Applied and Computational Mathematics
Volume 4, Issue 4, August 2015, Pages: 286-295
Received: Feb. 1, 2015;
Accepted: Feb. 1, 2015;
Published: Jul. 2, 2015
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H. Idder, Faculty of Science and Technology, Laboratory of Mechanics and Civil Engineering, Tangier, Morocco
M. Mabssout, Faculty of Science and Technology, Laboratory of Mechanics and Civil Engineering, Tangier, Morocco
M. I. Herreros, CEDEX, Madrid, Spain
This paper presents the basis and validation of the Taylor-SPH meshless method formulated in terms of stresses and velocities which can be applied to Solid Dynamic problems. The proposed method consists of applying first the time discretization by means of a Taylor series expansion in two steps and a corrected SPH method for the space discretization. In order to avoid numerical instabilities, two different sets of particles are used in the time discretization. To validate the Taylor-SPH method, it has been applied to solve the propagation of shock waves in elastic materials and the results have been compared with those obtained with a corrected SPH discretization combined with a 4th order Runge-Kutta time integration. The Taylor-SPH method is shown to be stable, robust and efficient and it provides more accurate results than those obtained with the standard SPH along with the Runge-Kutta time integration scheme. Numerical dispersion and diffusion are eliminated and only a reduced number of particles is required to obtain accurate results.
M. I. Herreros,
The Taylor-SPH Meshfree Method: Basis and Validation, Applied and Computational Mathematics.
Vol. 4, No. 4,
2015, pp. 286-295.
T. Belytschko, Y. Guo, W. K. Liu and S. P. Xiao, “A unified stability analysis of meshless particle methods”, Int. J. Numer. Meth. Engrg. (2000), 48: 1359–1400.
T. Belytschko, Y. Krongauz, Dolbow, Y , Gerlach, C. “On the completeness of meshfree particle methods”. Int. J. Numer. Methods Eng. (1998); 43:785-819.
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl. “Meshless methods: An overview and recent developments”. Comput. Methods. Appl. Mech. Engrg , (1996); 139:3-47.
J.C. Butcher. Numerical Methods for Ordinary Differential Equations. Second ed., Wiley, Chichester, England, 2003.
C. T. Dyka , R. P. Ingel, “An approach for tension instability in Smoothed Particle Hydrodynamics”, Computers and Structures, (1995); 57: 573–580.
C. T. Dyka, P. W. Randles and R. P. Ingel, “Stress points for tension instability in SPH”, Int. J. Numer. Meth. Engrg. (1997); 40: 2325–2341.
R.A. Gingold , J.J. Monaghan, “Smoothed particles hydrodynamics: Theory and application to non-spherical stars”, Monthly Notices of the Royal Astronomical Society, (1977); 181: 375-389.
M.I. Herreros, M. Mabssout, “A two-steps time discretization scheme using the SPH method for shock wave propagation”, Comput. Methods Appl. Mech. Eng. 200 (2011) 1833–1845.
G.R. Johnson, S.R. Beissel, “Normalized smoothing functions for SPH impact computations”, Int. J. Numer. Methods Engrg., (1996); 39:2725–2741.
L.D. Libersky , A.G.Petschek, “Smooth particle hydrodynamics with strength of materials, Advances in the Free Lagrange Method”, Lecture Notes in Physics, (1991); 395: 248-257.
W. K. Liu, S. Jun, Y. F. Zhang, “Reproducing kernel particle methods”, Int. J. Numer. Meth. Engng., (1995); 20(8-9):1081-1106.
L.B. Lucy, “A numerical approach to the testing of fusion process”, Astronomical Journal, 1977; 82:1013-1024.
M. Mabssout, M. Pastor, “A Taylor–Galerkin algorithm for shock wave propagation and strain localization failure of viscoplastic continua”, Comput. Methods. Appl. Mech. Engrg, (2003); 192: 955–971.
M. Mabssout, M. Pastor, “A two step Taylor–Galerkin algorithm for shock wave propagation in soils”, Int. J. Numer. Analytical Meth. Geomechanics, (2003); 27: 685–704.
M. Mabssout, M. Pastor, M.I. Herreros, M. Quecedo, “A Runge-Kutta, Taylor-Galerkin scheme for hyperbolic systems with source terms. Application to shock wave propagation in viscoplastic geomaterials”. Int. J. Numer. Analytical Meth. Geomechanics, (2006); 30(13): 1337-1355.
M. Mabssout, M. Herreros, “Taylor-SPH vs Taylor–Galerkin for shock waves in viscoplastic continua”, Eur. J. Comput. Mech. 20 (5–6) (2011) 281–308.
M. Mabssout, M. I. Herreros, “Runge-Kutta vs Taylor-SPH. Two time integration schemes for SPH with application to Soil Dynamics”, App. Math. Modelling, 37(5), pp. 3541-3563, 2013.
J.J. Monaghan , J.C. Lattanzio, “A refined particle method for astrophysical problems”, Astronomy and Astrophysics, (1985); 149: 135-143.
T. Rabczuk, T. Belytschko , S. P. Xiao, “Stable particle methods based on Lagrangian kernels”, Comput. Methods. Appl. Mech. Engrg, (2004); 193: 1035–1063.
P. W. Randles , L. D. Libersky, “ Recent improvements in SPH modelling of hypervelocity impact”. Int. J. Impact Engrg., (1997); 20: 525-532.
P.W. Randles , L.D. Libersky, “Normalized SPH with stress points”, Int. J. Numer. Meth. Engng., (2000); 48:1445-1462.
J.W. Swegle, D.A. Hicks, S.W. Attaway “Smooth particle hydrodynamics stability analysis” J. Comput. Phys. 116 (1995) 123–134.
S.P. Xiao, T. Belytschko, “Material stability analysis of particle methods”, Adv. Comput. Math. 23 (2005) 171–190.