Skip to main content
SpringerLink
Log in

Three-dimensional numerical manifold method for heat conduction problems with a simplex integral on the boundary

  • Article
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

The three-dimensional numerical manifold method (3D-NMM), which is based on the derivation of Galerkin’s variation, is a powerful calculation tool that uses two cover systems. The 3D-NMM can be used to handle continue-discontinue problems and extend to THM coupling. In this study, we extended the 3D-NMM to simulate both steady-state and transient heat conduction problems. The modelling was carried out using the raster methods (RSM). For the system equation, a variational method was employed to drive the discrete equations, and the crucial boundary conditions were solved using the penalty method. To solve the boundary integral problem, the face integral of scalar fields and two-dimensional simplex integration were used to accurately describe the integral on polygonal boundaries. Several numerical examples were used to verify the results of 3D steady-state and transient heat-conduction problems. The numerical results indicated that the 3D-NMM is effective for handling 3D both steady-state and transient heat conduction problems with high solution accuracy.

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. Tan F, Tong D, Liang J, et al. Two-dimensional numerical manifold method for heat conduction problems. Eng Anal Bound Elem, 2022, 137: 119–138

    Article  MathSciNet  Google Scholar 

  2. Han Z Y, Li D Y, Li X B. Effects of axial pre-force and loading rate on Mode I fracture behavior of granite. Int J Rock Mech Min, 2022, 157: 105172

    Article  Google Scholar 

  3. Fan L F, Yi X W, Ma G W. Numerical manifold method (NMM) simulation of stress wave propagation through fractured rock mass. Int J Appl Mech, 2013, 05: 1350022

    Article  Google Scholar 

  4. Fan L, Gao J, Du X, et al. Spatial gradient distributions of thermal shock-induced damage to granite. J Rock Mech Geotechnical Eng, 2020, 12: 917–926

    Article  Google Scholar 

  5. Fan L F, Wu Z J, Wan Z, et al. Experimental investigation of thermal effects on dynamic behavior of granite. Appl Thermal Eng, 2017, 125: 94–103

    Article  Google Scholar 

  6. Zhang H H, Han S Y, Fan L F, et al. The numerical manifold method for 2D transient heat conduction problems in functionally graded materials. Eng Anal Bound Elem, 2018, 88: 145–155

    Article  MathSciNet  Google Scholar 

  7. Bruch J C, Zyvoloski G. Transient two-dimensional heat conduction problems solved by the finite element method. Int J Numer Meth Engng, 1974, 8: 481–494

    Article  Google Scholar 

  8. Reddy J N, Gartling D K. The Finite Element Method in Heat Transfer and Fluid Dynamics. Boca Raton: CRC Press, 2010

    Book  Google Scholar 

  9. Brian P L T. A finite-difference method of high-order accuracy for the solution of three-dimensional transient heat conduction problems. AIChE J, 1961, 7: 367–370

    Article  Google Scholar 

  10. Wang C C. Application of the maximum principle for differential equations in combination with the finite difference method to find transient approximate solutions of heat equations and error analysis. Numer Heat Transfer Part B-Fundamentals, 2009, 55: 56–72

    Article  Google Scholar 

  11. Li W, Yu B, Wang X, et al. A finite volume method for cylindrical heat conduction problems based on local analytical solution. Int J Heat Mass Transfer, 2012, 55: 5570–5582

    Article  Google Scholar 

  12. Liu D, Cheng Y M. The interpolating element-free Galerkin method for three-dimensional transient heat conduction problems. Results Phys, 2020, 19: 103477

    Article  Google Scholar 

  13. Wrobel L C, Brebbia C A. The boundary element method for steady state and transient heat conduction. Numerical Methods in Thermal Problems, 1979. 58–73

  14. Gao X W. A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity. Int J Numer Meth Engng, 2006, 66: 1411–1431

    Article  Google Scholar 

  15. Shi G H. Manifold method of material analysis. In: Proceedings of the Transactions of the 9th Army Conference On Applied Mathematics and Computing. Minneapolis, 1991. 57–76

  16. Ma G, An X, He L. The numerical manifold method: A review. Int J Comput Methods, 2010, 07: 1–32

    Article  MathSciNet  Google Scholar 

  17. Yang S, Ma G, Ren X, et al. Cover refinement of numerical manifold method for crack propagation simulation. Eng Anal Bound Elem, 2014, 43: 37–49

    Article  Google Scholar 

  18. Liu F, Xia K. Structured mesh refinement in MLS-based numerical manifold method and its application to crack problems. Eng Anal Bound Elem, 2017, 84: 42–51

    Article  MathSciNet  Google Scholar 

  19. Liu F, Zhang K, Xu D D. Crack analysis using numerical manifold method with strain smoothing technique and corrected approximation for blending elements. Eng Anal Bound Elem, 2020, 113: 402–415

    Article  MathSciNet  Google Scholar 

  20. Chen G, Ohnishi Y, Ito T. Development of high-order manifold method. Int J Numer Meth Engng, 1998, 43: 685–712

    Article  Google Scholar 

  21. Zhang H H, Liu S M, Han S Y, et al. Modeling of2D cracked FGMs under thermo-mechanical loadings with the numerical manifold method. Int J Mech Sci, 2018, 148: 103–117

    Article  Google Scholar 

  22. Wu Z, Sun H, Wong L N Y. A cohesive element-based numerical manifold method for hydraulic fracturing modelling with voronoi grains. Rock Mech Rock Eng, 2019, 52: 2335–2359

    Article  Google Scholar 

  23. Yang Y, Chen T, Zheng H. Mathematical cover refinement of the numerical manifold method for the stability analysis of a soil-rock-mixture slope. Eng Anal Bound Elem, 2020, 116: 64–76

    Article  MathSciNet  Google Scholar 

  24. Chen T, Yang Y, Zheng H, et al. Numerical determination of the effective permeability coefficient of soil-rock mixtures using the numerical manifold method. Int J Numer Anal Methods Geomech, 2019, 43: 381–414

    Article  Google Scholar 

  25. He J, Liu Q, Wu Z, et al. Modelling transient heat conduction of granular materials by numerical manifold method. Eng Anal Bound Elem, 2018, 86: 45–55

    Article  MathSciNet  Google Scholar 

  26. He J, Liu Q, Wu Z, et al. Geothermal-related thermo-elastic fracture analysis by numerical manifold method. Energies, 2018, 11: 1380

    Article  Google Scholar 

  27. Zhang H H, Liu S M, Han S Y, et al. The numerical manifold method for crack modeling of two-dimensional functionally graded materials under thermal shocks. Eng Fract Mech, 2019, 208: 90–106

    Article  Google Scholar 

  28. Zhang H H, Ma G W, Ren F. Implementation of the numerical manifold method for thermo-mechanical fracture of planar solids. Eng Anal Bound Elem, 2014, 44: 45–54

    Article  MathSciNet  Google Scholar 

  29. Gao H, Wei G. Complex variable meshless manifold method for transient heat conduction problems. Int J Appl Mech, 2017, 09: 1750067

    Article  Google Scholar 

  30. Zhang L, Guo F, Zheng H. The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials. Int Commun Heat Mass Transfer, 2022, 139: 106428

    Article  Google Scholar 

  31. Tan F, Tong D F, Yi X W, et al. 3D numerical manifold element generation software. Version 1.0. Wuhan (CN): National Copyright Administration. 2021

    Google Scholar 

  32. Sun L, Zhao G, Ma X. Quality improvement methods for hexahedral element meshes adaptively generated using grid-based algorithm. Int J Numer Meth Engng, 2012, 89: 726–761

    Article  MathSciNet  Google Scholar 

  33. He L, An X M, Ma G W, et al. Development of three-dimensional numerical manifold method for jointed rock slope stability analysis. Int J Rock Mech Min Sci, 2013, 64: 22–35

    Article  Google Scholar 

  34. Yang Y T, Li J F. A practical parallel preprocessing strategy for 3D numerical manifold method. Sci China Tech Sci, 2022, 65: 2856–2865

    Article  Google Scholar 

  35. Liu F, Zhang K, Liu Z. Three-dimensional MLS-based numerical manifold method for static and dynamic analysis. Eng Anal Bound Elem, 2019, 109: 43–56

    Article  MathSciNet  Google Scholar 

  36. Liang J, Tong D, Tan F, et al. Two-Dimensional magnetotelluric modelling based on the numerical manifold method. Eng Anal Bound Elem, 2021, 124: 87–97

    Article  MathSciNet  Google Scholar 

  37. Zheng H, Xu D. New strategies for some issues of numerical manifold method in simulation of crack propagation. Int J Numer Meth Engng, 2014, 97: 986–1010

    Article  MathSciNet  Google Scholar 

  38. Yang Y, Tang X, Zheng H, et al. Three-dimensional fracture propagation with numerical manifold method. Eng Anal Bound Elem, 2016, 72: 65–77

    Article  MathSciNet  Google Scholar 

  39. Mills A F. Heat Transfer. Boca Raton: CRC Press, 1992

    Google Scholar 

  40. Logan D L. A First Course in the Finite Element Method. Hoboken: John Wiley & Sons, 2007

    Google Scholar 

  41. Lin S Z. Recursive formula for simplex integration. J Yangtze River Sci Res Inst, 2005, 22: 32

    Google Scholar 

  42. Wang X, Wu W, Zhu H, et al. Three-dimensional discontinuous deformation analysis derived from the virtual work principle with a simplex integral on the boundary. Comput Geotechnics, 2022, 146: 104710

    Article  Google Scholar 

  43. Huang H C, Wang Q L. Finite Element Analysis of Heat Conduction Problems. Beijing: Science Press, 2011

    Google Scholar 

  44. Huang Y, Jiang H. Analysis of characteristics of plastic zone and mechanical properties of anchor structure in hydraulic tunnels with high ground temperature. Eur J Comput Mech, 2021, 481–500

  45. Yan C, Wei D, Wang G. Three-dimensional finite discrete element-based contact heat transfer model considering thermal cracking in continuous-discontinuous media. Comput Methods Appl Mech Eng, 2022, 388: 114228

    Article  MathSciNet  Google Scholar 

  46. Wang T, Yan C, Wang G, et al. Numerical study on the deformation and failure of soft rock roadway induced by humidity diffusion. Tunnelling Underground Space Tech, 2022, 126: 104565

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, China University of Geoscience, Wuhan, 430074, China

    DeFu Tong, Fei Tan, YuYong Jiao & JiaWei Liang

  2. Department of Civil Engineering, Monash University, Melbourne, VIC, 3800, Australia

    DeFu Tong

  3. Badong National Observation and Research Station of Geohazards, China University of Geoscience, Wuhan, 430074, China

    DeFu Tong & XiongWei Yi

Authors
  1. DeFu Tong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. XiongWei Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fei Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. YuYong Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. JiaWei Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Fei Tan or YuYong Jiao.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 42277165, 41920104007 and 41731284), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Grant No. CUGCJ1821 and CUGDCJJ202234), and the National Overseas Study Fund (Grant No. 202106410040).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tong, D., Yi, X., Tan, F. et al. Three-dimensional numerical manifold method for heat conduction problems with a simplex integral on the boundary. Sci. China Technol. Sci. 67, 1007–1022 (2024). https://doi.org/10.1007/s11431-022-2321-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-022-2321-9

Search

Navigation