Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions

SIAM Journal on Numerical Analysis

Volumn 48, Issue 5, 2010, Pages 1759-1780

  Zhou, Kun ^a   Du, Qiang ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Condition numbers; Error estimates; Galerkin Ritz approximations; Nonlocal boundary conditions; Peridynamics model; Well posedness and regularity

Indexed keywords

CONDITION NUMBERS; ERROR ESTIMATES; GALERKIN; NON-LOCAL BOUNDARY CONDITIONS; PERIDYNAMICS; WELL-POSEDNESS AND REGULARITY;

GALERKIN METHODS; MATHEMATICAL OPERATORS; NUMBER THEORY;

BOUNDARY CONDITIONS;

EID: 78649500561     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/090781267     Document Type: Article

References (29)

1. B. Aksoylu and M. Parks, Towards Domain Decomposition for Nonlocal Problems, CCTTR-2009-12, Louisiana State University, LA, 2009.
2. B. Alali and R. Lipton, Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation, IMA Preprint 2241, Institute for Mathematics and its Applications, University of Minnesota, MN, 2009.
3. G. Aubert and P. Kornprobst, Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems? SIAM J. Numer. Anal., 47 (2009), pp. 844-860.
4. O. Axelsson and V. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation, Academic Press, London, 1983; reprinted as Classics Appl. Math. 35, SIAM, Philadelphia, 2001.
5. S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer, New York, 2002.
6. E. Askari, F. Bobaru, R. B. Lehoucq, M. L. Parks, S. A. Silling, and O. Weckner, Peridynamics for multiscale materials modeling, J. Phys. Conf. Ser., 125 (2008), article 012078.
7. F. Bobaru, M. Yang, L.F. Alves, S. A. Silling, E. Askari, and J. Xu, Convergence, adaptive refinement, and scaling in 1D peridynamics, Internat. J. Numer. Engrg., 77 (2009), pp. 852-877.
8. X. Chen and M. Gunzburger, Continuous and Discontinuous Finite Element Methods for a Peridynamics Model of Mechanics, Comp. Methods Appl. Mech. Eng., to appear.
9. K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, J. Mech. Phys. Solids, 54 (2006), pp. 1811-1842.
10. Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: M2AN Math. Mod. Numer. Anal., to appear.
11. Q. Du, D. Wang, and L. Zhu, On mesh geometry and stiffness matrix conditioning for general finite element spaces, SIAM J. Numer. Anal., 47 (2009), pp. 1421-1444.
12. T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp., 34 (1980), pp. 441-463.
13. E. Emmrich and O. Weckner, Analysis and numerical approximation of an integrodifferential equation modelling non-local effects in linear elasticity, Math. Mech. Solids, 12 (2007), pp. 363-384.
14. E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci., 5 (2007), pp. 851-864.
15. E. Emmrich and O. Weckner, The peridynamic equation and its spatial discretisation, Math. Model. Anal., 12 (2007), pp. 1-10.
16. M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), pp. 1581-1598.
17. R. B. Lehoucq and S. A. Silling, Statistical Coarse-graining of Molecular Dynamics into Peridynamics, Technical report SAND2007-6410, Sandia National Laboratories, Albuquerque, NM, 2007.
18. R. B. Lehoucq and S. A. Silling, Force flux and the peridynamic stress tensor, J. Mech. Phys. Solids, 56 (2008), pp. 1566-1577.
19. M. Parks, R. B. Lehoucq, S. Plimpton, and S. Silling, Implementing peridynamics within a molecular dynamics code, Comput. Phys. Comm., 179 (2008), pp. 777-783.
20. P. Seleson, M. Parks, M. Gunzburger, and R. B. Lehoucq, Peridynamics as an upscaling of molecular dynamics, Multiscale Model. Simul., 8 (2009), pp. 204-227.
21. S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), pp. 175-209.
22. S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity, 88 (2007), pp. 151-184.
23. S. A. Silling, Linearized theory of peridynamic states, J. Elasticity, 99 (2010), pp. 85-111.
24. S. A. Silling and E. Askari, A meshfree method based on the peridynamic model of solid mechanics, Comput. & Structures, 83 (2005), pp. 1526-1535.
25. S. A. Silling, O. Weckner, E. Askari, and F. Bobaru, Crack nucleation in a peridynamic solid, Internat. J. Fracture, 162 (2010), pp. 219-227.
26. S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory, J. Elasticity, 93 (2008), pp. 13-37.
27. S. A. Silling and R. B. Lehoucq, Peridynamic theory of solid mechanics, Adv. Appl. Mech., 44 (2010), pp. 73-168.
28. L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, Berlin, 2007.
29. O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar, J. Mech. Phys. Solids, 53 (2005), pp. 705-728.