Abstract
This work aims to provide a comprehensive treatment on how to enforce inhomogeneous local boundary conditions (BC) in nonlocal problems in 1D. In prior work, we have presented novel governing operators with homogeneous BC. Here, we extend the construction to inhomogeneous BC. The construction of the operators is inspired by peridynamics. The operators agree with the original peridynamic operator in the bulk of the domain and simultaneously enforce local Dirichlet or Neumann BC. We explain methodically how to construct forcing functions to enforce local BC and their relationship to initial values. We present exact solutions with both homogeneous and inhomogeneous BC and utilize the resulting error to verify numerical experiments. We explain the critical role of the Hilbert-Schmidt property in enforcing local BC rigorously. For the Neumann BC, we prescribe an interpolation strategy to find the appropriate value of the forcing function from its derivative. We also present numerical experiments with unknown solution and report the computed displacement and strain fields.
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Aksoylu B, Beyer HR, Celiker F (2017) Application and implementation of incorporating local boundary conditions into nonlocal problems. Numer Funct Anal Optim 38(9):1077–1114. https://doi.org/10.1080/01630563.2017.1320674
Aksoylu B, Beyer HR, Celiker F (2017) Theoretical foundations of incorporating local boundary conditions into nonlocal problems. Rep Math Phys 40(1):39–71. https://doi.org/10.1016/S0034-4877(17)30061-7
Aksoylu B, Celiker F (2016) Comparison of nonlocal operators utilizing perturbation analysis. In: BK et al. (eds) Numerical mathematics and advanced applications ENUMATH 2015, Lecture Notes in Computational Science and Engineering, vol 112. Springer, pp 589–606. https://doi.org/10.1007/978-3-319-39929-4_57
Aksoylu B, Celiker F (2017) Nonlocal problems with local Dirichlet and Neumann boundary conditions. J Mech Mater Struct 12(4):425–437. https://doi.org/10.2140/jomms.2017.12.425
Aksoylu B, Celiker F, Gazonas GA (2020) Higher order collocation methods for nonlocal problems and their asymptotic compatibility. Commun Appl Math Comput 2(2):261–303. https://doi.org/10.1007/s42967-019-00051-8
Aksoylu B, Celiker F, Kilicer O (2019) Nonlocal operators with local boundary conditions: An overview. Springer, Cham, pp 1293–1330. https://doi.org/10.1007/978-3-319-58729-5_34
Aksoylu B, Celiker F, Kilicer O (2019) Nonlocal problems with local boundary conditions in higher dimensions. Adv Comp Math 45(1):453–492. https://doi.org/10.1007/s10444-018-9624-6
Aksoylu B, Gazonas GA (2018) 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). Glasgow, UK. In press
Aksoylu B, Kaya A (2018) Conditioning and error analysis of nonlocal problems with local boundary conditions. J Comput Appl Math 335:1–19. https://doi.org/10.1016/j.cam.2017.11.023
Aksoylu B, Unlu Z (2014) Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J Numer Anal 52(2):653–677. https://doi.org/10.1137/13092407X
Beyer HR, Aksoylu B, Celiker F (2016) On a class of nonlocal wave equations from applications. J Math Phys 57(6):062902. https://doi.org/10.1063/1.4953252. Eid: 062902
Bitsadze A (1980) Equations of mathematical physics. Mir Publishers, Moscow
Du Q (2018) An invitation to nonlocal modeling, analysis and computation. In: Sirakov B, de Souza PN, Viana M (eds) Proceedings of the international congress of mathematicians 2018 (ICM 2018). World Scientific, vol 3, pp 3523–3552
Du Q, Gunzburger M, Lehoucq RB, Zhou K (2012) Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev 54:667–696
Du Q, Lipton R (2014) Peridynamics, fracture, and nonlocal continuum models. SIAM News 47(3)
Greenberg MD (1998) Advanced engineering mathematics, 2nd edn. Prentice Hall, Englewood Cliffs
Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, New York. https://doi.org/10.1007/978-1-4614-8465-3
Silling S (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209
Silling S, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73– 168
Strauss WA (2008) Partial differential equations: an introduction. Wiley, New Jersey
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Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-18-2-0090. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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Aksoylu, B., Gazonas, G.A. On Nonlocal Problems with Inhomogeneous Local Boundary Conditions. J Peridyn Nonlocal Model 2, 1–25 (2020). https://doi.org/10.1007/s42102-019-00022-w
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DOI: https://doi.org/10.1007/s42102-019-00022-w