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.
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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.
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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 (TÜB\(\dot {\mathrm {I}}\)TAK) MFAG 115F473 grant.
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Aksoylu, B., Celiker, F., Kilicer, O. (2018). 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/10.1007/978-3-319-22977-5_34-1
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