On the model reduction methods for dynamic analysis of large finite element models

Abstract

Recently, the demands for the dynamic response analysis of the large and complex structures have been in-creasing rapidly. Because the finite element (FE) models for those structures contain over several millions of degrees of freedoms (DOFs), considerable computing time would be required. This issue has inspired the needs for model reduction methods, and for decades, various model reduction methods have been developed to re-duce the computational cost with negligible accuracy loss. The model reduction methods can be categorized into two groups, mode based and DOFs based reduction methods. A representative mode based reduction method is component mode synthesis (CMS) method such as Craig-Bampton (CB) and automated multi-level substruc-turing (AMLS) methods, and a representative DOFs based reduction method are Guyan reduction and the im-proved reduced system (IRS) methods.

In model reduction methods, there are three critical issues remaining to be solved. One is how to evaluate the reliability of reduced models. To handle this, various error estimation methods have been proposed. In general, relative eigenvalue errors are used for the reliability evaluation of reduced models. However, basically, it is diffi-cult to estimate relative eigenvalue errors because the exact eigenvalues are unknown. Another is to develop a new reduction method suggesting the approximated eigensolutions more accurately than the previous methods. The other is how to effectively reduce the large FE models containing several millions DOFs without any limita-tions of computer memory and computational costs. In this thesis, these issues will be discussed, and as the solu-tions for those issues, we propose the error estimation methods to guarantee the solution reliability of the re-duced model, a new reduction method to improve the solution accuracy, and an efficient reduction method to improve the reduction efficiency.

For the first issue, we propose the simplified error estimation method developed for the CB method that is the most famous one in the CMS methods. The original formulation is simplified by neglecting insignificant terms, a new error estimator is obtained, and thus computational cost is significantly reduced with negligible accuracy loss. In addition, the contribution of a specific substructure to a relative eigenvalue error can be estimated using the new formulation, in which the estimated relative eigenvalue error is represented by a simple summation of the substructural errors estimated. Therefore, the new formulation can be employed for error control by using the detailed errors estimated for a certain substructure. Through various numerical examples, we verify the ac-curacy and computational efficiency of the new formulation, and demonstrate an error control strategy.

We also propose the error estimation method for the AMLS method. A new, enhanced transformation matrix for the AMLS method is derived from the original transformation matrix by properly considering the contribution of residual substructural modes. The enhanced transformation matrix is an important prerequisite to develop the error estimation method. Adopting the basic concept of the error estimation method developed for the CB method, an error estimation method is developed for the AMLS method.

For the second issue, to improve the solution accuracy, we propose an effective new CMS method based on the concept of the AMLS method. Herein, the original transformation matrix of the AMLS method is enhanced by considering the residual mode effect, and the resulting unknown eigenvalue in the formulation is approximated by employing the idea of the IRS method. Using the newly defined transformation matrix, we develop an en-hanced AMLS method by which original FE models can be more precisely approximated by reduced models, and their solution accuracy is significantly improved. The formulation details of the enhanced AMLS method is presented, and its accuracy and computational cost is investigated through numerical examples.

For the final issue, to improve the reduction efficiency, we develop a robust reduced-order modeling method, named algebraic dynamic condensation, which is based on the IRS method. Using algebraic substructuring, the global mass and stiffness matrices are divided into many small submatrices without considering the physical domain, and substructures and interface boundary are defined in the algebraic perspective. The reduced model is then constructed using three additional procedures: substructural stiffness condensation, interface boundary reduction, and substructural inertial effect condensation. The formulation of the reduced model is simply ex-pressed at a submatrix level without using a transformation matrix that induces huge computational cost. Through various numerical examples, the performance of the proposed method is demonstrated in terms of accuracy and computational cost.