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Weak forms for modelling of rotationally symmetric, multilayered structures, including anisotropic poro‐elastic media
Author(s) -
Östberg M.,
Göransson P.,
Hörlin NE.,
Kari L.
Publication year - 2012
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.3354
Subject(s) - curvilinear coordinates , cartesian coordinate system , cylindrical coordinate system , coordinate system , mathematical analysis , orthogonal coordinates , finite element method , basis function , biot number , fourier series , anisotropy , geometry , spherical coordinate system , boundary value problem , orthogonal basis , mathematics , barycentric coordinate system , physics , mechanics , optics , quantum mechanics , thermodynamics
SUMMARY A weak form of the anisotropic Biot's equation represented in a cylindrical coordinate system using a spatial Fourier expansion in the circumferential direction is presented. The original three dimensional Cartesian anisotropic weak formulation is rewritten in an arbitrary orthogonal curvilinear basis. Introducing a cylindrical coordinate system and expanding the circumferential wave propagation in terms of orthogonal harmonic functions, the original, geometrically rotationally symmetric three dimensional boundary value problem, is decomposed into independent two‐dimensional problems, one for each harmonic function. Using a minimum number of dependent variables, pore pressure and frame displacement, a computationally efficient procedure for vibro‐acoustic finite element modelling of rotationally symmetric three‐dimensional multilayered structures including anisotropic porous elastic materials is thus obtained. By numerical simulations, this method is compared with, and the correctness is verified against, a full three‐dimensional Cartesian coordinate system finite element model. Copyright © 2012 John Wiley & Sons, Ltd.