Acta Mechanica Solida Sinica

, Volume 30, Issue 5, pp 445–464 | Cite as

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

  • Gao Lin
  • Peng Li
  • Jun Liu
  • Pengchong Zhang


The Non-uniform rational B-spline (NURBS) enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a reduction of the spatial dimension by one with no fundamental solution required. Nevertheless, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often unavoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analysis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.


Transient heat conduction analysis Scaled boundary finite element method NURBS Isogeometric analysis Modified precise integration method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Y. Zhang, Z. Wan, B. Gu, H. Zhang, P. Zhou, Finite difference analysis of transient heat transfer in surrounding rock mass of high geothermal roadway, Math. Prob. Eng 2016 (2016) 8951524.Google Scholar
  2. 2.
    E. Li, Z. Zhang, Z.C. He, X. Xu, G.R. Liu, Q. Li, Smoothed finite element method with exact solutions in heat transfer problems, Int. J. Heat Mass Transf 78 (2014) 1219–1231.CrossRefGoogle Scholar
  3. 3.
    D. Prestini, G. Filippini, P.S.B. Zdanski, M. Vaz Jr., Fundamental approach to anisotropic heat conduction using the element-based finite volume method, Numer. Heat Transf. Part B: Fundam 71 (4) (2017) 327–345.CrossRefGoogle Scholar
  4. 4.
    M. Messner, M. Schanz, J. Tausch, An efficient Galerkin boundary element method for the transient heat equation, SIAM J. Sci. Comput 37 (3) (2015) A1554–A1576.MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Sun, H.L. Yi, H.P. Tan, Local RBF meshless scheme for coupled radiative and conductive heat transfer, Numer. Heat Transf Part A: Appl (2016) 1–15.CrossRefGoogle Scholar
  6. 6.
    W.X. Zhong, F.W. Williams, A precise time-step integration method, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci 208 (6) (1994) 427–430.CrossRefGoogle Scholar
  7. 7.
    W. Yao, B. Yu, X. Gao, Q. Gao, A precise integration boundary element method for solving transient heat conduction problems, Int J Heat Mass Transf 78 (2014) 883–891.CrossRefGoogle Scholar
  8. 8.
    Q.H. Li, S.S. Chen, G.X. Kou, Transient heat conduction analysis using the MLPG method and modified precise time-step integration method, J Comput Phys 230 (7) (2011) 2736–2750.MathSciNetCrossRefGoogle Scholar
  9. 9.
    F.L. Mei, G.L. Li, Simulation of transient heat transfer based on element-free Galerkin method and increment-dimensional precise integration method, Applied Mechanics and Materials, 204, Trans Tech Publications, 2012, pp. 4254–4259.CrossRefGoogle Scholar
  10. 10.
    J.P. Wolf, C. Song, Finite-element Modelling of Unbounded Media, Wiley, Chichester, 1996.zbMATHGoogle Scholar
  11. 11.
    E.T. Ooi, S. Natarajan, C. Song, E.H. Ooi, Dynamic fracture simulations using the scaled boundary finite element method on hybrid polygon–quadtree meshes, Int. J. Impact Eng 90 (2016) 154–164.CrossRefGoogle Scholar
  12. 12.
    A.A. Saputra, C. Birk, C. Song, Computation of three-dimensional fracture parameters at interface cracks and notches by the scaled boundary finite element method, Eng. Fract. Mech 148 (2015) 213–242.CrossRefGoogle Scholar
  13. 13.
    H. Xu, D. Zou, X. Kong, Z. Hu, Study on the effects of hydrodynamic pressure on the dynamic stresses in slabs of high CFRD based on the scaled boundary finite-element method, Soil Dyn. Earthq. Eng 88 (2016) 223–236.CrossRefGoogle Scholar
  14. 14.
    C. Song, The scaled boundary finite element method in structural dynamics, Int. J. Numer. Methods Eng 77 (8) (2009) 1139–1171.MathSciNetCrossRefGoogle Scholar
  15. 15.
    K. Chen, D. Zou, X. Kong, A. Chan, Z. Hu, A novel nonlinear solution for the polygon scaled boundary finite element method and its application to geotechnical structures, Comput. Geotech 82 (2017) 201–210.CrossRefGoogle Scholar
  16. 16.
    H. Xu, D. Zou, X. Kong, X. Su, Error study of Westergaard’s approximation in seismic analysis of high concrete-faced rockfill dams based on SBFEM, Soil Dyn. Earthq. Eng 94 (2017) 88–91.CrossRefGoogle Scholar
  17. 17.
    N.M. Syed, B.K. Maheshwari, Improvement in the computational efficiency of the coupled FEM–SBFEM approach for 3D seismic SSI analysis in the time domain, Comput. Geotech 67 (2015) 204–212.CrossRefGoogle Scholar
  18. 18.
    K. Chen, D. Zou, X. Kong, A nonlinear approach for the three-dimensional polyhedron scaled boundary finite element method and its verification using Koyna gravity dam, Soil Dyn. Earthq. Eng 96 (2017) 1–12.CrossRefGoogle Scholar
  19. 19.
    L. Lehmann, S. Langer, D. Clasen, Scaled boundary finite element method for acoustics, J. Comput. Acoust 14 (04) (2006) 489–506.MathSciNetCrossRefGoogle Scholar
  20. 20.
    E.T. Ooi, C. Song, F. Tin-Loi, A scaled boundary polygon formulation for elasto-plastic analyses, Comput. Methods Appl. Mech. Eng 268 (2014) 905–937.MathSciNetCrossRefGoogle Scholar
  21. 21.
    A.J. Deeks, L. Cheng, Potential flow around obstacles using the scaled boundary finite-element method, Int. J. Numer. Methods Fluids 41 (7) (2003) 721–741.CrossRefGoogle Scholar
  22. 22.
    Y. He, H. Yang, A.J. Deeks, On the use of cyclic symmetry in SBFEM for heat transfer problems, Int. J. Heat Mass Transf 71 (2014) 98–105.CrossRefGoogle Scholar
  23. 23.
    M.H. Bazyar, A. Talebi, Scaled boundary finite-element method for solving non-homogeneous anisotropic heat conduction problems, Appl. Math. Model 39 (23) (2015) 7583–7599.MathSciNetGoogle Scholar
  24. 24.
    C. Song, J.P. Wolf, The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion, Int. J. Numer. Methods Eng 45 (10) (1999) 1403–1431.CrossRefGoogle Scholar
  25. 25.
    W. Wang, Y. Peng, Y. Zhou, Q. Zhang, Liquid sloshing in partly-filled laterally-excited cylindrical tanks equipped with multi baffles, Appl. Ocean Res 59 (2016) 543–563.CrossRefGoogle Scholar
  26. 26.
    W. Wang, Z. Guo, Y. Peng, Q. Zhang, A numerical study of the effects of the T-shaped baffles on liquid sloshing in horizontal elliptical tanks, Ocean Eng 111 (2016) 543–568.CrossRefGoogle Scholar
  27. 27.
    W. Wang, G. Tang, X. Song, Y. Zhou, Transient sloshing in partially filled laterally excited horizontal elliptical vessels with T-shaped baffles, J. Press. Vessel Technol 139 (2) (2017) 021302.Google Scholar
  28. 28.
    A.J. Deeks, C.E. Augarde, A meshless local Petrov-Galerkin scaled boundary method, Comput. Mech 36 (3) (2005) 159–170.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Y. He, H. Yang, A.J. Deeks, An element-free Galerkin scaled boundary method for steady-state heat transfer problems, Numer Heat Transf Part B: Fundam 64 (3) (2013) 199–217.CrossRefGoogle Scholar
  30. 30.
    T.J. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput Methods Appl. Mech. Eng 194 (39) (2005) 4135–4195.MathSciNetzbMATHGoogle Scholar
  31. 31.
    Y.O. Tafa, G. Zhao, W. Wang, Isogeometric analysis prediction of stress concentration factor of Trivariate NURBS-based rectangular plate with central elliptical hole, Applied Mechanics and Materials, 556, Trans Tech Publications, 2014, pp. 742–746.CrossRefGoogle Scholar
  32. 32.
    L.V. Tran, C.H. Thai, H.T. Le, B.S. Gan, J. Lee, H. Nguyen-Xuan, Isogeometric analysis of laminated composite plates based on a four-variable refined plate theory, Eng. Anal. Bound. Elem 47 (2014) 68–81.MathSciNetCrossRefGoogle Scholar
  33. 33.
    R. Kolman, S. Sorokin, B. Bastl, J. Kopačka, J. Plešek, Isogeometric analysis of free vibration of simple shaped elastic samplesa), J. Acoust. Soc. Am 137 (4) (2015) 2089–2100.CrossRefGoogle Scholar
  34. 34.
    T. Yu, S. Yin, T.Q. Bui, S. Xia, S. Tanaka, S. Hirose, NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method, Thin-Walled Struct 101 (2016) 141–156.CrossRefGoogle Scholar
  35. 35.
    H. Al Akhras, T. Elguedj, A. Gravouil, M. Rochette, Isogeometric analysis-suitable trivariate NURBS models from standard B-Rep models, Comput. Methods Appl. Mech. Eng 307 (2016) 256–274.MathSciNetCrossRefGoogle Scholar
  36. 36.
    A. Cazzani, M. Malagù, E. Turco, F. Stochino, Constitutive models for strongly curved beams in the frame of isogeometric analysis, Math. Mech. Solids (2015) 1081286515577043.MathSciNetCrossRefGoogle Scholar
  37. 37.
    M.N. Nguyen, T.Q. Bui, T. Yu, S. Hirose, Isogeometric analysis for unsaturated flow problems, Comput. Geotech 62 (2014) 257–267.CrossRefGoogle Scholar
  38. 38.
    M. Dittmann, M. Franke, I. Temizer, C. Hesch, Isogeometric analysis and thermomechanical mortar contact problems, Comput. Methods Appl. Mech. Eng 274 (2014) 192–212.MathSciNetCrossRefGoogle Scholar
  39. 39.
    D. Fußeder, B. Simeon, A.V. Vuong, Fundamental aspects of shape optimization in the context of isogeometric analysis, Comput. Methods Appl. Mech. Eng 86 (2015) 313–331.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Y. Wang, D.J. Benson, Isogeometric analysis for parameterized LSM-based structural topology optimization, Comput. Mech 57 (1) (2016) 19–35.MathSciNetCrossRefGoogle Scholar
  41. 41.
    Y. Zhang, G. Lin, Z.Q. Hu, Isogeometric analysis based on scaled boundary finite element method, IOP Conference Series: Materials Science and Engineering, 10, IOP Publishing, 2010.CrossRefGoogle Scholar
  42. 42.
    S. Klinkel, L. Chen, W. Dornisch, A NURBS based hybrid collocation–Galerkin method for the analysis of boundary represented solids, Comput. Methods Appl Mech Eng 284 (2015) 689–711.MathSciNetCrossRefGoogle Scholar
  43. 43.
    S. Natarajan, J. Wang, C. Song, C. Birk, Isogeometric analysis enhanced by the scaled boundary finite element method, Comput. Methods Appl. Mech. Eng 283 (2015) 733–762.MathSciNetCrossRefGoogle Scholar
  44. 44.
    L. Piegl, W. Tiller, The NURBS Book, Springer Science & Business Media, Berlin Germany, 2012.zbMATHGoogle Scholar
  45. 45.
    J.P. Wolf, The Scaled Boundary Finite Element Method, John Wiley & Sons, Hoboken American, 2003.Google Scholar
  46. 46.
    M. Wang, X. Zhou, Modified precise time-step integration method of structural dynamic analysis, Earthq. Eng. Eng. Vib 4 (2) (2005) 287–293.Google Scholar
  47. 47.
    H.S. Carslaw, J.C. Jaeger, H. Feshbach, Conduction of heat in solids, Phys. Today 15 (11) (1962) 74–76.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2017

Authors and Affiliations

  • Gao Lin
    • 1
    • 2
  • Peng Li
    • 1
    • 2
  • Jun Liu
    • 1
    • 2
    • 3
  • Pengchong Zhang
    • 1
    • 2
  1. 1.School of Hydraulic Engineering, Faculty of Infrastructure EngineeringDalian University of TechnologyDalianChina
  2. 2.State Key Laboratory of Coastal and Offshore EngineeringDalian University of TechnologyDalianChina
  3. 3.Center for Offshore Foundation Systems, Faculty of EngineeringUniversity of Western AustraliaPerthAustralia

Personalised recommendations