Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T17:16:50.477Z Has data issue: false hasContentIssue false
  • English
  • Français

Matching Boundary Conditions for Scalar Waves in Body-Centered-Cubic Lattices

Published online by Cambridge University Press:  03 June 2015

Ming Fang ,
Xianming Wang ,
Zhihui Li  and
Shaoqiang Tang
Ming Fang*
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, P.O. Box 211, Mianyang 621000, Sichuan, China HEDPS, CAPT and LTCS, College of Engineering, Peking University, Beijing 100871, China
Xianming Wang*
Affiliation:
Center for Combustion Energy and Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
Zhihui Li*
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, P.O. Box 211, Mianyang 621000, Sichuan, China
Shaoqiang Tang*
Affiliation:
HEDPS, CAPT and LTCS, College of Engineering, Peking University, Beijing 100871, China
*
Email: fangm@pku.edu.cn
Email: wangxianming06@mails.tsinghua.edu.cn
Email: lizhihui@mail.tsinghua.edu.cn
Corresponding author. Email: maotang@pku.edu.cn
Get access

Abstract

Matching boundary conditions (MBC’s) are proposed to treat scalar waves in the body-centered-cubic lattices. By matching the dispersion relation, we construct MBC’s for normal incidence and incidence with an angle α. Multiplication of MBC operators then leads to multi-directional absorbing boundary conditions. The effectiveness are illustrated by the reflection coefficient analysis and wave packet tests. In particular, the designed M1M1 treats the scalar waves in a satisfactory manner.

Keywords

65Z0570-08Body-centered-cubic (BCC) crystalline solidsdispersion relationmatching boundary conditionreflection coefficientscalar waves
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 3 , June 2013 , pp. 337 - 350
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Armfield, S. W., Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids, Comput. Fluids, 20 (1991), pp. 1–17.Google Scholar
[2]Berenger, J. P., A perfectly matched layer for the absorption of the electromagnetic waves, J. Comput. Phys., 114 (1994), pp. 185–200.Google Scholar
[3]Born, M. and Huang, K., Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954.Google Scholar
[4]Bos, C., Sietsma, J. and Thijsse, B. J., Molecular dynamics simulation of interface dynamics during thefcc-bcc transformation of a martensitic nature, Phys. Rev. B, 73 (2006), 104117.CrossRefGoogle Scholar
[5]Chen, L., Wang, C. and Yu, T., Molecular dynamics simulation of kink in(100) edge dislocation in body centered cubic iron, Chin. Sci. Bull., 52 (2007), pp. 2291–2296.Google Scholar
[6]Clayton, R. and Engquist, B., Absorbing boundary condtions for acoustic and elastic wave equations, Bull. Seism. Soc. Am., 67 (1977), pp. 1529–1540.Google Scholar
[7]Cuitino, A. M., Stainier, L., Wang, G., Strachan, A., Cagin, T., Goddard, W. A. and Ortiz, M., A multiscale approach for moldeling crystalline solids, J. Comput. Aided Mater., 8 (2002), pp. 127–149.CrossRefGoogle Scholar
[8]Dreher, M. and Tang, S., Time history interfacial conditions in multiscale computations of lattice oscillations, Comput. Mech., 41 (2008), pp. 683–698.Google Scholar
[9]W., E and Huang, Z., Matching conditions in atomistic-continuum modeling of materials, Phys. Rev. Lett., 87 (2001), 135501.Google Scholar
[10]Engquist, B. and Majda, A., Radiation boundary conditions f or acoustic and elastic calculations, Comm. Pure Appl. Math., 32 (1979), pp. 313–357.Google Scholar
[11]Fang, M. and Tang, S., Efficient and robust design for absorbing boundary conditions in atomistic computations, Chin. Phys. Lett., 26 (2009), 116201.Google Scholar
[12]Fikar, J. and Schaublin, R., Molecular dynamics simulation of radiation damage in bcc tungsten, J. Nuc. Mater., 386-388 (2009), pp. 97–101.Google Scholar
[13]Guddati, M. N. and Thirunavukkarasu, S., Phonon absorbing boundary conditions for molecular dynamics, J. Comput. Phys., 228 (2009), pp. 8112–8134.Google Scholar
[14]Higdon, R. L., Absorbing boundary conditions for the wave equation, Math. Comput., 49 (1987), pp. 65–90.Google Scholar
[15]Karpov, E. G., Yu, H., Park, H. S., Liu, W. K., Wang, Q. J. and Qian, D., Multiscale boundary conditions in crystalline solids: theory and application to nanoindentation, Int. J. Solids Struct., 43 (2006), pp. 6359–6379.Google Scholar
[16]Kotrechko, S. A., Filatov, A. V. and Ovsjannikov, A. V., Molecular dynamics simulation of deformation and failure of nanocry stals of bcc metals, Theo. Appl. Frac. Mech., 45 (2006), pp. 92–99.CrossRefGoogle Scholar
[17]Li, X. and W., E, Multiscale modeling of the dynamics of solids at finite temperature, J. Mech. Phys. Solids, 53 (2005), pp. 1650–1685.Google Scholar
[18]Li, X. and E, W., Variational boundary conditions for molecular dynamics simulation of solids at low temperature, Comm. Comput. Phys., 1 (2006), pp. 135–175.Google Scholar
[19]Li, X. and E, W., Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: treatment of the thermal bath, Phys. Rev. B, 76 (2007), 104107.Google Scholar
[20]Lindman, E. L., “Free-space” boundary conditions for the time dependent wave equation, J. Comput. Phys., 18 (1975), pp. 66–78.CrossRefGoogle Scholar
[21]Liu, W. K., Karpov, E. G. and Park, H. S., Nano Mechanics and Materials: Theory, Multi-scale Methods and Applications, Wiley, New York, 2006.Google Scholar
[22]Pang, G. and Tang, S., Time history kernel functions for square lattice, Comput. Mech., 48 (2011), pp. 699–711.Google Scholar
[23]Park, H. S., Karpov, E. G., Klein, P. A. and Liu, W. K., Three-dimensional bridging scale analysis of dynamic fracture, J. Comput. Phys., 207 (2005), pp. 588–609.Google Scholar
[24]Park, H. S., Karpov, E. G. and Liu, W. K., Non-reflecting boundary conditions for atomistic, continuum and coupled atomistic/continuum simulations, Int. J. Numer. Methods Eng., 64 (2005), pp. 237–259.Google Scholar
[25]Rao, S., Hernandez, C., Simmons, J. P., Parthasarathy, T. A. and Woodward, C., Green’s function boundary conditions in two-dimensional and three-dimensional atomistic simulations of dislocations, Philos. Mag. A, 77 (1998), pp. 231–256.CrossRefGoogle Scholar
[26]Sinclair, J. E., Improved atomistic model of a bcc dislocation core, J. Appl. Phys., 42 (1971), pp. 5321–5329.CrossRefGoogle Scholar
[27]Sinclair, J. E., Gehlen, P. C., Hoagland, R. G. and Hirth, J. P., Flexible boundary conditions and nonlinear geometric effects in atomic dislocation modeling, J. Appl. Phys., 49 (1978), pp. 3890–3897.CrossRefGoogle Scholar
[28]Tang, S., A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids, J. Comput. Phys., 227 (2008), pp. 4038–4062.CrossRefGoogle Scholar
[29]Tang, S. and Fang, M., Unstable surface modes in finite chain computations: deficiency of reflection coefficient approach, Commun. Comput. Phys., 8 (2010), pp. 143–158.Google Scholar
[30]To, A. C. and Li, S., Perfectly matched multiscale simulations, Phys. Rev. B, 72 (2005), 035414.Google Scholar
[31]Verzicco, R. and Orlandi, P., A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates, J. Comput. Phys., 123 (1996), pp. 402–414.Google Scholar
[32]Wang, X. and Tang, S., Matching boundary conditions for diatomic chains, Comput. Mech., 46 (2010), pp. 813–826.Google Scholar
[33]Wang, X., Matching Boundary Conditions for Atomic Simulations of Crystalline Solids, PhD thesis, Tsinghua University, Beijing, 2011.Google Scholar
[34]Wang, X. and Tang, S., Matching boundary conditions for lattice dynamics, Int. J. Numer. Methods Eng., 93 (2013), pp. 1255–1285.CrossRefGoogle Scholar
[35]Xu, D. S., Yang, R., Li, J., Chang, J. P., Wang, H., Li, D. and Yip, S., Atomistic simulation of the influence of pressure on dislocation between nucleation in bcc Mo, Comput. Mater. Sci., 36 (2006), pp. 60–64.CrossRefGoogle Scholar