SOME ISSUES IN THE GENERALIZED NONLINEAR EIGENVALUE ANALYSIS OF TIME-DEPENDENT PROBLEMS IN THE SIMPLIFIED BOUNDARY ELEMENT METHOD
DOI:
https://doi.org/10.26512/ripe.v2i7.21716Keywords:
Boundary elements. Time-dependent problems. Generalized modal analysis. Quasi-symmetric problems. Deflation method.Abstract
The third author and collaborators have combined and extended Pian’s hybrid finite element formulation and Przemieniecki’s suggestion of displacement-based, frequencydependent elements to arrive at a hybrid boundary element method for the general modal analysis of transient problems. Starting from a frequency-domain formulation, it has been shown that there is an underlying symmetric, nonlinear eigenvalue problem related to the lambda-matrices of a free-vibration analysis, with an effective stiffness matrix expressed as the frequency power series of generalized stiffness and mass matrices. Although the formulation is undeniably advantageous in the analysis of framed structures, for which all coefficient matrices can be analytically obtained, its practical application as a general finite/boundary element analysis tool is questionable. In fact, dealing with large-scale problems calls for simplifications to speed up the numerical evaluations, which unavoidably occur at the cost of the symmetry ”“ or just positive-definitiveness ”“ of the involved matrices. These issues deserve a closer theoretical investigation both in terms of applicability of the method and of the further generalization of the underlying eigenvalue problem, whose efficient solution seems to demand the use of advanced eigenvalue-deflation techniques, among other manipulation possibilities. This is the subject of the present paper, which also includes some illustrative numerical examples.
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