for Computing in Science

# PhD Projects

### M1 (Thermal Convection) - Heat Transport in thermal convection

#### Principal Investigator Camilla Nobili

The research is concerned with temperature driven Rayleigh-Benard (RBC) and Benard-Marangoni convection (BMC). Related mathematical models are important for meteorological and industrial flows ranging from electronic cooling to large-scale power plants [Chilla & Schumacher (2012), Incropera (1999), Chilla et al. (2004)]. For both models, our objective is to derive bounds on the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensional parameters appearing in the models. In particular we aim at proving quantitative bounds which reproduce the predicted power-law scalings. The challenge here is at rigorously deriving the exponents and determining the transitions between scaling regimes which agree with scaling analysis, (direct) numerical simulation and experiments. In RBC the major goal is to produce upper bounds on Nu for low-viscosity fluids (i.e. low Pr-number fluids like liquid metals or engine oils), for which rigorous as well as experimental results are few and partial. Despite the widespread applications, heat transport properties of the shear-driven BMC have been less studied than the one of the buoyancy-driven RBC and our initial aim is to analyze a simplified model with negligible fluid inertia (i.e. Pr = 1). The novelty of the research refers to the employment of PDE-tools (e.g. regularity estimates) to derive predictions on a quantity of physical and engineering interest.

### M2 (Solar Plant Modeling) - Model-based design of solar thermal power plants

#### Principal Investigator Ingenuin Gasser

There are many ideas to design and construct solar thermal power plants: From well-known parabolic troughs to exotic solar power or updraft towers, see e.g. [El Jai & Chalqi (2013),Zhai Rongrong et al. (2013),Richter (2017)] or [von Allwöorden et al. (2018)]. In almost all of these concepts solar power is used to heat a fluid (special liquid, salt, water, gas, liquid gas mixtures etc.), the increased inner energy of the fluid is transported to a heat storage or directly to the energy production unit where heat energy is transformed into mechanical energy which is used to power a generator for electrical energy production. Some of these concepts only exist on paper, some are still in a first phase of development and testing, others - such as the parabolic troughs - have been developed and used already over decades. At the moment there is no final conclusion on what the most promising technologies could be. In this phase of development it is natural to aim at optimization: to design and to operate a power plant in order to maximize e.g. the net. When operating such a power plant, the flow velocity in the pipes is a typical control parameter. Low flow velocities in the pipes guarantee low frictional losses but imply high temperatures (due to the long exposure time of the fluid) and thus high energy losses by radiation and heat transfer. On the other hand high flow velocities in the pipes induce high frictional losses but with low temperatures and thus low energy losses by radiation and heat transfer. Therefore we expect an optimal velocity somewhere in between with a related (maximal) net power output. This example refers to the operational phase of the power plant. Similar issues have to be addressed in the planning phase. The aim is to design the "optimal" power station, i.e., maximal net power output with respect to a series of design parameters under certain given constraints. As described above, the main physical concepts of many solar thermal power stations are very similar. Therefore, similar optimization issues can be formulated for other types of solar thermal power plants. The case of solar updraft towers is meanwhile well understood [von Allwörden et al. (2018)]. In the solar power tower case promising results were obtained in [Richter (2017)]. We will develop modelbased mathematical approaches to design and to operate solar thermal power plants with a focus on parabolic troughs. This requires to advance mathematical models for alternative energy power plants towards optimization, and leads to mathematical challenging optimization problems for transient, coupled thermo-fluid dynamic processes in a large 1D-network of pipes, whose analytical and numerical treatment demands the development of new solution methods.

M3 (Adaptive Modeling) - Optimized refinement criteria for adaptive modeling of transient flows

#### Principal Investigator Jörn Behrens

Typical transient geoscientific fluid dynamics and wave propagation problems expose a large span of spatial and temporal scales and corresponding numerical methods need to resolve those. One approach to bridge the scale gap in transient geophysical fluid dynamics and wave propagation applications refers to adaptive meshing or order refinement [Becker & Rannacher (2001), Eriksson et al. (2017), Gerwing et al. (2017)]. While the general computational methods for adaptive simulation with appropriate numerical methods is well established [Behrens (2006)], an open issue is still to find efficient and robust criteria for determining the refinement. In this sub-project we focus on spatial adaptivity and seek to develop computationally efficient refinement indicators. A gold standard of physically motivated error estimation is the goal-oriented weighted residual adjoint method, originally proposed by [Becker & Rannacher (2001)]. It has been successfully applied for ocean modeling [Pflower et al. (2006)] and recently improvements for non-smooth situations as they appear in discontinuous Galerkin discretizations or along sharp interfaces have been studied in [Beckers (2017),Kröger et al. (2018)]. However, for nonlinear transient flow regimes such adjoint methods are extremely costly, requiring a forward-backward propagation of the equations, usually performed on coarse model grids or even with simplified model equations [Bauer et al. (2014)]. Therefore, this topic aims at developing novel error estimation techniques that achieve similar accuracy as sophisticated adjoint methods, but at lower computational cost. The strategies are based on discretization characteristics, e.g., the jumps at cell boundaries in flux-based methods, or the spectral coefficients in spectral methods [Vater et al. (2015)], and physics-induced error propagation. In particular Lagrangian flow paradigms as well as optimal transport methods are to be utilized in order to gain the necessary error propagation information for transient regimes. While efficient refinement strategies in adaptive mesh methods optimize the run-time behavior on a methodological or physical model level, efficiency is also required on an algorithmic level in order to enable solution of complex realistic problems. Therefore, parallelization of such possibly unstructured and dynamically changing problems is an important issue. Previous works [Kunst & Behrens (2014), Behrens & Bader (2009)] in this direction will need to be improved and extended, and developed refinement strategies be parallelized accordingly.

### S1 (Adaptive Kernels) - Adaptive kernel-based approaches for fluid flow simulations

#### Principal Investigator Armin Iske

Adaptive particle methods relying on meshfree kernels (e.g. radial basis functions) are popular tools for the numerical solution of partial differential equations [Fasshauer & McCourt (2015), Griebel & Schweitzer (2017)]. In particular, the utility of smoothed particle hydrodynamic (SPH) methods [Casulli (1990),Monaghan (2005), Vila (1999)] has been demonstrated for fluid flow simulations in the geosciences [Brecht et al. (2017),Fornberg & Flyer (2015), Iske & Randen (2005)] and other areas, including e.g. (mathematical) data analysis, and kernel-based (machine) learning. The application of adaptive kernel-based particle methods requires particular care, especially for their critical interaction with high order flux evaluations [Ben-Artzi & Falcovitz (2003),Goetz & Iske (2016),Toro (2009)]. The objective of this project is the development of novel concepts for the applicationoriented design of adaptive kernel methods to

satisfy the ever increasing demands of large-scale fluid dynamic computations [Ferrari et al. (2009)]. The innovation of this research refers to recent advances in the design of non-standard kernels providing highly flexible approximations, as they are required in contemporary applications of mathematical data science.

### S2 (Preconditioning) - Preconditioners for RBF-FD discretized fluid flow problems

#### Principal Investigator Sabine Le Borne

Recently, a new meshfree discretization technique for partial differential equations (PDEs) based on radial basis functions (RBF) has been introduced [Fornberg & Flyer (2015)]. Given a set of scattered nodes, RBF-interpolation is used to compute a (local) differentiation stencil (RBF-FD), leading to a sparse linear system of equations. The novelty and objective of this project is the development and analysis of rank-structured preconditioners for the linear systems of equations resulting from meshfree RBF-FD discretization of fluid flow problems

### S3 (Blood Flow) - Blood Flow modeling and estimation by magnetic particle imaging

#### Principal Investigator Christina Brandt

In cardiovascular diagnostics, magnetic particle imaging (MPI) has been developed to acquire blood flow patterns inside arteries. This new imaging modality offers benefits step and of the blood flow which requires efficient minimization algorithms for real-time computations. The problem will advance existing approaches for combined image and flow reconstruction dedicated to denoising and motion estimation to real-time image reconstruction and flow estimation for MPI.

### O1 (Data Assimilation) - Continous Data Assimilation for Ocean Modeling

#### 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) - CAD-free adjoint shape optimization of floating vessels

#### 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) - Scalable algorithms for shape design and interface identification

#### 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.