The MUSCL-Hancock finite volume method with different slope limiters has been analyzed in the context of a fast transient flow problem. A derivation and analysis of the axial forces inside a pipe system due to a flow transient is also performed.  The following slope limiters were implemented and compared: MC, van Leer, van Albada, Minmod and Superbee. The comparison was based on the method's ability to calculate the forces due to a flow transient inside a pipe system. The tests and comparisons in this thesis show that the MC, van Leer, van Albada and Minmod limiters behave very much the same for the flow transient problem. If one would rank these four limiters with respect to the numerical error, the order would be the one presented above, the MC limiter being the most accurate. The error the four limiters produce is mainly of diffusive nature and it is just the magnitude of the diffusion that seems to differ between the methods. One should also note that the workload rank of the four limiters is the same as the order presented above. The MC limiter being the least efficient of the four and the Minmod limiter the most efficient. In most of the tests performed the Superbee limiter display a rather negative unpredictable behavior. For some relatively simple cases this particular approach shows big difficulties maintaining the dynamical properties of the force. However, the upside of the Superbee limiter is its remarkable ability to maintain the maximum value of the forces present in the pipe system, preventing underestimation of the maximum magnitude of the force.

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This work develops finite element methods with high order stabilization, and robust and efficient adaptive algorithms for Large Eddy Simulation of turbulent compressible flows. The equations are approximated by continuous piecewise linear functions in space, and the time discretization is done in implicit/explicit fashion: the second order Crank-Nicholson method and third/fourth order explicit Runge-Kutta methods. The full residual of the system and the entropy residual, are used in the construction of the stabilization terms. These methods are consistent for the exact solution, conserves all the quantities, such as mass, momentum and energy, is accurate and very simple to implement. We prove convergence of the method for scalar conservation laws in the case of an implicit scheme. The convergence analysis is based on showing that the approximation is uniformly bounded, weakly consistent with all entropy inequalities, and strongly consistent with the initial data. The convergence of the explicit schemes is tested in numerical examples in 1D, 2D and 3D. To resolve the small scales of the flow, such as turbulence fluctuations, shocks, discontinuities and acoustic waves, the simulation needs very fine meshes. In this thesis, a robust adjoint based adaptive algorithm is developed for the time-dependent compressible Euler/Navier-Stokes equations. The adaptation is driven by the minimization of the error in quantities of interest such as stresses, drag and lift forces, or the mean value of some quantity. The implementation and analysis are validated in computational tests, both with respect to the stabilization and the duality based adaptation. / QC 20110627

Three-dimensional topological insulators are materials that have a bulk band gap like a traditional insulator, but which hold topologically protected conducting surface states. In this thesis we present a numerical analysis of the surface states of topological insulator nanowires in the tight-binding approximation. We carry out the calculations at zero temperature under the presence of coaxial and perpendicular magnetic fields using Dirac Hamiltonians to model the surface. The results are obtained using Kwant, a Python package first developed in 2014 by Groth et al. for the purpose of aiding in the creation of quantum transport simulations in tight-binding models. The main focus is the self-contained and complete study of the behaviour of the conductance in clean and disordered systems, as well as to serve as an introduction to Kwant. We first study the main properties of quantum transport in mesoscopic systems, and present the scattering problem in the tight-binding approximation, which is the one treated in Kwant. We review the main properties of topological insulators, as well as the history of their discovery. We then present Kwant in detail, and illustrate its inner workings by considering the example of a clean wire. We study clean wires and show the existence of the perfectly transmitted mode under a coaxial magnetic field, obtain the quantisation of the conductance expected from the Laundauer-Büttiker formalism, and recover Fabry-Pérot oscillations when considering highly doped leads. We discuss how disorder can be introduced in our systems to simulate more realistic models, analyse its effects in the period of the conductance oscillations, and recover the robustness to disorder of the perfectly transmitted mode. Finally, we comment on how this thesis can be expanded to cover a wider range of systems and phenomena. / Tredimensionella topologiska isolatorer är material som har ett bulkbandgap som traditionella isolatorer, men som har topologiskt skyddade ledande yttilstånd. I detta arbete presenterar vi en numerisk analys av yttilstånden hos topologiska isolator nanotrådar i tight-binding approximationen vid nolltemperatur, under närvaron av koaxiala och vinkelräta magnetfält med användning av Dirac-Hamiltonians för att modellera ytan. Resultaten erhålls med hjälp av Kwant, ett Python-paket som först utvecklades 2014 av Groth et al. i syfte att underlätta skapandet av simuleringar för kvanttransport i tight-binding modeller. Huvudfokus ligger på en självständig och komplett studie av beteendet hos konduktansen i rena och oordnade system, samt att fungera som en introduktion till Kwant. Vi studerar först de huvudsakliga egenskaperna hos kvanttransport i mesoskopiska system och presenterar spridningsproblemet i tight-binding approximationen, vilket är det som behandlas i Kwant. Dessutom går vi igenom de viktigaste egenskaperna hos topologiska isolatorer, samt deras upptäckthistoria. Sedan pre- senterar vi Kwant i detalj och illustrerar dess inre funktioner genom att titta på en ren tråd. Vi studerar rena trådar och visar förekomsten av det perfekt överförda läget under ett koaxialt magnetfält, erhåller kvantiseringen av den förväntade konduktansen från Laundauer-Büttiker-formalismen och återfår Fabry-Pérot-oscillationer när vi överväger starkt dopade ledare. Sedan diskuterar vi hur oordning kan införas i våra system för att simulera mer realistiska modeller, analysera dess effekter under tiden för oscillationer vid konduktans och återfå robustheten mot oordning av det perfekt överförda läget. Slutligen kommenterar vi hur detta arbete kan utvidgas för att täcka ett bredare spektrum av system och fenomen.

Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system occurs. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case. When applying an iterative solver to an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence performance of Landweber-type iteration, and propose new ways to specify the relaxation parameters. These are computed so as to control the propagated error. We also describe a class of stopping rules for Landweber-type iteration for solving linear inverse problems. The class includes the well known discrepancy principle, and the monotone error rule. We unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections. Kaczmarz's method, also called ART (Algebraic Reconstruction Technique) is often used for solving the linear system which appears in image reconstruction. This is a fully sequential method. We examine and compare ART and its symmetric version. It is shown that the cycles of symmetric ART, unlike ART, converge to a weighted least squares solution if and only if the relaxation parameter lies between zero and two. Further we show that ART has faster asymptotic rate of convergence than symmetric ART. Also a stopping criterion is proposed and evaluated for symmetric ART. We further investigate a class of block-iterative methods used in image reconstruction. The cycles of the iterative sequences are characterized in terms of the original linear system. We define symmetric block-iteration and compare the behavior of symmetric and non-symmetric block-iteration. The results are illustrated using some well-known methods. A stopping criterion is offered and assessed for symmetric block-iteration.

This dissertation addresses the development and use of novel software tools and environments for the computation and visualization of canonical information as well as stratification hierarchies for matrices and matrix pencils. The simplest standard shape to which a matrix pencil with a given set of eigenvalues can be reduced is called the Kronecker canonical form (KCF). The KCF of a matrix pencil is unique, and all pencils in the manifold of strictly equivalent pencils - collectively termed the orbit - can be reduced to the same canonical form and so have the same canonical structure. For a problem with fixed input size, all orbits are related under small perturbations. These relationships can be represented in a closure hierarchy with a corresponding graph depicting the stratification of these orbits. Since degenerate canonical structures are common in many applications, software tools to determine canonical information, especially under small perturbations, are central to understanding the behavior of these problems. The focus in this dissertation is the development of a software tool called StratiGraph. Its purpose is the computation and visualization of stratification graphs of orbits and bundles (i.e., union of orbits in which the eigenvalues may change) for matrices and matrix pencils. It also supports matrix pairs, which are common in control systems. StratiGraph is extensible by design, and a well documented plug-in feature enables it, for example, to communicate with Matlab(TM). The use and associated benefits of StratiGraph are illustrated via numerous examples. Implementation considerations such as flexible software design, suitable data representations, and good and efficient graph layout algorithms are also discussed. A way to estimate upper and lower bounds on the distance between an input S and other orbits is presented. The lower bounds are of Eckhart-Young type, based on the matrix representation of the associated tangent spaces. The upper bounds are computed as the Frobenius norm F of a perturbation such that S + F is in the manifold defining a specified orbit. Using associated plug-ins to StratiGraph this information can be computed in Matlab, while visualization alongside other canonical information remains within StratiGraph itself. Also, a proposal of functionality and structure of a framework for computation of matrix canonical structure is presented. Robust, well-known algorithms, as well algorithms improved and developed in this work, are used. The framework is implemented as a prototype Matlab toolbox. The intention is to collect software for computing canonical structures as well as for computing bounds and to integrate it with the theory of stratification into a powerful new environment called the MCS toolbox. Finally, a set of utilities for generating web computing environments related to mathematical and engineering library software is presented. The web interface can be accessed from a standard web browser with no need for additional software installation on the local machine. Integration with the control and systems library SLICOT further demonstrates the efficacy of this approach.

Rational Krylov methods for model order reduction are studied. A dual rational Arnoldi method for model order reduction and a rational Krylov method for model order reduction and eigenvalue computation have been implemented. It is shown how to deflate redundant or unwanted vectors and how to obtain moment matching. Both methods are designed for generalised state space systems---the former for multiple-input-multiple-output (MIMO) systems from finite element discretisations and the latter for single-input-single-output (SISO) systems---and applied to relevant test problems. The dual rational Arnoldi method is designed for generating real reduced order systems using complex shift points and stabilising a system that happens to be unstable. For the rational Krylov method, a forward error in the recursion and an estimate of the error in the approximation of the transfer function are studie. A stability analysis of a heat exchanger model is made. The model is a nonlinear partial differential-algebraic equation (PDAE). Its well-posedness and how to prescribe boundary data is investigated through analysis of a linearised PDAE and numerical experiments on a nonlinear DAE. Four methods for generating reduced order models are applied to the nonlinear DAE and compared: a Krylov based moment matching method, balanced truncation, Galerkin projection onto a proper orthogonal decomposition (POD) basis, and a lumping method. / QC 20101013

<p>Rational Krylov methods for model order reduction are studied. A dual rational Arnoldi method for model order reduction and a rational Krylov method for model order reduction and eigenvalue computation have been implemented. It is shown how to deflate redundant or unwanted vectors and how to obtain moment matching. Both methods are designed for generalised state space systems---the former for multiple-input-multiple-output (MIMO) systems from finite element discretisations and the latter for single-input-single-output (SISO) systems---and applied to relevant test problems. The dual rational Arnoldi method is designed for generating real reduced order systems using complex shift points and stabilising a system that happens to be unstable. For the rational Krylov method, a forward error in the recursion and an estimate of the error in the approximation of the transfer function are studie.</p><p>A stability analysis of a heat exchanger model is made. The model is a nonlinear partial differential-algebraic equation (PDAE). Its well-posedness and how to prescribe boundary data is investigated through analysis of a linearised PDAE and numerical experiments on a nonlinear DAE. Four methods for generating reduced order models are applied to the nonlinear DAE and compared: a Krylov based moment matching method, balanced truncation, Galerkin projection onto a proper orthogonal decomposition (POD) basis, and a lumping method.</p>

The dry deep mixing method is widely used to improve a soft clay soil to increase the shear strength as well as to reduce the time for consolidation. It is a mechanical mixing process that makes parts of the soil stiffer than its original strength. It is mainly applicable to soft clay or peat soil.   In this master thesis, the objective was set to perform a comparative analysis on the prediction of the settlements of a clay soil improved by lime/cement columns (LCC). The theoretical settlement predictions were made using two analytical and numerical modeling. A case study was carried out on a part of Stockholm bypass project where LCC was applied to improve soft clay for a foundation of a concrete trough. Field measurements of the vertical deformation of the improved soil were performed using settlement plates to compare the analytical and numerical results. The first analytical method was performed based on the recommendation of TK Geo 13 (2013) while the second method was performed based on the concept of a composite ground. In the case of the numerical method, FEA was performed using 2D plane strain model in Plaxis simulation. The performance of the geometry and combined matching models were investigated to convert the axisymmetric to plane strain model. The variation in stiffness of the columns were taken into consideration by applying two stiffness values 30 and 33 MPa for the upper and lower half of the column respectively. A preload of 58 kPa was applied on the improved clay soil to simulate the time-dependent consolidation settlement due to the stress addition. A comparison was carried out between the results obtained from the analysis and a field measurement. The two analytical methods produced a better agreement with the field measurement regarding long-term consolidation settlement and a reasonable agreement concerning the rate of consolidation. The numerical analysis showed a good agreement with the benchmark concerning both the long-term consolidation settlement as well as the rate of consolidation. The geometry matching model gave a reasonable result regarding correctness of the result compared with the combined matching. Based on the results obtained in this study, the numerical methods had a better agreement with the measurements. / Jordförstärkning med kalkcementpelare är en vanlig metod för förstärkning av lösa jordar genom ökning av den blandade jordens hållfasthet samt minskning av konsolideringstiden. Metoden är en mekanisk process som ökar jordens styvhet och är främst tillämpbar i lös leror men även organiska jordar.   Detta examensarbete har syftat till att jämföra sättningsberäkningar i lera som är förstärk med KC-pelare. De teoretiska beräkningarna har utförts genom två analytiska modeller samt numerisk modellering. En fallstudie har utförts på del av Förbifart Stockholm där jordförstärkning av lös lera med KC-pelare har använts inför grundläggning av ett betongtråg. Resultat från fältmätningar av installerade markpeglar har jämförts med resultat från de teoretiska sättningsberäkningarna.   Den första beräkningsmetoden utfördes i enlighet med rekommendationer från TK Geo 13 (2013) och den andra metoden är baserad på principer för kompositjordar. Den numeriska beräkningen har utgjorts av FEM-modellering i 2D i programmet Plaxis. För att anpassa en plan-töjningsmodell till en axialsymmetrisk modell har inverkan av geometrin samt kombinerad anpassning av modell studerats. Hänsyn har tagits till KC-pelarnas styvhet genom att använda två olika värden (30 resp. 33 MPa) för KC-pelarnas övre respektive undre del. En överlast om 58 kPa applicerades på KC-pelarförstärkt området för att påskynda den tidsberoende konsolideringssättningarnas förlopp som orsakas av överlastens tillskottspänningar.   Baserat på resultat från uppmätta sättningar jämfört med beräkningar, har följande slutsatser dragits. Jämförelser mellan resultaten har visat på en rimlig överrensstämmelse mellan de två analytiska metoderna och utförda fältmätningar avseende långtids konsolideringssättningar. Den numeriska beräkningen har visat en god överensstämmelse med fältmätningar med hänsyn till både konsolideringssättningar och konsolideringsgraden. Den geometriskt anpassade modellen visade ett rimligare resultat i förhållande till den kombinerade anpassade modellen. Sammanfattningsvis bedöms det att den numeriska modelleringen stämmer bättre överens med resultaten från uppmätta sättningar i förhållande till analytiska beräkningar.

This study explores the performance of porous Paptic paper materials composed of a mixture of softwood and lyocell fibers. The investigation involves laboratory experiments and numerical simulations to analyse the impact of various parameters on the paper's characteristics. Tensile and hygroexpansion tests were conducted on sheets with different binder quantities and drying methods. VTT aided in the analysis of mechanical properties using tomography images. The objective was to determine the optimal binder content, to understand the behaviour of the paper under different drying conditions and to optimise the pulp mixture through numerical simulations. Experimental tests involved producing paper sheets with varying binder amounts and different drying methods. Tensile tests were conducted to assess the elastic stiffness, strength, and strain at break. Constrained dried and freely dried papers were compared to evaluate the influence of drying conditions. Hygroexpansion tests were performed to examine the water storage behaviour of papers with added binders. Tomography measurements provided the density profile, which was replicated in the numerical sheets. A micro-mechanical model was employed for numerical simulations, representing each fiber as a beam. The model was calibrated using stress-strain data from VTT's tensile testing of the paper with the highest binder content. The influence of altering the amount and length of lyocell fibers was examined to optimise the pulp. From the tensile tests, an optimal binder content was identified that yielded the highest elastic stiffness while considering the density increase caused by binders. Further additions of binders did not enhance elastic stiffness. However, no optimal value was found for strength and strain at break, as both parameters continued to increase with additional binders. Tensile tests comparing constrained dried and freely dried papers showed similar behaviour, suggesting inadequate constraint in the former. Hygroexpansion tests confirmed the similarities between the two drying methods and revealed that papers with added binders stored less water at a given humidity. Additionally, the drying-moistening cycling exhibited an unusual behaviour not observed in conventional paper, with irreversible expansion occurring during the first drying cycle. Numerical simulations using a micro-mechanical model demonstrated that higher amounts of lyocell fibers improved performance, increasing strength and strain at break. However, varying fiber length did not yield significant improvements in these parameters, although stiffness showed a slight increase. While the literature suggests that the addition of long lyocell fibers decreases paper strength, this study found that when maintaining constant bulk, strength increased under the assumption that the bonding strength was unaffected by lyocell fibers. Furthermore, numerical simulations indicated that an even density profile throughout the paper thickness resulted in higher strength and strain at break. The tomography data revealed that the density profile is affected by the binder quantity. With the addition of binders, the thickness decreased even though the grammage increased. The density is high on the top and bottom surface of the papers which contain more binders while the density is lower in the middle. This difference in density is more pronounced with higher amounts of binders. / Denna studie undersöker prestandan hos porösa pappersmaterial från Paptic som består av en blandning av barrträd och lyocell-fibrer. Undersökningen innefattar laboratorieexperiment och numeriska simuleringar för att analysera effekten av olika parametrar på papperets egenskaper. Drag- och hygroexpansionstester utfördes på ark med olika mängder bindemedel och torkningsmetoder. VTT hjälpte till med analysen av mekaniska egenskaper genom tomografibilder. Målet var att bestämma optimalt innehåll av bindemedel, att förstå papperets beteende under olika torkningsförhållanden och att optimera pappersmassan genom numeriska simuleringar.  De experimentella testerna innefattade framställning av pappersark med varierande mängder bindemedel och olika torkningsmetoder. Dragtester utfördes för att bedöma elastisk styvhet, styrka och töjning vid brott. Inspännt samt fritt torkade papper jämfördes för att utvärdera inflytandet av torkningsförhållanden. Hygroexpansionstester utfördes för att undersöka vattenlagringsbeteendet hos papper med tillsatt bindemedel. Tomografimätningar gav papprens densitetsprofilen, som återskapades i de numeriska arken. En mikromekanisk modell användes för de numeriska simuleringarna, där varje fiber representeras som en balk. Modellen kalibrerades med hjälp av spännings-töjnings-data från VTT:s dragprovning av papperet med högst mängd bindemedel. Inflytandet av att ändra mängden och längden på lyocell-fibrer undersöktes för att optimera pappersmassan.  Från dragtesterna identifierades en optimal mängd bindemedel som gav högst elastisk styvhet när den densitetsökning som orsakas av bindemedel har tagits till hänsyn. Ytterligare tillsatser av bindemedel förbättrade inte den elastiska styvheten. Dock hittades ingen optimal värde för styrka och brottsträckning, eftersom båda parametrarna fortsatte öka med ytterligare bindemedel. Dragtester som jämförde inspännt och fritt torkat papper visade liknande beteende, vilket tyder på otillräcklig begränsning i det förstnämnda fallet. Hygroexpansionstester bekräftade likheterna mellan de två torkningsmetoderna och avslöjade att papper med tillsatt bindemedel lagrade mindre vatten vid en given luftfuktighet. Dessutom uppvisade torknings-fuktnings-cykeln ett ovanligt beteende som inte observerats i konventionellt papper, där irreversibel expansion inträffade under den första torkningscykeln. Numeriska simuleringar med en mikromekanisk modell visade att högre mängder lyocell-fibrer förbättrade prestandan genom att öka styrka och töjning vid brott. Ändringar i fibrernas längd gav dock inte märkvärda förbättringar i dessa parametrar, även om styvheten visade en liten ökning. Medan litteraturen föreslår att tillsatsen av långa lyocell-fibrer minskar papperets styrka, fann denna studie att när densiteten hålls konstant ökar styrkan under antagandet att bindestyrkan inte påverkas av lyocell-fibrer. Dessutom indikerade de numeriska simuleringarna att en jämn densitetsprofil genom papperets tjocklek resulterade i högre styrka och töjning vid brott. Tomografidata visade att densitetsprofilen påverkas av mängden bindemedel. När mängden bindemedel ökade, minskade tjockleken trots att ytvikten ökade. Densiteten är hög på papperets över- och undersida, medan densiteten är lägre i mitten. Skillnaden i densitet är mer uttalad med högre mängder bindemedel.