Abstract
We present novel nonlocal governing operators in 2D/3D for wave propagation and diffusion. The operators are inspired by peridynamics. They agree with the original peridynamics operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of separable kernel functions together with even and odd parts of bivariate functions on rectangular/box domains. The operators are bounded and self-adjoint. We present all possible 36 different types of BC in 2D which include pure and mixed combinations of Neumann, Dirichlet, periodic, and antiperiodic BC. Our construction is systematic and easy to follow. We provide numerical experiments that verify our theoretical findings. We also compare the solutions of the classical wave and heat equations to their nonlocal counterparts.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aksoylu, B., Beyer, H.R., Celiker, F.: Application and implementation of incorporating local boundary conditions into nonlocal problems. Numer. Funct. Anal. Optim. 38(9), 1077–1114 (2017). https://doi.org/10.1080/01630563.2017.1320674
Aksoylu, B., Beyer, H.R., Celiker, F.: Theoretical foundations of incorporating local boundary conditions into nonlocal problems. Rep. Math. Phys. 40(1), 39–71 (2017). https://doi.org/10.1016/S0034-4877(17)30061-7
Aksoylu, B., Celiker, F.: Comparison of Nonlocal Operators Utilizing Perturbation Analysis. In: Others, B.K. (ed.) Numerical Mathematics and Advanced Applications ENUMATH 2015, Lecture Notes in Computational Science and Engineering, vol. 112, pp 589–606. Springer (2016). https://doi.org/10.1007/978-3-319-39929-4_57
Aksoylu, B., Celiker, F.: Nonlocal problems with local Dirichlet and Neumann boundary conditions. J. Mech. Mater. Struct. 12(4), 425–437 (2017). https://doi.org/10.2140/jomms.2017.12.425
Aksoylu, B., Celiker, F., Kilicer, O.: Nonlocal Problems with Local Boundary Conditions: An Overview. In: Voyiadjis, G.Z. (ed.) Handbook on Nonlocal Continuum Mechanics for Materials and Structures, pp 1–38. Springer International Publishing, Cham (2018). https://doi.org/10.1007/978-3-319-22977-5_34-1
Aksoylu, B., Gazonas, G.A.: Inhomogeneous local boundary conditions in nonlocal problems. In: Proceedings of ECCOMAS2018, 6th European Conference on Computational Mechanics (ECCM 6) and 7th European Conference on Computational Fluid Dynamics (ECFD 7), 11-15. In press, Glasgow (2018)
Aksoylu, B., Gazonas, G.A.: On nonlocal problems with inhomogeneous local boundary conditions. Submitted
Aksoylu, B., Mengesha, T.: Results on nonlocal boundary value problems. Numer. Funct. Anal. Optim. 31(12), 1301–1317 (2010). https://doi.org/10.1080/01630563.2010.519136
Aksoylu, B., Parks, M.L.: Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comp. 217, 6498–6515 (2011). https://doi.org/10.1016/j.amc.2011.01.027
Aksoylu, B., Unlu, Z.: Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52(2), 653–677 (2014). https://doi.org/10.1137/13092407X
Andreu-Vaillo, F., Mazon, J.M., Rossi, J.D., Toledo-melero, J.: Nonlocal Diffusion problems, Mathematical Surveys and Monographs, vol. 165 American Mathematical Society and Real Socied Matematica Espanola (2010)
Beyer, H.R., Aksoylu, B., Celiker, F.: On a class of nonlocal wave equations from applications. J. Math. Phy. 57(6), 062902 (2016). https://doi.org/10.1063/1.4953252. Eid: 062902
Bobaru, F., Duangpanya, M.: The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53, 4047–4059 (2010)
Bobaru, F., Duangpanya, M.: A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J. Comput. Phys. 231, 2764–2785 (2012)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Diff. Eqs. 32, 1245–1260 (2007)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54, 667–696 (2012)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model Simul. 7(3), 1005–1028 (2008)
Grote, M.J., Schneebeli, A., Schötzau, D.: Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006)
Kamwal, R.P.: Linear Integral Equations: Theory and Technique. 2, Boston (1997)
Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer, New York (2014). https://doi.org/10.1007/978-1-4614-8465-3
Mitchell, J.A., Silling, S.A., Littlewood, D.J.: A position-aware linear solid constitutive model for peridynamics. J. Mech. Mater. Struct. 10(5), 539–557 (2015)
Moiseiwistch, B.: Integral Equations. Longman Inc., New York (1977)
Nochetto, R.H., Otarola, E., Salgado, A.J.: a PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15, 733–791 (2015)
Oterkus, S., Madenci, E., Agwai, A.: Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014)
Silling, S.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51(6), 3458–3482 (2013)
Funding
Burak Aksoylu was supported in part by the European Commission Marie Curie Career Integration 293978 grant, and Scientific and Technological Research Council of Turkey (TÜ BİTAK) MFAG 115F473 grant. Portion of his work was also supported in part by the Oak Ridge Institute for Science and Engineering (ORISE) contract 1120-1120-99 at the US Army Research Laboratory.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Ilaria Perugia
The original version of this article was revised: In the original publication, Figure 4 image should be Figure 5 and Figure 5 image was a repetition of Figure 6. The original article was updated by correcting the images of figures 4 and 5.
Rights and permissions
About this article
Cite this article
Aksoylu, B., Celiker, F. & Kilicer, O. Nonlocal operators with local boundary conditions in higher dimensions. Adv Comput Math 45, 453–492 (2019). https://doi.org/10.1007/s10444-018-9624-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-018-9624-6