A Thermal Conduction Comparative Study Between the FDM and SPH Methods with A Proposed C++ Home Code
Keywords:Thermal conduction, Finite difference method, Smoothed particle hydrodynamic, C code
The heat transfer phenomenon is modeled by the Finite Difference Method (FDM) and the Soothed Particle Hydrodynamic (SPH) approach. The numerical approach under investigation may be used to solve many complex problems of applied mechanics. The Finite Element Method (FEM) is generally used for the Lagrangian description, and the FDM is used for the Eulerian report. However, the SPH method, which is better than other approaches to solve some problems, may be used in many aspects. Numerical details on the SPH method are discussed in this paper, with a focus on its application on the heat equation. A simple two-dimensional heat conduction problem is simulated by using the SPH approximation procedure and the newly constructed quartic smoothing function. Besides, a comparison is made between both techniques. Finally, C++ code is proposed for SPH and FDM methods.
J.J. Monaghan. "Smoothed particle hydrodynamics." Institute of Physics Publishing, Reports on Progress in Physics 68 (2005) 1703-1759. https://doi.org/10.1088/0034-4885/68/8/r01
H. Ameur, and D. Sahel. "Effect of the baffle design and orientation on the efficiency of a membrane tube." Chemical Engineering Research and Design 117 (2017) 500-508. https://doi.org/10.1016/j.cherd.2016.11.005
H. Ameur. "Effect of the shaft eccentricity and rotational direction on the mixing characteristics in cylindrical tank reactors." Chinese Journal of Chemical Engineering 24 (2016)1647–1654, 2016. https://doi.org/10.1016/j.cjche.2016.05.011
H. Ameur. "Mixing of shear thinning fluids in cylindrical tanks: effect of the impeller blade design and operating conditions." International Journal of Chemical Reactor Engineering, 14 (2016) 1025-1034. https://doi.org/10.1515/ijcre-2015-0200
H. Ameur. "Investigation of the performance of V-cut turbines for stirring shear-thinning fluids in a cylindrical vessel." Periodica Polytechnica Mechanical Engineering, 64 (2020) 207–211, 2020. https://doi.org/10.3311/PPme.13359
H. Ameur. "Newly modified curved-bladed impellers for process intensification: Energy saving in the agitation of Hershel-Bulkley fluids." Chemical Engineering and Processing - Process Intensification 154 (2020) 108009. https://doi.org/10.1016/j.cep.2020.108009
H. Ameur. "Effect of the baffle inclination on the flow and thermal fields in channel heat exchangers." Results in Engineering, 3 (2019) 100021. https://doi.org/10.1016/j.rineng.2019.100021
W. Benz. "Smoothed particle hydrodynamics - a review." The Numerical Modelling of Nonlinear Stellar Pulsations, Springer, Dordrecht, 1990, pp. 269-288. https://doi.org/10.1007/978-94-009-0519-1_16
W. Benz, and E. Asphaug. "Impact simulations with fracture: I. Method and tests." Icarus 107 (1994) 98-116. https://doi.org/10.1006/icar.1994.1009
J. Bonet, and S. Kulasegaram. "Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations." International Journal for Numerical Methods in Engineering 47 (2000) 1189-1214. https://doi.org/10.1002/(sici)1097-0207(20000228)47:6<1189::aid-nme830>3.0.co;2-i
S. Børve, M. Omang, and J. Trulsen. "Regularized smoothed particle hydrodynamics: a new approach to simulating magneto hydrodynamic shocks." The Astrophysical Journal 561 (2001) 82-93. https://doi.org/10.1086/323228
M.L. Hosain, J.M. Domínguez, R. Bel Fdhila, and K. Kyprianidis. "Smoothed particle hydrodynamics modeling of industrial processes involving heat transfer." Applied Energy 252 (2019) 113441. https://doi.org/10.1016/j.apenergy.2019.113441
S. Hardi, M. Schreiner, and U. Janoske. "Enhancing smoothed particle hydrodynamics for shallow water equations on small scales by using the finite particle method." Computer Methods in Applied Mechanics and Engineering 344 (2019) 360-375. https://doi.org/10.1016/j.cma.2018.10.021
W. Hu, G. Guo, X. Hu, D. Negrut, Z. Xu, and W. Pan. "A consistent spatially adaptive smoothed particle hydrodynamics method for fluid-structure interactions." Computer Methods in Applied Mechanics and Engineering 347 (2019) 402-424. https://doi.org/10.1016/j.cma.2018.10.049
E. Francomano, and M. Paliaga. "The smoothed particle hydrodynamics method via residual iteration." Computer Methods in Applied Mechanics and Engineering 352 (2019) 237-245. https://doi.org/10.1016/j.cma.2019.04.004
R. Fatehi, A. Rahmat, N. Tofighi, M. Yildiz, and M.S. Shadloo. "Density-based smoothed particle hydrodynamics methods for incompressible flows." Computer Methods in Applied Mechanics and Engineering 185 (2019) 22-33. https://doi.org/10.1016/j.compfluid.2019.02.018
R. Tayeb, Y. Mao, and Y. Zhang. "Smooth particle hydrodynamics simulation of granular system under cyclic compression." Powder Technology 353 (2019) 84-97. https://doi.org/10.1016/j.powtec.2019.04.079
H. Wu, P.G.A. Njock, J. Chen, and S. Shen. "Numerical simulation of spudcan-soil interaction using an improved smoothed particle hydrodynamics (SPH) method." Marine Structures 66 (2019) 213-226. https://doi.org/10.1016/j.marstruc.2019.04.007
M. Hopp-Hirschler, J. Baz, N. Hansen , and U. Nieken. "Generalized Fickian approach for phase separating fluid mixtures in Smoothed Particle Hydrodynamics." Computers and Fluids 179 (2019) 78-90. https://doi.org/10.1016/j.compfluid.2018.10.020
Z. Ji, M. Stanic, E.A. Hartono, and V. Chernoray. "Numerical simulations of oil flow inside a gearbox by Smoothed Particle Hydrodynamics (SPH) method." Tribology International 127 (2018) 47-58. https://doi.org/10.1016/j.triboint.2018.05.034
Z. Mao, G.R. Liu, and X. Dong. "A comprehensive study on the parameters setting in smoothed particle hydrodynamics (SPH) method applied to hydrodynamics problems." Computers and Geotechnics 92 (2017) 77-95.
A.M.A. Nasar, B.D. Rogers, A. Revell, P.K. Stansby, and S.J. Lind. "Eulerian weakly compressible smoothed particle hydrodynamics (SPH) with the immersed boundary method for thin slender bodies." Journal of Fluids and Structures 84 (2019) 263-282. https://doi.org/10.1016/j.jfluidstructs.2018.11.005
H. Nonoyama, S. Moriguchi, K. Sawada, and A. Yashima. "Slope stability analysis using smoothed particle hydrodynamics (SPH) method." Soils and Foundations 55 (2015) 458-470. https://doi.org/10.1016/j.sandf.2015.02.019
R. Ray, K. Deb, and A. Shaw. "Pseudo-Spring smoothed particle hydrodynamics (SPH) based computational model for slope failure." Engineering Analysis with Boundary Elements 101 (2019) 139-148. https://doi.org/10.1016/j.enganabound.2019.01.005
L.V. Vela, J.M. Reynolds-Barredo, and R. Sánchez. "A positioning algorithm for SPH ghost particles in smoothly curved geometries." Journal of Computational and Applied Mathematics 353 (2019) 140-153. https://doi.org/10.1016/j.cam.2018.12.021
N. Zhang, X. Zheng, and Q. Ma. "Study on wave-induced kinematic responses and flexures of ice floe by Smoothed Particle Hydrodynamics." Computers and Fluids 189 (2019) 46-59. https://doi.org/10.1016/j.compfluid.2019.04.020
H. Basser, M. Rudman, and E. Daly. "Smoothed Particle Hydrodynamics modelling of fresh and salt water dynamics in porous media." Journal of Hydrology 576 (2019) 370–380. https://doi.org/10.1016/j.jhydrol.2019.06.048
Ng, K. C., Y. L. Ng, T. W. H. Sheu, and A. Alexiadis. "Assessment of Smoothed Particle Hydrodynamics (SPH) models for predicting wall heat transfer rate at complex boundary." Engineering Analysis with Boundary Elements 111 (2020): 195-205. https://doi.org/10.1016/j.enganabound.2019.10.017
Ng, Khai Ching, Yee Luon Ng, T. W. H. Sheu, and A. Mukhtar. "Fluid-solid conjugate heat transfer modelling using weakly compressible smoothed particle hydrodynamics." International Journal of Mechanical Sciences 151 (2019): 772-784. https://doi.org/10.1016/j.ijmecsci.2018.12.028
G. Fourtakas, J.M. Dominguez, R. Vacondio, and B.D. Rogers. "Local Uniform Stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models." Computers and Fluids 190 (2019) 346-361. https://doi.org/10.1016/j.compfluid.2019.06.009
X. Yang, and S.C. Kong. "Adaptive resolution for multiphase smoothed particle hydrodynamics." Computer Physics Communications 239 (2019) 112-125. https://doi.org/10.1016/j.cpc.2019.01.002
M. Mimault, M. Ptashnyk, G.W. Bassel, and L.X. Dupuy. "Smoothed particle hydrodynamics for root growth mechanics." Engineering Analysis with Boundary Elements 105 (2019) 20-30. https://doi.org/10.1016/j.enganabound.2019.03.025
S. Gharehdash, M. Barzegar, I.B. Palymskiy, and P.A. Fomin. "Blast induced fracture modelling using smoothed particle hydrodynamics." International Journal of Impact Engineering 135 (2020) 103235.
J. de Anda-Suárez, S. Jeyakumar, M. Carpio, H.J. Puga, A. Rojas-Domínguez, L. Cruz-Reyes, and J.F. Mosiño. "Parameter optimization for the smoothed-particle hydrodynamics method by means of evolutionary metaheuristics." Computer Physics Communications 243 (2019) 30-40. https://doi.org/10.1016/j.cpc.2019.05.008.
Liu, Gui-Rong, and Moubin B. Liu. Smoothed particle hydrodynamics: a meshfree particle method. World scientific, 2003.