This repository consists of a Python implementation of a frequency domain Reduced Order Model (ROM) for second-order systems with frequency dependent system matrices, called Frequency dependent second-order Rational Krylov (FSRK) [jith2018].

It can be applied to systems of the form

-\omega^2 M(\omega) x + i \omega C(\omega) x + K(\omega) x = f,

where M, C, K are frequency dependent mass, damping and stiffness matrices.

FSRK ROM is implemented as a class in the file fsrk.py. The current implementation was made for systems with non-viscous damping. Therefore, it only considers the damping matrix to be frequency dependent. It can be easily extended if needed (feel free to open up an issue).

The usage of the ROM has been demonstrated in main.py. A toy problem consisting of serially connected spring-mass-damper units (of order 500) is reduced using the FSRK ROM. The performance of the ROM is compared with the existing modal projection based multi-model ROM (multimodel.py) [balmes1997].

References

[jith2018]

Jith J. and Sarkar S. A Frequency Domain Model Order Reduction Technique for Second-Order Systems with Nonlinear Frequency Dependent Damping. Submitted to Computer methods in Applied Mechanics and Engineering, 2018.

[balmes1997]

Balmès E. Model Reduction for Systems with Frequency Dependent Damping Properties. Proceedings of the 15th International Modal Analysis Conference, vol. 3089, 1997, p. 223.