Skip to main content
SpringerLink
Log in

The improved element-free Galerkin method for three-dimensional transient heat conduction problems

  • Article
  • Published:
Science China Physics, Mechanics and Astronomy Aims and scope Submit manuscript

Abstract

With the improved moving least-squares (IMLS) approximation, an orthogonal function system with a weight function is used as the basis function. The combination of the element-free Galerkin (EFG) method and the IMLS approximation leads to the development of the improved element-free Galerkin (IEFG) method. In this paper, the IEFG method is applied to study the partial differential equations that control the heat flow in three-dimensional space. With the IEFG technique, the Galerkin weak form is employed to develop the discretized system equations, and the penalty method is applied to impose the essential boundary conditions. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the transient heat conduction equations and the boundary and initial conditions are time dependent, the scaling parameter, number of nodes and time step length are considered in a convergence study.

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

Similar content being viewed by others

References

  1. Wazwaz A M. Partial Differential Equations: Methods and Applications. Lisse: Swets & Zeitlinger B V, 2002

    Google Scholar 

  2. Zienkiewicz O C. The Finite Element Method. 3rd ed. London: McGraw-Hill, 1977

    MATH  Google Scholar 

  3. Bathe K J, Khoshgoftaar M R. Finite element formulation and solution of nonlinear heat transfer. Nucl Eng Des, 1979, 51: 389–401

    Article  Google Scholar 

  4. Donea J, Giuliani S. Finite element analysis of steady-state nonlinear heat transfer problems. Nucl Eng Des, 1974, 30: 205–213

    Article  Google Scholar 

  5. Skerget P, Alujevic A. Boundary element method for nonlinear transient heat transfer of reactor solids with convection and radiation on surfaces. Nucl Eng Des, 1983, 76: 47–54

    Article  Google Scholar 

  6. Belytschko T, Krongauz Y, Organ D, et al. Meshless methods: An overview and recent developments. Comput Method Appl M, 1996, 139: 3–47

    Article  MATH  Google Scholar 

  7. Li S, Liu W K. Meshless and particles methods and their applications. Appl Mech Rev, 2002, 55: 1–34

    Article  ADS  Google Scholar 

  8. Chen C S. A numerical method for heat transfer problems using collocation and radial basis functions. Int J Numer Meth Eng, 1998, 42: 1263–1278

    Article  MATH  Google Scholar 

  9. Singh I V. Meshless EFG method in three-dimensional heat transfer problems: A numerical comparison, cost and error analysis. Numer Heat Tr A-Appl, 2004, 46: 199–220

    Article  ADS  Google Scholar 

  10. Singh I V. A numerical study of weight functions, scaling and penalty parameters for heat transfer applications. Numer Heat Tr A-Appl, 2005, 47: 1025–1053

    Article  ADS  Google Scholar 

  11. Singh I V, Prakash R. The numerical solution of three dimensional transient heat conduction problems using element free Galerkin method. Int J Heat Tech, 2005, 21: 73–80

    Google Scholar 

  12. Singh I V, Jain P K. Parallel EFG algorithm for heat transfer problems. Adv Eng Softw, 2005, 36: 554–560

    Article  MATH  Google Scholar 

  13. Singh A, Singh I V, Prakash R. Meshless element free galerkin method for unsteady nonlinear heat transfer problems. Int J Heat Mass Tran, 2007, 50: 1212–1219

    Article  MATH  Google Scholar 

  14. Arefmanesh A, Najafi M, Abdi H. A meshless local Petrov-Galerkin method for fluid dynamics and heat transfer applications. J Fluid Eng-T Asme, 2005, 127: 647–655

    Article  Google Scholar 

  15. Batra R C, Porfiri M, Spinello D. Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction. Int J Numer Meth Eng, 2004, 61: 2461–2479

    Article  MATH  Google Scholar 

  16. Liu Y, Zhang X, Lu M W. A meshless method based on least-squares approach for steady- and unsteady-state heat conduction problems. Numer Heat Tr A-Appl, 2005, 47: 257–275

    Article  ADS  Google Scholar 

  17. Sladek J, Sladek V, Atluri S N. A pure contour formulation for the meshless local boundary integral equation method in thermoelasticity. Cmes-Comp Modelo Eng, 2001, 2: 423–433

    MathSciNet  MATH  Google Scholar 

  18. Sladek J, Sladek V, Hellmich C. Heat conduction analysis of 3-D axisymmetric and anisotropic FGM bodies by meshless local Petrov-Galerkin method. Comput Mechs, 2006, 38: 157–167

    Google Scholar 

  19. Marin L, Lesnic D. The method of fundamental solutions for nonlinear functionally graded materials. Int J Solids Struct, 2007, 44: 6878–6890

    Article  MATH  Google Scholar 

  20. Fu Z J, Chen W, Qin Q H. Boundary knot method for heat conduction in nonlinear functionally graded material. Eng Anal Bound Elem, 2011, 35: 729–734

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang Z, Zhao P, Liew K M. Analyzing three-dimensional potential problems with the improved element-free Galerkin method. Comput Mechs, 2009, 44: 273–284

    Article  MathSciNet  ADS  Google Scholar 

  22. Liu G R. Mesh free methods: Moving beyond the finite element method. Boca Raton, FL: CRC Press LLC, 2003

    Google Scholar 

  23. Nayroles B, Touzot G, Villon P. Generalizing the finite element method: Difference approximation and diffuse elements. Comput Mechs, 1992, 10: 583–594

    Google Scholar 

  24. Lancaster P, Salkauskas K. Surfaces generated by moving least-squares methods. Math Comput, 1981, 37: 141–158

    Article  MathSciNet  MATH  Google Scholar 

  25. Liew K M, Cheng Y, Kitipornchai S. Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform. Int J Numer Meth Eng, 2005, 64: 1610–1627

    Article  MATH  Google Scholar 

  26. Liew K M, Cheng Y, Kitipornchai S. Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems. Int J Numer Meth Eng, 2006, 65: 1310–1332

    Article  MATH  Google Scholar 

  27. Liew K M, Zou G P, Rajendran S. A spline strip kernel particle method and its application to two-dimensional elasticity problems. Int J Numer Meth Eng, 2003, 57: 599–616

    Article  MATH  Google Scholar 

  28. Cheng Y, Liew K M, Kitipornchai S. Reply to ‘Comments on ‘Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems”. Int J Numer Meth Eng, 2009, 78: 1258–1260

    Article  MATH  Google Scholar 

  29. Gunther F C, Liu W K. Implementation of boundary conditions for meshless methods. Comput Method Appl M, 1998, 163: 205–230

    Article  MathSciNet  Google Scholar 

  30. Mukherjee Y X, Mukherjee S. On boundary conditions in the element-free Galerkin method. Comput Mech, 1997, 19: 264–270

    Article  MathSciNet  MATH  Google Scholar 

  31. Ventura G. An augmented Lagrangian approach to essential boundary conditions in meshless methods. Int J Numer Meth Eng, 2002, 53: 825–843

    Article  Google Scholar 

  32. Krysl P, Belytschko T. Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions. Comput Method Appl M, 1997, 148: 257–277

    Article  MathSciNet  MATH  Google Scholar 

  33. Chung H J, Belytschko T. An error estimate in the EFG method. Comput Mech, 1998, 21: 91–100

    Article  MathSciNet  MATH  Google Scholar 

  34. Gavete L, Cuesta J L, Ruiz A. A procedure for approximation of the error in the EFG method. Int J Numer Meth Eng, 2002, 53: 677–690

    Article  MathSciNet  MATH  Google Scholar 

  35. Gavete L, Gavete M L, Alonso B, et al. A posteriori error approximation in EFG method. Int J Numer Meth Eng, 2003, 58: 2239–2263

    Article  MathSciNet  MATH  Google Scholar 

  36. Gavete L, Faclon S, Ruiz A. An error indicator for the element free Galerkin method. Eur J Mech A-Solid, 2001, 20: 327–341

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, East China University of Science and Technology, Shanghai, 200237, China

    Zan Zhang

  2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    JianFei Wang & YuMin Cheng

  3. Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong, China

    Kim Meow Liew

Authors
  1. Zan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JianFei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. YuMin Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kim Meow Liew
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YuMin Cheng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Z., Wang, J., Cheng, Y. et al. The improved element-free Galerkin method for three-dimensional transient heat conduction problems. Sci. China Phys. Mech. Astron. 56, 1568–1580 (2013). https://doi.org/10.1007/s11433-013-5135-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11433-013-5135-0

Keywords

Search

Navigation