Structure-Preserving Model Reduction for Dissipative Mechanical Systems

耗散机械系统的结构保持模型简化

• 批准号: 315077451
• 负责人: Professor Dr. Peter Benner
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/315077451?language=en
• 关键词: Structure; Preserving; Model; Reduction; Dissipative

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.

我们考虑耗散机械系统模型简化（MOR）的数值方法。在离散描述性弹性方程或使用有限元方法直接建模后，阻尼机械系统产生二阶微分方程系统。为了优化阻尼特性，外部阻尼器安装在机械结构的适当位置。对于 k 个阻尼器，可以使用它们的 k 个粘度和位置向量来参数化外部阻尼矩阵。为了抑制振动，编码可能振动的适当标准被最小化。外部阻尼矩阵的参数。这会产生（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。它们的收敛通常非常慢，例如用于 Nelder-Mead 方法或遗传算法。在这种方法的每次迭代中，都会评估最小化标准——通常是非常昂贵的计算。例如，当最小化系统所含的总能量时，这需要确定系统对应的李亚普诺夫方程的解的迹。这本身就是一项艰巨的计算，因此需要 MOR 来有效计算解决阻尼优化问题。第一位 PI (PB) 使用基于模态分析的简单方法对此进行了多篇文章的研究。然而，这些方法并不能保证在降阶系统中保留耗散性。因此，有必要为耗散机械系统构建新颖的结构保持MOR方法。或者，当使用标准 MOR 技术时，需要消散给定降阶系统的方法。这暗示了该提案的目标：我们的目标是开发基于最小扰动的机械系统消钝方法。 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属性。这些想法将扩展到二阶系统，以完成耗散的新任务。此外，还将开发用于耗散机械系统的新的结构保持 MOR 方法。为此，我们建议采用 4 种不同的方法，基于 PB 和 2nd PI (TR) 先前对二阶系统的 MOR 所做的工作，以及最近在博士论文中开发的耗散描述符系统的新微积分。 MV 论文：使用 Lure 方程进行平衡截断；制定为港哈密尔顿系统并为此开发新的 MOR 方法；通过将一阶系统的 MOR 方法应用于机械系统而获得降阶模型的二阶化； MOR 用于受限机械系统。所有方法均需针对真实世界的结构进行验证、测试和比较。