With respect to the effects of ports, dissipation, and non-linear parametrizations, we develop higher order efficient time integration schemes for port-Hamiltonian systems of ordinary differential equations and differential-algebraic equations. The schemes are based on variants of operator splitting, which extend and generalize concepts of forced gradients, Cayley transforms and multi-rate approaches. Moreover, we consider and investigate Galerkin and discrete gradient methods particularly tailored to the port-Hamiltonian structure.

Mathematical optimization is a discipline of high importance for economy, science, industry, policy support, and the public and private sector. Due to practical demand and the rise of computing capabilities, experts proficient in this field are needed in research and development. This research training group (RTG) therefore focuses on education in the field of algorithmic optimization.

The goal of the subproject is the development and analysis of structure-preserving approximations for low-temperature subnetworks with fixed flow direction and their application in state estimation for model predictive control. The targeted data (snapshot) based reduction under compatibility constraints include Galerkin projections for spatial discretization and model order reduction as well as complexity reduction for the nonlinearities.

The main objective of this mathematically oriented subproject is the development of methods and analysis of reduced models or model hierarchies and their application to dynamic state estimations for model-predictive control.

The subproject is concerned with the development and analysis of model hierarchies for district heating networks. Under the port-Hamiltonian framework efficient discretization methods, model reduction techniques, as well as error estimators are developed.

In view of the intended prognostic online simulation capability, this subproject aims at the development and analysis of reduced models by means of model order reduction (MOR) techniques. The combination of MOR and space-mapping promises a further increase in performance, which is assessed qualitatively and quantitatively for the specific application of MR-thermometry.

The objective of this project is the simulation of highly dynamic spinning processes, geared to the key challenge of understanding the melt-blowing process with its complex dependency on the driving turbulent hot air streams together with the arising large stretching of the liquid fiber jet. For the underlying instationary viscous Cosserat rod model, system of partial and ordinary differential equations with random source term, numerical schemes are developed with focus on appropriate refinement strategies.

The online (possible real-time) control and parametric optimization of gas transport requires the stable and fast simulation of large instationary networks that are modeled by huge coupled systems of partial differential and algebraic equations. This project aims at the development of an appropriate numerical discretization adapted to the network topology and in regard of a small perturbation index. Establishing a PDAE-surrogate model hierarchy by using model order reduction techniques promises to yield an efficient compromise between complexity and accuracy.

The manufacturing of modern lightweight components is often based on aerodynamic laydown processes. In the airlay-process a porous structure is formed by a fiber-loaded air flow on a conveyor belt and then bonded. The material properties of the finalized product, such as structural strength or bulk modulus, depend on the parameters of the production process. OPAL aims at the optimization of the production process and the material properties on basis of the consistent mathematical description of the process chain. The focus of this subproject lies on the kinetic modeling of the fiber-loaded flow.

In the production process of nonwoven materials filaments are spun from a polymer melt by aerodynamic stretching, then entangled by turbulent air flows and finally delivered onto a conveyor belt. The quality of the product is fundamentally influenced by the stochastic properties of the production process. ProFil contributes to innovations in this field with regard to material savings and quality improvements through mathematical modeling and numerical simulation of the process chain «melting-spinning-entangling-deposition». This subproject deals with fiber-fluid interactions and turbulence modeling.