# Numerical Implementation of Elastodynamic Green's Function for Anisotropic Media

http://hdl.handle.net/10150/623144
Title:
Numerical Implementation of Elastodynamic Green's Function for Anisotropic Media
Author:
Issue Date:
2016
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Displacement Green's function is the building block for some semi-analytical methods like Boundary Element Method (BEM), Distributed Point Source Method (DPCM), etc. In this thesis, the displacement Green`s function in anisotropic media due to a time harmonic point force is studied. Unlike the isotropic media, the Green's function in anisotropic media does not have a closed form solution. The dynamic Green's function for an anisotropic medium can be written as a summation of singular and non-singular or regular parts. The singular part, being similar to the result of static Green's function, is in the form of an integral over an oblique circular path in 3D. This integral can be evaluated either by a numerical integration technique or can be converted to a summation of algebraic terms via the calculus of residue. The other part, which is the regular part, is in the form of an integral over the surface of a unit sphere. This integral needs to be evaluated numerically and its evaluation is considerably more time consuming than the singular part. Obtaining dynamic Green's function and its spatial derivatives involves calculation of these two types of integrals. The spatial derivatives of Green's function are important in calculating quantities like stress and stain tensors. The contribution of this thesis can be divided into two parts. In the first part, different integration techniques including Gauss Quadrature, Simpson's, Chebyshev, and Lebedev integration techniques are tried out and compared for evaluation of dynamic Green’s function. In addition the solution from the residue theorem is included for the singular part. The accuracy and performance of numerical implementation is studied in detail via different numerical examples. Convergence plots are used to analyze the numerical error for both Green's function and its derivatives. The second part of contribution of this thesis relates to the mathematical derivations. As mentioned above, the regular part of dynamic Green's function, being an integral over the surface of a unit sphere, is responsible for the majority of computational time. From symmetry properties, this integration domain can be reduced to a hemisphere, but no more simplification seems to be possible for a general anisotropic medium. In this thesis, the integration domain for regular part is further reduced to a quarter of a sphere for the particular case of transversely isotropic material. This reduction proposed for the first time in this thesis nearly halves the number of integration points for the evaluation of regular part of dynamic Green's function. It significantly reduces the computational time.
Type:
text; Electronic Thesis
Keywords:
Elastodynamic Green’s function; Anisotropic medium; Time-harmonic solutio; Residue method; Chebyshev integration; Lebedev integration
Degree Name:
M.S.
Degree Level:
masters
Degree Program:
Degree Grantor:
University of Arizona
Kundu, Tribikram

DC FieldValue Language
dc.language.isoen_USen
dc.titleNumerical Implementation of Elastodynamic Green's Function for Anisotropic Mediaen_US
dc.date.issued2016-
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.description.abstractDisplacement Green's function is the building block for some semi-analytical methods like Boundary Element Method (BEM), Distributed Point Source Method (DPCM), etc. In this thesis, the displacement Green`s function in anisotropic media due to a time harmonic point force is studied. Unlike the isotropic media, the Green's function in anisotropic media does not have a closed form solution. The dynamic Green's function for an anisotropic medium can be written as a summation of singular and non-singular or regular parts. The singular part, being similar to the result of static Green's function, is in the form of an integral over an oblique circular path in 3D. This integral can be evaluated either by a numerical integration technique or can be converted to a summation of algebraic terms via the calculus of residue. The other part, which is the regular part, is in the form of an integral over the surface of a unit sphere. This integral needs to be evaluated numerically and its evaluation is considerably more time consuming than the singular part. Obtaining dynamic Green's function and its spatial derivatives involves calculation of these two types of integrals. The spatial derivatives of Green's function are important in calculating quantities like stress and stain tensors. The contribution of this thesis can be divided into two parts. In the first part, different integration techniques including Gauss Quadrature, Simpson's, Chebyshev, and Lebedev integration techniques are tried out and compared for evaluation of dynamic Green’s function. In addition the solution from the residue theorem is included for the singular part. The accuracy and performance of numerical implementation is studied in detail via different numerical examples. Convergence plots are used to analyze the numerical error for both Green's function and its derivatives. The second part of contribution of this thesis relates to the mathematical derivations. As mentioned above, the regular part of dynamic Green's function, being an integral over the surface of a unit sphere, is responsible for the majority of computational time. From symmetry properties, this integration domain can be reduced to a hemisphere, but no more simplification seems to be possible for a general anisotropic medium. In this thesis, the integration domain for regular part is further reduced to a quarter of a sphere for the particular case of transversely isotropic material. This reduction proposed for the first time in this thesis nearly halves the number of integration points for the evaluation of regular part of dynamic Green's function. It significantly reduces the computational time.en
dc.typetexten
dc.typeElectronic Thesisen
dc.subjectElastodynamic Green’s functionen
dc.subjectAnisotropic mediumen
dc.subjectTime-harmonic solutioen
dc.subjectResidue methoden
dc.subjectChebyshev integrationen
dc.subjectLebedev integrationen
thesis.degree.nameM.S.en
thesis.degree.levelmastersen
thesis.degree.disciplineMechanical Engineeringen
thesis.degree.grantorUniversity of Arizonaen