# Research Area O

**Optimization**

**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.

### PhD Projects:

## O1: Data Assimilation

**Principal Investigator: Peter Korn**

**PhD student: Kamal Sharma**

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.

## O1.2: Convex Integration

**Principal Investigator: Peter Korn**

In the past decade the method of convex integration has been introduced in mathematical fluid dynamics, leading to a number of very striking results, e.g. the resolution of the celebrated Onsager conjecture and the non-uniqueness of weak solutions of the 3D Euler and Navier-Stokes equations. Convex integration is a constructive algorithm that produces a sequence of increasingly oscillatory vector fields that converge in a very mild sense to a solution of the Euler or Navier-Stokes Equations. Efforts to formulate a finite-dimensional version of convex integration have begun, with the goal to actually compute these types of turbulent solutions. The topic of this PhD thesis is to contribute to this effort and investigates theoretical as well as numerical aspects.

## O2: Ship Optimization

**Principal Investigator: Thomas Rung**

**Marvin Müller**

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.

## O2.2: Unsteady simulation approaches for the shape optimization of Energy-Saving-Devices

**Principal Investigator: Thomas Rung**

**PhD student: Denis Andreev**

Unsteady simulation approaches for the shape optimization of Energy-Saving-Devices

**Aims & Motivation:** Merchant shipping is responsible for more than 90% of the global trade, and

the 50.000 merchant vessels have a significant environmental impact as regards CO2-, SOx- and NOxemissions. The project aims at improving the environmental impact of merchant vessels using simulationbased shape optimizations of Energy-Saving-Devices. Such devices favorably manipulate either the flow approaching or leaving the propeller. As indicated by the German Environmental Award 2022, they offer a huge potential for reducing the emissions. At the same time they increase the vessel’s fuel efficiency and improve the economy. The project will follow the path of previousphase projects to numerically optimize the resistance of free-floating vessels exposed to turbulent two-phase flows, using gradient-based adjoint shape optimization approaches [11, 12, 14]. These were also coupled to advanced descent strategies to improve parameter-free shapes [21] and successfully validated for realistic steady state configurations at large Reynolds- and Froude-numbers [16]. Future applications will address minimizing the power requirement for ESD-equipped vessels for which unsteady strategies will be developed. Emphasis will be given to ”smart” methods that describe the unsteady interaction of the rotating propeller, the ESD and the hull and thereby support related shape optimizations at moderate cost.

**Approach & Innovation:** The challenges of maritime two-phase flows refer to the large Reynoldsnumber (Re ≥ 108), possibly large Froude-numbers (Fn ≈ 0.4) and the free floatation of the vessel. Related simulation-driven shape optimizations – including an integrated shape description – were developed during previous efforts for massively parallel applications [10, 11, 12, 13, 14, 15, 16, 17, 20], but did not consider active propulsion or unsteady operations. Future efforts will address active propulsion which requires unsteady adjoint optimization methods, where difficulties arise from the oppositely directed information transport of the primal and adjoint procedures. In a trade-off between compute and memory expenses, check-pointing strategies were previously suggested [26, 6, 24, 27], but the related overheads often question their feasibility in an industrial design process. An innovative alternative refers to order reducing singular value decomposition (SVD) methods, which are often employed for reduced-order models [23, 8]. In the context of adjoint shape optimizations they must be implemented as time-incrementing [1, 2, 5, 3, 4], spatially parallel strategies to project the primal flow field into a compact formulation. Improving recent suggestions [25, 19] for smart field data reconstructions as regards the

attainable computational efficiency [18] and the independency of the parallelization will be a first aspect of the project. Moreover, the reconstructed data should obey to the physical realizability, e.g., adhere to plausible limits,which supports both the robustness of the optimization process and the predictive quality. The incremental SVD or related alternatives, e.g., [22], can be used as a smart ”interpolator” that avoids check-pointing while optimizing the ESD in unsteady flows. Moreover, mode-based (ROM) optimization methods are an attractive area of research, that will provide more extensive reductions of the computational cost and might alleviate stability issues frequently reported for adjoint methods.

## O3: Shape Design

**Principal Investigator: Martin Siebenborn**

**PhD student: Henrik Wyschka**

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.

## O4: Coupling finite elements and data driven models in identification - learning of flow laws

**Principal Investigator: Winnifried Wollner**

Coupling finite elements and data driven models in identification/learning of flow laws

**Aims & Motivation:** In the identification/learning of governing equations of a fluid, unknown parameters and material laws (such as the stress-strain relation) need to be identified from measurements by minimizing the mismatch/loss between (numerical) model predictions and measurement data. In some situations these unknown parameters may be just a finite set of real numbers. However, when coupling with different physical quantities such as temperature, or polymer densities [2], the material laws may also be unknown functions which need to be identified. Due to their known approximation quality neuronal networks (NN) may be used to approximate these unknown functions [1, 7, 5] or [8] for flow in porous

media, but fitting to preselected sets of functions is also possible. Similar ideas have also been used for improved turbulence modeling [4]. In order to simulate the flow based on such a data driven/empirical flow law the governing partial differential equation (PDE) needs to be discretized, e.g., by finite elements. When training the NN the accuracy of the discretization need to be carefully tuned to avoid excessive training cost and potential stalling of the training.

**Approach & Innovation:** In contrast to so called physics informed NN [6] where the unknown solution of the PDE is approximated by a NN, we want to approximate the unknown material laws by a NN while continuing to use standard approaches for the discretization. In this context the possible PhD topics will deal with the inherent difficulties associated with the coupling of a finite element discretization of the PDE with a NN for the material laws. This includes the efficient adjoint based gradient calculation for the coupled problem, the error assessment of the coupled system for the prediction and identification problem by means of adjoint based error estimation as well as the analysis and design of suitable optimization methods for the training of the material law NN.

**Exemplary PhD-Topics**:

• Inexact non-convex optimization methods for the simulation based identification of flow laws

• A posteriori error estimation in coupled finite element NN flow models

**References**:

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[2] E. Feireisl, Y. Lu, and E. Süli. Dissipative weak solutions to compressible navier–stokes–fokker–planck systems with variable viscosity coefficients. Journal of Mathematical Analysis and Applications, 443(1):322–351, 2016.

[3] C. Geiersbach and W. Wollner. A stochastic gradient method with mesh refinement for PDE constrained optimization under uncertainty. SIAM J. Sci. Comput., 42(5):A2750–A2772, 2020.

[4] J. N. Kutz. Deep learning in fluid dynamics. Journal of Fluid Mechanics, 814:1–4, 2017.

[5] O. Pantalé, P. Tize Mha, and A. Tongne. Efficient implementation of non-linear flow law using neural network into the abaqus explicit fem code. Finite Elements in Analysis and Design, 198:103647, 2022.

[6] M. Raissi, P. Perdikaris, and G. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019.

[7] H. Yang, Q. Xiang, S. Tang, and X. Guo. Learning material law from displacement fields by artificial neural network. Theoretical and Applied Mechanics Letters, 10(3):202–206, 2020.

[8] A. B. Zolotukhin and A. T. Gayubov. Machine learning in reservoir permeability prediction and modelling of fluid flow in porous media. IOP Conference Series: Materials Science and Engineering, 700:012023, 2019.