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.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

abstract: The canon of music performed in recitals by American trombonists contains very few works for trombone by composers from Russia and the Soviet Union. Trombonists in the United States periodically perform trombone solos by Nikolai Rimsky-Korsakov, Alexei Lebedev, Vladislav Blazhevich, Gregory Kalinkovich, Alexander Tcherepnin, and Eugene Reiche. But these works represent a very small percentage of trombone solos performed in recitals in the United States, and compositions written after 1960 by composers in the U.S.S.R. are completely absent from recital programs. The purpose of this project is to identify several Soviet-era compositions for trombonists that are worthy of introduction into trombone recital programs in the West. To support the thesis that Soviet-Russian trombone music has been disproportionally under-represented in American recital programs, a survey of over 3300 trombone recitals given in the United States from 1972 to 2013 was conducted. Once a body of significant works that had previously not been performed on American trombone recitals was identified, they were acquired, analyzed, and several were performed. The following compositions represent a list of Soviet-Russian solos not programmed on any of the 3300 recitals: German Grigoryevich Okunev, Adagio and Scherzo; Gregory Markovich Kalinkovich, Concertino for Trombone; Pavel Davidovich Saliman-Vladimirov, Concertino for Trombone; Vadim Veniaminovich Kulyov, Concertino for Trombone; Vladislav Alexanderovich Uspensky, Concertino for Trombone and Orchestra; Sergei Vasilyev and Vladimir Robertovich Enke, Concerto for Trombone and Orchestra; Sergei Borisovich Chebotaryov, Rondo for Tuba; Victor Nikolaevich Smirnov, Scherzo; Alfred Garievich Schnittke,“Schall und Hall”;and Tatyana Alexseyevna Chudova, Sonata for Trombone. / Dissertation/Thesis / Doctoral Dissertation Music 2015

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