Exact quantum Monte Carlo methods for correlated electron systems

Thema dieser Arbeit ist die Entwicklung und Kombination verschiedener numerischer Methoden, sowie deren Anwendung auf Probleme stark korrelierter Elektronensysteme. Solche Materialien zeigen viele interessante physikalische Eigenschaften, wie z.B. Supraleitung und magnetische Ordnung und spielen eine bedeutende Rolle in technischen Anwendungen. Es werden zwei verschiedene Modelle behandelt: das Hubbard-Modell und das Kondo-Gitter-Modell (KLM). In den letzten Jahrzehnten konnten bereits viele Erkenntnisse durch die numerische Lösung dieser Modelle gewonnen werden. Dennoch bleibt der physikalische Ursprung vieler Effekte verborgen. Grund dafür ist die Beschränkung aktueller Methoden auf bestimmte Parameterbereiche. Eine der stärksten Einschränkungen ist das Fehlen effizienter Algorithmen für tiefe Temperaturen.rnrnBasierend auf dem Blankenbecler-Scalapino-Sugar Quanten-Monte-Carlo (BSS-QMC) Algorithmus präsentieren wir eine numerisch exakte Methode, die das Hubbard-Modell und das KLM effizient bei sehr tiefen Temperaturen löst. Diese Methode wird auf den Mott-Übergang im zweidimensionalen Hubbard-Modell angewendet. Im Gegensatz zu früheren Studien können wir einen Mott-Übergang bei endlichen Temperaturen und endlichen Wechselwirkungen klar ausschließen.rnrnAuf der Basis dieses exakten BSS-QMC Algorithmus, haben wir einen Störstellenlöser für die dynamische Molekularfeld Theorie (DMFT) sowie ihre Cluster Erweiterungen (CDMFT) entwickelt. Die DMFT ist die vorherrschende Theorie stark korrelierter Systeme, bei denen übliche Bandstrukturrechnungen versagen. Eine Hauptlimitation ist dabei die Verfügbarkeit effizienter Störstellenlöser für das intrinsische Quantenproblem. Der in dieser Arbeit entwickelte Algorithmus hat das gleiche überlegene Skalierungsverhalten mit der inversen Temperatur wie BSS-QMC. Wir untersuchen den Mott-Übergang im Rahmen der DMFT und analysieren den Einfluss von systematischen Fehlern auf diesen Übergang.rnrnEin weiteres prominentes Thema ist die Vernachlässigung von nicht-lokalen Wechselwirkungen in der DMFT. Hierzu kombinieren wir direkte BSS-QMC Gitterrechnungen mit CDMFT für das halb gefüllte zweidimensionale anisotrope Hubbard Modell, das dotierte Hubbard Modell und das KLM. Die Ergebnisse für die verschiedenen Modelle unterscheiden sich stark: während nicht-lokale Korrelationen eine wichtige Rolle im zweidimensionalen (anisotropen) Modell spielen, ist in der paramagnetischen Phase die Impulsabhängigkeit der Selbstenergie für stark dotierte Systeme und für das KLM deutlich schwächer. Eine bemerkenswerte Erkenntnis ist, dass die Selbstenergie sich durch die nicht-wechselwirkende Dispersion parametrisieren lässt. Die spezielle Struktur der Selbstenergie im Impulsraum kann sehr nützlich für die Klassifizierung von elektronischen Korrelationseffekten sein und öffnet den Weg für die Entwicklung neuer Schemata über die Grenzen der DMFT hinaus.

Numerical and semi-analytical models of sliding masses: application to the 1783 Scilla tsunamigenic landslide

The Scilla rock avalanche occurred on 6 February 1783 along the coast of the Calabria region (southern Italy), close to the Messina Strait. It was triggered by a mainshock of the Terremoto delle Calabrie seismic sequence, and it induced a tsunami wave responsible for more than 1500 casualties along the neighboring Marina Grande beach. The main goal of this work is the application of semi-analtycal and numerical models to simulate this event. The first one is a MATLAB code expressly created for this work that solves the equations of motion for sliding particles on a two-dimensional surface through a fourth-order Runge-Kutta method. The second one is a code developed by the Tsunami Research Team of the Department of Physics and Astronomy (DIFA) of the Bologna University that describes a slide as a chain of blocks able to interact while sliding down over a slope and adopts a Lagrangian point of view. A wide description of landslide phenomena and in particular of landslides induced by earthquakes and with tsunamigenic potential is proposed in the first part of the work. Subsequently, the physical and mathematical background is presented; in particular, a detailed study on derivatives discratization is provided. Later on, a description of the dynamics of a point-mass sliding on a surface is proposed together with several applications of numerical and analytical models over ideal topographies. In the last part, the dynamics of points sliding on a surface and interacting with each other is proposed. Similarly, different application on an ideal topography are shown. Finally, the applications on the 1783 Scilla event are shown and discussed.

MODEL UPDATING AND STRUCTURAL HEALTH MONITORING OF HORIZONTAL AXIS WIND TURBINES VIA ADVANCED SPINNING FINITE ELEMENTS AND STOCHASTIC SUBSPACE IDENTIFICATION METHODS

Wind energy has been one of the most growing sectors of the nation’s renewable energy portfolio for the past decade, and the same tendency is being projected for the upcoming years given the aggressive governmental policies for the reduction of fossil fuel dependency. Great technological expectation and outstanding commercial penetration has shown the so called Horizontal Axis Wind Turbines (HAWT) technologies. Given its great acceptance, size evolution of wind turbines over time has increased exponentially. However, safety and economical concerns have emerged as a result of the newly design tendencies for massive scale wind turbine structures presenting high slenderness ratios and complex shapes, typically located in remote areas (e.g. offshore wind farms). In this regard, safety operation requires not only having first-hand information regarding actual structural dynamic conditions under aerodynamic action, but also a deep understanding of the environmental factors in which these multibody rotating structures operate. Given the cyclo-stochastic patterns of the wind loading exerting pressure on a HAWT, a probabilistic framework is appropriate to characterize the risk of failure in terms of resistance and serviceability conditions, at any given time. Furthermore, sources of uncertainty such as material imperfections, buffeting and flutter, aeroelastic damping, gyroscopic effects, turbulence, among others, have pleaded for the use of a more sophisticated mathematical framework that could properly handle all these sources of indetermination. The attainable modeling complexity that arises as a result of these characterizations demands a data-driven experimental validation methodology to calibrate and corroborate the model. For this aim, System Identification (SI) techniques offer a spectrum of well-established numerical methods appropriated for stationary, deterministic, and data-driven numerical schemes, capable of predicting actual dynamic states (eigenrealizations) of traditional time-invariant dynamic systems. As a consequence, it is proposed a modified data-driven SI metric based on the so called Subspace Realization Theory, now adapted for stochastic non-stationary and timevarying systems, as is the case of HAWT’s complex aerodynamics. Simultaneously, this investigation explores the characterization of the turbine loading and response envelopes for critical failure modes of the structural components the wind turbine is made of. In the long run, both aerodynamic framework (theoretical model) and system identification (experimental model) will be merged in a numerical engine formulated as a search algorithm for model updating, also known as Adaptive Simulated Annealing (ASA) process. This iterative engine is based on a set of function minimizations computed by a metric called Modal Assurance Criterion (MAC). In summary, the Thesis is composed of four major parts: (1) development of an analytical aerodynamic framework that predicts interacted wind-structure stochastic loads on wind turbine components; (2) development of a novel tapered-swept-corved Spinning Finite Element (SFE) that includes dampedgyroscopic effects and axial-flexural-torsional coupling; (3) a novel data-driven structural health monitoring (SHM) algorithm via stochastic subspace identification methods; and (4) a numerical search (optimization) engine based on ASA and MAC capable of updating the SFE aerodynamic model.

Numerical solution of optimal control problems with constant control delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a stateof- the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.

Comparison of experimental and numerical sloshing loads in partially filled tanks

Sloshing describes the movement of liquids inside partially filled tanks, generating dynamic loads on the tank structure. The resulting impact pressures are of great importance in assessing structural strength, and their correct evaluation still represents a challenge for the designer due to the high level of nonlinearities involved, with complex free surface deformations, violent impact phenomena and influence of air trapping. In the present paper, a set of two-dimensional cases, for which experimental results are available, is considered to assess the merits and shortcomings of different numerical methods for sloshing evaluation, namely two commercial RANS solvers (FLOW-3D and LS-DYNA), and two academic software (Smoothed Particle Hydrodynamics and RANS). Impact pressures at various critical locations and global moment induced by water motion in a partially filled rectangular tank, subject to a simple harmonic rolling motion, are evaluated and predictions are compared with experimental measurements. 2012 Copyright Taylor and Francis Group, LLC.

FE–BE Coupling Methods for Elastoplasticity

Non-linear physical systems of infinite extent are conveniently modelled using FE–BE coupling methods. By the combination of both methods, suitable use of the advantages of each one may be obtained. Several possibilities of FEM–BEM coupling and their performance in some practical cases are discussed in this paper. Parallelizable coupling algorithms based on domain decomposition are developed and compared with the most traditional coupling methods.

Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation

It is well known that the evaluation of the influence matrices in the boundary-element method requires the computation of singular integrals. Quadrature formulae exist which are especially tailored to the specific nature of the singularity, i.e. log(*- x0)9 Ijx- JC0), etc. Clearly the nodes and weights of these formulae vary with the location Xo of the singular point. A drawback of this approach is that a given problem usually includes different types of singularities, and therefore a general-purpose code would have to include many alternative formulae to cater for all possible cases. Recently, several authors1"3 have suggested a type independent alternative technique based on the combination of standard Gaussian rules with non-linear co-ordinate transformations. The transformation approach is particularly appealing in connection with the p.adaptive version, where the location of the collocation points varies at each step of the refinement process. The purpose of this paper is to analyse the technique in eference 3. We show that this technique is asymptotically correct as the number of Gauss points increases. However, the method possesses a 'hidden' source of error that is analysed and can easily be removed.

Numerical analysis of shell and spatial structures

Since the advent of the computer into the engineering field, the application of the numerical methods to the solution of engineering problems has grown very rapidly. Among the different computer methods of structural analysis the Finite Element (FEM) has been predominantly used. Shells and space structures are very attractive and have been constructed to solve a large variety of functional problems (roofs, industrial building, aqueducts, reservoirs, footings etc). In this type of structures aesthetics, structural efficiency and concept play a very important role. This class of structures can be divided into three main groups, namely continuous (concrete) shells, space frames and tension (fabric, pneumatic, cable etc )structures. In the following only the current applications of the FEM to the analysis of continuous shell structures will be discussed. However, some of the comments on this class of shells can be also applied to some extend to the others, but obviously specific computational problems will be restricted to the continuous shells. Different aspects, such as, the type of elements,input-output computational techniques etc, of the analysis of shells by the FEM will be described below. Clearly, the improvements and developments occurring in general for the FEM since its first appearance in the fifties have had a significative impact on the particular class of structures under discussion.

Nonlinear normal modes in mechanical systems: concept and computation with numerical continuation

The purpose of this Project is, first and foremost, to disclose the topic of nonlinear vibrations and oscillations in mechanical systems and, namely, nonlinear normal modes NNMs to a greater audience of researchers and technicians. To do so, first of all, the dynamical behavior and properties of nonlinear mechanical systems is outlined from the analysis of a pair of exemplary models with the harmonic balanced method. The conclusions drawn are contrasted with the Linear Vibration Theory. Then, it is argued how the nonlinear normal modes could, in spite of their limitations, predict the frequency response of a mechanical system. After discussing those introductory concepts, I present a Matlab package called 'NNMcont' developed by a group of researchers from the University of Liege. This package allows the analysis of nonlinear normal modes of vibration in a range of mechanical systems as extensions of the linear modes. This package relies on numerical methods and a 'continuation algorithm' for the computation of the nonlinear normal modes of a conservative mechanical system. In order to prove its functionality, a two degrees of freedom mechanical system with elastic nonlinearities is analized. This model comprises a mass suspended on a foundation by means of a spring-viscous damper mechanism -analogous to a very simplified model of most suspended structures and machines- that has attached a mass damper as a passive vibration control system. The results of the computation are displayed on frequency energy plots showing the NNMs branches along with modal curves and time-series plots for each normal mode. Finally, a critical analysis of the results obtained is carried out with an eye on devising what they can tell the researcher about the dynamical properties of the system.

Numerical modeling of electromagnetic coupling phenomena in the vicinities of overhead power transmission lines.

Electromagnetic coupling phenomena between overhead power transmission lines and other nearby structures are inevitable, especially in densely populated areas. The undesired effects resulting from this proximity are manifold and range from the establishment of hazardous potentials to the outbreak of alternate current corrosion phenomena. The study of this class of problems is necessary for ensuring security in the vicinities of the interaction zone and also to preserve the integrity of the equipment and of the devices there present. However, the complete modeling of this type of application requires the three- -dimensional representation of the region of interest and needs specific numerical methods for field computation. In this work, the modeling of problems arising from the flow of electrical currents in the ground (the so-called conductive coupling) will be addressed with the finite element method. Those resulting from the time variation of the electromagnetic fields (the so-called inductive coupling) will be considered as well, and they will be treated with the generalized PEEC (Partial Element Equivalent Circuit) method. More specifically, a special boundary condition on the electric potential is proposed for truncating the computational domain in the finite element analysis of conductive coupling problems, and a complete PEEC formulation for modeling inductive coupling problems is presented. Test configurations of increasing complexities are considered for validating the foregoing approaches. These works aim to provide a contribution to the modeling of this class of problems, which tend to become common with the expansion of power grids.

Models and methods for the direct simulation of rough and micropatterned surfaces

Friction in hydrodynamic bearings are a major source of losses in car engines ([69]). The extreme loading conditions in those bearings lead to contact between the matching surfaces. In such conditions not only the overall geometry of the bearing is relevant, but also the small-scale topography of the surface determines the bearing performance. The possibility of shaping the surface of lubricated bearings down to the micrometer ([57]) opened the question of whether friction can be reduced by mean of micro-textures, with mixed results. This work focuses in the development of efficient numerical methods to solve thin film (lubrication) problems down to the roughness scale of measured surfaces. Due to the high velocities and the convergent-divergent geometries of hydrodynamic bearings, cavitation takes place. To treat cavitation in the lubrication problem the Elrod- Adams model is used, a mass-conserving model which has proven in careful numerical ([12]) and experimental ([119]) tests to be essential to obtain physically meaningful results. Another relevant aspect of the modeling is that the bearing inertial effects are considered, which is necessary to correctly simulate moving textures. As an application, the effects of micro-texturing the moving surface of the bearing were studied. Realistic values are assumed for the physical parameters defining the problems. Extensive fundamental studies were carried out in the hydrodynamic lubrication regime. Mesh-converged simulations considering the topography of real measured surfaces were also run, and the validity of the lubrication approximation was assessed for such rough surfaces.

Numerical resolution of Emden's equation using Adomian polynomials

Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

Numerical solution of flood prediction and river regulation problems; report.

Reports 2 and 3 by E. Isaacson, J. J. Stoker, and A. Troesch.

Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Theoretical and numerical analysis of large-scale heat transfer problems with temperature-dependent pore-fluid densities

Purpose - In many scientific and engineering fields, large-scale heat transfer problems with temperature-dependent pore-fluid densities are commonly encountered. For example, heat transfer from the mantle into the upper crust of the Earth is a typical problem of them. The main purpose of this paper is to develop and present a new combined methodology to solve large-scale heat transfer problems with temperature-dependent pore-fluid densities in the lithosphere and crust scales. Design/methodology/approach - The theoretical approach is used to determine the thickness and the related thermal boundary conditions of the continental crust on the lithospheric scale, so that some important information can be provided accurately for establishing a numerical model of the crustal scale. The numerical approach is then used to simulate the detailed structures and complicated geometries of the continental crust on the crustal scale. The main advantage in using the proposed combination method of the theoretical and numerical approaches is that if the thermal distribution in the crust is of the primary interest, the use of a reasonable numerical model on the crustal scale can result in a significant reduction in computer efforts. Findings - From the ore body formation and mineralization points of view, the present analytical and numerical solutions have demonstrated that the conductive-and-advective lithosphere with variable pore-fluid density is the most favorite lithosphere because it may result in the thinnest lithosphere so that the temperature at the near surface of the crust can be hot enough to generate the shallow ore deposits there. The upward throughflow (i.e. mantle mass flux) can have a significant effect on the thermal structure within the lithosphere. In addition, the emplacement of hot materials from the mantle may further reduce the thickness of the lithosphere. Originality/value - The present analytical solutions can be used to: validate numerical methods for solving large-scale heat transfer problems; provide correct thermal boundary conditions for numerically solving ore body formation and mineralization problems on the crustal scale; and investigate the fundamental issues related to thermal distributions within the lithosphere. The proposed finite element analysis can be effectively used to consider the geometrical and material complexities of large-scale heat transfer problems with temperature-dependent fluid densities.