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An efficient boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials under heat source load

  • Qiang Xi
  • Zhuo-Jia FuEmail author
  • Timon Rabczuk
Original Paper

Abstract

This paper presents a boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials (FGMs) with heat source load. In the proposed scheme, Laplace transformation and the numerical inverse Laplace transformation (NILT) are implemented to avoid the troublesome time-stepping effect on numerical efficiency. The collocation Trefftz method (CTM) coupled with composite multiple reciprocity method is used to obtain the high accurate results in the solution of nonhomogeneous problems in Laplace-space domain. The extended precision arithmetic is introduced to overcome the ill-posed issues generated from the CTM simulation, the NILT process and the large-size-ratio FGM. Heuristic error analysis and numerical investigation are presented to demonstrate the effectiveness of the proposed scheme for transient thermal analysis. Several benchmark examples are considered under large-size-ratio FGMs with some specific spatial variations (quadratic, exponential and trigonometric functions). The proposed scheme is validated in comparison with known analytical solutions and COMSOL simulation.

Keywords

Collocation Trefftz scheme Numerical inverse Laplace transformation Extended precision arithmetic Transient thermal analysis Large size ratio 

Notes

Acknowledgements

The authors thank the anonymous reviewers of this article for their very helpful comments and suggestions to significantly improve the academic quality of this article. The work described in this paper was supported by the National Science Funds of China (Grant No. 11772119), the Fundamental Research Funds for the Central Universities (Grant No. 2016B06214), the Foundation for Open Project of State Key Laboratory of Structural Analysis for Industrial Equipment (Grant No. GZ1707), Qing Lan Project.

References

  1. 1.
    Zhao X, Liew KM (2010) A mesh-free method for analysis of the thermal and mechanical buckling of functionally graded cylindrical shell panels. Comput Mech 45:297–310CrossRefzbMATHGoogle Scholar
  2. 2.
    Tian JH, Jiang K (2018) Heat conduction investigation of the functionally graded materials plates with variable gradient parameters under exponential heat source load. Int J Heat Mass Transf 122:22–30CrossRefGoogle Scholar
  3. 3.
    Thai CH, Tran DT, Nguyen-Xuan H (2017) A naturally stabilized nodal integration meshfree formulation for thermo-mechanical analysis of functionally graded material plates. In: International conference on advances in computational mechanics, pp 615–629Google Scholar
  4. 4.
    Miao Y, Wang Q, Zhu H, Li Y (2014) Thermal analysis of 3D composites by a new fast multipole hybrid boundary node method. Comput Mech 53:77–90MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Qian LF, Batra RC (2005) Three-Dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov–Galerkin method. Comput Mech 35:214–226CrossRefzbMATHGoogle Scholar
  6. 6.
    Zhang HH, Han SY, Fan LF, Huang D (2018) The numerical manifold method for 2D transient heat conduction problems in functionally graded materials. Eng Anal Bound Elem 88:145–155MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wang H, Qin QH, Kang YL (2006) A meshless model for transient heat conduction in functionally graded materials. Comput Mech 38:51–60CrossRefzbMATHGoogle Scholar
  8. 8.
    Krahulec S, Sladek J, Sladek V, Hon YC (2016) Meshless analyses for time-fractional heat diffusion in functionally graded materials. Eng Anal Bound Elem 62:57–64MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhou SW, Zhuang XY, Zhu HH, Rabczuk T (2018) Phase field modelling of crack propagation, branching and coalescence in rocks. Theor Appl Fract Mec 96:174–192CrossRefGoogle Scholar
  10. 10.
    Fu ZJ, Chen W, Yang HT (2013) Boundary particle method for Laplace transformed time fractional diffusion equations. J Comput Phys 235:52–66MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sutradhar A, Paulino GH, Gray LJ (2002) Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. Eng Anal Bound Elem 26:119–132CrossRefzbMATHGoogle Scholar
  12. 12.
    Fu ZJ, Reutskiy S, Sun HG, Ma J, Khan MA (2019) A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains. Appl Math Lett 94:105–111MathSciNetCrossRefGoogle Scholar
  13. 13.
    Abreu AI, Canelas A, Mansur WJ (2013) A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs. Appl Math Model 37:776–792MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kielhorn L, Schanz M (2010) Convolution quadrature method-based symmetric Galerkin boundary element method for 3-d elastodynamics. Int J Numer Methods Eng 76:1724–1746MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li Y, Zhang JM, Xie GZ, Zheng XS, Guo SP (2014) Time-domain BEM analysis for three-dimensional elastodynamic problems with initial conditions. Comput Model Eng Sci 101:187–206MathSciNetzbMATHGoogle Scholar
  16. 16.
    Cho JR, Ha DY (2002) Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM. Comput Methods Appl Mech Eng 191:3195–3211CrossRefzbMATHGoogle Scholar
  17. 17.
    Brian PLT (2010) A finite-difference method of high-order accuracy for the solution of three-dimensional transient heat conduction problems. AIChE J 7:367–370CrossRefGoogle Scholar
  18. 18.
    Wang H, Lei YP, Wang JS, Qin QH, Xiao Y (2015) Theoretical and computational modeling of clustering effect on effective thermal conductivity of cement composites filled with natural hemp fibers. J Compos Mater 50:1509–1521CrossRefGoogle Scholar
  19. 19.
    Mijuca D, Žiberna A, Medjo B (2007) A novel primal-mixed finite element approach for heat transfer in solids. Comput Mech 39:367–379MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Olatunji-Ojo AO, Boetcher SKS, Cundari TR (2012) Thermal conduction analysis of layered functionally graded materials. Comput Mater Sci 54:329–335CrossRefGoogle Scholar
  21. 21.
    Sarler B, Mencinger J (1999) Solution of temperature field in DC cast aluminium alloy billet by the dual reciprocity boundary element method. Int J Numer Methods Heat Fluid Flow 9:269–297CrossRefzbMATHGoogle Scholar
  22. 22.
    Feng WZ, Yang K, Cui M, Gao XW (2016) Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems. Int Commun Heat Mass Transf 79:21–30CrossRefGoogle Scholar
  23. 23.
    Yang K, Peng HF, Wang J, Xing CH, Gao XW (2017) Radial integration BEM for solving transient nonlinear heat conduction with temperature-dependent conductivity. Int J Heat Mass Transf 108:1551–1559CrossRefGoogle Scholar
  24. 24.
    Qu WZ, Chen W (2015) Solution of two-dimensional stokes flow problems using improved singular boundary method. Adv Appl Math Mech 7:13–30MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li JP, Fu ZJ, Chen W (2016) Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method. Comput Math Appl 71:381–390MathSciNetCrossRefGoogle Scholar
  26. 26.
    Fu ZJ, Chen W, Wen PH, Zhang CZ (2018) Singular boundary method for wave propagation analysis in periodic structures. J Sound Vib 425:170–188CrossRefGoogle Scholar
  27. 27.
    Sarler B (1996) Boundary integral formulation of general source-based method for convective-diffusive solid-liquid phase change problems. Bound Elem 18:551–560zbMATHGoogle Scholar
  28. 28.
    Sun Y (2017) Indirect boundary integral equation method for the Cauchy problem of the Laplace equation. J Sci Comput 71:469–498MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Chen LC, Li XL (2019) Boundary element-free methods for exterior acoustic problems with arbitrary and high wavenumbers. Appl Math Model 72:85–103MathSciNetCrossRefGoogle Scholar
  30. 30.
    Chen LC, Liu X, Li XL (2019) The boundary element-free method for 2D interior and exterior Helmholtz problems. Comput Math Appl 77:846–864MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhou FL, Yuan L, Zhang JM, Cheng H, Lu CJ (2015) A time step amplification method in boundary face method for transient heat conduction. Int J Heat Mass Transf 84:671–679CrossRefGoogle Scholar
  32. 32.
    Li M, Chen CS, Chu CC, Young DL (2014) Transient 3D heat conduction in functionally graded materials by the method of fundamental solutions. Eng Anal Bound Elem 45:62–67MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lin J, Zhang CZ, Sun LL, Lu J (2018) Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method. Adv Appl Math Mech 10:322–342MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tang ZC, Fu ZJ, Zheng DJ, Huang JD (2018) Singular boundary method to simulate scattering of SH wave by the canyon topography. Adv Appl Math Mech 10:912–924MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wang FJ, Hua QS, Liu CS (2018) Boundary function method for inverse geometry problem in two-dimensional anisotropic heat conduction equation. Appl Math Lett 84:130–136MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Li JP, Fu ZJ, Chen W, Liu XT (2019) A dual-level method of fundamental solutions in conjunction with kernel-independent fast multipole method for large-scale isotropic heat conduction problems. Adv Appl Math Mech 11:501–517CrossRefGoogle Scholar
  37. 37.
    O’Hara P, Duarte CA, Eason T, Garzon J (2013) Efficient analysis of transient heat transfer problems exhibiting sharp thermal gradients. Comput Mech 51:743–764MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gu Y, He XQ, Chen W, Zhang CZ (2017) Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method. Comput Math Appl 75:33–44MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gao XW, Zhang JB, Zheng BJ, Zhang C (2016) Element-subdivision method for evaluation of singular integrals over narrow strip boundary elements of super thin and slender structures. Eng Anal Bound Elem 66:145–154MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhou H, Niu Z, Cheng C, Guan Z (2008) Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems. Comput Struct 86:1656–1671CrossRefGoogle Scholar
  41. 41.
    Sarra SA, Cogar S (2017) An examination of evaluation algorithms for the RBF method. Eng Anal Bound Elem 75:36–45MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kansa EJ, Holoborodko P (2017) On the ill-conditioned nature of C ∞ RBF strong collocation. Eng Anal Bound Elem 78:26–30MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Fu ZJ, Xi Q, Chen W, Cheng AH-D (2018) A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations. Comput Math Appl 76:760–773MathSciNetCrossRefGoogle Scholar
  44. 44.
    Fu ZJ, Chen W, Qin QH (2012) Three boundary meshless methods for heat conduction analysis in nonlinear FGMs with Kirchhoff and Laplace transformation. Adv Appl Math Mech 4:519–542MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Movahedian B, Boroomand B, Soghrati S (2013) A Trefftz method in space and time using exponential basis functions: application to direct and inverse heat conduction problems. Eng Anal Bound Elem 37:868–883MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wei X, Chen W, Chen B, Sun LL (2015) Singular boundary method for heat conduction problems with certain spatially varying conductivity. Comput Math Appl 69:206–222MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sutradhar A, Paulino GH (2004) The simple boundary element method for transient heat conduction in functionally graded materials. Comput Methods Appl Mech Eng 193:4511–4539CrossRefzbMATHGoogle Scholar
  48. 48.
    Sladek J, Sladek V, Zhang C (2004) A local BIEM for analysis of transient heat conduction with nonlinear source terms in FGMs. Eng Anal Bound Elem 28:1–11CrossRefzbMATHGoogle Scholar
  49. 49.
    Chen W, Fu ZJ, Qin QH (2009) Boundary particle method with high-order Trefftz functions. Comput Mater Contin 13:201–217MathSciNetzbMATHGoogle Scholar
  50. 50.
    Li GY, Guo SP, Zhang JM, Li Y, Han L (2015) Transient heat conduction analysis of functionally graded materials by a multiple reciprocity boundary face method. Eng Anal Bound Elem 60:81–88MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Valkó PP, Abate J (2005) Numerical inversion of 2-D Laplace transforms applied to fractional diffusion equations. Appl Numer Math 53:73–88MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Abate J, Valkó PP (2004) Multi-precision Laplace transform inversion. Int J Numer Meth Eng 60:979–993CrossRefzbMATHGoogle Scholar
  53. 53.
    Abate J, Whitt W (2006) A unified framework for numerically inverting Laplace transforms. INFORMS J Comput 18:408–421MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Li ZC, Lu TT, Huang HT, Cheng HD (2009) Error analysis of Trefftz methods for Laplace’s equations and its applications. Comput Model Eng Sci 5252:39–8139MathSciNetzbMATHGoogle Scholar
  55. 55.
    Gilbarg D, Trudinger NS (1977) Elliptic partial differential equations of second order. Springer, New YorkCrossRefzbMATHGoogle Scholar
  56. 56.
    Li ZC (2008) The Trefftz method for the Helmholtz equation with degeneracy. Appl Numer Math 58:131–159MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    http://www.advanpix.com. Multi-precision computing toolbox for MATLAB. In: Advanpix LLC, Yokohama, 2008–2018

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Key Laboratory of Coastal Disaster and Defence, Ministry of EducationHohai UniversityNanjingChina
  2. 2.Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and MaterialsHohai UniversityNanjingChina
  3. 3.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianChina
  4. 4.Institute of Structural MechanicsBauhaus-University WeimarWeimarGermany

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