A Multiscale Thermo-Fluid Computational Model for a Two-Phase Cooling System

In this paper, we describe a mathematical model and a numerical simulation method for the condenser component of a novel two-phase thermosyphon cooling system for power electronics applications. The condenser consists of a set of roll-bonded vertically mounted fins among which air flows by either natural or forced convection. In order to deepen the understanding of the mechanisms that determine the performance of the condenser and to facilitate the further optimization of its industrial design, a multiscale approach is developed to reduce as much as possible the complexity of the simulation code while maintaining reasonable predictive accuracy. To this end, heat diffusion in the fins and its convective transport in air are modeled as 2D processes while the flow of the two-phase coolant within the fins is modeled as a 1D network of pipes. For the numerical solution of the resulting equations, a Dual Mixed-Finite Volume scheme with Exponential Fitting stabilization is used for 2D heat diffusion and convection while a Primal Mixed Finite Element discretization method with upwind stabilization is used for the 1D coolant flow. The mathematical model and the numerical method are validated through extensive simulations of realistic device structures which prove to be in excellent agreement with available experimental data

Computational Fluid Dynamics Methods for Gas Pipeline System Control

At the present level of development of long, branched gas transmission networks (GTN), solving the problems of improving safety, efficiency and environmental soundness of operation of industrial pipeline systems calls for the application of methods of numerical simulation. The development of automated devices for technical inspection and process control, and availability of high-performance computer hardware have created a solid technical basis to introduce numerical simulation methods into the industrial practice of GTN analysis and operation. One of the promising approaches for numerical analysis of GTN operating is the development and application of high-accuracy computational fluid dynamics (CFD) simulators of modes of gas mixture transmission through long, branched pipeline systems (CFD-simulator) (Seleznev, 2007). Actually, a CFD-simulator is a special-purpose software simulating, in online and real time modes with a high similarity and in sufficient detail, the physical processes of gas mixture transmission through a particular GTN. The development of a CFD-simulator focuses much attention to correctness of simulation of gas flows in the pipelines and to the impact produced by operation of relevant GTN gas pumping equipment (including gas compressor unit (GCU), valves, gas pressure reducers, etc.) and the environment upon the physical processes under study. From the standpoint of mathematical physics, a CFD-simulator performs numerical simulation of steady and transient, non-isothermal processes of a gas mixture flow in long, branched, multi-line, multi-section gas pipeline network. Such simulation is aimed at obtaining high-accuracy estimates of the actual distribution (over time and space) of fluid dynamics parameters for the full range of modes of gas mixture transmission through the specific GTN in normal and emergency conditions of its operation, as well as of the actual (temporal) distribution of main parameters of GTN equipment operation, which can be Document type: Part of book or chapter of boo

Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks

Zuweilen wird gefördertes Erdgas als eine Brückentechnologie noch eine Weile erhalten bleiben, aber unsere Gasnetzinfrastruktur hat auch in einer Ära post-fossiler Brennstoffe eine Zukunft, um Klima-neutral erzeugtes Methan, Ammoniak oder Wasserstoff zu transportieren. Damit die Dispatcher der Zukunft, in einer sich fortwährend dynamisierenden Marktsituation, mit sich beständig wechselnden Kleinstanbietern, auch weiterhin einen sicheren Gasnetzbetrieb ermöglichen und garantieren können, werden sie auf moderne, schnelle Simulations- sowie performante Optimierungstechnologie angewiesen sein. Der Schlüssel dazu liegt in einem besseren Verständnis zur numerischen Behandlung nicht differenzierbarer Funktionen und diese Arbeit möchte einen Beitrag hierzu leisten. Wir werden stückweise differenzierbare Funktionen in sog. Abs-Normalen Form betrachten. Durch einen Prozess, der Abs-Linearisierung genannt wird, können wir stückweise lineare Approximationsmodelle erster Ordnung, mittels Techniken der algorithmischen Differentiation erzeugen. Jene Modelle können über Matrizen und Vektoren mittels gängiger Software-Bibliotheken der numerischen linearen Algebra auf Computersystemen ausgedrückt, gespeichert und behandelt werden. Über die Generalisierung der Formel von Faà di Bruno können auch Splinefunktionen höherer Ordnung generiert werden, was wiederum zu Annäherungsmodellen mit besserer Güte führt. Darauf aufbauend lassen sich gemischte Taylor-Kollokationsmethoden, darunter die mit Ordnung zwei konvergente generalisierte Trapezmethode, zur Integration von Gasnetzen, in Form von nicht glatten Algebro-Differentialgleichungssystemen, definieren. Numerische Experimente demonstrieren das Potential. Da solche implizite Integratoren auch nicht lineare und in unserem Falle zugleich auch stückweise differenzierbare Gleichungssysteme erzeugen, die es als Unterproblem zu lösen gilt, werden wir uns auch die stückweise differenzierbare, sowie die stückweise lineare Newtonmethode betrachten.As of yet natural gas will remain as a bridging technology, but our gas grid infrastructure does have a future in a post-fossil fuel era for the transportation of carbon-free produced methane, ammonia or hydrogen. In order for future dispatchers to continue to enable and guarantee safe gas network operations in a continuously changing market situation with constantly switching micro-suppliers, they will be dependent on modern, fast simulation as well as high-performant optimization technology. The key to such a technology resides in a better understanding of the numerical treatment of non-differentiable functions and this work aims to contribute here. We will consider piecewise differentiable functions in so-called abs-normal form. Through a process called abs-linearization, we can generate piecewise linear approximation models of order one, using techniques of algorithmic differentiation. Those models can be expressed, stored and treated numerically as matrices and vectors via common software libraries of numerical linear algebra. Generalizing the Faà di Bruno's formula yields higher order spline functions, which in turn leads to even higher order approximation models. Based on this, mixed Taylor-Collocation methods, including the generalized trapezoidal method converging with an order of two, can be defined for the integration of gas networks represented in terms of non-smooth system of differential algebraic equations. Numerical experiments will demonstrate the potential. Since those implicit integrators do generate non-linear and, in our case, piecewise differentiable systems of equations as sub-problems, it will be necessary to consider the piecewise differentiable, as well as the piecewise linear Newton method in advance

Numerical Methods for Uncertainty Quantification in Gas Network Simulation

When modeling the gas flow through a network, some elements such as pressure control valves can cause kinks in the solution. In this thesis we modify the method of simplex stochastic collocation such that it is applicable to functions with kinks. First, we derive a system of partial differential and algebraic equations describing the gas flow through different elements of a network. Restricting the gas flow to an isothermal and stationary one, the solution can be determined analytically. After introducing some common methods for the forward propagation of uncertainty, we present the method of simplex stochastic collocation to approximate functions of uncertain parameters. By utilizing the information whether a pressure regulator is active or not in the current simulation, we improve the method such that we can prove algebraic convergence rates for functions with kinks. Moreover, we derive two new error estimators for an adaptive refinement and show that multiple refinements are possible. Conclusively, several numerical results for a real gas network are presented and compared with standard methods to demonstrate the significantly better convergence results

Coupling, Conservation, and Performance in Numerical Simulations

This thesis considers three aspects of the numerical simulations, which arecoupling, conservation, and performance. We conduct a project and addressone challenge from each of these aspects.We propose a novel penalty force to enforce contacts with accurate Coulombfriction. The force is compatible with fully-implicit time integration and theuse of optimization-based integration. In addition to processing collisionsbetween deformable objects, the force can be used to couple rigid bodies todeformable objects or the material point method. The force naturally leads tostable stacking without drift over time, even when solvers are not run toconvergence. The force leads to an asymmetrical system, and we provide apractical solution for handling these.Next we present a new technique for transferring momentum and velocity betweenparticles and MAC grids based on the Affine-Particle-In-Cell (APIC) frameworkpreviously developed for co-locatedgrids. We extend the original APIC paper and show thatthe proposed transfers preserve linear and angular momentum and also satisfyall of the original APIC properties.Early indications in the original APIC paper suggested that APIC might besuitable for simulating high Reynolds fluids due to favorable retention ofvortices, but these properties were not studied further. We use twodimensional Fourier analysis to investigate dissipation in the limit \dt=0.We investigate dissipation and vortex retention numerically to quantify theeffectiveness of APIC compared with other transfer algorithms.Finally we present an efficient solver for problems typically seen inmicrofluidic applications.Microfluidic ``lab on a chip'' devices are small devices that operate on smalllength scales on small volumes of fluid. Designs for microfluidic chips aregenerally composed of standardized and often repeated components connected bylong, thin, straight fluid channels. We propose a novel discretizationalgorithm for simulating the Stokes equations on geometry with these features,which produces sparse linear systems with many repeated matrix blocks. Thediscretization is formally third order accurate for velocity and second orderaccurate for pressure in the $L^\infty$ norm. We also propose a novel linearsystem solver based on cyclic reduction, reordered sparse Gaussian elimination,and operation caching that is designed to efficiently solve systems withrepeated matrix blocks

Numerical Techniques for Optimization Problems with PDE Constraints

[no abstract available

Recurring automated model calibration for dynamically adaptive water distribution networks

The ability to build and maintain accurate hydraulic models in the water industry is a challenge of increasing importance as the applications of these models are used for short-term operational management, as well as strategic infrastructure investment and repair decisions. However, uncertainties are introduced into hydraulic models as water networks evolve over time, both in terms of physical degradation and changing consumer demand within urban environments. The process of calibrating and validating the hydraulic model can bring back confidence to the end user, but it is traditionally only undertaken on an ad hoc basis as it is a labour intensive and expensive exercise with frequently questionable results. This thesis utilises the ever-increasing quantity and quality of hydraulic data to develop robust and efficient model fitting methods for the purpose of hydraulic model calibration. Furthermore, data from long-term telemetry systems is used, together with advanced control, to study automatic and recurrent model validation for the continuous maintenance of dynamically adaptive networks. Methods for model calibration have been extensively discussed in previously published literature. However, there have been key limitations, which were exacerbated by using theoretical water networks with fictitious data. These case studies often disregard the constraints of an operational model from the water industry, hence an underlying principle of this work is to implement advanced calibration procedures on operational networks and investigate the level of accuracy that can be achieved. Investigations into reducing the ill-posedness of the problem and its scalability are also undertaken. Passive and active data sampling approaches are developed for continuously maintaining and improving the accuracy of the hydraulic model. Passive data sampling approaches use machine learning techniques to sample hydraulic data streams over an extended period, utilising natural variations and changes in control to improve the prediction accuracy of the hydraulic model. Active data sampling approaches involve optimally modifying hydraulic conditions via remotely actuated valves within dynamically adaptive networks for the purpose of improving the prediction accuracy of the hydraulic model. Three large case studies that provide hydraulic data with high temporal and spatial resolution are used as unique test areas for implementing the methods presented in the thesis. Using these methods, water companies can now recurrently validate and maintain their models, and as data and control become more ubiquitous, the process of automatic recurrent model validation will be further enhanced.Open Acces