Multiscale Computational Method for Dynamic Thermo-Mechanical Problems of Composite Structures with Diverse Periodic Configurations in Different Subdomains

  • Hao DongEmail author
  • Xiaojing Zheng
  • Junzhi Cui
  • Yufeng Nie
  • Zhiqiang YangEmail author
  • Zihao Yang


In this paper, a novel multiscale computational method is presented to simulate and analyze dynamic thermo-mechanical problems of composite structures with diverse periodic configurations in different subdomains. In these composite structures, thermo-mechanical coupled behaviors at microscale have an important impact on the macroscopic displacement and temperature fields. Firstly, the novel second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on multiscale asymptotic analysis. Then, the error analysis in the pointwise sense is given to illustrate the importance of developing the SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method, finite difference method and decoupling method. Finally, some numerical examples are shown, which demonstrate the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for dynamic thermo-mechanical problems of composite structures with diverse periodic configurations in different subdomains.


Dynamic thermo-mechanical problems Multiscale asymptotic analysis Diverse periodic configurations Error estimate SOTS numerical algorithm 



This research was supported by the Fundamental Research Funds for the Central Universities (No. JB180703), the National Natural Science Foundation of China (Nos. 51739007, 11471262 and 11501449), the National Basic Research Program of China (No. 2012CB025904), the State Scholarship Fund of China Scholarship Council (File No. 201606290191), and also supported by the Key Technology Research of FRP-Concrete Composite Structure and Center for high performance computing of Northwestern Polytechnical University.


  1. 1.
    Chen, F., Liu, H., Zhang, S.T.: Coupled heat transfer and thermo-mechanical behavior of hypersonic cylindrical leading edges. Int. J. Heat Mass Transf. 122, 846–862 (2018)CrossRefGoogle Scholar
  2. 2.
    Tsalis, D., Chatzigeorgiou, G., Charalambakis, N.: Effective behavior of thermo-elastic tubes with wavy layers. Compos. Part B Eng. 99, 173–187 (2016)CrossRefGoogle Scholar
  3. 3.
    Cui, J.Z.: The two-scale expression of the solution for the structure with several sub-domains of small periodic configurations. Workshop Sci. Comput. 99, 27–30 (1996)Google Scholar
  4. 4.
    Cui, J.Z., Shan, Y.J.: Computational Techniques for Materials, Composites and Composite Structures, pp. 255–264. Civil-Comp Press, Edinburgh (2000)CrossRefGoogle Scholar
  5. 5.
    Cui, J.Z.: Proceedings on Computational Mechanics in Science and Engineering, pp. 33–43. Peking University Press, Beijing (2001)Google Scholar
  6. 6.
    Terada, K., Kurumatani, M., Ushida, T., Kikuchi, N.: A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer. Comput. Mech. 46, 269–285 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Yang, Z.H., Cui, J.Z.: The statistical second-order two-scale analysis for dynamic thermo-mechanical performances of the composite structure with consistent random distribution of particles. Comput. Mater. Sci. 69, 359–373 (2013)CrossRefGoogle Scholar
  8. 8.
    Wang, X., Cao, L.Q., Wong, Y.S.: Multiscale computation and convergence for coupled thermoelastic system in composite materials. SIAM Multiscale Model. Simul. 13, 661–690 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Savatorova, V.L., Talonov, A.V., Vlasov, A.N.: Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities. Z. Angew. Math. Mech. 93, 575–596 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, S., Yang, D.S., Zhang, H.W., Zheng, Y.G.: Coupling extended multiscale finite element method for thermoelastic analysis of heterogeneous multiphase materials. Comput. Struct. 121, 32–49 (2013)CrossRefGoogle Scholar
  11. 11.
    Nasution, M.R.E., Watanabe, N., Kondo, A., Yudhanto, A.: Thermo-mechanical properties and stress analysis of 3-D textile composites by asymptotic expansion homogenization method. Compos. Part B Eng. 60, 378–391 (2014)CrossRefGoogle Scholar
  12. 12.
    Wu, Y.T., Nie, Y.F., Yang, Z.H.: Comparison of four multiscale methods for elliptic problems. CMES-Comput. Model. Eng. Sci. 99, 297–325 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dong, H., Nie, Y.F., Yang, Z.H., Wu, Y.T.: The numerical accuracy analysis of asymptotic homogenization method and multiscale finite element method for periodic composite materials. CMES-Comput. Model. Eng. Sci. 111, 395–419 (2016)Google Scholar
  14. 14.
    Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization, pp. 13–38. North-Holland, Amsterdam (1992)zbMATHGoogle Scholar
  15. 15.
    Xing, Y.F., Du, C.Y.: An improved multiscale eigenelement method of periodical composite structures. Compos. Struct. 118, 200–207 (2014)CrossRefGoogle Scholar
  16. 16.
    Francfort, G.A.: Homogenization and linear thermoelasticity. SIAM J. Math. Anal. 14, 696–708 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yang, Z.H., Cui, J.Z., Wu, Y.T., Wang, Z.Q., Wan, J.J.: Second-order two-scale analysis method for dynamic thermo-mechanical problems in periodic structure. Int. J. Numer. Anal. Model. 12, 144–161 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Yang, Z.Q., Cui, J.Z., Sun, Y., Liang, J., Yang, Z.H.: Multiscale analysis method for thermo-mechanical performance of periodic porous materials with interior surface radiation. Int. J. Numer. Methods Eng. 105, 323–350 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Feng, Y.P., Cui, J.Z.: Multi-scale analysis and FE computation for the structure of composite materials with small periodic configuration under condition of coupled thermoelasticity. Int. J. Numer. Methods Eng. 60, 1879–1910 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ma, Q., Cui, J.Z.: Second-order two-scale analysis method for the quasi-periodic structure of composite materials under condition of coupled thermo-elasticity. Adv. Mater. Res. 629, 160–164 (2013)CrossRefGoogle Scholar
  21. 21.
    Chatzigeorgiou, G., Efendiev, Y., Charalambakis, N., Lagoudas, D.C.: Effective thermoelastic properties of composites with periodicity in cylindrical coordinates. Int. J. Solids Struct. 49, 2590–2603 (2012)CrossRefGoogle Scholar
  22. 22.
    Li, Z.H., Ma, Q., Cui, J.Z.: Second-order two-scale finite element algorithm for dynamic thermoCmechanical coupling problem in symmetric structure. J. Comput. Phys. 314, 712–748 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dong, H., Cui, J.Z., Nie, Y.F., Yang, Z.H.: Second-order two-scale computational method for damped dynamic thermo-mechanical problems of quasi-periodic composite materials. J. Comput. Appl. Math. 343, 575–601 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dong, H., Nie, Y.F., Cui, J.Z., Yang, Z.H., Wang, Z.Q.: Second-order two-scale analysis method for dynamic thermo-mechanical problems of composite structures with cylindrical periodicity. Int. J. Numer. Anal. Model. 15, 834–863 (2018)MathSciNetGoogle Scholar
  25. 25.
    Dong, H., Cui, J.Z., Nie, Y.F., Ma, Q., Yang, Z.H.: Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Appl. Math. Model. 60, 634–660 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Cioranescu, D., Donato, P.: An Introduction to Homogenization, pp. 221–238. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  27. 27.
    Dong, Q.L., Cao, L.Q.: Multiscale asymptotic expansions methods and numerical algorithms for the wave equations in perforated domains. Appl. Math. Comput. 232, 872–887 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Dong, Q.L., Cao, L.Q.: Multiscale asymptotic expansions methods and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. Appl. Numer. Math. 59, 3008–3032 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Dong, Q.L., Cao, L.Q., Wang, X., Huang, J.Z.: Multiscale numerical algorithms for elastic wave equations with rapidly oscillating coefficients. Appl. Math. Comput. 336, 16–35 (2018)MathSciNetGoogle Scholar
  30. 30.
    Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures, pp. 165–173. American Mathematical Society, Rhode Island (2011)zbMATHGoogle Scholar
  31. 31.
    Cao, L.Q.: Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains. Numer. Math. 103, 11–45 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Cao, L.Q., Luo, J.L.: Multiscale numerical algorithm for the elliptic eigenvalue problem with the mixed boundary in perforated domains. Appl. Numer. Math. 58, 1349–1374 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, X., Wang, T.S., Liu, Y.: Computational Dynamics, pp. 94–118. Tsinghua University Press, Beijing (2007)Google Scholar
  34. 34.
    Lin, Q., Zhu, Q.D.: The Preprocessing snd Preprocessing for the Finite Element Method, pp. 48–71. Shanghai Scientific & Technical Publishers, Shanghai (1994)Google Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China
  2. 2.School of Mechano-Electronic EngineeringXidian UniversityXi’anPeople’s Republic of China
  3. 3.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.Department of Applied Mathematics, School of ScienceNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  5. 5.Department of Astronautic Science and MechanicsHarbin Institute of TechnologyHarbinPeople’s Republic of China

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