Adaptive model reduction strategies for fast prediction and optimization of the vibro-acoustic performance of dynamical systems with damping

Over the past years, the dynamic and acoustic qualities of products have become an increasingly important design criterion in many industrial sectors. Moreover, in order to improve the cost-efficiency of products, lightweight designs are emerging. However, their high stiffness-to-mass ratio can often be the cause of noise and vibration problems. Therefore, it is customary to use complex damping materials such as viscoelastic and porous materials in the passive control of structural vibration and noise radiation. The underlying physical mechanisms behind these kinds of damping materials are typically quite complex and highly frequency-dependent, which requires efficient constitutive models able to simply approximate the mechanisms. 
 
Numerical methods are often applied to approximately solve the resulting dynamic equations of complex systems. As the most commonly used predictive tool, the finite element (FE) method is often selected in the design phase to obtain detailed information on the performance of complex vibro-acoustic systems with damping treatments. Its use, however, inevitably leads to very large, complex-valued and frequency-dependent numerical models, which makes original full-order model (FOM) evaluation intractable due to time and memory limitations. Furthermore, the vibro-acoustic optimization problems based on FE model updating require frequently iterative prediction of responses of the large-scale FOM before an optimized solution is arrived, which further increases the computational complexity.

In order to ease these problems, model order reduction (MOR) techniques have become indispensable. Most of them, however, do not handle complex-valued and especially frequency-dependent system matrices well. Due to frequency dependencies of the damping material properties, their FE equations of motion are not of a standard second-order form as that for regular elastic FE models. Another practical problem with the use of MOR is the adaptive determination of the dimension of the reduced-order model (ROM) to avoid different trial-errors tests with the FOM for validation. 

In this dissertation, an adaptive MOR strategy is proposed to reduce the number of degrees of freedom (DOF) involved such that the required computational cost can be largely alleviated while a desired high accuracy can be achieved in a controllable way. This technique consists of making use of Taylor's theorem to the frequency-dependent scalar function(s) coming from the complex material behavior, and then performing the structure-preserving second-order Arnoldi algorithm to solve the underlying FE model in the frequency domain. In support, a relative error indicator is developed to iteratively enrich the reduced model and determine its final order. It should be mentioned that the proposed adaptive MOR process can also be used for standard second-order vibro-acoustic dynamic systems with Rayleigh damping.

In addition to rapid prediction of dynamical responses of vibro-acoustic FE systems with damping, the availability of the proposed MOR technique can be exploited in the context of optimization problems with a multi-parameter space. 

In order to theoretically analyze and design structural systems, the material parameters in the mathematical model of viscoelastic materials need to be known, which can be derived through an inverse identification process. Especially to speed up the exploration of the parametric space, a parametric model order reduction (pMOR) technique is used, where a sampling strategy is introduced. The local orthonormal bases around the selected sampling points are obtained with the proposed adaptive reduction algorithm. A global orthonormal basis can then be constructed by non-weighted singular value decomposition on all local bases. Since the parameter- and frequency dependency can be suitably preserved, the generated single ROM in conjunction with optimization algorithms is very useful to fast identify the material parameters of viscoelastic damping.

In order to effectively enhance damping properties under economical- and practical constraints, both the location and the geometry of viscoelastic patches should be optimized. Therefore, the proposed MOR technique is embedded in the explicit moving morphable component (MMC) topology optimization framework to seek the optimal layouts of damping patches under a prescribed area constraint. With the MMC to reduce the number of design variables in the topology formulation and the MOR to reduce the number of DOFs in the FE model, the optimization simulation can be largely sped up.