Numerical Evaluation of Integrals in Laplace Domain Anisotropic Elastic Fundamental Solutions for High Frequencies
In this paper, we address a problem of computing the integrals appearing in integral expressions of Laplace domain anisotropic elastic displacement fundamental solutions. The essence of the problem is that these integrals can become highly oscillatory for
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Numerical Evaluation of Integrals in Laplace Domain Anisotropic Elastic Fundamental Solutions for High Frequencies Ivan P. Markov and Marina V. Markina Abstract In this paper, we address a problem of computing the integrals appearing in integral expressions of Laplace domain anisotropic elastic displacement fundamental solutions. The essence of the problem is that these integrals can become highly oscillatory for high values of frequency or large distance between source and observation points. The modified integral expressions for displacement fundamental solutions and their first derivative are given. We propose a procedure based on the quadrature rule developed by Evans and Webster for the evaluation of rapidly oscillatory integrals. For a triclinic anisotropic elastic material, we consider an illustrative numerical example which involves phase functions with stationary points. Keywords Anisotropic elasticity · Laplace transform · Fundamental solutions · Oscillatory integrals · Evans-Webster quadrature rule · Boundary element method
11.1 Introduction In modern science, computational work is an important complement to both experiments and theory, and nowadays, a vast majority of both experimental and theoretical papers involve some numerical calculations, simulations, or computer modeling. One of the most interesting subjects in modern continuum mechanics and engineering worth mentioning is the development of newly (scientifically) conceived materials (‘metamaterials’) with mechanical properties that cannot be found in nature (Del Vescovo and Giorgio 2014; Barchiesi et al. 2018). These (macroscopic) properties are mainly determined by the micro- or nanostructure of the considered metamaterial rather than by the chemical and physical properties of the materials constituting it at the microscopic level. Designing of such metamaterials is based on high gradient continuum approaches (Alibert et al. 2003; Sciarra et al. 2007; dell’Isola et al. 2012, I. P. Markov (B) · M. V. Markina Research Institute for Mechanics, National Research Lobachevsky State University of Nizhny Novgorod, 23, Bldg. 6, Prospekt Gagarina (Gagarin Avenue), Nizhny Novgorod 603950, Russian Federation e-mail: [email protected] © Springer Nature Switzerland AG 2021 F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability in Multiscale Mechanics, Advanced Structured Materials 137, https://doi.org/10.1007/978-3-030-53755-5_11
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2016a, b, c; Auffray et al. 2013; dell’Isola et al. 2015; Rahali et al. 2015). Pantographic structures are an example of the mechanical metamaterials (Placidi et al. 2016; dell’Isola et al. 2019a, b, c). Identification of constitutive parameters and validation of theoretical predictions of built models are only a few subjects where numerical simulations are used on everyday basis (dell’Isola et al. 2016a, b, c, 2017; Placidi et al. 2017). Green’s functions or fundamental solutions are extensively used in the solution of boundary value problems, and they have
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