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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: [email protected]
Email: [email protected]
Email: [email protected]
Corresponding author. Email: [email protected]
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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

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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