Open AccessRayleigh's Principle in Finite Element Calculations of Seismic Wave Response
Author(s) -
Smith Warwick D.,
Bolt Bruce A.
Publication year - 1958
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1958.tb05348.x
Subject(s) - eigenfunction , eigenvalues and eigenvectors , rayleigh quotient , context (archaeology) , mathematics , order (exchange) , series (stratigraphy) , rayleigh scattering , moduli , mathematical analysis , physics , geology , quantum mechanics , paleontology , finance , economics
Summary Rayleigh's principle in the context of finite element modelling is shown to provide a powerful and convenient method for estimating the seismic eigenfrequencies of irregular Earth structures. It is not necessary to solve the eigenvalue problem completely, but instead to construct the elastic moduli and density matrices for the irregular structure, multiply them by an approximate eigenfunction vector, and form the Rayleigh quotient. The resulting error in frequency is of second order in the error of the eigenfunction. In order to conserve computer storage for large models the matrices need not be constructed and stored in their entirety, but multiplications can be accumulated one element at a time. Calculations for an inhomogeneous vibrating string and Rayleigh waves in a layered Earth model illustrate the technique.