Chapter 11
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: markov@mech.unn.ru
© 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
153
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: markov@mech.unn.ru
© 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
153