Determinant Search Method for the Large Structural Systems With Small Bandwidth

Abstract In a solution to large eigenvalue problems, the number of required eigenvalues and corresponding vectors is usually much smaller than the order of matrices. For small bandwidth problems, the determinant search method is very efficient and the oldest. Its from a family of nonlinear Eigen solution techniques also referred to as frequency scanning methods. By using the shift technique, it can also calculate the largest eigenvalues. This calculation process involves three steps: polynomial iteration, inverse iteration, and polynomial deflation. The polynomial iteration determines the eigenvalues of interest by looking at a plot of characteristic polynomial versus the eigenvalue λ, which involves evaluating the determinant |K–λM| at fine intervals of λ selected from accelerated secant iteration. The method was developed in the 1970s and was extensively used for only small-size problems due to its limitations. During that time, the memory available was very low for the in-core solver. Also, the numbers generated in polynomial iteration overflows the float point variables in a computer program. Of course, scaling was the obvious choice. However, the method still suffered the numerical scaling difficulties as a determinant of the matrix |K–λM| in structural problems is generally a very fast varying function. Despite different scaling, the number gets overflows after dozens of eigenvalues which made the technique unattractive. The variables type float, double and long double in a computer program can store values up to 1049, 10308, and 104932, respectively, which is not comparatively much smaller than the determinant of small size structural models. In the present paper, a power number representation is proposed in the polynomial iteration. The determinant |K–λM| can be stored in the power form ρD¯n, where prefactor ρ stores the determinant sign, n is the size of the matrix, and D¯ can be calculated and stored accordingly. The order analysis shows the storage of D¯ requires a variable of range 108 to 1010, which can be stored by float or double type variable. The determinant search method gets to be free from limitations and may easily estimate each large eigenpair at higher frequencies independently from all those previously calculated. The determinant search method can also estimate the clustered eigenvalues accurately. The models’ practical significance is in the FEA analysis of beams, frames, buckling analysis, and stress analysis of pipes in the industry. In the present work, the proposed approach has been applied to the piping model. The results show the determinant search algorithm can extract modes in a higher frequency range without any limitations. The modal analysis is the basis of dynamic analyses of structures such as Response spectrum analysis. The inclusion of higher modes ensures that more than 95% of modal mass has been included in the calculation.