATHGoogle Scholar
  15. Hematiyan MR, Mohammadi M, Aliabadi MH (2011) Boundary element analysis of two- and three-dimensional thermo-elastic problems with various concentrated heat sources. J Strain Anal Eng Des 46:227–242CrossRefGoogle Scholar
  16. Hematiyan MR, Haghighi A, Khosravifard A (2018) A two-constrained method for appropriate determination of the configuration of source and collocation points in the method of fundamental solutions for 2D Laplace equation. Adv Appl Math Mech 10:496–522CrossRefGoogle Scholar
  17. Hon YC, Li M (2009) A discrepancy principle for the source points location in using the MFS for solving the BHCP. Int J Comput Methods 6:181–197MathSciNetCrossRefGoogle Scholar
  18. Johansson BT, Lesnic D, Reeve T (2011) A method of fundamental solutions for two-dimensional heat conduction. Int J Comput Math 88:1697–1713MathSciNetCrossRefGoogle Scholar
  19. Lefèvre F, Le Niliot C (2002) The BEM for point heat source estimation: application to multiple static sources and moving sources. Int J Therm Sci 41:536–545CrossRefGoogle Scholar
  20. Levin P (2008) A general solution of 3-D quasi-steady state problem of a moving heat source on a semi-infinite solid. Mech Res Commun 35:151–157MathSciNetCrossRefGoogle Scholar
  21. Mohammadi M, Hematiyan MR, Marin L (2010) Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions. Eng Anal Bound Elem 34:655–665MathSciNetCrossRefGoogle Scholar
  22. Mohammadi M, Hematiyan MR, Shiah YC (2018) An efficient analysis of steady-state heat conduction involving curved line/surface heat sources in two/three-dimensional isotropic media. J Theor Appl Mech 56:1123–1137CrossRefGoogle Scholar
  23. Mohammadi M, Hematiyan MR, Khosravifard A (2019) Analysis of two- and three-dimensional steady-state thermo-mechanical problems including curved line/surface heat sources using the method of fundamental solutions. Eur J Comput Mech 28:51–80Google Scholar
  24. Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New YorkzbMATHGoogle Scholar
  25. Nowak AJ (1989) The multiple reciprocity method of solving heat conduction problems. In: 11th BEM conference, Cambridge, Massachusetts, USA. Springer, vol 2, pp 81–95Google Scholar
  26. Ochiai Y, Kitayama Y (2009) Three-dimensional unsteady heat conduction analysis by triple-reciprocity boundary element method. Eng Anal Bound Elem 33:789–795MathSciNetCrossRefGoogle Scholar
  27. Partridge PW, Sensale B (2000) The method of fundamental solutions with dual reciprocity for diffusion and diffusion-convection using subdomains. Eng Anal Bound Elem 24:633–641CrossRefGoogle Scholar
  28. Reeve T, Johansson BT (2013) The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem. Eng Anal Bound Elem 37:569–578MathSciNetCrossRefGoogle Scholar
  29. Rizzo FJ, Shippy DJ (1970) A method of solution for certain problems of transient heat conduction. AIAA J 8:2004–2009CrossRefGoogle Scholar
  30. Shiah YC, Guao TL, Tan CL (2005) Two-dimensional BEM thermoelastic analysis of anisotropic media with concentrated heat sources. Comput Model Eng Sci 7:321–338MathSciNetzbMATHGoogle Scholar
  31. Shiah YC, Hwang PW, Yang RB (2006) Heat conduction in multiply adjoined anisotropic media with embedded point heat sources. J Heat Transf 128:207–214CrossRefGoogle Scholar
  32. Shiah YC, Chaing YC, Matsumoto T (2016) Analytical transformation of volume integral for the time-stepping BEM analysis of 2D transient heat conduction in anisotropic media. Eng Anal Bound Elem 64:101–110MathSciNetCrossRefGoogle Scholar
  33. Sladek J, Sladek V, Zhangb Ch (2004) A local BIEM for analysis of transient heat conduction with nonlinear source terms in FGMs. Eng Anal Bound Elem 28:1–11CrossRefGoogle Scholar
  34. Sobamowo GM, Jaiyesimi L, Waheed AO (2017) Transient three-dimensional thermal analysis of a slab with internal heat generation and heated by a point moving heat source. Iran J Mech Eng 18:43–64Google Scholar
  35. Valtchev SS, Roberty NC (2008) A time-marching MFS scheme for heat conduction problems. Eng Anal Boundary Elem 32:480–493CrossRefGoogle Scholar
  36. Walker SP (1997) Diffusion problems using transient discrete source superposition. Int J Numer Methods Eng 35:165–178CrossRefGoogle Scholar
  37. Wrobel LC, Brebbia CA (1979) The boundary element method for steady state and transient heat conduction. In: Numerical methods in thermal problems, first international conference, Swansea, Wales. Pineridge PressGoogle Scholar
  38. Wrobel LC, Brebbia CA (1987) The dual reciprocity boundary element formulation for nonlinear diffusion problems. Comput Methods Appl Mech Eng 65:147–164CrossRefGoogle Scholar
  39. Young DL, Tsai CC, Murugesan K, Fan CM, Chen CW (2004) Time dependent fundamental solutions for homogeneous diffusion problems. Eng Anal Bound Elem 28:1463–1473CrossRefGoogle Scholar
  40. Zhu S, Zhang Y, Marchant TR (1995) A DRBEM model for microwave heating problems. Appl Math Model 19:287–297CrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, College of Engineering, Shiraz BranchIslamic Azad UniversityShirazIran

Personalised recommendations