Research Area O
Coordination: Prof. Dr. T. Rung
The projects in research area O cover the two essential ingredients of an optimization process. On the one hand adjusting PDE models to measured data and on the other improving designs towards optimality based on these calibrated models. In particular, the projects aim at developing novel mathematical concepts related to PDE constrained optimization techniques for shape design and data-assimilation.
Shape optimization examples refer to two-phase flows around free- floating ships, modeled with diffusive phase-field approaches which will include fluid-structure interaction (O2), as well as aerodynamic shape design based on Euler and Navier-Stokes equations using efficient hardware architectures (O3). Data-assimilation is devoted to meteorological applications, in particular for global ocean modeling (O1). With the composition of the research team in area O, we combine mathematical expertise in the field of PDE constrained optimization with engineering and geophysical competence in challenging fluid dynamic applications.
O1 (Data Assimilation) - Continous Data Assimilation for Ocean Modeling
O2 (Ship Optimization) - CAD-free adjoint shape optimization of floating vessels
O3 (Shape Design) - Scalable algorithms for shape design and interface identification
O1: Data Assimilation
Principal Investigator: Peter Korn
Data assimilation blends observational data of a physical process with a dynamical model of this process [Wunsch (1997)]. It constitutes a fundamental technique for modeling real world flows. Numerical weather prediction and ocean state estimation are prominent examples of scientific disciplines that rely on data assimilation with the goal to derive an accurate initialization that leads to an accurate prediction. For this purpose assimilation methods such as the Kalman Filter, or variational approaches like the adjoint method (4D-Var) [Kalnay (2003)] are employed. These assimilation algorithms of enormous complexity are applied to the nonlinear PDEs of atmosphere and ocean dynamics. This combination of complex data assimilation algorithm and dynamical equations with its highly nonlinear multi-scale interaction impedes a mathematical analysis. One essentially relies on experimental studies to gain understanding of such system and measures their success by their impact on the prediction horizon. Mathematical analysis of fundamental properties has been applied to sophisticated assimilation algorithms only when they are used in conjunction with idealized models such as the Lorenz model [Lorenz (1963)]. In this project we take a different route. We assimilates observations into a complex dynamical equation but compromise on the complexity of the assimilation algorithm by using the conceptually relatively simple method of continuous data assimilation.
O2: Ship Optimization
Principal Investigator: Thomas Rung
Due to the low fuel consumption per tonne-km, merchant shipping covers the transport of around 90% of the global trade with about 45000 vessels. The associated pollution as well as approximately half of the direct operating costs can be attributed to the fuel consumption, which is primarily governed by the hydrodynamic performance. This sets the focus of our research into adjoint-based shape optimization of ship hulls. Whilst industrial applications of adjoint methods to optimize fluid dynamic shapes have recently achieved an impressive level of maturity for single-phase flows [Papoutsis-Kiachagias & Giannakoglou (2016), Othmer (2014)], only few contributions with limited capabilities were published for maritime two-phase flows [Palacios et al. (2012), Kroger et al. (2018), Springer & Urban (2015)]. We therefore intend to advance novel primal & adjoint Reynolds-averaged Cahn-Hilliard Navier-Stokes formulations to simulate immiscible two-phase flows that are coupled to a 6-DoF oating vessel motion model, and assess their shape sensitivities. A CAD-free, kernel based Lagrangian shape parameterization will be utilized to translate the sensitivities into suitable deformations of the surface mesh and drive the related volume mesh deformation. The innovations refer to consistent primal & adjoint models for immiscible two-phase flows with an embedded motion model and the shape parameterization. Particular challenges are associated to the high Reynolds number (Re> 108), the description of immiscible fluids, the coupled influence of the flow field and the floatation on the sensitivities and an appropriate description of the discrete shape evolution.
O3: Shape Design
Principal Investigator: Martin Siebenborn
This topic is concerned with the mathematical development of efficient, shape optimization algorithms for aerodynamic shapes in Euler and Navier-Stokes flows. Besides high performance solvers for the fluid dynamics and adjoint equations, fast convergence of the outer optimization loop is desirable. Quasi-Newton methods and robust shape deformations lead to mesh-independent convergence, which is mandatory to achieve scalability on supercomputers. Especially, for computationally attractive, higher order methods (e.g. discontinuous Galerkin [Kaland et al. (2015)]) maintaining the mesh quality during the optimization is essential. In order to achieve a high order of approximation of the overall discretization also shape representations have to be chosen in appropriate spaces. This makes the process of discretizing the geometry computationally costly. It is thus crucial to develop solid deformation techniques for the mesh rather than a remeshing after each design update. Interpreting descent directions as deformations to the finite element mesh are a vivid field of research and there are various approaches available (e.g. [Gangl et al. (2015),Schmidt et al. (2016),Kheuten & Ulbrich (2015),Kroger & Rung (2016)]). This is tightly connected to the definition of a suitable space of feasible shapes and its tangent space, which is still an open question and will be addressed in this research. From a computational point of view it turns out to be attractive to define shapes, tangent vectors and shape metrics in terms of the surrounding space and not the surface on its own. Thereby, it is possible to merge quality of the fluid mesh, scalability of the optimization and computational efficiency into one unit. Moreover, novel scalable optimization algorithms will be developed for implicit geometry representations and mesh-free discretization techniques like SPH. In contrast to approaches based on mesh deformations, a high level of algorithmic efficiency can be achieved by resolving the fluid dynamics on a structured background mesh, while the aerodynamic shapes are only implicitly described on that mesh. Thus, also design updates are implicit and do not affect the fluid discretization. This enables to speed up computations by the application of a wide range of modern many-core compute architectures.