Meshless analyses for time-fractional heat diffusion in functionally graded materials. (English) Zbl 1403.74305
Summary: The meshless local radial basis function method is applied to solve stationary and transient heat conduction problems in 2-D and 3-D bodies with functionally graded material properties. Time fractional derivative using Caputo definition is considered to describe anomalous diffusion phenomena. For temporal discretization, the Caputo time fractional derivative is approximated within each time interval \(\langle t_k, t_{k+1} \rangle\) by series of derivatives of integer order. The spatial discretization is performed by using the local radial basis collocation method. Numerical analyses are given on square (2D) and cubic (3D) domains to show the influence of the temporal fractional derivative parameter and gradation material parameter on the temperature distribution and temperature evolution in transient heat conduction problem.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 74A15 | Thermodynamics in solid mechanics |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
Keywords:
fractional diffusion; transient heat conduction problem; functionally graded material; Caputo derivative; local radial basis functionReferences:
| [1] | Miller, K.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley & Sons New York · Zbl 0789.26002 |
| [2] | Podlubny, I., Fractional differential equations, (1999), Academic Press London · Zbl 0918.34010 |
| [3] | Caputo, M.; Mainardi, F., A new dissipation model based on a memory mechanism, Pure Appl Geophys, 8, 91, 134-147, (1971) |
| [4] | Bagley, R.; Torvik, P., On the appearance of the fractional derivative in the behavior of real materials, J Appl Mech, 51, 294-298, (1984) · Zbl 1203.74022 |
| [5] | Koeller, R., Applications of fractional calculus to the theory of viscoelasticity, J Appl Mech, 51, 299-307, (1984) · Zbl 0544.73052 |
| [6] | Sun, H.; Chen, W.; Li, C.; Chen, Y., Fractional differential models for anomalous diffusion, Physica A, 389, 14, 2719-2724, (2010) |
| [7] | Chen, W.; Sun, H.; Zhang, X.; Koroak, D., Anomalous diffusion modeling by fractal and fractional derivatives, Comput Math Appl, 59, 5, 1754-1758, (2010) · Zbl 1189.35355 |
| [8] | Baeumer, B.; Meerschaert, M.; Benson, D.; Wheatcraft, S., Subordinal advection-dispersion equation for contaminant transport, Water Resour Res, 37, 1543-1550, (2001) |
| [9] | Babouskos, N.; Katsikadelis, J., Nonlinear vibrations of viscoelastic plates of fractional derivative type: an AEM solution, Open Mech J, 4, 8-20, (2010) |
| [10] | Katsikadelis, J., The BEM for numerical solution of partial fractional differential equations, Comput Math Appl, 62, 3, 891-901, (2011) · Zbl 1228.74103 |
| [11] | Katsikadelis, J.; Babouskos, N., Post-buckling analysis of viscoelastic plates with fractional derivative models, Eng Anal Bound Elem, 34, 12, 1038-1048, (2010) · Zbl 1244.74058 |
| [12] | Nerantzaki, M.; Babouskos, N., Analysis of inhomogeneous anisotropic viscoelastic bodies described by multi-parameter fractional differential constitutive models, Comput Math Appl, 62, 3, 945-960, (2011) · Zbl 1228.74018 |
| [13] | Katsikadelis, J., Numerical solution of multi-term fractional differential equations, ZAMM—J Appl Math Mech, 89, 7, 593-608, (2009) · Zbl 1175.26013 |
| [14] | Shirzadi, A.; Ling, L.; Abbasbandy, S., Meshless simulations of the two-dimensional fractional-time convection-diffusion-reaction equations, Eng Anal Bound Elem, 36, 1522-1527, (2012) · Zbl 1352.65263 |
| [15] | Gu, Y. T.; Zhuang, P.; Liu, F., An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES - Comput Model Eng Sci, 56, 3, 303-333, (2010) · Zbl 1231.65178 |
| [16] | Dou, F. F.; Hon, Y. C., Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation, Eng Anal Bound Elem, 36, 1344-1352, (2012) · Zbl 1352.65309 |
| [17] | Dou, F. F.; Hon, Y. C., Numerical computation for backward time-fractional diffusion equation, Eng Anal Bound Elem, 40, 138-146, (2014) · Zbl 1297.65112 |
| [18] | Wen, P. H.; Hon, Y. C., Meshless computation for partial differential equations of fractional order, WIT Trans Model Simul, 52, 333-343, (2011) · Zbl 1244.76093 |
| [19] | Suresh, S.; Mortensen, A., Fundamentals of functionally graded materials, (1998), Institute of Materials London |
| [20] | Paulino, G.; Jin, Z.; Dodds, R., Failure of functionally graded materials, (Milne, I.; Ritchie, R. O.; Karihaloo, B., Comprehensive structural integrity, 2, (2003)), 607-644 |
| [21] | Chen, W.; Fu, Z.; Chen, C., Recent advances in radial basis function collocation methods, (2014), Springer Heidelberg · Zbl 1282.65160 |
| [22] | Kansa, E., Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates, Comput Math Appl, 19, 8-9, 127-145, (1990) · Zbl 0692.76003 |
| [23] | Kansa, E., Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput Math Appl, 19, 8-9, 147-161, (1990) · Zbl 0850.76048 |
| [24] | Hon, Y. C.; Sarler, B.; Yun, D. F., Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface, Eng Anal Bound Elem, 57, 2-8, (2015), 10.1016/j.enganabound.2014.11.006 · Zbl 1403.76140 |
| [25] | Mramor, K.; Vertnik, R.; Sarler, B., Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method, Eng Anal Bound Elem, 49, 37-47, (2014) · Zbl 1403.76196 |
| [26] | Siraj-Ul-Islam, R.; Vertnik; Sarler, B., Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations, Appl Numer Math, 67, 136-151, (2013) · Zbl 1263.65099 |
| [27] | Yao Siraj-Ul-Islam, G.; Sarler, B., Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions, Eng Anal Bound Elem, 36, 11, 1640-1648, (2012) · Zbl 1352.65403 |
| [28] | Wei, S.; Chen, W.; Hon, Y. C., Implicit local radial basis function method for solving two-dimensional constant- and variable-order time fractional diffusion equations, Thermal Science, 19, S59-S67, (2015), 10.2298/TSCI15S1S59W |
| [29] | Zhuang, P.; Gu, Y. T.; Liu, F.; Turner, I. W.; Yarlagadda, P. K., Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method, Int J Numer Methods Eng, 88, 13, 1346-1362, (2011) · Zbl 1242.76262 |
| [30] | Liu, Q.; Gu, Y. T.; Zhuang, P.; Liu, F.; Nie, Y., An implicit RBF meshless approach for time fractional diffusion equations, Comput Mech, 48, 1, 1-12, (2011) · Zbl 1377.76025 |
| [31] | Hardy, R., Multiquadric equations of topography and other irregular surfaces, J Geophys Res, 76, 1905-1915, (1971) |
| [32] | Sladek, V.; Sladek, J.; Tanaka, M.; Zhang, C., Transient heat conduction in anisotropic and functionally graded media by local integral equations, Eng Anal Bound Elem, 29, 11, 1047-1065, (2005) · Zbl 1182.80016 |
| [33] | Sladek, V.; Sladek., J.; Zhang, Ch., Comparative study of meshless approximations in local integral equation method, CMC: Comput Mater Contin, 4, 177-188, (2006) |
| [34] | Sladek, V.; Sladek, J.; Zhang, Ch., Computation of stresses in non-homogeneous elastic solids by local integral equation method: a comparative study, Comput Mech, 41, 827-845, (2008) · Zbl 1142.74051 |
| [35] | Sladek, V.; Sladek, J.; Zhang, Ch, On increasing computational efficiency of local integral equation method combined with meshless implementations, CMES - Comput Model Eng Sci, 63, 243-263, (2010) · Zbl 1231.65221 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
