Certified Model Order Reduction for Coupled Mechanical Systems

• Applicants: Professor Dr.-Ing. Jörg Fehr; Professor Dr. Bernard Haasdonk
• Subject Area: Mechanics
• Term: from 2016 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 314733389

• Final Report Year: 2023

Final Report Abstract

Modeling, simulation, and optimization is nowadays an integral part of the development and life cycle of complex technical systems. Such systems typically consist of multiple coupled domains and different coupled physical processes. The simulation of one physical domain can already be computationally challenging, e.g. using the well known FE method, where the models can have more than 107 degrees of freedom (DOF), based on the meshing of 3D-data from design or CT-scans. However, for correct prediction, optimization, and control of complex systems the different physical simulation domains and their respective substructures need to be connected with each other. This leads to even higher computational challenges and requires fundamental improvements in simulation performance. Frequently, this combination is only possible by using advanced and modern model order reduction (MOR) techniques, where the large scale system is approximated with a system of much smaller dimension. Those reduced models can then be simulated efficiently in multi-query and/or real-time scenarios or be used in model-based control applications. The gain in simulation efficiency often comes at the cost of the lower fidelity of the reduced model. It is essential to certify the reduction method by estimating or rigorously bounding the incurred error to use the simulation for management decisions and engineering applications. In particular, error estimation is important for further acceptance and usage of MOR to speed up simulations and to allow the performance optimization of technical systems. Additionally, error estimation is a crucial enabler for adaptive simulation solutions. To ensure a higher quality in the reduced models, rewriting the system as a port Hamiltonian (pH) system and preserving this structure in the reduced model were investigated. This can be applied to multi-physics scenarios and, in particular, techniques were developed using the model of a classical guitar as an acousto-structural problem and a disc brake as a thermo-mechanical system. Error quantification was performed by further developing techniques such as hierarchical estimators and auxiliary linear problem (ALP) based error estimates. These were then used for further improvement of the reduced basis building process by means of a greedy technique. This can be time intensive and thus further basis generating techniques relying on random sampling were developed to further reduce the time required to obtain a reduced basis/model. Both state approximation and error estimation were extended to output approximation and output error estimation. All computations were performed using the software package CCMOR which was extended to include the mentioned structure preserving techniques.

Publications

• Hybrid Digital Twins Using FMUs to Increase the Validity and Domain of Virtual Commissioning Simulations. ARENA2036, 200-209. Springer International Publishing.
  Pfeifer, Denis; Baumann, Andreas; Giani, Marco; Scheifele, Christian & Fehr, Jörg
  
• Port-Hamiltonian fluid–structure interaction modelling and structure-preserving model order reduction of a classical guitar. Mathematical and Computer Modelling of Dynamical Systems, 29(1), 116-148.
  Rettberg, Johannes; Wittwar, Dominik; Buchfink, Patrick; Brauchler, Alexander; Ziegler, Pascal; Fehr, Jörg & Haasdonk, Bernard
  
• Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds. Comptes Rendus. Mathématique, 362(G13), 1881-1891.
  Buchfink, Patrick; Glas, Silke & Haasdonk, Bernard
  
• Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems. Advances in Computational Mathematics, 50(1).
  Herkert, Robin; Buchfink, Patrick & Haasdonk, Bernard
  
• Improved a posteriori error bounds for reduced port-Hamiltonian systems. Advances in Computational Mathematics, 50(5).
  Rettberg, Johannes; Wittwar, Dominik; Buchfink, Patrick; Herkert, Robin; Fehr, Jörg & Haasdonk, Bernard
  
• Model reduction on manifolds: A differential geometric framework. Physica D: Nonlinear Phenomena, 468, 134299.
  Buchfink, Patrick; Glas, Silke; Haasdonk, Bernard & Unger, Benjamin
  
• Randomized Symplectic Model Order Reduction for Hamiltonian Systems. Lecture Notes in Computer Science, 99-107. Springer Nature Switzerland.
  Herkert, R.; Buchfink, P.; Haasdonk, B.; Rettberg, J. & Fehr, J.