A new algorithm for solving large eigenproblems is presented, which is to increase efficiency of computation and is more adapted to the parallel computing environments.
In this study substructuring technique is used for modelling a structure, which has parallelism in itself. The whole domain is divided into several substructures and each substructure is modelled independently with other substructures. The interface degrees of freedom are selected as masters and the internal slave degrees of freedom are eliminated by condensation. Unlike the conventional approaches the exact condensation form retaining the nonlinear term is used to increase the accuracy. Therefore the reduced eigenproblem becomes nonlinear with respect to eigenvalues of the system and is difficult to solve by conventional solution techniques.
Homotopy continuation method is proposed to solve the reduced nonlinear eigenproblem. Because homotopy method is globally convergent and each eigenpath can be followed separately, it is suitable for parallel processing. Guyan reduction form is used as an initial problem and convex homotopy is constructed. To follow an eigenpath a predictor-corrector method is used. In prediction Nelson's method is used to calculate the derivatives of eigenvalue and eigenvector. The Rayleigh quotient iteration is used as a corrector. Because the valid range of homotopy method is below the cut-off frequency, some additional masters are selected automatically to increase the cut-off frequency. If the initial problem has multiple roots the subspace iteration method is used on the subspace of eigenvalue cluster.
Some simple 2D plane problems are solved to check the validity and accuracy of the algorithm. All the eigenvalues below given frequency are converged and the accuracy is improved considerably. To increase the accuracy and the speed of convergency, it is needed making the interface DOFs describing the modes of whole structure modes more precisely. Also the number of modes used in condensation may be increased.
Parallel application of the algorithm is performed on EWS network using PVM[2]. Some 2D problems are solved to validate the efficiency of the algorithm. In case of 1 processor more time is spent than existing sequential algorithm. As using more processors the wall clock time is reduced than the sequential one. The efficiency of parallel processing is fairly good.
The efficiency depends mainly on the load balancing between processors. If the load balancing is not adjusted well, some processors are idle and the performance becomes worse. In this case the efficiency is degraded severely.
In summary the proposed method of eigenvalue analysis is accurate and competitive in parallel computational environments.
