Efficient dynamic reanalysis of structures

The combined approximations approach, developed originally for linear static reanalysis, is used for dynamic reanalysis of structures. The approach is based on the integration of several concepts and methods, including series expansion, reduced basis, matrix factorization, and Gram-Schmidt orthogonalizations. The advantage is that efficient local approximations and accurate global approximations are combined to achieve an effective solution procedure. Considering modal analysis, the computational effort involved in reanalysis is significantly reduced. However, the accuracy of the higher mode shapes might be insufficient. A procedure intended to improve the accuracy of the results is developed. A reduced eigenproblem is introduced, and a method for determining the basis vectors is presented. Improved basis vectors, using the concepts of shifts and Gram-Schmidt orthogonalizations, are developed, and eigenproblem reanalysis by combined approximations using inverse iteration with shifts is introduced. Numerical examples demonstrate the quality of the results. It is shown that high accuracy is achieved efficiently for various types of changes in the structure.