Structure-Preserving Model Reduction for Dissipative Mechanical Systems

我们考虑耗散力学系统模型降阶的数值方法。将描述的弹性力学方程离散化或用有限元方法直接建模后，阻尼力系统可转化为二阶微分方程组。为了优化减振性能，在机械结构的适当位置安装了外部阻尼器。对于k个阻尼器，可以利用其k个粘性系数和位置向量来参数化外部阻尼器矩阵。为了抑制振动，对可能的振动进行编码的适当标准被最小化。外部阻尼阵的参数。这就产生了一个(通常)非凸优化问题，可以使用全局优化的启发式搜索算法来解决。它们的收敛速度往往很慢，例如对于内尔德-米德方法或遗传算法。在这种方法的每一次迭代中，都要评估最小化标准--这通常是一种非常昂贵的计算。例如，当最小化系统中包含的总能量时，这需要确定与该系统对应的Lyapunov方程的解的迹。这本身就是一个令人敬畏的计算，因此需要MOR来有效地计算求解阻尼优化问题。在第一个PI(PB)的几篇文章中，使用基于模态分析的简单方法对此进行了研究。然而，这些方法并不能保证在降阶系统中保持耗散性。因此，有必要为耗散力学系统构造新的保结构静力方法。或者，当使用标准的MOR技术时，需要对给定的降阶系统进行耗散钝化。这意味着这项提议的目标：我们的目标是开发基于最小扰动的机械系统耗散激活的方法。PB和第三PI(MV)已经在不同的背景下研究了使用最小扰动来施加动力系统的系统性质。这些想法将被推广到二阶系统，以完成耗散活化的新任务。此外，对于耗散力学系统，新的保结构静力方法正在被开发。为此，我们提出了四种不同的方法，基于PB和二阶PI(Tr)以前对二阶系统的MOR所做的工作，以及最近在MV的博士论文中发展的耗散广义系统的新的计算：使用Lure方程的平衡截断；将其表示为端口-哈密顿系统并发展新的MOR方法；将一阶系统的MOR方法应用于机械系统而得到的降阶模型的二阶化；对约束力学系统的MOR。所有方法都要经过验证、测试，并与真实世界的结构进行比较。

We consider numerical methods for model reduction (MOR) of dissipative mechanical systems. After discretizing the descriptive elasticity equations or direct modeling using the finite element method, damped mechanical systems lead to systems of 2nd order differential equations. For optimized damping properties, external dampers are attached to the mechanical structure at appropriate positions. For k dampers, the external damping matrix can be parameterized using their k viscosities and position vectors. In order to suppress vibrations, an appropriate criterion encoding the possible vibrations is minimized w.r.t. the parameters of the external damping matrix. This yields a (usually) nonconvex optimization problem that can be solved using heuristic search algorithms of global optimization. Their convergence is more often than not very slow, e.g. for the Nelder-Mead method or genetic algorithms. In each iteration of such a method, the minimization criterion is to be evaluated - a usually very expensive computation. For example, when minimizing the total energy contained in the system, this requires determining the trace of the solution of the Lyapunov equation corresponding to the system. This by itself is a formidable computation so that MOR is required for efficient computational solution of the damping optimization problem. This has been investigated in several articles by the first PI (PB), using simple approaches based on modal analysis. Nevertheless, these methods do not guarantee preservation of dissipativity in the reduced-order system. Therefore, it is necessary to construct novel structure-preserving MOR methods for dissipative mechanical systems. Alternatively, methods for dissipassivation of a given reduced-order system are required when standard MOR techniques are used. This implies the goals of this proposal: We aim at developing methods for the dissipassivation of mechanical systems based on minimal perturbations. Imposing system properties of dynamical systems using minimal perturbations has already been investigated by PB and the 3rd PI (MV) in a different context. These ideas are to be extended to 2nd order systems for the new task of dissipassivation. Moreover, new structure-preserving MOR methods for dissipative mechanical systems are to be developed. For this, we suggest 4 different approaches, based on previous work for MOR of 2nd order systems by PB and the 2nd PI (TR), and on the new calculus for dissipative descriptor systems recently developed in the Ph.D. thesis of MV: balanced truncation using the Lure equations; formulation as port-Hamiltonian system and developing new MOR methods for these; 2nd-orderization of reduced-order models obtained by applying MOR methods for 1st order systems to mechanical systems; MOR for constrained mechanical systems. All methods are to be validated, tested and compared for real-world structures.