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In contrast to the spatial motion setting, the material motion setting of continuum mechanics is concerned with the response to variations of material placements of particles with respect to the ambient material. The material motion point of view is thus extremely prominent when dealing with defect mechanics to which it has originally been introduced by Eshelby more than half a century ago. Its primary unknown, the material deformation map is governed by the material motion balance of momentum, i.e. the balance of material forces on the material manifold in the sense of Eshelby. Material (configurational) forces are concerned with the response to variations of material placements of 'physical particles' with respect to the ambient material. Opposed to that, the common spatial (mechanical) forces in the sense of Newton are considered as the response to variations of spatial placements of 'physical particles' with respect to the ambient space. Material forces as advocated by Maugin are especially suited for the assessment of general defects as inhomogeneities, interfaces, dislocations and cracks, where the material forces are directly related to the classical J-Integral in fracture mechanics, see also Gross & Seelig. Another classical example of a material - or rather configurational - force is emblematized by the celebrated Peach-Koehler force, see e.g. the discussion in Steinmann. The present work is mainly divided in four parts. In the first part we will introduce the basic notions of the mechanics and numerics of material forces for a quasi-static conservative mechanical system. In this case the internal potential energy density per unit volume characterizes a hyperelastic material behaviour. In the first numerical example we discuss the reliability of the material force method to calculate the vectorial J-integral of a crack in a Ramberg-Osgood type material under mode I loading and superimposed T-stresses. Secondly, we study the direction of the single material force acting as the driving force of a kinked crack in a geometrically nonlinear hyperelastic Neo-Hooke material. In the second part we focus on material forces in the case of geometrically nonlinear thermo-hyperelastic material behaviour. Therefore we adapt the theory and numerics to a transient coupled problem, and elaborate the format of the Eshelby stress tensor as well as the internal material volume forces induced by the gradient of the temperature field. We study numerically the material forces in a bimaterial bar under tension load and the time dependent evolution of material forces in a cracked specimen. The third part discusses the material force method in the case of geometrically nonlinear isotropic continuum damage. The basic equations are similar to those of the thermo-hyperelastic problem but we introduce an alternative numerical scheme, namely an active set search algorithm, to calculate the damage field as an additional degree of freedom. With this at hand, it is an easy task to obtain the gradient of the damage field which induces the internal material volume forces. Numeric examples in this part are a specimen with an elliptic hole with different semi-axis, a center cracked specimen and a cracked disc under pure mode I loading. In the fourth part of this work we elaborate the format of the Eshelby stress tensor and the internal material volume forces for geometrically nonlinear multiplicative elasto-plasticity. Concerning the numerical implementation we restrict ourselves to the case of geometrically linear single slip crystal plasticity and compare here two different numerical methods to calculate the gradient of the internal variable which enters the format of the internal material volume forces. The two numerical methods are firstly, a node point based approach, where the internal variable is addressed as an additional degree of freedom, and secondly, a standard approach where the internal variable is only available at the integration points level. Here a least square projection scheme is enforced to calculate the necessary gradients of this internal variable. As numerical examples we discuss a specimen with an elliptic inclusion and an elliptic hole respectively and, in addition, a crack under pure mode I loading in a material with different slip angles. Here we focus on the comparison of the two different methods to calculate the gradient of the internal variable. As a second class of numerical problems we elaborate and implement a geometrically linear von Mises plasticity with isotropic hardening. Here the necessary gradients of the internal variables are calculated by the already mentioned projection scheme. The results of a crack in a material with different hardening behaviour under various additional T-stresses are given.
Matter-wave Optics of Dark-state Polaritons: Applications to Interferometry and Quantum Information
(2006)
The present work "Materwave Optics with Dark-state Polaritons: Applications to Interferometry and Quantum Information" deals in a broad sense with the subject of dark-states and in particular with the so-called dark-state polaritons introduced by M. Fleischhauer and M. D. Lukin. The dark-state polaritons can be regarded as a combined excitation of electromagnetic fields and spin/matter-waves. Within the framework of this thesis the special optical properties of the combined excitation are studied. On one hand a new procedure to spatially manipulate and to increase the excitation density of stored photons is described and on the other hand the properties are used to construct a new type of Sagnac Hybrid interferometer. The thesis is devided into four parts. In the introduction all notions necessary to understand the work are described, e.g.: electromagnetically induced transparency (EIT), dark-state polaritons and the Sagnac effect. The second chapter considers the method developed by A. Andre and M. D. Lukin to create stationary light pulses in specially dressed EIT-media. In a first step a set of field equations is derived and simplified by introducing a new set of normal modes. The absorption of one of the normal modes leads to the phenomenon of pulse-matching for the other mode and thereby to a diffusive spreading of its field envelope. All these considerations are based on a homogeneous field setup of the EIT preparation laser. If this restriction is dismissed one finds that a drift motion is superimposed to the diffusive spreading. By choosing a special laser configuration the drift motion can be tailored such that an effective force is created that counteracts the spreading. Moreover, the force can not only be strong enough to compensate the diffusive spreading but also to exceed this dynamics and hence to compress the field envelope of the excitation. The compression can be discribed using a Fokker-Planck equation of the Ornstein-Uhlenbeck type. The investigations show that the compression leads to an excitation of higher-order modes which decay very fast. In the last section of the chapter this exciation will be discussed in more detail and conditions will be given how the excitation of higher-order modes can be avoided or even suppressed. All results given in the chapter are supported by numerical simulatons. In the third chapter the matterwave optical properties of the dark-state polaritons will be studied. They will be used to construct a light-matterwave hybrid Sagnac interferometer. First the principle setup of such an interferometer will be sketched and the relevant equations of motion of light-matter interaction in a rotating frame will be derived. These form the basis of the following considerations of the dark-state polariton dynamics with and without the influence of external trapping potentials on the matterwave part of the polariton. It will be shown that a sensitivity enhancement compared to a passive laser gyroscope can be anticipated if the gaseous medium is initially in a superfluid quantum state in a ring-trap configuration. To achieve this enhancement a simultaneous coherence and momentum transfer is furthermore necessary. In the last part of the chapter the quantum sensitivity limit of the hybrid interferometer is derived using the one-particle density matrix equations incorporating the motion of the particles. To this end the Maxwell-Bloch equations are considered perturbatively in the rotation rate of the noninertial frame of reference and the susceptibility of the considered 3-level \(\Lambda\)-type system is derived in arbitrary order of the probe-field. This is done to determine the optimum operation point. With its help the anticipated quantum sensitivity of the light-matterwave hybrid Sagnac interferometer is calculated at the shot-noise limit and the results are compared to state-of-the-art laser and matterwave Sagnac interferometers. The last chapter of the thesis originates from a joint theoretical and experimental project with the AG Bergmann. This chapter does no longer consider the dark-state polaritons of the last two chapters but deals with the more general concept of dark states and in particular with the transient velocity selective dark states as introduced by E. Arimondo et al. In the experiment we could for the first time measure these states. The chapter starts with an introduction into the concept of velocity selective dark states as they occur in a \(\Lambda\)-configuration. Then we introduce the transient velocity selective dark-states as they occur in an particular extension of the \(\Lambda\)-system. For later use in the simulations the relevant equations of motion are derived in detail. The simulations are based on the solution of the generalized optical Bloch equations. Finally the experimental setup and procedure are explained and the theoretical and experimental results are compared.
In this thesis, we have dealt with two modeling approaches of the credit risk, namely the structural (firm value) and the reduced form. In the former one, the firm value is modeled by a stochastic process and the first hitting time of this stochastic process to a given boundary defines the default time of the firm. In the existing literature, the stochastic process, triggering the firm value, has been generally chosen as a diffusion process. Therefore, on one hand it is possible to obtain closed form solutions for the pricing problems of credit derivatives and on the other hand the optimal capital structure of a firm can be analysed by obtaining closed form solutions of firm's corporate securities such as; equity value, debt value and total firm value, see Leland(1994). We have extended this approach by modeling the firm value as a jump-diffusion process. The choice of the jump-diffusion process was a crucial step to obtain closed form solutions for corporate securities. As a result, we have chosen a jump-diffusion process with double exponentially distributed jump heights, which enabled us to analyse the effects of jump on the optimal capital structure of a firm. In the second part of the thesis, by following the reduced form models, we have assumed that the default is triggered by the first jump of a Cox process. Further, by following Schönbucher(2005), we have modeled the forward default intensity of a firm as a geometric Brownian motion and derived pricing formulas for credit default swap options in a more general setup than the ones in Schönbucher(2005).
The fast development of the financial markets in the last decade has lead to the creation of a variety of innovative interest rate related products that require advanced numerical pricing methods. Examples in this respect are products with a complicated strong path-dependence such as a Target Redemption Note, a Ratchet Cap, a Ladder Swap and others. On the other side, the usage of the standard in the literature one-factor Hull and White (1990) type of short rate models allows only for a perfect correlation between all continuously compounded spot rates or Libor rates and thus are not suited for pricing innovative products depending on several Libor rates such as for example a "steepener" option. One possible solution to this problem deliver the two-factor short rate models and in this thesis we consider a two-factor Hull and White (1990) type of a short rate process derived from the Heath, Jarrow, Morton (1992) framework by limiting the volatility structure of the forward rate process to a deterministic one. In this thesis, we often choose to use a variety of modified (binomial, trinomial and quadrinomial) tree constructions as a main numerical pricing tool due to their flexibility and fast convergence and (when there is no closed-form solution) compare their results with fine grid Monte Carlo simulations. For the purpose of pricing the already mentioned innovative short-rate related products, in this thesis we offer and examine two different lattice construction methods for the two-factor Hull-White type of a short rate process which are able to deal easily both with modeling of the mean-reversion of the underlying process and with the strong path-dependence of the priced options. Additionally, we prove that the so-called rotated lattice construction method overcomes the typical for the existing two-factor tree constructions problem with obtaining negative "risk-neutral probabilities". With a variety of numerical examples, we show that this leads to a stability in the results especially in cases of high volatility parameters and negative correlation between the base factors (which is typically the case in reality). Further, noticing that Chan et al (1992) and Ritchken and Sankarasubramanian (1995) showed that option prices are sensitive to the level of the short rate volatility, we examine the pricing of European and American options where the short rate process has a volatility structure of a Cheyette (1994) type. In this relation, we examine the application of the two offered lattice construction methods and compare their results with the Monte Carlo simulation ones for a variety of examples. Additionally, for the pricing of American options with the Monte Carlo method we expand and implement the simulation algorithm of Longstaff and Schwartz (2000). With a variety of numerical examples we compare again the stability and the convergence of the different lattice construction methods. Dealing with the problems of pricing strongly path-dependent options, we come across the cumulative Parisian barrier option pricing problem. We notice that in their classical form, the cumulative Parisian barrier options have been priced both analytically (in a quasi closed form) and with a tree approximation (based on the Forward Shooting Grid algorithm, see e.g. Hull and White (1993), Kwok and Lau (2001) and others). However, we offer an additional tree construction method which can be seen as a direct binomial tree integration that uses the analytically calculated conditional survival probabilities. The advantage of the offered method is on one side that the conditional survival probabilities are easier to calculate than the closed-form solution itself and on the other side that this tree construction is very flexible in the sense that it allows easy incorporation of additional features such as e.g a forward starting one. The obtained results are better than the Forward Shooting Grid tree ones and are very close to the analytical quasi closed form solution. Finally, we pay our attention to pricing another type of innovative interest rate alike products - namely the Longevity bond - whose coupon payments depend on the survival function of a given cohort. Due to the lack of a market for mortality, for the pricing of the Longevity bonds we develop (following Korn, Natcheva and Zipperer (2006)) a framework that contains principles from both Insurance and Financial mathematic. Further on, we calibrate the existing models for the stochastic mortality dynamics to historical German data and additionally offer new stochastic extensions of the classical (deterministic) models of mortality such as the Gompertz and the Makeham one. Finally, we compare and analyze the results of the application of all considered models to the pricing of a Longevity bond on the longevity of the German males.
The topic of this thesis is the coupling of an atomistic and a coarse scale region in molecular dynamics simulations with the focus on the reflection of waves at the interface between the two scales and the velocity of waves in the coarse scale region for a non-equilibrium process. First, two models from the literature for such a coupling, the concurrent coupling of length scales and the bridging scales method are investigated for a one dimensional system with harmonic interaction. It turns out that the concurrent coupling of length scales method leads to the reflection of fine scale waves at the interface, while the bridging scales method gives an approximated system that is not energy conserving. The velocity of waves in the coarse scale region is in both models not correct. To circumvent this problems, we present a coupling based on the displacement splitting of the bridging scales method together with choosing appropriate variables in orthogonal subspaces. This coupling allows the derivation of evolution equations of fine and coarse scale degrees of freedom together with a reflectionless boundary condition at the interface directly from the Lagrangian of the system. This leads to an energy conserving approximated system with a clear separation between modeling errors an errors due to the numerical solution. Possible approximations in the Lagrangian and the numerical computation of the memory integral and other numerical errors are discussed. We further present a method to choose the interpolation from coarse to atomistic scale in such a way, that the fine scale degrees of freedom in the coarse scale region can be neglected. The interpolation weights are computed by comparing the dispersion relations of the coarse scale equations and the fully atomistic system. With this new interpolation weights, the number of degrees of freedom can be drastically reduced without creating an error in the velocity of the waves in the coarse scale region. We give an alternative derivation of the new coupling with the Mori-Zwanzig projection operator formalism, and explain how the method can be extended to non-zero temperature simulations. For the comparison of the results of the approximated with the fully atomistic system, we use a local stress tensor and the energy in the atomistic region. Examples for the numerical solution of the approximated system for harmonic potentials are given in one and two dimensions.
This thesis introduces so-called cone scalarising functions. They are by construction compatible with a partial order for the outcome space given by a cone. The quality of the parametrisations of the efficient set given by the cone scalarising functions are then investigated. Here, the focus lies on the (weak) efficiency of the generated solutions, the reachability of effiecient points and continuity of the solution set. Based on cone scalarising functions Pareto Navigation a novel, interactive, multiobjective optimisation method is proposed. It changes the ordering cone to realise bounds on partial tradeoffs. Besides, its use of an equality constraint for the changing component of the reference point is a new feature. The efficiency of its solutions, the reachability of efficient solutions and continuity is then analysed. Potential problems are demonstrated using a critical example. Furthermore, the use of Pareto Navigation in a two-phase approach and for nonconvex problems is discussed. Finally, its application for intensity-modulated radiotherapy planning is described. Thereby, its realisation in a graphical user interface is shown.
This thesis deals with modeling aspects of generalized Newtonian and of non-Newtonian fluids, as well as with development and validation of algorithms used in simulation of such fluids. The main contribution in the modeling part are the introduction and analysis of a new model for the generalized Newtonian fluids, where constitutive equation is of an algebraic form. Distinction between shear and extensional viscosities leads to anisotropic viscosity model. It can be considered as a natural extension of the well known (isotropic viscosity) Carreau model, which deals only with shear viscosity properties of the fluid. The proposed model takes additionally into account extensional viscosity properties. Numerical results show that the anisotropic viscosity model gives much better agreement with experimental observations than the isotropic one. Another contribution of the thesis consists of the development and analysis of robust and reliable algorithms for simulation of generalized Newtonian fluids. For such fluids the momentum equations are strongly coupled through mixed derivatives appearing in the viscous term (unlike the case of Newtonian fluids). It is shown in this thesis, that a careful treatment of those derivatives is essential in deriving robust algorithms. A modification of a standard SIMPLE-like algorithm is given, where all the viscous terms from the momentum equations are discretized in an implicit manner. Moreover, it is shown that a block diagonal preconditioner to the viscous operator is good enough to be used in simulations. Furthermore, different solution techniques, namely projection type methods (consists of solving momentum equations and pressure correction equation) and fully coupled methods (momentum and continuity equations are solved together), are compared. It is shown, that explicit discretization of the mixed derivatives lead to stability problems. Further, analytical estimates of eigenvalue distribution for three different preconditioners, applied to the transformed system arising after discretization and linearization of the momentum and continuity equations, are provided. We propose to apply a block Gauss-Seidel preconditioner to the transformed system. The analysis shows, that this preconditioner is able to cluster eigenvalues around unity independent of the transformation step. It is not the case for other preconditioners applied to the transformed system as discussed in the thesis. The block Gauss-Seidel preconditioner has also shown the best behavior (among all preconditioners discussed in the thesis) in numerical experiments. Further contribution consists of comparison and validation of numerical algorithms applied in simulations of non-Newtonian fluids modeled by time integral constitutive equations. Numerical results from simulations of dilute polymer solutions, described by the integral Oldroyd B model, have shown very good quantitative agreement with the results obtained by differential Oldroyd B counterpart in 4:1 planar contraction domain at low Weissenberg numbers. In this case, the Weissenberg number is changed by changing the relaxation time. However, contrary to the differential Oldroyd B model, the integral one allows to perform stable simulations also in the range of high Weissenberg numbers. Moreover, very good agreement with experimental observations has been achieved. Simulations of concentrated polymer solutions (polystyrene and polybutadiene solutions), modeled by the integral Doi Edwards model, supplemented by chain length fluctuations, have shown very good qualitative agreement with the results obtained by its differential approximation in 4:1:4 constriction domain. Again, much higher Weissenberg numbers can be achieved when the integral model is used. Moreover, very good quantitative results with experimental data of polystyrene solution for the first normal stress difference and shear viscosity defined here as the quotient of a shear stress and a shear rate. Finally, comparison of the two methods used for approximating the time integral constitutive equation, namely Deformation Field Method (DFM) and Backward Lagrangian Particle Method (BLPM), is performed. In BLPM the particle paths are recalculated at every time step of the simulations, what has never been tried before. The results have shown, that in the considered geometries both methods give similar results.
Biological Soil Crusts (BSCs), composed of lichens, mosses, green algae, microfungi and cyanobacteria are an ecological important part of the perennial landcover of many arid and semiarid regions (Belnap et al. 2001a), (Büdel 2002). In many arid and hyperarid areas BSCs form the only perennial "vegetation cover" largely due to their extensive resistance to drought (Lange et al. 1975). For the Central Namib Desert (Namibia), BSCs consisting of extraordinary vast lichen communities were recently mapped and classified into six morphological classes for a coastal area of 350 km x 60 km. Embedded into the project "BIOTA" (www.biota-africa.org) financed by the German Federal Ministry of Education and Research the study was undertaken in the framework of the PhD thesis by Christoph Schultz. Some of these lichen communities grouped together in so called "lichen fields" have already been studied concerning their ecology and diversity in the past (Lange et al. 1994), (Loris & Schieferstein 1992), (Loris et al. 2004), (Ullmann & Büdel 2001a), (Wessels 1989). Multispectral LANDSAT 7 ETM+ and LANDSAT 5 TM satellite imagery was utilized for an unitemporal supervised classification as well as for the establishment of a monitoring based on a combined retrospective supervised classification and change detection approach (Bock 2003), (Weiers et al. 2003). Results comprise the analysis of the mapped distribution of lichen communities for the Central Namib Desert as of 2003 as well as reconstructed distributions for the years 2000, 1999, 1992 and 1991 derived from retrospective supervised classification. This allows a first monitoring of the disturbance, destruction and recovery of the lichen communities in these arid environments including the analysis of the major abiotic processes involved. Further analysis of these abiotic processes is key for understanding the influence of Namib lichen communities on overall aeolian and water induced erosion rates, nutrient cycles, water balance and pedogenic processes (Belnap & Gillette 1998), (Belnap et al. 2001b), (Belnap 2001c), (Evans & Lange 2001), (McKenna Neumann & Maxwell 1999). In order to aid the understanding of these processes SRTM digital elevation model data as well as climate data sets were used as reference. Good correlation between geomorphological form elements as well as hydrological drainage system and the disturbance patterns derived from individual post classification change comparisons between the timeframes could be observed. Conjoined with the climate data sets sporadic foehn-like windstorms as well as extraordinary precipitation events were identified to largely affect the distribution patterns of lichen communities. Therefore the analysis and monitoring of the diversity, distribution and spatiotemporal change of Central Namib BSCs with the means of Remote Sensing and GIS applications proof to be important tools to create further understanding of desertification and degradation processes in these arid regions.
Discontinuities can appear in different fields of mechanics. Some examples where discontinuities arise are more obvious such as the formation of cracks. Other sources of discontinuities are less apparent such as interfaces between different materials. Furthermore continuous fields with steep gradients can also be considered as discontinuous fields. This work aims at the inclusion of arbitrary discontinuities within the finite element method. Although the finite element method is the most sophisticated numerical tool in modern engineering, the inclusion of discontinuities is still a challenging task. Traditionally within finite the framework of FE methods discontinuities are modeled explicitely by the construction of the mesh. Thus, when a fixed mesh is used, the position of the discontinuity is prescribed by the location of interelement boundaries and not by the physical situation. The simulation of crack growth requires a frequent adaption of the mesh and that can be a difficult and computationally expensive task. Thus a more flexible numerical approach is needed which leads to the mesh-independent representation of the discontinuity. A challenging field where the accurate description of discontinuities is of vital importance is the modeling of failure in engineering materials. The load capacity of a structure is limited by the material strength. If the load limit is exceeded failure zones arise and increase. Representative examples of failure mechanisms are are cracks in brittle materials or shear bands in metals or soils. Failure processes are often accompanied by a strain softening material behaviour (decreasing load carrying capacity with increasing strain at a material point). It is known that the inclusion of strain softening material behaviour within a continuum description requires regularization techniques to preserve the well- posedness of the governing equations. One possibility is the consideration of non-local or gradient terms in the constitutive equations but these approaches require a sufficiently fine discretization in the localization zone, which leads to a high numerical effort. If the extent of the failure zone and the failure process to the point of the development of discrete cracks is considered it seems reasonable to include strong discontinuities. In the framework of fracture mechanics the inclusion of displacement jumps is intuitively comprehensible. However, the modeling of localized failure processes demands the consideration of inelastic material behaviour. Cohesive zone models represent an approach which is especially suited for the incorporation within the finite element framework. It is supposed that cohesive tractions are transmitted between the discontinuity surfaces. These tractions are constitutively prescribed by a phenomenological traction separation law and thus allow for the modeling of different inelastic mechanisms, like micro-crack evolution, initiation of voids, plastic flow or crack bridging. The incorporation of a displacement discontinuity in combination with a cohesive traction separation law leads to a sound model to describe failure processes and crack propagation. Another area where the existence of discontinuities is not as obvious is the occurence of material interfaces, inclusions or holes. The accurate modeling of such internal interfaces is important to predict the mechanical behaviour of components. The present discontinuity is of different nature: the displacement field is continuous but there is a jump in the strains, which is denoted by the expression weak discontinuity. Usually in FE methods material interfaces are taken into account by the mesh construction. But if the structure exhibits multiple inclusions of complex geometry it can be advantageous if the interface does not have to be meshed. And when we look at at problems where the interface moves with time, e. g. phase transformation, the mesh-independent modeling of the weak discontinuities naturally holds major advantages. The greatest challenge in the modeling of discontinuities is their incorporation into numerical methods. The focus of the present work is the development, analysis and application of a finite element approach to model mesh-independent discontinuities. The method shall be robust and flexible to be applicable to both, strong and weak discontinuities.
This work deals with the mathematical modeling and numerical simulation of the dynamics of a curved inertial viscous Newtonian fiber, which is practically applicable to the description of centrifugal spinning processes of glass wool. Neglecting surface tension and temperature dependence, the fiber flow is modeled as a three-dimensional free boundary value problem via instationary incompressible Navier-Stokes equations. From regular asymptotic expansions in powers of the slenderness parameter leading-order balance laws for mass (cross-section) and momentum are derived that combine the unrestricted motion of the fiber center-line with the inner viscous transport. The physically reasonable form of the one-dimensional fiber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms. For the numerical simulation of the derived model a finite volume code is developed. The results of the numerical scheme for high Reynolds numbers are validated by comparing them with the analytical solution of the inviscid problem. Moreover, the influence of parameters, like viscosity and rotation on the fiber dynamics are investigated. Finally, an application based on industrial data is performed.