Convergence Acceleration of Iterative Modal Reduction Methods

Abstract

An accelerated method is presented for the iterative condensation of eigenproblems. The present study was motivated by the improved reduction system and the succession-level approximate reduction (SAR). The reduction procedures are supplemented with the second-order approximation in the series expansion of the system transformation. The reduced equation of an equivalent system and the transformation matrix are updated in an iterative manner.In addition,systematicderivation andcomparisonoftheequationsinvolvedin variouscondensationshave been sought.Thematrixupdateincorporatesnotonlyinverseiteration butalsosubspacetransformation implicitly. Theseriesexpansion canbeconsideredasrepeatedupdatesofthetransformationmatrixthroughinverseiteration. The solution accuracy is sensitive to the selection of the degrees of freedom, for which sequential elimination or energy method may be preferable. When a poor selection causes a sudden failure of the updatemethod, the hybrid dynamic condensation can be used. The method of SAR is closely related to the hybrid dynamic condensation. Nomenclature [As] = [kss] i 1 [mss] in Eq. (10) [Bpp],[Bsp] = submatrices for inertia force in Eq. (68) [DG] = [MG] i 1 [KG] dynamic matrix in Eq. (13) [Ke], [Me] = equivalent structural matrices in Eq. (26) [KG],[MG] = reduced matrices in Guyan’ s condensation [Kr], [Mr] = structural matrices reduced through [Tr] [KX], [MX] = structural matrices reduced through [TX] [k], [ka b ] = stiffness matrix and submatrices [m], [ma b ] = mass matrix and submatrices p = number of primary degrees of freedom [Q pp] = modal matrix for generalized coordinates {R} = residual error in eigenproblem s = number of secondary degrees of freedom [T] = exact transformation matrix in Eq. (22) [Ti] = matrices in series of Eqs. (15) and (18) [Tp] = system transformation matrix [Tr] = approximate transformation matrix [TX] = improved transformation matrix in Eq. (40) [T0] = transformation in Guyan’ s condensation [T(k )] = frequency-dependent transformation in Eq. (3) [X p] = improved modal matrix [X pp],[Xsp] = submatrices of [X p] e = convergence tolerance [K p]