Moving contact line problems appear in many natural and industrial processes. A contact line is formed where the interface between two immiscible fluids meets a solid wall. Examples from everyday life include raindrops falling on a window and water bugs resting on water surfaces. In many cases the dynamics of the contact line affects the overall behavior of the system. Industrial applications where the contact line behavior is important include gas and oil recovery in porous media, lubrication, inkjet printing and microfluidics. Computer simulations are fundamental tools to understand and predict the behavior.   In this thesis we look at numerical simulations of dynamic contact line problems. Despite their importance, the physics of moving contact lines is poorly understood. The standard Navier-Stokes equations together with the conventional no-slip boundary condition predicts a singularity in the shear stresses at the contact line. Atomistic processes at the contact line come into play, and it is necessary to include these processes in the model to resolve the singularity. In the case of capillary driven flows for example, it has been observed that the microscopic contact line dynamics has a large impact on the overall macroscopic flow. In Paper I we present a new multiscale model for numerical simulation of flow of two immiscible and incompressible fluids in the presence of moving contact points (i.e. two-dimensional problems). The paper presents a new boundary methodology based on combining a relation between the apparent contact angle and the contact point velocity, and a similarity solution for Stokes flow at a planar interface (the analytic Huh and Scriven velocity). The relation between the angle and the velocity is determined by performing separate microscopic simulations. The classical Huh and Scriven solution is only valid for flow over flat walls. In Paper II we use perturbation analysis to extend the solution to flow over curved walls. Paper III presents the parallel finite element solver that is used to perform the numerical experiments presented in this thesis. Finally, the new multiscale model (presented in Paper I) is applied to a relevant microfluidic research problem in Paper IV. For this problem it is very important to have a model that accurately takes the atomistic effects at contact lines into account.

To ensure a high level of safety and reliability of electronic/electric systems EMC (electromagnetic compatibility) tests together with computational techniques are used. In this thesis, mathematical modeling and computational electromagnetics are applied to mainly two case studies. In the first case study, electromagnetic modeling of electric networks and antenna structures above, and buried in, the ground are studied. The ground has been modelled either as a perfectly conducting or as a dielectric surface.  The second case study is focused on mathematical modeling and algorithms to solve the direct and inverse electromagnetic scattering problem for providing a model-based illustration technique. This electromagnetic scattering formulation is applied to describe a microwave imaging system called Breast Phantom. The final goal is to simulate and detect cancerous tissues in the human female breast by this microwave technique.   The common issue in both case studies has been the long computational time required for solving large systems of equations numerically. This problem has been dealt with using approximation methods, numerical analysis, and also parallel processing of numerical data. For the first case study in this thesis, Maxwell’s equations are solved for antenna structures and electronic networks by approximation methods and parallelized algorithms implemented in a LAN (Local Area Network). In addition, PMM (Point-Matching Method) has been used for the cases where the ground is assumed to act like a dielectric surface. For the second case study, FDTD (Finite-Difference Time Domain) method is applied for solving the electromagnetic scattering problem in two dimensions. The parallelized numerical FDTD-algorithm is implemented in both Central Processing Units (CPUs) and Graphics Processing Units (GPUs). / För att säkerställa människors säkerhet och tillförlitligheten hos elektriska/elektroniska system används EMC (elektromagnetisk kompatibilitet)-tester i kombination med matematisk modellering. För att undersöka biologiska vävnaders egenskaper används så kallade elektromagnetiska spridningsmetoder vid sidan om elektromagnetisk modellering. I denna avhandling har matematisk modellering och beräkningsmetoder använts för huvudsakligen två fallstudier. Den första fallstudien handlar om att analysera antennstrukturer och elektriska nät ovanför, och nergrävda i marken. Marken har modellerats antingen som en elektriskt ledande yta eller en dielektrisk yta. Den andra fallstudien fokuserar på matematisk modellering och algoritmer för att lösa ett elektromagnetiskt spridningsproblem för att beskriva en modellbaserad illustrationsteknik. Spridningsformuleringen tillämpas för att modellera ett avbildningssystem som använder mikrovågor, kallat Bröstfantomen. Det slutliga målet är att upptäcka cancervävnader i kvinnobröst genom denna mikrovågsteknik. Flaskhalsen i de båda fallstudierna har visat sig vara de långa beräkningstider som krävs för att lösa stora numeriska system. För att lösa problemet har approximationsmetoder, numerisk analys och även parallella beräkningar genomförts i detta arbete. För den första fallstudien har Maxwells ekvationer lösts genom CEM (Complex Image Methods) och med parallellisering i ett LAN (Local Area Network). I de fall där marken betraktas som en dielektrisk yta, har PMM (Point-Matching Method) tillämpats. I samband med den andra fallstudien har FDTD (Finite-Difference Time Domain) metoder tillämpats för att lösa ett elektromagnetiskt spridningsproblem i två dimensioner. En parallelliserad FDTD-algoritm har implementerats i både CPU:s (Central Processing Units) och GPU:s (Graphics Processing Units). / RALF3

In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.

In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems. The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times. In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid. In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.

In this thesis, techniques for efficient utilization of modern computer hardwarefor numerical simulation are considered. In particular, we study techniques for improving the performance of computations using the finite element method. One of the main difficulties in finite-element computations is how to perform the assembly of the system matrix efficiently in parallel, due to its complicated memory access pattern. The challenge lies in the fact that many entries of the matrix are being updated concurrently by several parallel threads. We consider transactional memory, an exotic hardware feature for concurrent update of shared variables, and conduct benchmarks on a prototype multicore processor supporting it. Our experiments show that transactions can both simplify programming and provide good performance for concurrent updates of floating point data. Secondly, we study a matrix-free approach to finite-element computation which avoids the matrix assembly. In addition to removing the need to store the system matrix, matrix-free methods are attractive due to their low memory footprint and therefore better match the architecture of modern processors where memory bandwidth is scarce and compute power is abundant. Motivated by this, we consider matrix-free implementations of high-order finite-element methods for execution on graphics processors, which have seen a revolutionary increase in usage for numerical computations during recent years due to their more efficient architecture. In the implementation, we exploit sum-factorization techniques for efficient evaluation of matrix-vector products, mesh coloring and atomic updates for concurrent updates, and a geometric multigrid algorithm for efficient preconditioning of iterative solvers. Our performance studies show that on the GPU, a matrix-free approach is the method of choice for elements of order two and higher, yielding both a significantly faster execution, and allowing for solution of considerably larger problems. Compared to corresponding CPU implementations executed on comparable multicore processors, the GPU implementation is about twice as fast, suggesting that graphics processors are about twice as power efficient as multicores for computations of this kind.

This thesis focuses on computationally efficient methods for flow in fractured porous media. Two approaches are presented where the interface is embedded on the underlying finite element mesh. The methods allow for representation of the interface geometry from the underlying discretization and with discontinuities across the interface. However, embedding interfaces raises stability concerns in which we alleviate using stabilization terms. The aim of this thesis is to present the basics of the two main approaches and to provide brief details on the mathematics involved. / Denna avhandling fokuserar på effektiva beräkningsmetoder för flöde i porösa media med sprickor. Två tillvägagångssätt presenteras där sprickan tillåts skära det underliggande finita elementnätet. Sprickans inverkan på flödet tas om hand med hjälp av den underliggande diskretiseringen som tillåter diskontinuiteter. Med andra ord kan flöden modelleras med olika egenskaper; på var sida av sprickan, samt längs sprickan. Metoden tar även hand om instabilitet som uppstår dels på grund av godtyckliga skärningar av bakgrundselementen och dels på grund av olika materialegenskaper. Syftet med denna avhandling är att presentera grunderna för dessa metoder och ge grundläggande matematiska förklaringar.

Underground mining is a complex operation that requires careful planning. The short-term scheduling, which is the scheduling of the tasks involved in the excavation process, is an important part of the planning process. In this master thesis we propose a new method for short-term scheduling of a cut-and-fill mine operated by the mining company Boliden AB. We include a new aspect of the problem by incorporating a priority between the excavation locations of the mine. The priority feature allows the user to control the output of the scheduling and to direct resources to the locations where they are most needed according to the long-term plans. Our solution method consists of two components: a constructive heuristic method that construct a complete solution by solving partial scheduling problems containing subsets of tasks, and a logic-based Benders decomposition scheme for solving these partial problems. The computational performance of the proposed method is evaluated on industrially relevant largescale instances generated from data provided by Boliden. Comparisons are made with applying a constraint programming solver on the complete problem and with replacing the logic-based Benders scheme by applying a constraint programming solver on the partial scheduling problems, respectively. Results show that the heuristic method combined with the logic-based Benders decomposition scheme outperforms the other two methods on all instances.

This thesis introduces a Large Neighborhood Search (LNS) method to solve a multi-processor avionics scheduling problem. In a typical scheduling problem, tasks are scheduled with exact starting times. In this thesis however, tasks will instead be assigned to disjoint time segments, called buckets. For an assignment to be feasible, precedence relations and capacity constraints related to network and computing resources need to be fulfilled. The introduced LNS method relies on solving Mixed-Integer Programming (MIP)-models. To make progress in the search for a feasible assignment, we construct a MIP-model that allows violation of the problem constraints at a cost of increased objective value. The LNS method uses two operators, a destroy operator that chooses a set of tasks that are allowed to change buckets, and a repair operator that through solving the MIP-model creates a new schedule. This thesis develops 11 types of destroy operators and 30 (concrete) variants of them. The MIP-based LNS is evaluated on a set of 60 instances with up to 84 000 tasks and 21 processors. The instances belongs to six categories of varying difficulty. The MIP-based LNS solves 50 instances within our time limit, and the largest instance solved has 77 757 tasks. This is significantly better than solving the complete MIP-model in a single step. With this approach only 36 instances can be solved within our time limit and the largest instance solved has 48554 tasks.

There is a large potential for automation and optimization in radio access networks (RANs) using a data-driven approach to efficiently handle the increase in complexity due to the steep growth in traffic and new technologies introduced with the development of 5G. Reinforcement learning (RL) has natural applications in RAN control loops such as link adaptation, interference management and power control at different timescales commonly occurring in the RAN context. Elevating the status of data-driven solutions in RAN and building a new, scalable, distributed and data-friendly RAN architecture will be needed to competitively tackle the challenges of coming 5G networks. In this work, we propose a systematic, efficient and robust methodology for applying RL on different control problems. Firstly, the proposed methodology is evaluated using a well-known control problem. Then, it is adapted to a real-world RAN scenario. Extensive simulation results are provided to show the effectiveness and potential of the proposed approach. The methodology was successfully created but results on a RAN-simulator were not mature / Det finns en stor potential automatisering och optimering inom radionätverk (RAN, radio access network) genom att använda datadrivna lösningar för att på ett effektivt sätt hantera den ökade komplexiteten på grund av trafikökningar and nya teknologier som introducerats i samband med 5G. Förstärkningsinlärning (RL, reinforcement learning) har naturliga kopplingar till reglerproblem i olika tidsskalor, såsom länkanpassning, interferenshantering och kraftkontroll, vilket är vanligt förekommande i radionätverk. Att förhöja statusen på datadrivna lösningar i radionätverk kommer att vara nödvändigt för att hantera utmaningarna som uppkommer med framtida 5G nätverk. I detta arbete föreslås vi en syetematisk metodologi för att applicera RL på ett reglerproblem. I första hand används den föreslagna metodologin på ett välkänt reglerporblem. Senare anpassas metodologin till ett äkta RAN-scenario. Arbetet inkluderar utförliga resultat från simuleringar för att visa effektiviteten och potentialen hos den föreslagna metoden. En lyckad metodologi skapades men resultaten på RAN-simulatorn saknade mognad.

This thesis investigates dierent types of Monte Carlo estimators for use in computationof chemical system, mainly to be used in calculating surface growthand evolution of SiC. Monte Carlo methods are a class of algorithms using randomsampling to numerical solve problems and are used in many cases. Threedierent types of Monte Carlo methods are studied, a simple Monte Carlo estimatorand two types of Markov chain Monte Carlo Metropolis algorithm MonteCarlo and kinetic Monte Carlo. The mathematical background is given for allmethods and they are tested both on smaller system, with known results tocheck their mathematical and chemical soundness and on larger surface systemas an example on how they could be used