In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

Gulizzi, V., Benedetti, I., Milazzo, A. (2017). Spherical harmonics expansion of fundamental solutions and their derivatives for homogeneous elliptic operators. JOURNAL OF MULTISCALE MODELLING, 8(3-4) [10.1142/S1756973717400066].

Spherical harmonics expansion of fundamental solutions and their derivatives for homogeneous elliptic operators

Vincenzo Gulizzi;Ivano Benedetti;Alberto Milazzo
2017-01-01

Abstract

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.
2017
Gulizzi, V., Benedetti, I., Milazzo, A. (2017). Spherical harmonics expansion of fundamental solutions and their derivatives for homogeneous elliptic operators. JOURNAL OF MULTISCALE MODELLING, 8(3-4) [10.1142/S1756973717400066].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/246509
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