Structure-Preserving Model Reduction for Dissipative Mechanical Systems

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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.