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Ozone is integral to tropospheric chemistry, and understanding the processes controlling its distribution is important in climate and air pollution contexts. The GEOS-Chem global chemical transport model and its adjoint are used to interpret the impacts of midlatitude precursor emissions and atmospheric transport on the tropospheric ozone distribution at middle and high northern latitudes. In the Arctic, the model reproduces seasonal cycles of peroxyacetyl nitrate (PAN) and ozone measured at the surface, and observed ozone abundances in the summer free troposphere. Source attribution analysis suggests that local photochemical production, ≤ 0.25 ppbv/day, driven by PAN decomposition accounts for more than 50% of ozone in the summertime Arctic boundary layer. In the mid-troposphere, photochemical production accounts for 30-40% of ozone, while ozone transported from midlatitudes contributes 25-35%. Adjoint sensitivity studies link summertime ozone production to anthropogenic, biomass burning, soil, and lightning emissions between 50N-70N. Over Alert, Nunavut, the sensitivity of mid-tropospheric ozone to lightning emissions sometimes exceeds that to anthropogenic emissions. Over the eastern U.S., numerous models overestimate ozone in the summertime boundary layer. An inversion analysis, using the GEOS-Chem four-dimensional variational data assimilation system, optimizes emissions of NOx and isoprene. Inversion results suggest the model bias cannot be explained by discrepancies in these precursor emissions. A separate inversion optimizes rates of key chemical reactions including ozone deposition rates, which are parameterized and particularly uncertain. The inversion suggests a factor of 2-3 increase in deposition rates in the northeastern U.S., decreasing the ozone bias from 17.5 ppbv to 6.0 ppbv. This analysis, however, is sensitive to the model boundary layer mixing scheme. Several inversion analyses are conducted to estimate lightning NOx emissions over North America in August 2006, using ozonesonde data. The high-resolution nested version of GEOS-Chem is used to better capture variability in the ozonesonde data. The analyses suggest North American lightning NOx totals between 0.076-0.204 Tg N. A major challenge is that the vertical distribution of the lightning source is not optimized, but the results suggest a bias in the vertical distribution. Reliably optimizing the three-dimensional distribution of lightning NOx emissions requires more information than the ozonesonde dataset contains.

The current work concerns the study and the implementation of a modern algorithm for a posteriori error estimation in Computational Fluid Dynamics (CFD) simulations based on partial differential equations (PDEs). The estimate involves the use of duality argument and proper consistent discretisation of primal and dual problem.A key element is the construction of the adjoint form of the primal differential operators where the data term is a quantity of interest depending on the application. In engineering, this is typically a physical functional of the solution. So, by solving this adjoint problem, it is possible to obtain important information about local sensitivity of the error with respect to the current target quantity and thereby, we are able to perform an a posteriori error representation based on adjoint data. Through this, we provide local error indicators which can drive an adaptive meshing algorithm in order to optimally reduce the target error. Therefore, we first derive and solve the discrete primal problem in agreementwith the chosen numerical method. According to consistency and compatibility conditions, we can use the same discretisation for solving the adjoint problem, simply by swapping the position of the unknowns and the test functions in the linearised variational operator. Remembering that the corresponding adjoint problem always remains linear, the computational cost for obtaining these data is limited compared to the effort needed to solve the primal nonlinear problem.This procedure, fully developed for Discontinuous Galerkin (DG) and Finite Volume (FV) methods, is here for the first time applied in a fully consistent way for Petrov-Galerkin (PG) discretisations. Differently from the latter, the biggest issue for the PG method becomes the need to handle two different functional spaces in the discretisation, one of which is often not even continuous. Stabilized finite element schemes such as Streamline Upwind (SUPG), bubble stabilized (BUBBLE) Petrov-Galerkin and stabilized Residual Distribution (RD) have been selected for implementation and testing. Indeed, based on local advection information, these schemes are naturally more suitable for solving hyperbolic problems and therefore, interesting alternatives for fluid dynamics applications.A scalar linear advection equation is used as a model problem for convergence rate of both primal and adjoint solutions and target quantity. In addition, it is also applied in order to verify the accuracy of the adjoint-based a posteriori error estimate. Next, we apply the methods to a complete collection of numerical examples, starting from scalar Burgers’ problem till 2D compressible Euler equations. Through suited quantities of interest, we illustrate aspects of the adjoint mesh refinement by comparing its efficiency with respect to the standard a posteriori error estimation. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18081号 / 理博第3959号 / 新制||理||1571(附属図書館) / 30939 / 京都大学大学院理学研究科地球惑星科学専攻 / (主査)准教授 宮﨑 真一, 教授 福田 洋一, 教授 平原 和朗 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Improving the design of flow channels in a liquid-cooled heat sink is critical for boosting the capabilities of electronic components as well as reducing energy usage by the pump. This work explores the use of topology optimization to minimize the pressure difference across a heat sink and consequently, the energy used to supply the liquid. Topology optimization involves solving mathematical equations to obtain the optimal design for a defined cost function, here the total pressure difference between the inlet and outlet. A design variable called the porosity is defined for each cell in the mesh. The porosity features in a sink term in the momentum equation, which 'solidifies' cells by velocity suppression when deemed to be counterproductive to the cost function. The adjoint method of topology optimization, in particular, is a well-established tool for use in flow network problems and includes non-physical parameters such as the adjoint velocity and pressure. The method isn't without its drawbacks, such as the numerical instability of the adjoint equations, and the absence of boundary layers or wall functions at the interface of high and low porosity. The strength of the adjoint method lies in the ease with which it calculates the gradient of the cost function with respect to the porosity. When applied to the geometries in this work, it is observed that the problem is non-convex and results in multiple optimums with similar cost values. Thus the objective becomes seeking solutions with the simplest shape and at the same time having a minimized pressure difference. Interesting techniques are tested, namely an interpolation function, a velocity tolerance, and a volume constraint. The work is accomplished by modifying an existing adjoint optimization solver in the open-source CFD software, OpenFOAM.

Accurate prediction of tsunami waves requires complete boundary and initial condition data, coupled with the appropriate mathematical model. However, necessary data is often missing or inaccurate, and may not have sufficient resolution to capture the dynamics of such nonlinear waves accurately. In this thesis we demonstrate that variational data assimilation for the continuous shallow water equations (SWE) is a feasible approach for recovering both initial conditions and bathymetry data from sparse observations. Using a Sadourny finite-difference finite volume discretisation for our numerical implementation, we show that convergence to true initial conditions can be achieved for sparse observations arranged in multiple configurations, for both isotropic and anisotropic initial conditions, and with realistic bathymetry data in two dimensions. We demonstrate that for the 1-D SWE, convergence to exact bathymetry is improved by including a low-pass filter in the data assimilation algorithm designed to remove scale-scale noise, and with a larger number of observations. A necessary condition for a relative L2 error less than 10% in bathymetry reconstruction is that the amplitude of the initial conditions be less than 1% of the bathymetry height. We perform Second Order Adjoint Sensitivity Analysis and Global Sensitivity Analysis to comprehensively assess the sensitivity of the surface wave to errors in the bathymetry and perturbations in the observations. By demonstrating low sensitivity of the surface wave to the reconstruction error, we found that reconstructing the bathymetry with a relative error of about 10% is sufficiently accurate for surface wave modelling in most cases. These idealised results with simplified 2-D and 1-D geometry are intended to be a first step towards more physically realistic settings, and can be used in tsunami modelling to (i) maximise accuracy of tsunami prediction through sufficiently accurate reconstruction of the necessary data, (ii) attain a priori knowledge of how different bathymetry and initial conditions can affect the surface wave error, and (iii) provide insight on how these can be mitigated through optimal configuration of the observations. / Thesis / Candidate in Philosophy

Many important engineering phenomena such as turbulent flow, fluid-structure interactions, and climate diagnostics are chaotic and sensitivity analysis of such systems is a challenging problem. Computational methods have been proposed to accurately and efficiently estimate the sensitivity analysis of these systems which is of great scientific and engineering interest. In this thesis, a new approach is applied to compute the direct and adjoint sensitivities of time-averaged quantities defined from the chaotic response of the Lorenz system and the double pendulum system. A stabilized time-integrator with adaptive time-step control is used to maintain stability of the sensitivity calculations. A study of convergence of a quantity of interest and its square is presented. Results show that the approach computes accurate sensitivity values with a computational cost that is multiple orders-of-magnitude lower than competing approaches based on least-squares-shadowing approach.

This study seeks to reduce the degree of uncertainty that often arises in computational fluid dynamics simulations about the computed accuracy of functional outputs. An error estimation methodology based on discrete adjoint sensitivity analysis is developed to provide a quantitative measure of the error in computed outputs. The developed procedure relates the local residual errors to the global error in output function via adjoint variables as weight functions. The three major steps in the error estimation methodology are: (1) development of adjoint sensitivity analysis capabilities; (2) development of an efficient error estimation procedure; (3) implementation of an output-based grid adaptive scheme. Each of these steps are investigated. For the first step, parallel discrete adjoint capabilities are developed for the variable Mach version of the U2NCLE flow solver. To compare and validate the implementation of adjoint solver, this study also develops direct sensitivity capabilities. A modification is proposed to the commonly used unstructured flux-limiters, specifically, those of Barth-Jespersen and Venkatakrishnan, to make them piecewise continuous and suitable for sensitivity analysis. A distributed-memory message-passing model is employed for the parallelization of sensitivity analysis solver and the consistency of linearization is demonstrated in sequential and parallel environments. In the second step, to compute the error estimates, the flow and adjoint solutions are prolongated from a coarse-mesh to a fine-mesh using the meshless Moving Least Squares (MLS) approximation. These error estimates are used as a correction to obtain highlyurate functional outputs and as adaptive indicators in an iterative grid adaptive scheme to enhance the accuracy of the chosen output to a prescribed tolerance. For the third step, an output-based adaptive strategy that takes into account the error in both the primal (flow) and dual (adjoint) solutions is implemented. A second adaptive strategy based on physics-based feature detection is implemented to compare and demonstrate the robustness and effectiveness of the output-based adaptive approach. As part of the study, a general-element unstructured mesh adaptor employing h-refinement is developed using Python and C++. Error estimation and grid adaptation results are presented for inviscid, laminar and turbulent flows.

Turbulent flows occur in various fields and are a central, yet an extremely complex, topic in fluid dynamics. Understanding the behaviour of fluids is vital for multiple research areas including, but not limited to: biological models, weather prediction, and engineering design models for automobiles and aircraft. In this thesis, we study a number of fundamental problems that arise in 2D turbulent flows, using the 2D Navier-Stokes system. Introducing optimization techniques for systems described by partial differential equations (PDE), we frame these problems such that they can be solved using computational methods. We utilize adjoint calculus to build the computational framework to be implemented in an iterative gradient flow procedure, using the "optimize-then-discretize" approach. Pseudospectral methods are employed for solving PDEs in a numerically efficient manner. The use of optimization methods together with computational mathematics in this work provides an illuminating perspective on fluid mechanics. We first apply these techniques to better understand enstrophy dissipation in 2D Navier-Stokes flows, in the limit of vanishing viscosity. By defining an optimization problem to determine optimal initial conditions, multiple branches of local maximizers were obtained each corresponding to a different mechanism producing maximum enstrophy dissipation. Viewing this quantity as a function of viscosity revealed quantitative agreement with an analytic bound, demonstrating the sharpness of this bound. We also introduce an extension of this problem, where enstrophy dissipation is maximized in the context of kinetic theory using the Boltzmann equation. Secondly, these PDE-constrained optimization techniques were used to probe the fundamental limitations on the performance of the Leith eddy-viscosity closure model for 2D Large-Eddy Simulations of the Navier-Stokes system. Obtained by solving an optimization problem with a non-standard structure, the results demonstrate the optimal eddy viscosities do not converge to a well-defined limit as regularization and discretization parameters are refined, hence the problem of determining an optimal eddy viscosity is ill-posed. Further extending the problem of finding optimal eddy-viscosity closures, we consider imposing an additional nonlinear constraint on the control variable in the problem, in the form of requiring the time-averaged enstrophy be preserved. To address this problem in a novel way, we employ adjoint calculus to characterize a subspace tangent to the constraint manifold, which allows one to approximately enforce the constraint. Not only do we demonstrate that this produces better results when compared to the case without constraints, but this also provides a flexible computational framework for approximate enforcement of general nonlinear constraints. Lastly in this thesis, we introduce an optimization problem to study the Kolmogorov-Richardson energy cascade, where a pathway towards solutions is outlined. / Thesis / Doctor of Philosophy (PhD)

Representation theory is a field of study within abstract algebra that originated around the turn of the 19th century in the work of Frobenius on representations of finite groups. More recently, Hopf algebras -- a class of algebras that includes group algebras, enveloping algebras of Lie algebras, and many other interesting algebras that are often referred to under the collective name of ``quantum groups'' -- have come to the fore. This dissertation will discuss generalizations of certain results from group representation theory to the setting of Hopf algebras. Specifically, our focus is on the following two areas: Frobenius divisibility and Kaplansky's sixth conjecture, and the adjoint representation and the Chevalley property. / Mathematics