Free Access
Issue
ESAIM: M2AN
Volume 54, Number 4, July-August 2020
Page(s) 1373 - 1413
DOI https://doi.org/10.1051/m2an/2019089
Published online 18 June 2020
  1. X. Antoine and H. Barucq, Approximation by generalized impedance boundary conditions of a transmission problem in acoustic scattering. ESAIM: M2AN 39 (2005) 1041–1059. [CrossRef] [EDP Sciences] [Google Scholar]
  2. B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems. Numer. Funct. Anal. Optim. 31 (2010) 1301–1317. [Google Scholar]
  3. B. Alali and M. Gunzburger, Peridynamics and material interfaces. J. Elast. 120 (2015) 225–248. [Google Scholar]
  4. E. Askari, J. Xu and S. Silling, Peridynamic analysis of damage and failure in composites. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. AIAA, Reston, VA (2006). [Google Scholar]
  5. M. Astorino, F. Chouly and M.A. Fernández, Robin based semi-implicit coupling in fluid-structure interaction: stability analysis and numerics. SIAM J. Sci. Comput. 31 (2009) 4041–4065. [Google Scholar]
  6. B. Baeumer, M. Kovács, M.M. Meerschaert and H. Sankaranarayanan, Boundary conditions for fractional diffusion. J. Comput. Appl. Math. 336 (2018) 408–424. [Google Scholar]
  7. S. Badia, F. Nobile and C. Vergara, Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227 (2008) 7027–7051. [Google Scholar]
  8. G. Barles, C. Georgelin and E.R. Jakobsen, On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations. J. Diff. Equ. 256 (2014) 1368–1394. [CrossRef] [Google Scholar]
  9. Z.P. Bažant and M. Jirásek, Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128 (2002) 1119–1149. [Google Scholar]
  10. M.A. Bessa, J.T. Foster, T. Belytschko and W.K. Liu, A meshfree unification: reproducing kernel peridynamics. Comput. Mech. 53 (2014) 1251–1264. [Google Scholar]
  11. F. Bobaru and Y.D. Ha, Adaptive refinement and multiscale modeling in 2D peridynamics. Int. J. Multiscale Comput. Eng. 9 (2011) 635–659. [Google Scholar]
  12. J.P. Borthagaray, W. Li and R.H. Nochetto, Finite element discretizations of nonlocal minimal graphs: convergence. Preprint arXiv:1905.06395 (2019). [Google Scholar]
  13. J. Bourgain, H. Brezis and P. Mironescu, Another Look at Sobolev Spaces. IOS Press (2001). [Google Scholar]
  14. S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. Springer Science & Business Media, 15 (2007). [Google Scholar]
  15. C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications. Springer 20 (2016). [CrossRef] [Google Scholar]
  16. N. Burch and R. Lehoucq, Classical, nonlocal, and fractional diffusion equations on bounded domains. Int. J. Multiscale Comput. Eng. 9 (2011) 661–674. [Google Scholar]
  17. F.A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. ESAIM: M2AN 52 (2018) 163–180. [CrossRef] [EDP Sciences] [Google Scholar]
  18. C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion. J. Differ. Equ. 234 (2007) 360–390. [Google Scholar]
  19. C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Ration. Mech. Anal. 187 (2008) 137–156. [Google Scholar]
  20. K. Dayal and K. Bhattacharya, A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries. Acta Mater. 55 (2007) 1907–1917. [Google Scholar]
  21. O. Defterli, M. D’Elia, Q. Du, M. Gunzburger, R. Lehoucq and M.M. Meerschaert, Fractional diffusion on bounded domains. Fract. Calc. Appl. Anal. 18 (2015) 342–360. [Google Scholar]
  22. P. Demmie and S. Silling, An approach to modeling extreme loading of structures using peridynamics. J. Mech. Mater. Struct. 2 (2007) 1921–1945. [Google Scholar]
  23. S. Dipierro, X. Ros Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoamericana 33 (2017) 377–416. [CrossRef] [Google Scholar]
  24. S. Dipierro, O. Savin and E. Valdinoci, Boundary behavior of nonlocal minimal surfaces. J. Funct. Anal. 272 (2017) 1791–1851. [Google Scholar]
  25. Q. Du and R. Lipton, Peridynamics, fracture, and nonlocal continuum models. SIAM News 47 (2014). [Google Scholar]
  26. Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM: M2AN 45 (2011) 217–234. [CrossRef] [EDP Sciences] [Google Scholar]
  27. Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012) 667–696. [CrossRef] [MathSciNet] [Google Scholar]
  28. Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23 (2013) 493–540. [Google Scholar]
  29. Q. Du, Z. Huang and R.B. Lehoucq, Nonlocal convection-diffusion volume-constrained problems and jump processes. Disc. Cont. Dyn. Syst. B 19 (2014) 961–977. [Google Scholar]
  30. Q. Du, R.B. Lehoucq and A.M. Tartakovsky, Integral approximations to classical diffusion and smoothed particle hydrodynamics. Comput. Methods Appl. Mech. Eng. 286 (2015) 216–229. [Google Scholar]
  31. Q. Du, R. Lipton and T. Mengesha, Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media. ESAIM: M2AN 50 (2016) 1425–1455. [CrossRef] [EDP Sciences] [Google Scholar]
  32. Q. Du, Y. Tao and X. Tian, A peridynamic model of fracture mechanics with bond-breaking. J. Elast. 132 (2018) 197–218. [Google Scholar]
  33. E. Emmrich and O. Weckner, Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity. Math. Mech. Solids 12 (2007) 363–384. [Google Scholar]
  34. 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) 851–864. [Google Scholar]
  35. E. Emmrich and D. Puhst, Survey of existence results in nonlinear peridynamics in comparison with local elastodynamics. Comput. Methods Appl. Math. 15 (2015) 483–496. [CrossRef] [Google Scholar]
  36. H.A. Erbay, S. Erbay and A. Erkip, Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations. ESAIM: M2AN 52 (2018) 803–826. [CrossRef] [EDP Sciences] [Google Scholar]
  37. J.T. Foster. Dynamic crack initiation toughness: experiments and peridynamic modeling. . Ph.D. thesis. Purdue University (2009). [CrossRef] [Google Scholar]
  38. W. Gerstle, N. Sau and S. Silling, Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237 (2007) 1250–1258. [CrossRef] [Google Scholar]
  39. G. Grubb, Local and nonlocal boundary conditions for #-transmission and fractional elliptic pseudodifferential operators. Anal. PDE 7 (2014) 1649–1682. [CrossRef] [Google Scholar]
  40. Y.D. Ha and F. Bobaru, Characteristics of dynamic brittle fracture captured with peridynamics. Eng. Fract. Mech. 78 (2011) 1156–1168. [Google Scholar]
  41. J.F. Kelly, H. Sankaranarayanan and M.M. Meerschaert, Boundary conditions for two-sided fractional diffusion. J. Comput. Phys. 376 (2019) 1089–1107. [Google Scholar]
  42. R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117 (2014) 21–50. [Google Scholar]
  43. A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth and G.E. Karniadakis, What is the fractional laplacian?. Preprint arXiv:1801.09767 (2018). [Google Scholar]
  44. D.J. Littlewood, S.A. Silling, J.A. Mitchell, P.D. Seleson, S.D. Bond, M.L. Parks, D.Z. Turner, D.J. Burnett, J. Ostien and M. Gunzburger, Strong local-nonlocal coupling for integrated fracture modeling. Technical report, Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA (United States) (2015). [CrossRef] [Google Scholar]
  45. E. Madenci and E. Oterkus, Peridynamic Theory and its Applications. Springer (2016). [Google Scholar]
  46. E. Madenci, M. Dorduncu, A. Barut and N. Phan, Weak form of peridynamics for nonlocal essential and natural boundary conditions. Comput. Methods Appl. Mech. Eng. 337 (2018) 598–631. [Google Scholar]
  47. R.L. Magin, Fractional Calculus in Bioengineering. Begell House Publishers Inc., Redding, CT (2006). [Google Scholar]
  48. C. Mantegazza and A.C. Mennucci, Hamilton-jacobi equations and distance functions on riemannian manifolds. Appl. Math. Optim. 47 (2003) 1–25. [Google Scholar]
  49. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific (2010). [Google Scholar]
  50. T. Mengesha and Q. Du, Analysis of a scalar peridynamic model with a sign changing kernel. Disc. Cont. Dyn. Sys. B 18 (2013) 1415–1437. [Google Scholar]
  51. T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic navier equation. J. Elast. 116 (2014) 27–51. [Google Scholar]
  52. T. Mengesha and Q. Du, Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140 (2016) 82–111. [CrossRef] [Google Scholar]
  53. E. Montefusco, B. Pellacci and G. Verzini, Fractional diffusion with Neumann boundary conditions: the logistic equation. Disc. Cont. Dyn. Sys. B 18 (2013) 2175–2202. [Google Scholar]
  54. M.L. Parks, P. Seleson, S.J. Plimpton, R.B. Lehoucq and S.A. Silling, Peridynamics with Lammps: A User Guide v0.2 Beta. Sandia National Laboraties (2008). [Google Scholar]
  55. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press 198 (1998). [Google Scholar]
  56. A.C. Ponce, An estimate in the spirit of poincaré’s inequality. J. Eur. Math. Soc. 6 (2004) 1–15. [CrossRef] [Google Scholar]
  57. J. Ren, Z.-Z. Sun and X. Zhao, Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232 (2013) 456–467. [Google Scholar]
  58. E.W. Sachs and M. Schu, A priori error estimates for reduced order models in finance. ESAIM: M2AN 47 (2013) 449–469. [CrossRef] [EDP Sciences] [Google Scholar]
  59. P. Seleson, M. Gunzburger and M.L. Parks, Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains. Comput. Methods Appl. Mech. Eng. 266 (2013) 185–204. [Google Scholar]
  60. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209. [Google Scholar]
  61. Y. Tao, X. Tian and Q. Du, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations. Appl. Math. Comput. 305 (2017) 282–298. [Google Scholar]
  62. M. Taylor and D.J. Steigmann, A two-dimensional peridynamic model for thin plates. Math. Mech. Solids 20 (2015) 998–1010. [Google Scholar]
  63. X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52 (2014) 1641–1665. [Google Scholar]
  64. N. Trask, H. You, Y. Yu and M.L. Parks, An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics. Comput. Methods Appl. Mech. Eng. 343 (2019) 151–165. [Google Scholar]
  65. O. Weckner, A. Askari, J. Xu, H. Razi and S.A. Silling, Damage and failure analysis based on peridynamics—theory and applications. In: 48th AIAA Structures, Structural Dynamics, and Materials Conf (2007). [Google Scholar]
  66. H. Wendland, Scattered Data Approximation. Cambridge University Press 17 (2004). [CrossRef] [Google Scholar]
  67. J. Xu, A. Askari, O. Weckner and S. Silling, Peridynamic analysis of impact damage in composite laminates. J. Aerosp. Eng. 21 (2008) 187–194. [Google Scholar]
  68. Y. Yu, F. Bargos, H. You, M.L. Parks, M.L. Bittencourt and G.E. Karniadakis, A partitioned coupling framework for peridynamics and classical theory: analysis and simulations. Comput. Methods Appl. Mech. Eng. 340 (2018) 905–931. [Google Scholar]
  69. K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48 (2010) 1759–1780. [Google Scholar]
  70. M. Zimmermann, A continuum theory with long-range forces for solids. Ph.D. thesis. Massachusetts Institute of Technology (2005). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you