Essentially Analytical Theory Closure for Space Filtered Thermal-Incompressible Navier-Stokes Partial Differential Equation System on Bounded Domains

Numerical simulation of turbulent flows is identified as one of the grand challenges in high-performance computing. The straight forward approach of solving the Navier-Stokes (NS) equations is termed Direct Numerical Simulation (DNS). In DNS the majority of computational effort is spent on resolving the smallest scales of turbulence, which makes this approach impractical for most industrial applications even on present-day supercomputers. A more feasible approach termed Large Eddy Simulation (LES) has evolved over the last five decades to facilitate turbulent flow predictions for reasonable Reynolds (Re) numbers and domain sizes. LES theory uses the concept of convolution with a spatial filter, which allows it to compute only the major scales of turbulence as determined by the diameter of the filter. The rest of the length scales are not resolved posing the so-called closure problem of LES. For bounded domains, besides the closure problem, an equally challenging issue of LES is that of prescribing the suitable boundary conditions for the resolved-scale state variables. Additional problems arise because the convolution operation does not generally commute with differentiation in the presence of boundaries. This dissertation details derivation of an essentially analytical closure theory for the unsteady three-dimensional space filtered thermal-incompressible NS partial differential equation (PDE) system on bounded domains. This is accomplished by the union of rational LES theory, Galdi and Layton, with modified continuous Galerkin theory of Kolesnikov with specific focus on correct adaptation of a constant measure filter near the Dirichlet type boundary. The analytical closure theory state variable organization is guided by classic fluid mechanics perturbation theory. Derivation and implementation of suitable boundary conditions (BCs) as well as the boundary commutation error (BCE) integral is accomplished using the ideas of approximate deconvolution (AD) theory. Non-homogeneous BCs for the auxiliary problem of arLES theory are derived

Ensembles for the Predictability of Average Temperatures

The instability of the atmosphere places an upper bound on the predictability of instantaneous weather patterns. The lack of complete periodicity in the atmosphere's behavior is sufficient evidence for instability (Lorenz, 1963), but it does not reveal the range at which the uncertainty in prediction must become large. Most estimates of this range have been based on numerical integrations of systems of equations of varying degrees of complexity, starting from two or more rather similar initial states. It has become common practice to measure the error which would be made by assuming one of these states to be correct, when in fact another is correct, by the root-mean-square difference between the two fields of wind, temperature, or some other element, and to express the rate of amplification of small errors in terms of a doubling time (Lorenz, 1963). The purpose of this thesis is to build tools with rigorous support useful for studying predictability of average temperatures. We apply our tools to a simple Earth-like example and make use of the Bred Vector algorithm to generate initial perturbations. The numerical model used is that of the Natural Convection problem. The analysis is done in steps, first by analyzing the turbulent natural convection problem then by introducing a fast calculation of an ensemble of solutions of the Navier-Stokes equations coupled with the temperature equation. Complete stability and convergence analysis of the methods are presented. The turbulent Earth model and its stability conditions are introduced at the end of the thesis

Ontwikkeling van een dynamische eindige differentiemethode voor Large-Eddy Simulatie

In de afgelopen decennia erkenden vele onderzoekers de noodzaak om de numerieke kwaliteit van Directe Numerieke Simulaties (DNS) en vooral voor Large-Eddy Simulaties (LES) voor turbulente stromingen te waarborgen. In tegenstelling tot DNS, worden in LES enkel de belangrijkste grootschalige turbulente wervelstructuren in de stroming berekend. Dit impliceert dat de kleinste structuren die berekend worden in LES, relatief belangrijk zijn voor de evolutie van de stromingsoplossing in LES. Bijgevolg dient de numerieke kwaliteit ook voor deze kleine wervels te worden gewaarborgd in LES. Echter, de klassieke eindige differentiebenaderingen, waarbij de nauwkeurigheid van de grootst geresolveerde structuren op het rekenrooster primeert ten koste van de kleinst geresolveerde structuren, zijn vaak suboptimaal voor LES, waar de grootteorde van de kleinste structuren in vele gevallen vergelijkbaar is met deze van de mazen van het rekenrooster. In het huidige proefschrift wordt een familie van dynamische eindige differentiemethoden ontwikkeld, die toelaat de globale numerieke fout op de stromingsoplossing onmiddellijk te minimaliseren tijdens de simulatie. Deze dynamische eindige differentiemethoden bevatten dus het intrinsiek vermogen zich optimaal aan te passen aan de fysische kenmerken van de berekende stroming in relatie tot het rekenrooster. Deze eindige differentiestrategie maakt het mogelijk om steeds een quasi optimale numerieke methode te waarborgen, in overeenstemming met de karakteristieken van de stromingsoplossing op dat moment

Uncertainty quantification for an electric motor inverse problem - tackling the model discrepancy challenge

In the context of complex applications from engineering sciences the solution of identification problems still poses a fundamental challenge. In terms of Uncertainty Quantification (UQ), the identification problem can be stated as a separation task for structural model and parameter uncertainty. This thesis provides new insights and methods to tackle this challenge and demonstrates these developments on an industrial benchmark use case combining simulation and real-world measurement data. While significant progress has been made in development of methods for model parameter inference, still most of those methods operate under the assumption of a perfect model. For a full, unbiased quantification of uncertainties in inverse problems, it is crucial to consider all uncertainty sources. The present work develops methods for inference of deterministic and aleatoric model parameters from noisy measurement data with explicit consideration of model discrepancy and additional quantification of the associated uncertainties using a Bayesian approach. A further important ingredient is surrogate modeling with Polynomial Chaos Expansion (PCE), enabling sampling from Bayesian posterior distributions with complex simulation models. Based on this, a novel identification strategy for separation of different sources of uncertainty is presented. Discrepancy is approximated by orthogonal functions with iterative determination of optimal model complexity, weakening the problem inherent identifiability problems. The model discrepancy quantification is complemented with studies to statistical approximate numerical approximation error. Additionally, strategies for approximation of aleatoric parameter distributions via hierarchical surrogate-based sampling are developed. The proposed method based on Approximate Bayesian Computation (ABC) with summary statistics estimates the posterior computationally efficient, in particular for large data. Furthermore, the combination with divergence-based subset selection provides a novel methodology for UQ in stochastic inverse problems inferring both, model discrepancy and aleatoric parameter distributions. Detailed analysis in numerical experiments and successful application to the challenging industrial benchmark problem -- an electric motor test bench -- validates the proposed methods

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library