Abstract
In a semi-discretized Euler-Bernoulli beam equation, the non-nearest neighboring interaction and large span of temporal scales for wave propagations pose challenges to the effectiveness and stability for artificial boundary treatments. With the discrete equation regarded as an atomic lattice with a three-atom potential, two accurate artificial boundary conditions are first derived here. Reflection coefficient and numerical tests illustrate the capability of the proposed methods. In particular, the time history treatment gives an exact boundary condition, yet with sensitivity to numerical implementations. The ALEX (almost EXact) boundary condition is numerically more effective.
Similar content being viewed by others
References
Brekhovskikh, L., Goncharov, V.: Mechanics of Continua and Wave Dynamics. Heidelberg, Springer (1985)
Ruge, P., Birk, C.: Comparison of infinite Timoshenko and Euler-Bernoulli beam models on Winkler foundation in the frequency- and time-domain. J. Sound and Vibration 304, 932–947 (2007)
Uzzal, R.U.A., Bhat, R.B., Ahmed, W.: Dynamical response of a beam subjected to moving load and moving mass supported by Pasternak foundation. Shock and Vibration 19, 205–220 (2012)
Gonella, S., To, A.C., Liu, W.K.: Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting J. Mech. Phys. Solids 57, 621–633 (2009)
Liu, L., Hussein, M.I.: Wave motion in periodic flexual beams and characterization of the transition between Bragg scattering and local resonance. J. Appl. Mech. 79, 011003 (2012)
Liu, W.K., Karpov, E.G., Park, H.S.: Nano mechanics and materials: theory, multi-scale methods and applications. Wiley: New York (2006)
Karpov, E.G., Wagner, G.J., Liu, W.K.: A Green’s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations Int. J. Numer. Methods Eng. 62, 1250–1262 (2005)
Karpov, E.G., Park, H.S., Liu, W.K.: A phonon heat bath approach for the atomistic and multiscale simulation of solids. Int. J. Numer. Meth. Eng. 70, 351–378 (2007)
Wang, X., Tang, S.: Matching boundary conditions for lattice dynamics. Int. J. Numer. Methods Eng. 93, 1255–1285 (2013)
Tang, S.: A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids. J. Comput. Phys. 227, 4038–4062 (2008)
Pang, G.: Accurate boundary conditions for atomic simulations of simple lattices with applications. [Ph.D. Thesis]. Beijing, Peking University (2013)
Pang, G., Bian, L., Tang, S.: Almost exact boundary condition for one-dimensional Schrodinger equation. Phys. Rev. E 86, 066709 (2012)
Antoine, X., Arnold, A., Besse, C., et al.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4, 729–796 (2008)
Author information
Authors and Affiliations
Corresponding authors
Additional information
The project was supported by the National Natural Science Foundation of China (11272009), National Basic Research Program of China (2010CB731503), and U.S. National Science Foundation (0900498).
Rights and permissions
About this article
Cite this article
Tang, SQ., Karpov, E.G. Artificial boundary conditions for Euler-Bernoulli beam equation. Acta Mech Sin 30, 687–692 (2014). https://doi.org/10.1007/s10409-014-0089-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-014-0089-7