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Frontiers of Structural and Civil Engineering

, Volume 13, Issue 2, pp 429–455 | Cite as

Multiscale RBF-based central high resolution schemes for simulation of generalized thermoelasticity problems

  • Hassan YousefiEmail author
  • Alireza Taghavi Kani
  • Iradj Mahmoudzadeh Kani
Research Article

Abstract

In this study, average-interpolating radial basis functions (RBFs) are successfully integrated with central high-resolution schemes to achieve a higher-order central method. This proposed method is used for simulation of generalized coupled thermoelasticity problems including shock (singular) waves in their solutions. The thermoelasticity problems include the LS (systems with one relaxation parameter) and GN (systems without energy dissipation) theories with constant and variable coefficients. In the central high resolution formulation, RBFs lead to a reconstruction with the optimum recovery with minimized roughness on each cell: this is essential for oscillation-free reconstructions. To guarantee monotonic reconstructions at cell-edges, the nonlinear scaling limiters are used. Such reconstructions, finally, lead to the total variation bounded (TVB) feature. As RBFs work satisfactory on non-uniform cells/grids, the proposed central scheme can handle adapted cells/grids. To have cost effective and accurate simulations, the multiresolution–based grid adaptation approach is then integrated with the RBF-based central scheme. Effects of condition numbers of RBFs, computational complexity and cost of the proposed scheme are studied. Finally, different 1-D coupled thermoelasticity benchmarks are presented. There, performance of the adaptive RBF-based formulation is compared with that of the adaptive Kurganov-Tadmor (KT) second-order central high-resolution scheme with the total variation diminishing (TVD) property.

Keywords

central high resolution schemes RBFs higher order accuracy generalized thermoelasticity multiresolution-based adaptation 

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Notes

Acknowledgements

The authors gratefully acknowledge the financial support of Iran National Science Foundation (INSF).

References

  1. 1.
    Chandrasekharaiah D S. Thermoelasticity with second sound: a review. Applied Mechanics Reviews, 1986, 39(3): 355–376zbMATHCrossRefGoogle Scholar
  2. 2.
    Mallik S H, Kanoria M. Generalized thermoelastic functionally graded solid with a periodically varying heat source. International Journal of Solids and Structures, 2007, 44(22–23): 7633–7645zbMATHCrossRefGoogle Scholar
  3. 3.
    Tamma K K, Namburu R R. Computational approaches with applications to non-classical and classical thermomechanical problems. Applied Mechanics Reviews, 1997, 50(9): 514–551CrossRefGoogle Scholar
  4. 4.
    Mitra K, Kumar S, Vedevarz A, Moallemi M. Experimental evidence of hyperbolic heat conduction in processed meat. Journal of Heat Transfer, 1995, 117(3): 568–573CrossRefGoogle Scholar
  5. 5.
    Lord H W, Shulman Y. A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids, 1967, 15(5): 299–309zbMATHCrossRefGoogle Scholar
  6. 6.
    Green A, Lindsay K. Thermoelasticity. Journal of Elasticity, 1972, 2(1): 1–7zbMATHCrossRefGoogle Scholar
  7. 7.
    Green A, Naghdi P. Thermoelasticity without energy dissipation. Journal of Elasticity, 1993, 31(3): 189–208MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Green A E, Naghdi P M. A re-examination of the basic postulates of thermomechanics. Proceedings of the Royal Society of London. Series A, 1991, 432(1885): 171–194MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Green A, Naghdi P. On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 1992, 15(2): 253–264MathSciNetCrossRefGoogle Scholar
  10. 10.
    Das B. Problems and Solutions in Thermoelasticity and Magnetothermoelasticity. Springer, 2017Google Scholar
  11. 11.
    Povstenko Y. Fractional Cattaneo-type equations and generalized thermoelasticity. Journal of Thermal Stresses, 2011, 34(2): 97–114CrossRefGoogle Scholar
  12. 12.
    Povstenko Y. Fractional thermoelasticity. In: Hetnarski R B, ed. Encyclopedia of Thermal Stresses. Springer, 2014, 1778–1787CrossRefGoogle Scholar
  13. 13.
    Ezzat M A, El-Karamany A S, Samaan A A. The dependence of the modulus of elasticity on reference temperature in generalized thermoelasticity with thermal relaxation. Applied Mathematics and Computation, 2004, 147(1): 169–189MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Youssef H M. Dependence of modulus of elasticity and thermal conductivity on reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity. Applied Mathematics and Mechanics, 2005, 26(4): 470–475zbMATHCrossRefGoogle Scholar
  15. 15.
    Aouadi M. Generalized thermo-piezoelectric problems with temperature-dependent properties. International Journal of Solids and Structures, 2006, 43(21): 6347–6358zbMATHCrossRefGoogle Scholar
  16. 16.
    Othman M I, Kumar R. Reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity. International Communications in Heat and Mass Transfer, 2009, 36(5): 513–520CrossRefGoogle Scholar
  17. 17.
    Allam M N, Elsibai K A, Abouelregal A E. Magneto-thermoelasticity for an infinite body with a spherical cavity and variable material properties without energy dissipation. International Journal of Solids and Structures, 2010, 47(20): 2631–2638zbMATHCrossRefGoogle Scholar
  18. 18.
    Abbas I A. Eigenvalue approach in a three-dimensional generalized thermoelastic interactions with temperature-dependent material properties. Computers & Mathematics with Applications (Oxford, England), 2014, 68(12): 2036–2056MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Xiong Q L, Tian X G. Transient magneto-thermoelastic response for a semi-infinite body with voids and variable material properties during thermal shock. International Journal of Applied Mechanics, 2011, 3(4): 881–902CrossRefGoogle Scholar
  20. 20.
    He T, Shi S. Effect of temperature-dependent properties on thermoelastic problems with thermal relaxations. Chinese Journal of Solid Mechanics, 2014, 27: 412–419Google Scholar
  21. 21.
    Sherief H, Abd El-Latief A M. Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity. International Journal of Mechanical Sciences, 2013, 74: 185–189CrossRefGoogle Scholar
  22. 22.
    Wang Y, Xue J. Asymptotic analysis of thermoelastic response in a functionally graded solid based on LS theory. International Journal of Material Science, 2016, 6(1): 35–40CrossRefGoogle Scholar
  23. 23.
    Wang Y, Liu D, Wang Q, Zhou J. Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock. Journal of Thermal Stresses, 2016, 39(4): 460–473CrossRefGoogle Scholar
  24. 24.
    Wang Y, Liu D, Wang Q, Zhou J. Asymptotic solutions for generalized thermoelasticity with variable thermal material properties. Archives of Mechanics, 2016, 68: 181–202MathSciNetzbMATHGoogle Scholar
  25. 25.
    Liang W, Huang S, Tan W, Wang Y. Asymptotic approach to transient thermal shock problem with variable material properties. Mechanics of Advanced Materials and Structures, 2017: 1–9Google Scholar
  26. 26.
    Youssef H, El-Bary A. Thermal shock problem of a generalized thermoelastic layered composite material with variable thermal conductivity. Math Probl Eng, 2006, 2006: 1–14MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Yousefi H, Noorzad A, Farjoodi J, Vahidi M. Multiresolutionbased adaptive simulation of wave equation. Applied Mathematics & Information Sciences, 2012, 6: 47S–58SzbMATHGoogle Scholar
  28. 28.
    Yousefi H, Ghorashi S S, Rabczuk T. Directly simulation of second order hyperbolic systems in second order form via the regularization concept. Communications in Computational Physics, 2016, 20 (01): 86–135MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Latifi M, Kharazi M, Ovesy H. Nonlinear dynamic response of symmetric laminated composite beams under combined in-plane and lateral loadings using full layerwise theory. Thin-walled Structures, 2016, 104: 62–70CrossRefGoogle Scholar
  30. 30.
    Latifi M, Farhatnia F, Kadkhodaei M. Buckling analysis of rectangular functionally graded plates under various edge conditions using Fourier series expansion. European Journal of Mechanics. A, Solids, 2013, 41: 16–27MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Latifi M, Kharazi M, Ovesy H. Effect of integral viscoelastic core on the nonlinear dynamic behaviour of composite sandwich beams with rectangular cross sections. International Journal of Mechanical Sciences, 2017, 123: 141–150CrossRefGoogle Scholar
  32. 32.
    Yousefi H, Noorzad A, Farjoodi J. Simulating 2D waves propagation in elastic solid media using wavelet based adaptive method. Journal of Scientific Computing, 2010, 42(3): 404–425MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Godunov S K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Matematicheskii Sbornik, 1959, 89: 271–306MathSciNetzbMATHGoogle Scholar
  34. 34.
    Harten A, Engquist B, Osher S, Chakravarthy S R. Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of Computational Physics, 1987, 71(2): 231–303MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Liu X D, Osher S, Chan T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 1994, 115(1): 200–212MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    LeVeque R J. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002Google Scholar
  37. 37.
    Kurganov A, Tadmor E. New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. Journal of Computational Physics, 2000, 160(1): 241–282MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Liu X D, Tadmor E. Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer Math, 1998, 79(3): 397–425MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Van Leer B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics, 1979, 32(1): 101–136zbMATHGoogle Scholar
  40. 40.
    Levy D, Puppo G, Russo G. Central WENO schemes for hyperbolic systems of conservation laws. Modélisation Mathématique et Analyse Numérique, 1999, 33(3): 547–571MathSciNetzbMATHGoogle Scholar
  41. 41.
    Levy D, Puppo G, Russo G. Compact central WENO schemes for multidimensional conservation laws. SIAM Journal on Scientific Computing, 2000, 22(2): 656–672MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Dehghan M, Jazlanian R. On the total variation of a third-order semi-discrete central scheme for 1D conservation laws. Journal of Vibration and Control, 2011, 17(9): 1348–1358MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Levy A, Ben-Dor G, Sorek S. Numerical investigation of the propagation of shock waves in rigid porous materials: development of the computer code and comparison with experimental results. Journal of Fluid Mechanics, 1996, 324: 163–179zbMATHCrossRefGoogle Scholar
  44. 44.
    Levy A, Ben-Dor G, Sorek S. Numerical investigation of the propagation of shock waves in rigid porous materials: flow field behavior and parametric study. Shock Waves, 1998, 8(3): 127–137zbMATHCrossRefGoogle Scholar
  45. 45.
    Heuzé T. Lax-Wendroff and TVD finite volume methods for unidimensional thermomechanical numerical simulations of impacts on elastic-plastic solids. Journal of Computational Physics, 2017, 346: 369–388MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Berezovski A, Maugin G. Simulation of thermoelastic wave propagation by means of a composite wave-propagation algorithm. Journal of Computational Physics, 2001, 168(1): 249–264MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Berezovski A, Maugin G. Application of wave-propagation algorithm to two-dimensional thermoelastic wave propagation in inhomogeneous media. In: Toro E F, ed. Godunov Methods: Theory and Applications. Boston: Springer Science & Business Media, 2001, 109–116CrossRefGoogle Scholar
  48. 48.
    Dehghan M, Abbaszadeh M. The space-splitting idea combined with local radial basis function meshless approach to simulate conservation laws equations. Alexandria Eng J, 2017, https://doi.org/10.1016/j.aej.2017.02.024 Google Scholar
  49. 49.
    Shokri A, Dehghan M. A meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation. Computer Modeling in Engineering & Sciences, 2012, 84: 333–358MathSciNetzbMATHGoogle Scholar
  50. 50.
    Guo J, Jung J H. Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters. Journal of Scientific Computing, 2017, 70(2): 551–575MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Dehghan M, Shokri A. A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Mathematics and Computers in Simulation, 2008, 79(3): 700–715MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Bigoni C, Hesthaven J S. Adaptive WENO methods based on radial basis function reconstruction. Journal of Scientific Computing, 2017, 72(3): 986–1020MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Sonar T. Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws. IMA Journal of Numerical Analysis, 1996, 16(4): 549–581MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Golomb M, Weinberger H F. Optimal approximations and error bounds. In: Langer R E, ed. On Numerical Approximation. Madison: The University of Wisconsin Press, 1958, 117–190Google Scholar
  55. 55.
    Micchelli C A, Rivlin T J. A survey of optimal recovery. In: Micchelli C A, Rivlin T J, eds. Optimal Estimation in Approximation Theory. Springer, 1977, 1–54CrossRefGoogle Scholar
  56. 56.
    Hickernell F J, Hon Y. Radial basis function approximations as smoothing splines. Applied Mathematics and Computation, 1999, 102(1): 1–24MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Iske A, Sonar T. On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions. Numer Math, 1996, 74(2): 177–201MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Aboiyar T, Georgoulis E H, Iske A. High order WENO finite volume schemes using polyharmonic spline reconstruction. In: Agratini O, Blaga P, eds. International Conference on Numerical Analysis and Approximation Theory. Cluj-Napoca: Babeş–Bolyai University, 2006, 113–126Google Scholar
  59. 59.
    Guo J, Jung J H. A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method. Applied Numerical Mathematics, 2017, 112: 27–50MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Schaback R. Error estimates and condition numbers for radial basis function interpolation. Advances in Computational Mathematics, 1995, 3(3): 251–264MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Mallat S. A Wavelet Tour of Signal Processing. New Delhi: Academic Press, 1999Google Scholar
  62. 62.
    Alves M, Cruz P, Mendes A, Magalhaes F, Pinho F, Oliveira P. Adaptive multiresolution approach for solution of hyperbolic PDEs. Computer Methods in Applied Mechanics and Engineering, 2002, 191(36): 3909–3928zbMATHCrossRefGoogle Scholar
  63. 63.
    Santos J, Cruz P, Alves M, Oliveira P, Magalhães F, Mendes A. Adaptive multiresolution approach for two-dimensional PDEs. Computer Methods in Applied Mechanics and Engineering, 2004, 193(3–5): 405–425zbMATHCrossRefGoogle Scholar
  64. 64.
    Cohen A, Kaber S, Müller S, Postel M. Fully adaptive multiresolution finite volume schemes for conservation laws. Mathematics of Computation, 2003, 72(241): 183–225MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Dahmen W, Gottschlich–Müller B, Müller S. Multiresolution schemes for conservation laws. Numer Math, 2001, 88(3): 399–443MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Harten A. Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Communications on Pure and Applied Mathematics, 1995, 48(12): 1305–1342MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Gottschlich-Miiller B, Miiller S. Application of multiscale techniques to hyperbolic conservation laws. In: Chen Z, Li Y, Micchelli C, Xu Y, eds. Advances in Computational Mathematics, Lecture Notes in Pure & Applied Mathematics. Gaungzhou: Marcel Dekker, Inc., 1998, 113–138Google Scholar
  68. 68.
    Berres S, Burger R, Kozakevicius A. Numerical approximation of oscillatory solutions of hyperbolic-elliptic systems of conservation laws by multiresolution schemes. Advances in Applied Mathematics and Mechanics, 2009, 1: 581–614MathSciNetCrossRefGoogle Scholar
  69. 69.
    Holmström M. Solving hyperbolic PDEs using interpolating wavelets. SIAM Journal on Scientific Computing, 1999, 21(2): 405–420MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Yousefi H, Noorzad A, Farjoodi J. Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems. Applied Mathematical Modelling, 2013, 37(12–13): 7095–7127MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Donoho D L, Johnstone J M. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 1994, 81(3): 425–455MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Wang J, Liu G. Radial point interpolation method for elastoplastic problems. In. ICSSD 2000: 1 st Structural Conference on Structural Stability and Dynamics. 2000, 703–708Google Scholar
  73. 73.
    Wang J, Liu G. On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering, 2002, 191(23–24): 2611–2630MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Fasshauer G E. Meshfree Approximation Methods with Matlab. World Scientific Publishing Co Inc., 2007Google Scholar
  75. 75.
    Liu G R. Gu Y-T. An Introduction to Meshfree Methods and Their Programming. Springer Science & Business Media, 2005Google Scholar
  76. 76.
    Wendland H. Scattered Data Approximation. Cambridge University Press, 2004Google Scholar
  77. 77.
    Driscoll T A, Fornberg B. Interpolation in the limit of increasingly flat radial basis functions. Computers & Mathematics with Applications (Oxford, England), 2002, 43(3–5): 413–422MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Fornberg B, Larsson E, Flyer N. Stable computations with Gaussian radial basis functions. SIAM Journal on Scientific Computing, 2011, 33(2): 869–892MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Wright G B, Fornberg B. Stable computations with flat radial basis functions using vector-valued rational approximations. Journal of Computational Physics, 2017, 331: 137–156MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Fornberg B, Wright G. Stable computation of multiquadric interpolants for all values of the shape parameter. Computers & Mathematics with Applications (Oxford, England), 2004, 48(5–6): 853–867MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Fasshauer G E, Zhang J G. Scattered data approximation of noisy data via iterated moving least squares. Curves and Surfaces: Avignon, 2006Google Scholar
  82. 82.
    Fasshauer G E, Zhang J G. On choosing “optimal” shape parameters for RBF approximation. Numer Algor, 2007, 45(1–4): 345–368MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Aboiyar T, Georgoulis E H, Iske A. Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction. SIAM Journal on Scientific Computing, 2010, 32(6): 3251–3277MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Kansa E, Carlson R. Improved accuracy of multiquadric interpolation using variable shape parameters. Computers & Mathematics with Applications (Oxford, England), 1992, 24(12): 99–120MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Fornberg B, Zuev J. The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Computers & Mathematics with Applications (Oxford, England), 2007, 54(3): 379–398MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Kansa E J. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications (Oxford, England), 1990, 19(8–9): 147–161MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Hardy R L. Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 1971, 76(8): 1905–1915CrossRefGoogle Scholar
  88. 88.
    Rippa S. An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics, 1999, 11(2/3): 193–210MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Sanyasiraju Y, Satyanarayana C. On optimization of the RBF shape parameter in a grid-free local scheme for convection dominated problems over non-uniform centers. Applied Mathematical Modelling, 2013, 37(12–13): 7245–7272MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Fjordholm U S, Ray D. A sign preserving WENO reconstruction method. Journal of Scientific Computing, 2016, 68(1): 42–63MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Gottlieb S, Shu C W, Tadmor E. Strong stability-preserving highorder time discretization methods. SIAM Review, 2001, 43(1): 89–112MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Franke R. Scattered data interpolation: tests of some methods. Mathematics of Computation, 1982, 38: 181–200MathSciNetzbMATHGoogle Scholar
  93. 93.
    Powell M J D. Tabulation of thin plate splines on a very fine twodimensional grid. In: Numerical Methods in Approximation Theory, Vol. 9. Springer, 1992, 221–244zbMATHCrossRefGoogle Scholar
  94. 94.
    Arad N, Dyn N, Reisfeld D, Yeshurun Y. Image warping by radial basis functions: application to facial expressions. Graphical Models, 1994, 56(2): 161–172CrossRefGoogle Scholar
  95. 95.
    Powell M. Truncated Laurent expansions for the fast evaluation of thin plate splines. Numer Algor, 1993, 5(2): 99–120MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Roussos G, Baxter B J. Rapid evaluation of radial basis functions. Journal of Computational and Applied Mathematics, 2005, 180(1): 51–70MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Müller S. Adaptive Multiscale Schemes for Conservation Laws. Springer Science & Business Media, 2012Google Scholar
  98. 98.
    Cueto-Felgueroso L, Colominas I. High-order finite volume methods and multiresolution reproducing kernels. Archives of Computational Methods in Engineering, 2008, 15(2): 185–228MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Iske A. Multiresolution Methods in Scattered Data Modelling. Springer Science & Business Media, 2004Google Scholar
  100. 100.
    Abd El-Latief A M, Khader S E. Exact solution of thermoelastic problem for a one-dimensional bar without energy dissipation. ISRN Mech Eng, 2014, 2014: 1–6CrossRefGoogle Scholar
  101. 101.
    Guo P, Wu W H, Wu Z G. A time discontinuous Galerkin finite element method for generalized thermo-elastic wave analysis, considering non-Fourier effects. Acta Mechanica, 2014, 225(1): 299–307MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Hassan Yousefi
    • 1
    Email author
  • Alireza Taghavi Kani
    • 2
  • Iradj Mahmoudzadeh Kani
    • 1
  1. 1.School of Civil Engineering, College of EngineeringUniversity of TehranTehranIran
  2. 2.Department of Civil EngineeringArak University of TechnologyArakIran

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