A Component-Based Parametric Reduced-Order Modeling Method Combined with Substructural Matrix Interpolation and Automatic Sampling

Ying Liu,Yun Li,Huanyu Du,Hongguang Li

doi:10.1155/2019/6407437

Abstract

An efficient parametric reduced-order modeling method combined with substructural matrix interpolation and automatic sampling procedure is proposed. This approach is based on the fixed-interface Craig-Bampton component mode synthesis method (CMS). The novel parametric reduced-order models (PROMs) are developed by interpolating substructural reduced-order matrices. To guarantee the compatibility of the coordinates, we develop a three-step adjustment procedure by reducing the local interface degrees of freedom (DOFs) and performing congruence transformation for the normal modes and interface reduced basis, respectively. In addition, an automatic sampling process is also introduced to dynamically fulfill the predefined error limits. It proceeds by first exploring the parameter space and identifying the sampling points with maximum error indicators for all the parameter-dependent substructures. The exact error of the assembled model at the optimal parameter point is subsequently calculated to determine whether the automatic sampling procedure reaches a desired error tolerance. The proposed framework is then applied to the moving coil of electrical-dynamic shaker to illustrate the advantage and validity. The results indicate that this new approach can significantly reduce both the offline database construction time and online calculation time. Besides, the automatic procedure can sample the parameter space efficiently and fulfill the stopping criterion dynamically with assurance of the resulting PROM accuracy.

Highlights

Reduced-Order Modeling with CB-component mode synthesis method (CMS) MethodThe FE equations of motion can be written in a matrix form as
There is no guaranty that the parameter space can be sampled effectively and that the parametric reduced-order models (PROMs) can satisfy the predefined error limits. erefore, we propose an automatic sampling procedure for the component-based parametric reduced-order modeling method
The automatic sampling method can sample the parameter space according to the error indicator and terminate the procedure automatically when the maximum actual error becomes smaller than a speci ed tolerance. is method can place the sampling points where the error reaches the maximum. erefore the approach can provide a better performance with smaller relative errors and cheaper o ine and online computational cost

Summary

Reduced-Order Modeling with CB-CMS Method

The FE equations of motion can be written in a matrix form as. By assuming no external force on the interior DOFs, i.e., FOi 0, the following constraint modes can be derived through the upper part in equation (4): ΨOi A −􏼐KOi O􏼑−1KOi A. en, we can obtain the reduced stiffness and mass matrices of the ith substructure by multiplying the transformation matrix to the original matrices: M􏽥 i. Where the structural damping is assumed in the form C􏽥 αM􏽥 + βK􏽥 , the force vector F􏽥 is generated based on the interface DOFs, and the assembled mass and stiffness matrices M􏽥 and K􏽥 are written as. The entire calculation flow of fixed-interface CB-CMS method can be summarized as: partition the entire structure, generate the reduced-order substructures, synthesize the components, and calculate the reduced responses

Proposed Parametric Reduced-Order Modeling Method

Online Calculation Using the Reduced-Order Component

Numerical Example

Findings

Conclusions