Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations. (English) Zbl 1411.74058
Summary: This paper studies nonlocal diffusion models associated with a finite nonlocal horizon parameter \(\delta\) that characterizes the range of nonlocal interactions. The focus is on the variational formulation associated with Neumann type constraints and its numerical approximations. We establish the well-posedness for some variational problems associated and study their local limit as \(\delta\rightarrow 0\). A main contribution is to derive a second order convergence to the local limit. We then discuss the numerical approximations including standard finite element methods and quadrature based finite difference methods. We study their convergence in the nonlocal setting and in the local limit.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 45K05 | Integro-partial differential equations |
| 74A45 | Theories of fracture and damage |
Keywords:
nonlocal diffusion; peridynamic model; nonlocal operator; Neumann problems; finite element; finite differenceReferences:
| [1] | Aksoylu, B.; Mengesha, T., Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim., 31, 1301-1317 (2010) · Zbl 1206.35251 |
| [2] | Alali, B.; Gunzburger, M., Peridynamics and material interfaces, Journal of Elasticity, 2015, 1-24 (2010) |
| [3] | Andreu-Vaillo, F.; Mazn, J. M.; Rossi, J. D.; Toledo-Melero, J. J., Nonlocal diffusion problems, American Mathematical Society. Mathematical Surveys and Monographs, 165 (2010), Providence, RI · Zbl 1214.45002 |
| [4] | Askari, E.; Bobaru, F.; Lehoucq, R. B.; Parks, M. L.; Silling, S. A.; Weckner, O., Peridynamics for multiscale materials modeling, J. Phys.: Conf. Ser., 125, 012078 (2008) |
| [5] | Barles, G.; Chasseigne, E.; Georgelin, C.; Jakobsen, E., On Neumann type problems for nonlocal equations set in a half space, Trans. Am. Math. Soc., 366, 9, 4873-4917 (2014) · Zbl 1323.35192 |
| [6] | Barles, G.; Georgelin, C.; Jakobsen, E. R., On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations, J. Differ. Equ., 256, 1368-1394 (2014) · Zbl 1285.35122 |
| [7] | Bessa, M.; Foster, J.; Belytschko, T.; Liu, W. K., A meshfree unification: reproducing kernel peridynamics, Comput. Mech., 53, 1251-1264 (2014) · Zbl 1398.74452 |
| [8] | Bobaru, F.; Yang, M.; Alves, L.; Silling, S.; Askari, E.; Xu, J., Convergence, adaptive refinement, and scaling in 1d peridynamics, Int. J. Numer. Methods Eng., 77, 852-877 (2009) · Zbl 1156.74399 |
| [9] | Bourgain, J.; Brézis, H.; Mironescu, P., Another look at sobolev spaces, (Menaldi, J. L.; Rofman, E.; Sulem, A., Optimal Control and Partial Differential Equations (2001), IOS Press), 439-455 · Zbl 1103.46310 |
| [10] | Burch, N.; Lehoucq, R., Classical, nonlocal, and fractional diffusion equations on bounded domains, Int. J. Multiscale Comput. Eng., 9, 661 (2011) |
| [11] | Chen, X.; Gunzburger, M., Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Methods .Appl. Mech. Eng., 200, 1237-1250 (2011) · Zbl 1225.74082 |
| [12] | Cortazar, C.; Elgueta, M.; Rossi, J. D.; Wolanski, N., How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive Ration. Mech. Anal., 187, 137-156 (2008) · Zbl 1145.35060 |
| [13] | Dayal, K.; Bhattacharya, K., Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, J. Mech. Phys. Solids, 54, 1811-1842 (2006) · Zbl 1120.74690 |
| [14] | Dipierro, S.; Ros-Oton, X.; Valdinoci, E., Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam. (2017) · Zbl 1371.35322 |
| [15] | Du, Q.; Gunzburger, M.; Lehoucq, R.; Zhou, K., Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elast., 113.2, 193-217 (2013) · Zbl 1277.35007 |
| [16] | Du, Q.; Gunzburger, M.; Lehoucq, R.; Zhou, K., A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23, 493-540 (2013) · Zbl 1266.26020 |
| [17] | Du, Q.; Gunzburger, M.; Lehoucq, R.; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54, 667-696 (2012) · Zbl 1422.76168 |
| [18] | Du, Q.; Huang, Z.; Lehoucq, R., Nonlocal convection-diffusion volume-constrained problems and jump processes, Discret. Contin. Dyn. Syst. Ser. B (DCDS-B), 19.2, 373-389 (2014) · Zbl 1284.35223 |
| [19] | Du, Q.; Lehoucq, R.; Tartakovsky, A., Integral approximations to classical diffusion and smoothed particle hydrodynamics, Comput. Meth. Appl. Mech. Eng., 286, 216-229 (2015) · Zbl 1423.76363 |
| [20] | Du, Q.; Tao, Y.; Tian, X.; Yang, J., Robust a posteriori stress analysis for approximations of nonlocal models via nonlocal gradient, Comput. Methods Appl. Mech. Eng., 310, 605-627 (2016) · Zbl 1439.74011 |
| [21] | Mengesha, T.; Du, Q., The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. R. Soc. Edinb. A, 144, 161-186 (2014) · Zbl 1381.35177 |
| [22] | Mengesha, T.; Du, Q., Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elasticity, 116.1, 27-51 (2014) · Zbl 1297.74051 |
| [23] | Mengesha, T.; Du, Q., Analysis of a scalar nonlocal peridynamic model with a sign changing kernel, Discret. Cont. Dyn . Sys. B., 18, 1415-1437 (2013) · Zbl 1278.45014 |
| [24] | T. Mengesha, Q. Du, On the variational limit of some nonlocal convex functionals of vector fields, Nonlinearity, 2015, 28, 3999-4035.; T. Mengesha, Q. Du, On the variational limit of some nonlocal convex functionals of vector fields, Nonlinearity, 2015, 28, 3999-4035. · Zbl 1330.45011 |
| [25] | Mengesha, T.; Du, Q., Characterization of function spaces of vector fields and an application in nonlinear peridynamics, Nonlinear Anal.: Theory Methods Appl., 140, 82-111 (2016) · Zbl 1353.46027 |
| [26] | Mitchell, J.; Stewart, S.; Littlewood, D., A position-aware linear solid constitutive model for peridynamics, J. Mech. Mater. Struct., 10.5, 539-557 (2015) |
| [27] | Oterkus, E.; Madenci, E., Peridynamic analysis of fiber-reinforced composite materials, J. Mech. Mater. Struct., 7, 45-84 (2012) |
| [28] | Ponce, A., An estimate in the spirit of poincare’s inequality, J. Eur. Math. Soc., 6, 1-15 (2004) · Zbl 1051.46019 |
| [29] | Silling, S., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 175-209 (2000) · Zbl 0970.74030 |
| [30] | Silling, S.; Lehoucq, R., Peridynamic theory of solid mechanics, Adv. Appl. Mech., 44, 73-168 (2010) |
| [31] | Silling, S.; Weckner, O.; Askari, E.; Bobaru, F., Crack nucleation in a peridynamic solid, Int. J. Fract., 162, 219-227 (2010) · Zbl 1425.74045 |
| [32] | Taylor, M.; Steigmann, D., A two-dimensional peridynamic model for thin plates, Math. Mech. Solids, 20, 998-1010 (2015) · Zbl 1330.74111 |
| [33] | Tian, X.; Du, Q., Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numerical Analysis, 51, 3458-3482 (2013) · Zbl 1295.82021 |
| [34] | Tian, X.; Du, Q., Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal., 52.4, 1641-1665 (2014) · Zbl 1303.65098 |
| [35] | Tian, X.; Du, Q., Trace theorems for some nonlocal function spaces with heterogeneous localization, SIAM J. Math. Anal. (2017) · Zbl 1373.46030 |
| [36] | Wang, H.; Tian, H., A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model, J. Comput. Phys.s, 231, 7730-7738 (2012) · Zbl 1254.74112 |
| [37] | Zhou, K.; Du, Q., Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary, SIAM J. Numer. Anal., 48, 1759-1780 (2010) · Zbl 1220.82074 |
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