An Adaptive Physics-Based Method for the Solution of One-Dimensional Wave Motion Problems

Document Type : Research Papers


1 Ph.D., Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran

2 Professor, Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran.


In this paper, an adaptive physics-based method is developed for solving wave motion problems in one dimension (i.e., wave propagation in strings, rods and beams). The solution of the problem includes two main parts. In the first part, after discretization of the domain, a physics-based method is developed considering the conservation of mass and the balance of momentum. In the second part, adaptive points are determined using the wavelet theory. This part is done employing the Deslauries-Dubuc (D-D) wavelets. By solving the problem in the first step, the domain of the problem is discretized by the same cells taking into consideration the load and characteristics of the structure. After the first trial solution, the D-D interpolation shows the lack and redundancy of points in the domain. These points will be added or eliminated for the next solution. This process may be repeated for obtaining an adaptive mesh for each step. Also, the smoothing spline fit is used to eliminate the noisy portion of the solution. Finally, the results of the proposed method are compared with the results available in the literature. The comparison shows excellent agreement between the obtained results and those already reported.


Chopard, B. (1990). “A cellular automata model of large-scale moving objects”, Journal of Physics A, 23(10), 1671-1687.
Chopard, B. and Droz, M. (1988). “Cellular automata approach to non-equilibrium phase transitions in surface reaction model: static and dynamic properties”, Journal of Physics A, 21(1), 205-211.
Chopard, B., Droz, M. and Kolb, M. (1989). “Cellular automata a roach to non-equilibrium diffusion and gradient percolation”, Journal of Physics A, 22, 1609-1619.
Chopard, B., Luthi, P.O. and Queloz, P.A. (1996). “Cellular automata model for car traffic in two-dimensional street networks”, Journal of Physics A, 29, 2325-2336. Courant, R., Friedrichs, K. and Lewy, H. (1928). “About the partial differential equations of mathematical physics (in German)”, Mathematische Annalen, 100, 32-74.
Cruz, P., Mendes, A. and Magalhães, F.D. (2001). “Using wavelets for solving PDEs: an adaptive collocation method”, Chemical Engineering Science, 56(10), 3305-3309.
Deslauriers, G. and Dubuc, S. (1989). “Symmetric iterative interpolation processes”, Constructive Approximation, 5, 49-68.
Donoho, D.L. (1992). Interpolating wavelet transforms tech, Report 408, Department of Statistics, Stanford University, Stanford, CA, U.S.A. Fourier, J.B. (1822). The analytical theory of heat, Chez Firmin Didot, pere et fils, Paris, France, (in French).
Frisch, U., d’Humieres, D., Hasslacher, B., Lallemand, P., Pomeau, Y. and Rivet, J.P. (1987). “Lattice gas hydrodynamics in two and three dimensions”, Complex Systems, 1(4), 649-707.
Jameson, L.M. (1998). “A wavelet-optimized, very high order adaptive grid and order numerical method”, SIAM Journal of Scientific Computing, 19(6), 1980-2013.
Holmström, M. (1999). “Solving hyperbolic PDEs using interpolating wavelets”, SIAM Journal of Scientific Computing, 21(2), 405-420.
Hopman, R.K. and Leamy, M.J. (2011). “Triangular cellular automata for computing two-dimensional elastodynamic response on arbitrary domains”, Journal of Applied Mechanics, 78(2), 1115-1132.
Kawamura, S., Shirashige, M. and Iwatsubo, T. (2005). “Simulation of the nonlinear vibration of a string using the cellular automation method”, Journal of Applied Acoustics, 66(1), 77-87.
Kawamura, S., Yoshida, T., Minamoto, H. and Hossain, Z. (2006). “Simulation of the nonlinear vibration of a string using the cellular automata based on the reflection rule”, Journal of Applied Acoustics, 67(2), 93-105.
Kwon, Y.W. and Hosoglu, S. (2008). “Application of lattice Boltzmann method, finite element method, and cellular automata and their coupling to wave propagation problems”, Computers and Structures, 86(7), 663-670.
Leamy, M.J. (2008). “Application of cellular automata modeling to seismic elastodynamics”, International Journal of Solids and Structures, 45(17), 4835-4849.
Liu, Y., Cameron, I.T. and Wang, F.Y. (2000). “The wavelet collocation method for transient problems with steep gradients”, Chemical Engineering Science, 55(9), 1729-1734. Mallat, S. (1999). A wavelet tour of signal processing, Academic Press, New York, U.S.A.
Reinsch, C.H. (1971). “Smoothing by spline functions II”, Numerische Mathematik, 16(5), 451–454.
Rothman, D.H. (1987). “Modeling seismic P-waves with cellular automata”, Geophysical Research Letters, 14(1), 17-20.
Schreckenburg, M., Schadschneider, A., Nagel, K. and Ito, N. (1995). “Discrete stochastic models for traffic flow”, Physical Review E, 51(4), 2939-2949. Timoshenko, S. (1953). History of strength of materials, McGraw-Hill, New York, U.S.A.
Timoshenko, S., Young, D.H. and Weaver, W. (1974). Vibration problems in engineering, Wiley, New York, U.S.A.
Vasilyev, O.V. and Paolucci, S. (1996). “A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain”, Journal of Computational Physics, 125(2), 498-512.
von Neumann, J. (1966). Theory of self-reproducing automata, University of Illinois Press, Urbana, IL, U.S.A.
Wolfram, S. (2002). A new kind of science, Wolfram media, U.S.A.
Yang, K. (2008). “A unified solution for longitudinal wave propagation in an elastic rod”, Journal of Sound and Vibration, 314(1), 307-329.
Volume 48, Issue 2 - Serial Number 2
December 2015
Pages 217-234
  • Receive Date: 20 January 2014
  • Revise Date: 17 January 2015
  • Accept Date: 10 March 2015
  • First Publish Date: 01 December 2015