Abstract
We present novel governing operators in arbitrary dimension for nonlocal diffusion in homogeneous media. The operators are inspired by the theory of peridynamics (PD). They agree with the original PD operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of kernel functions together with even and odd parts of bivariate functions. We present different types of BC in 2D which include pure and mixed combinations of Neumann and Dirichlet BC. Our construction is systematic and easy to follow. We provide numerical experiments that validate our theoretical findings. When our novel operators are extended to vector-valued functions, they will allow the extension of PD to applications that require local BC. Furthermore, we hope that the ability to enforce local BC provides a remedy for surface effects seen in PD.
We recently proved that the nonlocal diffusion operator is a function of the classical operator. This observation opened a gateway to incorporate local BC to nonlocal problems on bounded domains. The main tool we use to define the novel governing operators is functional calculus, in which we replace the classical governing operator by a suitable function of it. We present how to apply functional calculus to general nonlocal problems in a methodical way.
Mathematics Subject Classification (2000): 35L05, 74B99, 47G10.
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Notes
- 1.
We do not explicitly denote the dimension of the domain Ω. The dimension is implied by the number of iterated integrals present in the operator. The domain Ω is equal to [−1, 1], [−1, 1] × [−1, 1], and [−1, 1] × [−1, 1] × [−1, 1] in 1D, 2D, and 3D, respectively.
References
B. Aksoylu, H.R. Beyer, F. Celiker, Application and implementation of incorporating local boundary conditions into nonlocal problems. Numer. Funct. Anal. Optim. 38(9), 1077–1114 (2017a). https://doi.org/10.1080/01630563.2017.1320674
B. Aksoylu, H.R. Beyer, F. Celiker, Theoretical foundations of incorporating local boundary conditions into nonlocal problems. Rep. Math. Phys. 40(1), 39–71 (2017b). https://doi.org/10.1016/S0034-4877(17)30061-7
B. Aksoylu, F. Celiker, Comparison of nonlocal operators utilizing perturbation analysis, in Numerical Mathematics and Advanced Applications ENUMATH 2015, vol. 112, ed. by B.K. et al. Lecture Notes in Computational Science and Engineering (Springer, 2016), pp. 589–606. https://doi.org/10.1007/978-3-319-39929-4_57
B. Aksoylu, F. Celiker, 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
B. Aksoylu, F. Celiker, O. Kilicer, Nonlocal problems with local boundary conditions in higher dimensions (Submitted)
B. Aksoylu, A. Kaya, Conditioning and error analysis of nonlocal problems with local boundary conditions. J. Comput. Appl. Math. 335, 1–19 (2018). https://doi.org/10.1016/j.cam.2017.11.023
B. Aksoylu, T. Mengesha, Results on nonlocal boundary value problems. Numer. Funct. Anal. Optim. 31(12), 1301–1317 (2010)
B. Aksoylu, M.L. Parks, Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comput. 217, 6498–6515 (2011). https://doi.org/10.1016/j.amc.2011.01.027
B. Aksoylu, Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52(2), 653–677 (2014)
F. Andreu-Vaillo, J.M. Mazon, J.D. Rossi, J. Toledo-Melero, Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. (American Mathematical Society and Real Socied Matematica Espanola, 2010)
H.R. Beyer, B. Aksoylu, F. Celiker, On a class of nonlocal wave equations from applications. J. Math. Phys. 57(6), 062902 (2016). https://doi.org/10.1063/1.4953252. Eid:062902
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Part. Diff. Eqs. 32, 1245–1260 (2007)
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54, 667–696 (2012)
Q. Du, R. Lipton, Peridynamics, fracture, and nonlocal continuum models. SIAM News 47(3) (2014)
E. Emmrich, O. Weckner, The peridynamic equation and its spatial discretization. Math. Model. Anal. 12(1), 17–27 (2007)
G. Gilboa, S. Osher, Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)
B. Kilic, Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials. Ph.D. thesis, Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson (2008)
E. Madenci, E. Oterkus, Peridynamic Theory and Its Applications (Springer, New York/Heidelberg/Dordrecht/London, 2014). https://doi.org/10.1007/978-1-4614-8465-3
T. Mengesha, Q. Du, Analysis of a scalar peridynamic model for sign changing kernel. Disc. Cont. Dyn. Sys. B 18, 1415–1437 (2013)
J.A. Mitchell, S.A. Silling, D.J. Littlewood, A position-aware linear solid constitutive model for peridynamics. J. Mech. Mater. Struct. 10(5), 539–557 (2015)
R.H. Nochetto, E. Otarola, A.J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15, 733–791 (2015)
P. Seleson, M. Gunzburger, M.L. Parks, Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains. Comput. Methods Appl. Mech. Eng. 266, 185–204 (2013)
P. Seleson, M.L. Parks, On the role of the influence function in the peridynamic theory. Internat. J. Multiscale Comput. Eng. 9(6), 689–706 (2011)
S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
S. Silling, R.B. Lehoucq, Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)
S.A. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003)
X. Tian, Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51(6), 3458–3482 (2013)
O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)
Acknowledgements
Burak Aksoylu was supported in part by the European Commission Marie Curie Career Integration 293978 grant, and Scientific and Technological Research Council of Turkey (TÃœB\(\dot {\mathrm {I}}\)TAK) MFAG 115F473 grant.
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Aksoylu, B., Celiker, F., Kilicer, O. (2019). Nonlocal Operators with Local Boundary Conditions: An Overview. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-58729-5_34
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