On the Choice of Kernel Function in Nonlocal Wave Propagation
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On the Choice of Kernel Function in Nonlocal Wave Propagation Burak Aksoylu1,2
· George A. Gazonas1
Received: 12 November 2019 / Accepted: 17 April 2020 / © Springer Nature Switzerland AG 2020
Abstract It is a challenge to choose the appropriate kernel function in nonlocal problems. We tackle this challenge from the aspect of nonlocal wave propagation and study the dispersion relation at the analytical level. The kernel function enters the formulation as an input. Any effort to narrow down this function family is valuable. Dispersion relations of the nonlocal governing operators are identified. Using a Taylor expansion, a selection criterion is devised to determine the kernel function that provides the best approximation to the classical (linear) dispersion relation. The criterion is based on selecting the smallest coefficient in magnitude of the dominant term in the Taylor expansion after the constant term. The governing operators are constructed using functional calculus, which provides the explicit expression of the eigenvalues of the operators. The ability to express eigenvalues explicitly allows us to obtain dispersion relations at the analytical level, thereby isolating the effect of discretization on the dispersion relation. With the presence of expressions of eigenvalues of the governing operator, the analysis is clear and accessible. The choices made to obtain the best approximation to the classical dispersion relation become completely transparent. We find that the truncated Gaussian family is the most effective compared to power and rational function families. Keywords Nonlocal wave equation · Dispersion relation · Peridynamics · Functional calculus · Local boundary condition
1 Introduction It is a challenge to choose the appropriate kernel function in nonlocal problems; this choice may be best motivated by using experimental data [14, 19] or phonon dispersion curves [15, 34]. However, in this study, our goal is to find the kernel function that provides the best approximation to the local wave propagation. We study the dispersion relation at the analytical level, i.e., without discretizing the governing operator, and devise a selection criterion to determine the kernel function that provides the best approximation to the local Burak Aksoylu
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Extended author information available on the last page of the article.
Journal of Peridynamics and Nonlocal Modeling
(linear) dispersion relation. Our selection criterion is devised by the parallelism observed between the dispersion curve plots and the magnitude of the coefficient of the quadratic term in the Taylor expansion; the closer the dispersion curve gets to linear relation, the smaller the magnitude of the coefficient becomes. In nonlocal problems, the kernel function C enters the formulation as an input. Since the theory used to construct the operators applies to kernels in L2 , any C ∈ L2 is admissible. The choice of appropriate kernel functions remains a challenging open problem in nonlocal theories [11, p. 2]. Any criterion used
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