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In this work, we analyze two important and simple models of short rates, namely Vasicek and CIR models. The models are described and then the sensitivity of the models with respect to changes in the parameters are studied. Finally, we give the results for the estimation of the model parameters by using two different ways.

In this paper mathematical models for liquid films generated by impinging jets are discussed. Attention is stressed to the interaction of the liquid film with some obstacle. S. G. Taylor [Proc. R. Soc. London Ser. A 253, 313 (1959)] found that the liquid film generated by impinging jets is very sensitive to properties of the wire which was used as an obstacle. The aim of this presentation is to propose a modification of the Taylor's model, which allows to simulate the film shape in cases, when the angle between jets is different from 180°. Numerical results obtained by discussed models give two different shapes of the liquid film similar as in Taylors experiments. These two shapes depend on the regime: either droplets are produced close to the obstacle or not. The difference between two regimes becomes larger if the angle between jets decreases. Existence of such two regimes can be very essential for some applications of impinging jets, if the generated liquid film can have a contact with obstacles.

Granular systems in solid-like state exhibit properties like stiffness
dependence on stress, dilatancy, yield or incremental non-linearity
that can be described within the continuum mechanical framework.
Different constitutive models have been proposed in the literature either based on relations between some components of the stress tensor or on a quasi-elastic description. After a brief description of these
models, the hyperelastic law recently proposed by Jiang and Liu [1]
will be investigated. In this framework, the stress-strain relation is
derived from an elastic strain energy density where the stable proper-
ties are linked to a Drucker-Prager yield criteria. Further, a numerical method based on the finite element discretization and Newton-
Raphson iterations is presented to solve the force balance equation.
The 2D numerical examples presented in this work show that the stress
distributions can be computed not only for triangular domains, as previoulsy done in the literature, but also for more complex geometries.
If the slope of the heap is greater than a critical value, numerical instabilities appear and no elastic solution can be found, as predicted by
the theory. As main result, the dependence of the material parameter
Xi on the maximum angle of repose is established.

Radiotherapy is one of the major forms in cancer treatment. The patient is irradiated with high-energetic photons or charged particles with the primary goal of delivering sufficiently high doses to the tumor tissue while simultaneously sparing the surrounding healthy tissue. The inverse search for the treatment plan giving the desired dose distribution is done by means of numerical optimization [11, Chapters 3-5]. For this purpose, the aspects of dose quality in the tissue are modeled as criterion functions, whose mathematical properties also affect the type of the corresponding optimization problem. Clinical practice makes frequent use of criteria that incorporate volumetric and spatial information about the shape of the dose distribution. The resulting optimization problems are of global type by empirical knowledge and typically computed with generic global solver concepts, see for example [16]. The development of good global solvers to compute radiotherapy optimization problems is an important topic of research in this application, however, the structural properties of the underlying criterion functions are typically not taken into account in this context.

This report reviews selected image binarization and segmentation methods that have been proposed and which are suitable for the processing of volume images. The focus is on thresholding, region growing, and shape–based methods. Rather than trying to give a complete overview of the field, we review the original ideas and concepts of selected methods, because we believe this information to be important for judging when and under what circumstances a segmentation algorithm can be expected to work properly.

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented.

The stationary heat equation is solved with periodic boundary conditions in geometrically complex composite materials with high contrast in the thermal conductivities of the individual phases. This is achieved by harmonic averaging and explicitly introducing the jumps across the material interfaces as additional variables. The continuity of the heat flux yields the needed extra equations for these variables. A Schur-complent formulation for the new variables is derived that is solved using the FFT and BiCGStab methods. The EJ-HEAT solver is given as a 3-page Matlab program in the Appendix. The C++ implementation is used for material design studies. It solves 3-dimensional problems with around 190 Mio variables on a 64-bit AMD Opteron desktop system in less than 6 GB memory and in minutes to hours, depending on the contrast and required accuracy. The approach may also be used to compute effective electric conductivities because they are governed by the stationary heat equation.

Four aspects are important in the design of hydraulic lters. We distinguish between two cost factors and two performance factors. Regarding performance, filter eciencynd lter capacity are of interest. Regarding cost, there are production considerations such as spatial restrictions, material cost and the cost of manufacturing the lter. The second type of cost is the operation cost, namely the pressure drop. Albeit simulations should and will ultimately deal with all 4 aspects, for the moment our work is focused on cost. The PleatGeo Module generates three-dimensional computer models of a single pleat of a hydraulic lter interactively. PleatDict computes the pressure drop that will result for the particular design by direct numerical simulation. The evaluation of a new pleat design takes only a few hours on a standard PC compared to days or weeks used for manufacturing and testing a new prototype of a hydraulic lter. The design parameters are the shape of the pleat, the permeabilities of one or several layers of lter media and the geometry of a supporting netting structure that is used to keep the out ow area open. Besides the underlying structure generation and CFD technology, we present some trends regarding the dependence of pressure drop on design parameters that can serve as guide lines for the design of hydraulic lters. Compared to earlier two-dimensional models, the three-dimensional models can include a support structure.

A fully automatic procedure is proposed to rapidly compute the permeability of porous materials from their binarized microstructure. The discretization is a simplified version of Peskin’s Immersed Boundary Method, where the forces are applied at the no-slip grid points. As needed for the computation of permeability, steady flows at zero Reynolds number are considered. Short run-times are achieved by eliminating the pressure and velocity variables using an Fast Fourier Transform-based and 4 Poisson problembased fast inversion approach on rectangular parallelepipeds with periodic boundary conditions. In reference to calling it a fast method using fictitious or artificial forces, the implementation is called FFF-Stokes. Large scale computations on 3d images are quickly and automatically performed to estimate the permeability of some sample materials. A matlab implementation is provided to allow readers to experience the automation and speed of the method for realistic three-dimensional models.

In the presented work, we make use of the strong reciprocity between kinematics and geometry to build a geometrically nonlinear, shearable low order discrete shell model of Cosserat type defined on triangular meshes, from which we deduce a rotation–free Kirchhoff type model with the triangle vertex positions as degrees of freedom. Both models behave physically plausible already on very coarse meshes, and show good
convergence properties on regular meshes. Moreover, from the theoretical side, this deduction provides a
common geometric framework for several existing models.

Die Simulation von Prüfständen und insbesondere von Baugruppen und Gesamtfahrzeugen auf Prüfständen durch Kopplung von Mehrkörpersimulation mit Modellen für Regelung und Aktuatorik leistet einen wesentlichen Beitrag zur Entwicklungszeitverkürzung. In diesem Beitrag wird ein Kooperationsprojekt vorgestellt, in dem ein Co- Simulationsmodell für die beweglichen Massen sowie die Regelung und Hydraulik eines Gesamtfahrzeugprüfstands erstellt wurde. Es wird sowohl auf die Validierung des Fahrzeugmodells durch Straßenmessungen als auch auf die Identifikation und Validierung des Prüfstandsmodells einschließlich Servohydraulik und Regelung eingegangen.

Worldwide the installed capacity of renewable technologies for electricity production is
rising tremendously. The German market is particularly progressive and its regulatory
rules imply that production from renewables is decoupled from market prices and electricity
demand. Conventional generation technologies are to cover the residual demand
(defined as total demand minus production from renewables) but set the price at the
exchange. Existing electricity price models do not account for the new risks introduced
by the volatile production of renewables and their effects on the conventional demand
curve. A model for residual demand is proposed, which is used as an extension of
supply/demand electricity price models to account for renewable infeed in the market.
Infeed from wind and solar (photovoltaics) is modeled explicitly and withdrawn from
total demand. The methodology separates the impact of weather and capacity. Efficiency
is transformed on the real line using the logit-transformation and modeled as a stochastic process. Installed capacity is assumed a deterministic function of time. In a case study the residual demand model is applied to the German day-ahead market
using a supply/demand model with a deterministic supply-side representation. Price trajectories are simulated and the results are compared to market future and option
prices. The trajectories show typical features seen in market prices in recent years and the model is able to closely reproduce the structure and magnitude of market prices.
Using the simulated prices it is found that renewable infeed increases the volatility of forward prices in times of low demand, but can reduce volatility in peak hours. Prices
for different scenarios of installed wind and solar capacity are compared and the meritorder effect of increased wind and solar capacity is calculated. It is found that wind
has a stronger overall effect than solar, but both are even in peak hours.

Continuously improving imaging technologies allow to capture the complex spatial
geometry of particles. Consequently, methods to characterize their three
dimensional shapes must become more sophisticated, too. Our contribution to
the geometric analysis of particles based on 3d image data is to unambiguously
generalize size and shape descriptors used in 2d particle analysis to the spatial
setting.
While being defined and meaningful for arbitrary particles, the characteristics
were actually selected motivated by the application to technical cleanliness. Residual
dirt particles can seriously harm mechanical components in vehicles, machines,
or medical instruments. 3d geometric characterization based on micro-computed
tomography allows to detect dangerous particles reliably and with
high throughput. It thus enables intervention within the production line. Analogously
to the commonly agreed standards for the two dimensional case, we
show how to classify 3d particles as granules, chips and fibers on the basis of
the chosen characteristics. The application to 3d image data of dirt particles is
demonstrated.

We present the application of a meshfree method for simulations of interaction between fluids and flexible structures. As a flexible structure we consider a sheet of paper. In a two-dimensional framework this sheet can be modeled as curve by the dynamical Kirchhoff-Love theory. The external forces taken into account are gravitation and the pressure difference between upper and lower surface of the sheet. This pressure difference is computed using the Finite Pointset Method (FPM) for the incompressible Navier-Stokes equations. FPM is a meshfree, Lagrangian particle method. The dynamics of the sheet are computed by a finite difference method. We show the suitability of the meshfree method for simulations of fluid-structure interaction in several applications.

A Lattice Boltzmann Method for immiscible multiphase flow simulations using the Level Set Method
(2008)

We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young-Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences.

It is commonly believed that not all degrees of freedom are needed to produce good solutions for the treatment planning problem in intensity modulated radiotherapy treatment (IMRT). However, typical methods to exploit this fact have either increased the complexity of the optimization problem or were heuristic in nature. In this work we introduce a technique based on adaptively refining variable clusters to successively attain better treatment plans. The approach creates approximate solutions based on smaller models that may get arbitrarily close to the optimal solution. Although the method is illustrated using a specific treatment planning model, the components constituting the variable clustering and the adaptive refinement are independent of the particular optimization problem.

It has been empirically verified that smoother intensity maps can be expected to produce shorter sequences when step-and-shoot collimation is the method of choice. This work studies the length of sequences obtained by the sequencing algorithm by Bortfeld and Boyer using a probabilistic approach. The results of this work build a theoretical foundation for the up to now only empirically validated fact that if smoothness of intensity maps is considered during their calculation, the solutions can be expected to be more easily applied.

We present a parsimonious multi-asset Heston model. All single-asset submodels follow the well-known Heston dynamics and their parameters are typically calibrated on implied market volatilities. We focus on the calibration of the correlation structure between the single-asset marginals in the absence of sucient liquid cross-asset option price data. The presented model is parsimonious in the sense that d(d􀀀1)=2 asset-asset cross-correlations are required for a d-asset Heston model. In order to calibrate the model, we present two general setups corresponding to relevant practical situations: (1) when the empirical cross-asset correlations in the risk neutral world are given by the user and we need to calibrate the correlations between the driving Brownian motions or (2) when they have to be estimated from the historical time series. The theoretical background, including the ergodicity of the multidimensional CIR process, for the proposed estimators is also studied.

For the last decade, optimization of beam orientations in intensitymodulated radiation therapy (IMRT) has been shown to be successful in improving the treatment plan. Unfortunately, the quality of a set of beam orientations depends heavily on its corresponding beam intensity proles. Usually, a stochastic selector is used for optimizing beam orientation, and then a single objective inverse treatment planning algorithm is used for the optimization of beam intensity proles. The overall time needed to solve the inverse planning for every random selection of beam orientations becomes excessive. Recently, considerable improvement has been made in optimizing beam intensity proles by using multiple objective inverse treatment planning. Such an approach results in a variety of beam intensity proles for every selection of beam orientations, making the dependence between beam orientations and its intensity proles less important. We take advantage of this property to present a dynamic algorithm for beam orientation in IMRT which is based on multicriteria inverse planning. The algorithm approximates beam intensity proles iteratively instead of doing it for every selection of beam orientation, saving a considerable amount of calculation time. Every iteration goes from an N-beam plan to a plan with N + 1 beams. Beam selection criteria are based on a score function that minimizes the deviation from the prescribed dose, in addition to a reject-accept criterion. To illustrate the eciency of the algorithm it has been applied to an articial example where optimality is trivial and to three real clinical cases: a prostate carcinoma, a tumor in the head and neck region and a paraspinal tumor. In comparison to the standard equally spaced beam plans, improvements are reported in all of the three clinical examples, even, in some cases with a fewer number of beams.

Forderungen nach kürzeren Entwicklungszyklen bei gleichzeitig höherer Produktqualität führen in allen Bereichen der Nutzfahrzeugtechnik und insbesondere auch bei Baumaschinen zum verstärkten Einsatz von Simulationssoftware. Um in diesem Sinne Lebensdauerberechnungen durchführen zu können, sind jedoch genaue Kenntnisse über die im Kundeneinsatz auftretenden Betriebslasten und Beanspruchungen erforderlich. Für deren Ermittlung hat der Baumaschinenhersteller VOLVO Construction Equipment einen Mobilbagger umfassend mit Messtechnik ausgestattet, die neben den mechanischen Belastungen an der Arbeitsausrüstung auch wesentliche Kenndaten des Hydrauliksystems und des Fahrantriebs erfasst. Dieser Messbagger wurde bereits bei unterschiedlichen Kunden in Europa eingesetzt. Der Artikel beschreibt die methodische Vorgehensweise zur Verarbeitung der erfassten Daten und zur Generierung von repräsentativen Nutzungsprofilen am Beispiel der mechanischen Belastungen an der Arbeitseinrichtung, die im Wesentlichen vom Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM) erarbeitet wurde.

In this article, a new model predictive control approach to nonlinear stochastic systems will be presented. The new approach is based on particle filters, which are usually used for estimating states or parameters. Here, two particle filters will be combined, the first one giving an estimate for the actual state based on the actual output of the system; the second one gives an estimate of a control input for the system. This is basically done by adopting the basic model predictive control strategies for the second particle filter. Later in this paper, this new approach is applied to a CSTR (continuous stirred-tank reactor) example and to the inverted pendulum.

Bei der Erprobung sicherheitsrelevanter Bauteile von Nutzfahrzeugen steht man vor der Aufgabe, die sehr vielfältige Belastung durch die Kunden abschätzen zu müssen und daraus ein Prüfprogramm für die Bauteile abzuleiten, das mehreren gegenläufigen Anforderungen gerecht werden muss: Das Programm muss scharf genug sein, damit bei erfolgreicher Prüfung ein Ausfall im Feld im Rahmen eines bestimmungsgemäßen Gebrauchs ausgeschlossen werden kann, es soll aber nicht zu einer Überdimensionierung der Bauteile führen, und es soll mit relativ wenigen Bauteilversuchen eine ausreichende Aussagesicherheit erreicht werden. Wegen der hohen Anforderungen bzgl. Sicherheit müssen bei der klassischen statistischen Vorgehensweise – Schätzen der Verteilung der Kundenbeanspruchung aus Messdaten, Schätzen der Verteilung der Bauteilfestigkeit aus Versuchsergebnissen und Ableiten einer Ausfallwahrscheinlichkeit – die Verteilungen in den extremen Rändern bekannt sein. Dazu reicht aber das Datenmaterial in der Regel bei weitem nicht aus. Bei der klassischen „empirischen“ Vorgehensweise werden Kennwerte der Beanspruchung und der Festigkeit verglichen und ein ausreichender Sicherheitsabstand gefordert. Das hier vorgeschlagene Verfahren kombiniert beide Methoden, setzt dabei die Möglichkeiten der statistischen Modellierung soweit aufgrund der Datenlage vertretbar ein und ergänzt die Ergebnisse durch empirisch begründete Sicherheitsfaktoren. Dabei werden bei der Lastfestlegung die im Versuch vorhandenen Möglichkeiten berücksichtigt. Hauptvorteile dieses Verfahrens sind a) die Transparenz bzgl. der mit statistischen Mitteln erreichbaren Aussagen und des Zusammenspiels zwischen Lastermittlung und Versuch und b) die Möglichkeit durch entsprechenden Aufwand bei Messungen und Erprobung die empirischen zugunsten der statistischen Anteile zu reduzieren.

In the ground vehicle industry it is often an important task to simulate full vehicle models based on the wheel forces and moments, which have been measured during driving over certain roads with a prototype vehicle. The models are described by a system of differential algebraic equations (DAE) or ordinary differential equations (ODE). The goal of the simulation is to derive section forces at certain components for a durability assessment. In contrast to handling simulations, which are performed including more or less complex tyre models, a driver model, and a digital road profile, the models we use here usually do not contain the tyres or a driver model. Instead, the measured wheel forces are used for excitation of the unconstrained model. This can be difficult due to noise in the input data, which leads to an undesired drift of the vehicle model in the simulation.

Die Erprobung neuer Fahrzeugachsen oder Achsvarianten auf Basis von Lastdaten aus dem Fahrbetrieb erfolgt meist mit Hilfe komplexer mehrkanaliger Prüfstände. Bei solchen Erprobungen sollen im Allgemeinen die im Fahrbetrieb gemessenen Radnabenkräfte und Momente vom Prüfstand reproduziert werden. Aufgrund der komplexen Wechselwirkungen zwischen Prüfling und Prüfmaschine stellt sich bei jedem neuen Konzept die Frage, ob der gewünschte Test mit einem vorgegebenen Prüfsystemaufbau durchführbar ist, bzw. welche Konfiguration des Prüfsystems für den geplanten Test geeignet erscheint. In dieser Arbeit wird die Modellierung eines neuartigen Achsprüfsystemkonzeptes beschrieben, das auf zwei Hexapoden basiert. Die Modellierung umfasst neben der geometrischen Anordnung des Prüfsystems auch die Hydraulik sowie den internen Controller. Das Prüfsystemmodell wurde als so genanntes Template innerhalb des Fahrzeugsimulationsprogramms ADAMS/Car entwickelt und kann mit verschiedenen Achsmodellen zu einem Gesamtsystem gekoppelt werden. An diesem Gesamtmodell können alle am realen Prüfsystem auftretenden Arbeitsschritte wie Controllereinstellung, Drive-File-Iteration und Simulation durchgeführt werden. Geometrische oder hydraulische Parameter können auf einfache Weise geändert werden, um eine optimale Anpassung des Prüfsystems an den Prüfling und die vorgegebenen Lastdaten zu ermöglichen. Das im Rahmen des Projektes entwickelte Modell unterstützt und begleitet einerseits die Einführung des neuen Achsprüfsystemkonzeptes und kann andererseits zur virtuellen Vorbereitung von Testläufen eingesetzt werden. Am Beispiel einer Vorder- und einer Hinterachse wird die allgemeine Vorgehensweise erläutert und die neuen Möglichkeiten aufgezeigt, die sich durch die Prüfsystemsimulation ergeben.

Testing a new suspension based on real load data is performed on elaborate multi channel test rigs. Usually wheel forces and moments measured during driving maneuvers are reproduced on the rig. Because of the complicated interaction between rig and suspension each new rig configuration has to prove its efficiency with respect to the requirements and the configuration might be subject to optimization. This paper deals with modeling a new rig concept based on two hexapods. The real physical rig has been designed and meanwhile built by MOOG-FCS for VOLKSWAGEN. The aim of the simulation project reported here was twofold: First the simulation of the rig together with real VOLKSWAGEN suspension models at a time where the design was not yet finalized was used to verify and optimize the desired properties of the rig. Second the simulation environment was set up in a way that it can be used to prepare real tests on the rig. The model contains the geometric configuration as well as the hydraulics and the controller. It is implemented as an ADAMS/Car template and can be combined with different suspension models to get a complete assembly representing the entire test rig. Using this model, all steps required for a real test run such as controller adaptation, drive file iteration and simulation can be performed. Geometric or hydraulic parameters can be modified easily to improve the setup and adapt the system to the suspension and the load data.

One approach to multi-criteria IMRT planning is to automatically calculate a data set of Pareto-optimal plans for a given planning problem in a first phase, and then interactively explore the solution space and decide for the clinically best treatment plan in a second phase. The challenge of computing the plan data set is to assure that all clinically meaningful plans are covered and that as many as possible clinically irrelevant plans are excluded to keep computation times within reasonable limits. In this work, we focus on the approximation of the clinically relevant part of the Pareto surface, the process that consititutes the first phase. It is possible that two plans on the Parteto surface have a very small, clinically insignificant difference in one criterion and a significant difference in one other criterion. For such cases, only the plan that is clinically clearly superior should be included into the data set. To achieve this during the Pareto surface approximation, we propose to introduce bounds that restrict the relative quality between plans, so called tradeoff bounds. We show how to integrate these trade-off bounds into the approximation scheme and study their effects.

A simple transformation of the Equation of Motion (EoM) allows us to directly integrate nonlinear structural models into the recursive Multibody System (MBS) formalism of SIMPACK. This contribution describes how the integration is performed for a discrete Cosserat rod model which has been developed at the ITWM. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.

In this paper we address the improvement of transfer quality in public mass transit networks. Generally there are several transit operators offering service and our work is motivated by the question how their timetables can be altered to yield optimized transfer possibilities in the overall network. To achieve this, only small changes to the timetables are allowed. The set-up makes it possible to use a quadratic semi-assignment model to solve the optimization problem. We apply this model, equipped with a new way to assess transfer quality, to the solution of four real-world examples. It turns out that improvements in overall transfer quality can be determined by such optimization-based techniques. Therefore they can serve as a first step towards a decision support tool for planners of regional transit networks.

Open cell foams are a promising and versatile class of porous materials. Open metal foams serve as crash absorbers and catalysts, metal and ceramic foams are used for filtering, and open polymer foams are hidden in every-day-life items like mattresses or chairs. Due to their high porosity, classical 2d quantitative analysis can give only very limited information about the microstructure of open foams. On the other hand, micro computed tomography (μCT) yields high quality 3d images of open foams. Thus 3d imaging is the method of choice for open cell foams. In this report we summarise a variety of methods for the analysis of the resulting volume images of open foam structures developed or refined and applied at the Fraunhofer ITWM over a course of nearly ten years: The model based determination of mean characteristics like the mean cell volume or the mean strut thickness demanding only a simple binarisation as well as the image analytic cell reconstruction yielding empirical distributions of cell characteristics.

In order to optimize the acoustic properties of a stacked fiber non-woven, the microstructure of the non-woven is modeled by a macroscopically homogeneous random system of straight cylinders (tubes). That is, the fibers are modeled by a spatially stationary random system of lines (Poisson line process), dilated by a sphere. Pressing the non-woven causes anisotropy. In our model, this anisotropy is described by a one parametric distribution of the direction of the fibers. In the present application, the anisotropy parameter has to be estimated from 2d reflected light microscopic images of microsections of the non-woven. After fitting the model, the flow is computed in digitized realizations of the stochastic geometric model using the lattice Boltzmann method. Based on the flow resistivity, the formulas of Delany and Bazley predict the frequency-dependent acoustic absorption of the non-woven in the impedance tube. Using the geometric model, the description of a non-woven with improved acoustic absorption properties is obtained in the following way: First, the fiber thicknesses, porosity and anisotropy of the fiber system are modified. Then the flow and acoustics simulations are performed in the new sample. These two steps are repeatedc for various sets of parameters. Finally, the set of parameters for the geometric model leading to the best acoustic absorption is chosen.

IMRT planning on adaptive volume structures – a significant advance of computational complexity
(2004)

In intensity-modulated radiotherapy (IMRT) planning the oncologist faces the challenging task of finding a treatment plan that he considers to be an ideal compromise of the inherently contradictive goals of delivering a sufficiently high dose to the target while widely sparing critical structures. The search for this a priori unknown compromise typically requires the computation of several plans, i.e. the solution of several optimization problems. This accumulates to a high computational expense due to the large scale of these problems - a consequence of the discrete problem formulation. This paper presents the adaptive clustering method as a new algorithmic concept to overcome these difficulties. The computations are performed on an individually adapted structure of voxel clusters rather than on the original voxels leading to a decisively reduced computational complexity as numerical examples on real clinical data demonstrate. In contrast to many other similar concepts, the typical trade-off between a reduction in computational complexity and a loss in exactness can be avoided: the adaptive clustering method produces the optimum of the original problem. This flexible method can be applied to both single- and multi-criteria optimization methods based on most of the convex evaluation functions used in practice

In this paper, the analysis of one approach for the regularization of pure Neumann problems for second order elliptical equations, e.g., Poisson’s equation and linear elasticity equations, is presented. The main topic under consideration is the behavior of the condition number of the regularized problem. A general framework for the analysis is presented. This allows to determine a form of regularization term which leads to the “natural” asymptotic of the condition number of the regularized problem with respect to mesh parameter. Some numerical results, which support theoretical analysis are presented as well. The main motivation for the presented research is to develop theoretical background for an efficient and robust implementation of the solver for pure Neumann problems for the linear elasticity equations. Such solvers usually are needed in a number of domain decomposition methods, e.g. FETI. Developed approaches are planed to be used in software, developing in ITWM, e.g. KneeMech simulation software.

Virtual material design is the microscopic variation of materials in the computer, followed by the numerical evaluation of the effect of this variation on the material‘s macroscopic properties. The goal of this procedure is an in some sense improved material. Here, we give examples regarding the dependence of the effective elastic moduli of a composite material on the geometry of the shape of an inclusion. A new approach on how to solve such interface problems avoids mesh generation and gives second order accurate results even in the vicinity of the interface. The Explicit Jump Immersed Interface Method is a finite difference method for elliptic partial differential equations that works on an equidistant Cartesian grid in spite of non-grid aligned discontinuities in equation parameters and solution. Near discontinuities, the standard finite difference approximations are modified by adding correction terms that involve jumps in the function and its derivatives. This work derives the correction terms for two dimensional linear elasticity with piecewise constant coefficients, i.e. for composite materials. It demonstrates numerically convergence and approximation properties of the method.

We introduce a refined tree method to compute option prices using the stochastic volatility model of Heston. In a first step, we model the stock and variance process as two separate trees and with transition probabilities obtained by matching tree moments up to order two against the Heston model ones. The correlation between the driving Brownian motions in the Heston model is then incorporated by the node-wise adjustment of the probabilities. This adjustment, leaving the marginals fixed, optimizes the match between tree and model correlation. In some nodes, we are even able to further match moments of higher order. Numerically this gives convergence orders faster than 1/N, where N is the number of dis- cretization steps. Accuracy of our method is checked for European option prices against a semi closed-form, and our prices for both European and American options are compared to alternative approaches.

We study global and local robustness properties of several estimators for shape and scale in a generalized Pareto model. The estimators considered in this paper cover maximum likelihood estimators, skipped maximum likelihood estimators, moment-based estimators, Cramér-von-Mises Minimum Distance estimators, and, as a special case of quantile-based estimators, Pickands Estimator as well as variants of the latter tuned for higher finite sample breakdown point (FSBP), and lower variance. We further consider an estimator matching population median and median of absolute deviations to the empirical ones (MedMad); again, in order to improve its FSBP, we propose a variant using a suitable asymmetric Mad as constituent, and which may be tuned to achieve an expected FSBP of 34%. These estimators are compared to one-step estimators distinguished as optimal in the shrinking neighborhood setting, i.e., the most bias-robust estimator minimizing the maximal (asymptotic) bias and the estimator minimizing the maximal (asymptotic) MSE. For each of these estimators, we determine the FSBP, the influence function, as well as statistical accuracy measured by asymptotic bias, variance, and mean squared error—all evaluated uniformly on shrinking convex contamination neighborhoods. Finally, we check these asymptotic theoretical findings against finite sample behavior by an extensive simulation study.

We present some optimality results for robust Kalman filtering. To this end, we introduce the general setup of state space models which will not be limited to a Euclidean or time-discrete framework. We pose the problem of state reconstruction and repeat the classical existing algorithms in this context. We then extend the ideal-model setup allowing for outliers which in this context may be system-endogenous or -exogenous, inducing the somewhat conflicting goals of tracking and attenuation. In quite a general framework, we solve corresponding minimax MSE-problems for both types of outliers separately, resulting in saddle-points consisting of an optimally-robust procedure and a corresponding least favorable outlier situation. Still insisting on recursivity, we obtain an operational solution, the rLS filter and variants of it. Exactly robust-optimal filters would need knowledge of certain hard-to-compute conditional means in the ideal model; things would be much easier if these conditional means were linear. Hence, it is important to quantify the deviation of the exact conditional mean from linearity. We obtain a somewhat surprising characterization of linearity for the conditional expectation in this setting. Combining both optimal filter types (for system-endogenous and -exogenous situation) we come up with a delayed hybrid filter which is able to treat both types of outliers simultaneously. Keywords: robustness, Kalman Filter, innovation outlier, additive outlier

We are concerned with modeling and simulation of the pressing section of a paper machine. We state a two-dimensional model of a press nip which takes into account elasticity and flow phenomena. Nonlinear filtration laws are incorporated into the flow model. We present a numerical solution algorithm and a numerical investigation of the model with special focus on inertia effects.

This work deals with the optimal control of a free surface Stokes flow which responds to an applied outer pressure. Typical applications are fiber spinning or thin film manufacturing. We present and discuss two adjoint-based optimization approaches that differ in the treatment of the free boundary as either state or control variable. In both cases the free boundary is modeled as the graph of a function. The PDE-constrained optimization problems are numerically solved by the BFGS method, where the gradient of the reduced cost function is expressed in terms of adjoint variables. Numerical results for both strategies are finally compared with respect to accuracy and efficiency.

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system
of equations which describe filtration process in the pressing section of a
paper machine. Two flow regimes appear in the modeling of this problem.
The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards approach together with a dynamic capillary pressure model. The finite
volume method is used to approximate the system of PDEs. Then the existence of a discrete solution to proposed finite difference scheme is proven.
Compactness of the set of all discrete solutions for different mesh sizes is
proven. The main Theorem shows that the discrete solution converges
to the solution of continuous problem. At the end we present numerical
studies for the rate of convergence.

Numerical modeling of electrochemical process in Li-Ion battery is an emerging topic of great practical interest. In this work we present a Finite Volume discretization of electrochemical diffusive processes occurring during the operation of Li-Ion batteries. The system of equations is a nonlinear, time-dependent diffusive system, coupling the Li concentration and the electric potential. The system is formulated at length-scale at which two different types of domains are distinguished, one for the electrolyte and one for the active solid particles in the electrode. The domains can be of highly irregular shape, with electrolyte occupying the pore space of a porous electrode. The material parameters in each domain differ by several orders of magnitude and can be non-linear functions of Li ions concentration and/or the electrical potential. Moreover, special interface conditions are imposed at the boundary separating the electrolyte from the active solid particles. The field variables are discontinuous across such an interface and the coupling is highly non- linear, rendering direct iteration methods ineffective for such problems. We formulate a Newton iteration for an purely implicit Finite Volume discretization of the coupled system. A series of numerical examples are presented for different type of electrolyte/electrode configurations and material parameters. The convergence of the Newton method is characterized both as function of nonlinear material parameters as well as the nonlinearity in the interface conditions.

The paper at hand presents a slender body theory for the dynamics of a curved inertial viscous Newtonian ber. Neglecting surface tension and temperature dependence, the ber ow 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 ber center-line with the inner viscous transport. The physically reasonable form of the one-dimensional ber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms.

We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micropeaks and valleys is very small compared with the macrosize of the bodies’ domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method involves the twoscale asymptotic homogenization procedure that takes into account the microgeometry of the interface layer and the stiffnesses of materials of both domains. The macrocontact condition can then be used in a FEM model for the contact problem on the macrolevel. The averaged contact stiffness obtained allows the replacement of the interface layer in the macromodel by the macrocontact condition.

The theory of the two-scale convergence was applied to homogenization of elasto-plastic composites with a periodic structure and exponential hardening law. The theory is based on the fact that the elastic as well as the plastic part of the stress field two-scale converges to a limit, which is factorized by parts, depending only on macroscopic characteristics, represented in terms of corresponding part of the homogenised stress tensor and only on stress concentration tensor, related to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to metallic matrix material with Ludwik and Hocket-Sherby hardening law and pure elastic inclusions in two numerical examples. Results were compared with results of mechanical averaging based on the self-consistent methods.

A spectral theory for constituents of macroscopically homogeneous random microstructures modeled as homogeneous random closed sets is developed and provided with a sound mathematical basis, where the spectrum obtained by Fourier methods corresponds to the angular intensity distribution of x-rays scattered by this constituent. It is shown that the fast Fourier transform applied to three-dimensional images of microstructures obtained by micro-tomography is a powerful tool of image processing. The applicability of this technique is is demonstrated in the analysis of images of porous media.

Two approaches for determining the Euler-Poincaré characteristic of a set observed on lattice points are considered in the context of image analysis { the integral geometric and the polyhedral approach. Information about the set is assumed to be available on lattice points only. In order to retain properties of the Euler number and to provide a good approximation of the true Euler number of the original set in the Euclidean space, the appropriate choice of adjacency in the lattice for the set and its background is crucial. Adjacencies are defined using tessellations of the whole space into polyhedrons. In R 3 , two new 14 adjacencies are introduced additionally to the well known 6 and 26 adjacencies. For the Euler number of a set and its complement, a consistency relation holds. Each of the pairs of adjacencies (14:1; 14:1), (14:2; 14:2), (6; 26), and (26; 6) is shown to be a pair of complementary adjacencies with respect to this relation. That is, the approximations of the Euler numbers are consistent if the set and its background (complement) are equipped with this pair of adjacencies. Furthermore, sufficient conditions for the correctness of the approximations of the Euler number are given. The analysis of selected microstructures and a simulation study illustrate how the estimated Euler number depends on the chosen adjacency. It also shows that there is not a uniquely best pair of adjacencies with respect to the estimation of the Euler number of a set in Euclidean space.

Home Health Care (HHC) services are becoming increasingly important in Europe’s aging societies. Elderly people have varying degrees of need for assistance and medical treatment. It is advantageous to allow them to live in their own homes as long as possible, since a long-term stay in a nursing home can be much more costly for the social insurance system than a treatment at home providing assistance to the required level. Therefore, HHC services are a cost-effective and flexible instrument in the social system. In Germany, organizations providing HHC services are generally either larger charities with countrywide operations or small private companies offering services only in a city or a rural area. While the former have a hierarchical organizational structure and a large number of employees, the latter typically only have some ten to twenty nurses under contract. The relationship to the patients (“customers”) is often long-term and can last for several years. Therefore acquiring and keeping satisfied customers is crucial for HHC service providers and intensive competition among them is observed.

In this paper, a multi-period supply chain network design problem is addressed. Several aspects of practical relevance are considered such as those related with the financial decisions that must be accounted for by a company managing a supply chain. The decisions to be made comprise the location of the facilities, the flow of commodities and the investments to make in alternative activities to those directly related with the supply chain design. Uncertainty is assumed for demand and interest rates, which is described by a set of scenarios. Therefore, for the entire planning horizon, a tree of scenarios is built. A target is set for the return on investment and the risk of falling below it is measured and accounted for. The service level is also measured and included in the objective function. The problem is formulated as a multi-stage stochastic mixed-integer linear programming problem. The goal is to maximize the total financial benefit. An alternative formulation which is based upon the paths in the scenario tree is also proposed. A methodology for measuring the value of the stochastic solution in this problem is discussed. Computational tests using randomly generated data are presented showing that the stochastic approach is worth considering in these type of problems.

»Denn nichts ist für den Menschen als Menschen etwas wert, was er nicht mit Leidenschaft tun kann«
(2001)

Vortrag anlässlich der Verleihung des Akademiepreises des Landes Rheinland-Pfalz am 21.11.2001 Was macht einen guten Hochschullehrer aus? Auf diese Frage gibt es sicher viele verschiedene, fachbezogene Antworten, aber auch ein paar allgemeine Gesichtspunkte: es bedarf der »Leidenschaft« für die Forschung (Max Weber), aus der dann auch die Begeisterung für die Lehre erwächst. Forschung und Lehre gehören zusammen, um die Wissenschaft als lebendiges Tun vermitteln zu können. Der Vortrag gibt Beispiele dafür, wie in angewandter Mathematik Forschungsaufgaben aus praktischen Alltagsproblemstellungen erwachsen, die in die Lehre auf verschiedenen Stufen (Gymnasium bis Graduiertenkolleg) einfließen; er leitet damit auch zu einem aktuellen Forschungsgebiet, der Mehrskalenanalyse mit ihren vielfältigen Anwendungen in Bildverarbeitung, Materialentwicklung und Strömungsmechanik über, was aber nur kurz gestreift wird. Mathematik erscheint hier als eine moderne Schlüsseltechnologie, die aber auch enge Beziehungen zu den Geistes- und Sozialwissenschaften hat.

No doubt: Mathematics has become a technology in its own right, maybe even a key technology. Technology may be defined as the application of science to the problems of commerce and industry. And science? Science maybe defined as developing, testing and improving models for the prediction of system behavior; the language used to describe these models is mathematics and mathematics provides methods to evaluate these models. Here we are! Why has mathematics become a technology only recently? Since it got a tool, a tool to evaluate complex, "near to reality" models: Computer! The model may be quite old - Navier-Stokes equations describe flow behavior rather well, but to solve these equations for realistic geometry and higher Reynolds numbers with sufficient precision is even for powerful parallel computing a real challenge. Make the models as simple as possible, as complex as necessary - and then evaluate them with the help of efficient and reliable algorithms: These are genuine mathematical tasks.

On a multigrid solver for the threedimensional Biot poroelasticity system in multilayered domains
(2006)

In this paper, we present problem–dependent prolongation and problem–dependent restriction for a multigrid solver for the three-dimensional Biot poroelasticity system, which is solved in a multilayered domain. The system is discretized on a staggered grid using the finite volume method. During the discretization, special care is taken of the discontinuous coefficients. For the efficient multigrid solver, a need in operator-dependent restriction and/or prolongation arises. We derive these operators so that they are consistent with the discretization. They account for the discontinuities of the coefficients, as well as for the coupling of the unknowns within the Biot system. A set of numerical experiments shows necessity of use of the operator-dependent restriction and prolongation in the multigrid solver for the considered class of problems.

In this paper we propose a finite volume discretization for the threedimensional Biot poroelasticity system in multilayered domains. For the stability reasons, staggered grids are used. The discretization accounts for discontinuity of the coefficients across the interfaces between layers with different physical properties. Numerical experiments, based on the proposed discretization showed second order convergence in the maximum norm for the primary as well as flux unknowns of the system. A certain application example is presented as well.

Background and purpose Inherently, IMRT treatment planning involves compromising between different planning goals. Multi-criteria IMRT planning directly addresses this compromising and thus makes it more systematic. Usually, several plans are computed from which the planner selects the most promising following a certain procedure. Applying Pareto navigation for this selection step simultaneously increases the variety of planning options and eases the identification of the most promising plan. Material and methods Pareto navigation is an interactive multi-criteria optimization method that consists of the two navigation mechanisms “selection” and “restriction”. The former allows the formulation of wishes whereas the latter allows the exclusion of unwanted plans. They are realized as optimization problems on the so-called plan bundle – a set constructed from precomputed plans. They can be approximately reformulated so that their solution time is a small fraction of a second. Thus, the user can be provided with immediate feedback regarding his or her decisions.

Asymptotic homogenisation technique and two-scale convergence is used for analysis of macro-strength and fatigue durability of composites with a periodic structure under cyclic loading. The linear damage accumulation rule is employed in the phenomenological micro-durability conditions (for each component of the composite) under varying cyclic loading. Both local and non-local strength and durability conditions are analysed. The strong convergence of the strength and fatigue damage measure as the structure period tends to zero is proved and their limiting values are estimated.

A general multi-period network redesign problem arising in the context of strategic supply chain planning (SCP) is studied. Several aspects of practical relevance in SCP are captured namely, multiple facility layers with different types of facilities, flows between facilities in the same layer, direct shipments to customers, and facility relocation. An efficient two-phase heuristic approach is proposed for obtaining feasible solutions to the problem, which is initially modeled as a large-scale mixed-integer linear program. In the first stage of the heuristic, a linear programming rounding strategy is applied to second initial values for the binary location variables in the model. The second phase of the heuristic uses local search to correct the initial solution when feasibility is not reached or to improve the solution when its quality does not meet given criteria. The results of an extensive computational study performed on randomly generated instances are reported.

Structuring global supply chain networks is a complex decision-making process. The typical inputs to such a process consist of a set of customer zones to serve, a set of products to be manufactured and distributed, demand projections for the different customer zones, and information about future conditions, costs (e.g. for production and transportation) and resources (e.g. capacities, available raw materials). Given the above inputs, companies have to decide where to locate new service facilities (e.g. plants, warehouses), how to allocate procurement and production activities to the variousmanufacturing facilities, and how to manage the transportation of products through the supply chain network in order to satisfy customer demands. We propose a mathematical modelling framework capturing many practical aspects of network design problems simultaneously. For problems of reasonable size we report on computational experience with standard mathematical programming software. The discussion is extended with other decisions required by many real-life applications in strategic supply chain planning. In particular, the multi-period nature of some decisions is addressed by a more comprehensivemodel, which is solved by a specially tailored heuristic approach. The numerical results suggest that the solution procedure can identify high quality solutions within reasonable computational time.

Facility location decisions play a critical role in the strategic design of supply chain networks. In this paper, an extensive literature review of facility location models in the context of supply chain management is given. Following a brief review of core models in facility location, we identify basic features that such models must capture to support decision-making involved in strategic supply chain planning. In particular, the integration of location decisions with other decisions relevant to the design of a supply chain network is discussed. Furthermore, aspects related to the structure of the supply chain network, including those specific to reverse logistics, are also addressed. Significant contributions to the current state-of-the-art are surveyed taking into account numerous factors. Supply chain performance measures and optimization techniques are also reviewed. Applications of facility location models to supply chain network design ranging across various industries are discussed. Finally, a list of issues requiring further research are highlighted.

In this paper we focus on the strategic design of supply chain networks. We propose a mathematical modeling framework that captures many practical aspects of network design problems simultaneously but which have not received adequate attention in the literature. The aspects considered include: dynamic planning horizon, generic supply chain network structure, external supply of materials, inventory opportunities for goods, distribution of commodities, facility configuration, availability of capital for investments, and storage limitations. Moreover, network configuration decisions concerning the gradual relocation of facilities over the planning horizon are considered. To cope with fluctuating demands, capacity expansion and reduction scenarios are also analyzed as well as modular capacity shifts. The relation of the proposed modeling framework with existing models is discussed. For problems of reasonable size we report on our computational experience with standard mathematical programming software. In particular, useful insights on the impact of various factors on network design decisions are provided.

The understanding of the motion of long slender elastic fibers in turbulent flows is of great interest to research, development and production in technical textiles manufacturing. The fiber dynamics depend on the drag forces that are imposed on the fiber by the fluid. Their computation requires in principle a coupling of fiber and flow with no-slip interface conditions. However, theneeded high resolution and adaptive grid refinement make the direct numerical simulation of the three-dimensional fluid-solid-problem for slender fibers and turbulent flows not only extremely costly and complex, but also still impossible for practically relevant applications. Embedded in a slender body theory, an aerodynamic force concept for a general drag model was therefore derived on basis of a stochastic k-o; description for a turbulent flow field in [23]. The turbulence effects on the fiber dynamics were modeled by a correlated random Gaussian force and its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow-dependent amplitude. The concept was numerically studied under the conditions of a melt-spinning process for nonwoven materials in [24] – for the specific choice of a non-linear Taylor drag model. Taylor [35] suggested the heuristic model for high Reynolds number flows, Re in [20, 3 · 105], around inclined slender objects under an angle of attack of alpha in (pi/36, pi/2] between flow and object tangent. Since the Reynolds number is considered with respect to the relative velocity between flow and fiber, the numerical results lackaccuracy evidently for small Re that occur in cases of flexible light fibers moving occasionally with the flow velocity. In such a regime (Re << 1), linear Stokes drag forces were successfully applied for the prediction of small particles immersed in turbulent flows, see e.g. [25, 26, 32, 39], a modifiedStokes force taking also into account the particle oscillations was presented in [14]. The linear drag relation was also conferred to longer filaments by imposing free-draining assumptions [29, 8]. Apart from this, the Taylor drag suffers from its non-applicability to tangential incident flow situations (alpha = 0) that often occur in fiber and nonwoven production processes.

In this paper we extend the slender body theory for the dynamics of a curved inertial viscous Newtonian fiber [23] by the inclusion of surface tension in the systematic asymptotic framework and the deduction of boundary conditions for the free fiber end, as it occurs in rotational spinning processes of glass fibers. The fiber ow is described by a three-dimensional free boundary value problem in terms of instationary incompressible Navier-Stokes equations under the neglect of temperature dependence. From standard regular expansion techniques in powers of the slenderness parameter we derive asymptotically leading-order balance laws for mass and momentum combining the inner viscous transport with unrestricted motion and shape of the fiber center-line which becomes important in the practical application. For the numerical investigation of the effects due to surface tension, viscosity, gravity and rotation on the fiber behavior we apply a fnite volume method with implicit flux discretization.

Fiber Dynamics in Turbulent Flows -Part I: General Modeling Framework -Part II: Specific Taylor Drag
(2005)

Part I: General Modeling Framework The paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber. Independent of the choice of the drag model, a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow. Its construction as centered differentiable Gaussian field complies thereby with the requirements of the stochastic k-turbulence model and Kolmogorov’s universal equilibrium theory on local isotropy. Part II: Specific Taylor Drag In [12], an aerodynamic force concept for a general air drag model is derived on top of a stochastic k-epsilon description for a turbulent flow field. The turbulence effects on the dynamics of a long slender elastic fiber are particularly modeled by a correlated random Gaussian force and in its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow - dependent amplitude. The paper at hand now presents quantitative similarity estimates and numerical comparisons for the concrete choice of a Taylor drag model in a given application.

An easy numerical handling of time-dependent problems with complicated geometries, free moving boundaries and interfaces, or oscillating solutions is of great importance for many applications, e.g., in fluid dynamics (free surface and multiphase flows, fluid-structure interactions [22, 18, 24]), failure mechanics (crack growth and propagation [4]), magnetohydrodynamics (accretion disks, jets and cloud simulation [6]), biophysics and -chemistry. Appropriate discretizations, so-called mesh-less methods, have been developed during the last decades to meet these challenging demands and to relieve the burden of remeshing and successive mesh generation being faced by the conventional mesh-based methods, [16, 10, 3]. The prearranged mesh is an artificial constraint to ensure compatibility of the mesh-based interpolant schemes, that often conflicts with the real physical conditions of the continuum model. Then, remeshing becomes inevitable, which is not only extremely time- and storage consuming but also the source for numerical errors and hence the gradual loss of computational accuracy. Apart from this advantage, mesh-less methods also lead to fundamentally better approximations regarding aspects, such as smoothness, nonlocal interpolation character, flexible connectivity, refinement and enrichment procedures, [16]. The common idea of mesh-less methods is the discretization of the domain of interest by a finite set of independent, randomly distributed particles moving with a characteristic velocity of the problem. Location and distribution of the particles then account for the time-dependent description of the geometry, data and solution. Thereby, the global solution is linearly superposed from the local information carried by the particles. In classical particle methods [20, 21], the respective weight functions are Dirac distributions which yield solutions in a distributional sense.

This paper disscuses the minimal area rectangular packing problem of how to pack a set of specified, non-overlapping rectangels into a rectangular container of minimal area. We investigate different mathematical programming approaches of this and introduce a novel approach based on non-linear optimization and the \\\"tunneling effect\\\" achieved by a relaxation of the non-overlapping constraints.

The Folgar-Tucker equation (FTE) is the model most frequently used for the prediction of fiber orientation (FO) in simulations of the injection molding process for short-fiber reinforced thermoplasts. In contrast to its widespread use in injection molding simulations, little is known about the mathematical properties of the FTE: an investigation of e.g. its phase spaceMFT has been presented only recently. The restriction of the dependent variable of the FTE to the setMFT turns the FTE into a differential algebraic system (DAS), a fact which is commonly neglected when devising numerical schemes for the integration of the FTE. In this article1 we present some recent results on the problem of trace stability as well as some introductory material which complements our recent paper.

Summary. We present a model of exible rods | based on Kirchhoff\\\'s geometrically exact theory | which is suitable for the fast simulation of quasistatic deformations within VR or functional DMU applications. Unlike simple models of \\\"mass & spring\\\" type typically used in VR applications, our model provides a proper coupling of bending and torsion. The computational approach comprises a variational formulation combined with a nite dierence discretization of the continuum model. Approximate solutions of the equilibrium equations for sequentially varying boundary conditions are obtained by means of energy minimization using a nonlinear CG method. The computational performance of our model proves to be sucient for the interactive manipulation of exible cables in assembly simulation.

Recently we developed a discrete model of elastic rods with symmetric cross section suitable for a fast simulation of quasistatic deformations [33]. The model is based on Kirchhoff’s geometrically exact theory of rods. Unlike simple models of “mass & spring” type typically used in VR applications, our model provides a proper coupling of bending and torsion. The computational approach comprises a variational formulation combined with a finite difference discretization of the continuum model. Approximate solutions of the equilibrium equations for sequentially varying boundary conditions are obtained by means of energy minimization using a nonlinear CG method. As the computational performance of our model yields solution times within the range of milliseconds, our approach proves to be sufficient to simulate an interactive manipulation of such flexible rods in virtual reality applications in real time.

We present the derivation of a simple viscous damping model of Kelvin–Voigt type for geometrically exact
Cosserat rods from three–dimensional continuum theory. Assuming a homogeneous and isotropic material,
we obtain explicit formulas for the damping parameters of the model in terms of the well known stiffness
parameters of the rod and the retardation time constants defined as the ratios of bulk and shear viscosities to
the respective elastic moduli. We briefly discuss the range of validity of our damping model and illustrate
its behaviour with a numerical example.

This paper introduces methods for the detection of anisotropies which are caused by compression of regular three-dimensional point patterns. Isotropy tests based on directional summary statistics and estimators for the compression factor are developed. These allow not only for the detection of anisotropies but also for the estimation of their strength. Using simulated data the power of the methods and the dependence of the power on the intensity, the degree of regularity, and the compression strength are studied. The motivation of this paper is the investigation of anisotropies in the structure of polar ice. Therefore, our methods are applied to the point patterns of centres of air pores extracted from tomographic images of ice cores. This way the presence of anisotropies in the ice caused by the compression of the ice sheet as well as an increase of their strength with increasing depth are shown.

Modeling of species and charge transport in Li-Ion Batteries based on non-equilibrium thermodynamics
(2010)

In order to improve the design of Li ion batteries the complex interplay of various physical phenomena in the active particles of the electrodes and in the electrolyte has to be balanced. The separate transport phenomena in the electrolyte and in the active particle as well as their coupling due to the electrochemical reactions at the interfaces between the electrode particles and the electrolyte will inuence the performance and the lifetime of a battery. Any modeling of the complex phenomena during the usage of a battery has therefore to be based on sound physical and chemical principles in order to allow for reliable predictions for the response of the battery to changing load conditions. We will present a modeling approach for the transport processes in the electrolyte and the electrodesbased on non-equilibrium thermodynamics and transport theory. The assumption of local charge neutrality, which is known to be valid in concentrated electrolytes, is explicitly used to identify the independent thermodynamic variables and uxes. The theory guarantees strictly positive entropy production. Dierences to other theories will be discussed.

We will present a rigorous derivation of the equations and interface conditions for ion, charge and heat transport in Li-ion insertion batteries. The derivation is based exclusively on universally accepted principles of nonequilibrium thermodynamics and the assumption of a one step intercalation reaction at the interface of electrolyte and active particles. Without loss of generality the transport in the active particle is assumed to be isotropic. The electrolyte is described as a fully dissociated salt in a neutral solvent. The presented theory is valid for transport on a spatial scale for which local charge neutrality holds i.e. beyond the scale of the diffuse double layer. Charge neutrality is explicitely used to determine the correct set of thermodynamically independent variables. The theory guarantees strictly positive entropy production. The various contributions to the Peltier coeficients for the interface between the active particles and the electrolyte as well as the contributions to the heat of mixing are obtained as a result of the theory.

One of the main goals of an organization developing software is to increase the quality of the software while at the same time to decrease the costs and the duration of the development process. To achieve this, various decisions e.ecting this goal before and during the development process have to be made by the managers. One appropriate tool for decision support are simulation models of the software life cycle, which also help to understand the dynamics of the software development process. Building up a simulation model requires a mathematical description of the interactions between di.erent objects involved in the development process. Based on experimental data, techniques from the .eld of knowledge discovery can be used to quantify these interactions and to generate new process knowledge based on the analysis of the determined relationships. In this paper blocked neuronal networks and related relevance measures will be presented as an appropriate tool for quanti.cation and validation of qualitatively known dependencies in the software development process.

In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (‘stiff’ dof), bending and torsion (‘soft’ dof). For inner dissipation, a consistent damping potential proposed by Antman is chosen. We parametrise the rotational dof by unit quaternions and directly use the quaternionic evolution differential equation for the discretisation of the Cosserat rod curvature. The discrete version of our rod model is obtained via a finite difference discretisation on a staggered grid. After an index reduction from three to zero, the right-hand side function f and the Jacobian \(\partial f/\partial(q, v, t)\) of the dynamical system \(\dot{q} = v, \dot{v} = f(q, v, t)\) is free of higher algebraic (e. g. root) or transcendental (e. g. trigonometric or exponential) functions and therefore cheap to evaluate. A comparison with Abaqus finite element results demonstrates the correct mechanical behavior of our discrete rod model. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computational times within milliseconds, it is suitable for interactive applications in ‘virtual reality’ as well as for multibody dynamics simulation.

In this paper, we present a viscoelastic rod model that is suitable for fast and sufficiently accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (’stiff ’ dof), bending and torsion (’soft’ dof). For inner dissipation, a consistent damping potential from Antman is chosen. Our discrete model is based on a finite difference discretisation on a staggered grid. The right-hand side function f and the Jacobian ∂f/∂(q, v, t) of the dynamical system q˙ = v, v˙ = f(q, v, t) – after index reduction from three to zero – is free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore cheap to evaluate. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computation times within milliseconds, it is suitable for interactivemanipulation in ’virtual reality’ applications. In contrast to fast common VR rod models, our model reflects the structural mechanics solutions sufficiently correct, as comparison with ABAQUS finite element results shows.

In this paper, the model of Köttgen, Barkey and Socie, which corrects the elastic stress and strain tensor histories at notches of a metallic specimen under non-proportional loading, is improved. It can be used in connection with any multiaxial s -e -law of incremental plasticity. For the correction model, we introduce a constraint for the strain components that goes back to the work of Hoffmann and Seeger. Parameter identification for the improved model is performed by Automatic Differentiation and an established least squares algorithm. The results agree accurately both with transient FE computations and notch strain measurements.

In this article, we consider the quasistatic boundary value problems of linear elasticity and nonlinear elastoplasticity, with linear Hooke’s law in the elastic regime for both problems and with the linear kinematic hardening law for the plastic regime in the latter problem. We derive expressions and estimates for the difference of the solutions of both models, i.e. for the stresses, the strains and the displacements. To this end, we use the stop and play operators of nonlinear functional analysis. Further, we give an explicit example of a homotopy between the solutions of both problems.

Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrised by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests.

In this article, we summarise the rotation-free and quaternionic parametrisation of a rigid body. We derive and explain the close interrelations between both parametrisations. The internal constraints due to the redundancies in the parametrisations, which lead to DAEs, are handled with the null space technique. We treat both single rigid bodies and general multibody systems with joints, which lead to external joint constraints. Several numerical examples compare both formalisms to the index reduced versions of the corresponding standard formulations.

We give an analytical and geometrical treatment of what it means to sepa rate a Gaussian kernel along arbitrary axes in Rn, and we present a separation scheme that allows to efficiently implement anisotropic Gaussian convolution filters in arbitrary dimension. Based on our previous analysis we show that this scheme is optimal with regard to the number of memory accesses and nterpolation operations needed. Our method relies on non-orthogonal convolution axes and works com- pletely in image space. Thus, it avoids the need for an FFT-subroutine. Depending on the accuracy and speed requirements, different interpolation schemes and methods to implement the one-dimensional Gaussian (FIR, IIR) can be integrated. The algorithm is also feasible for hardware that does not contain a floating-point unit. Special emphasis is laid on analyzing the performance and accuracy of our method. In particular, we show that withot any special optimization of the source code, our method can perform anisotropic Gaussian filtering faster than methods relyin on the Fast Fourier Transform.

Inverse treatment planning of intensity modulated radiothrapy is a multicriteria optimization problem: planners have to find optimal compromises between a sufficiently high dose in tumor tissue that garantuee a high tumor control, and, dangerous overdosing of critical structures, in order to avoid high normal tissue complcication problems. The approach presented in this work demonstrates how to state a flexible generic multicriteria model of the IMRT planning problem and how to produce clinically highly relevant Pareto-solutions. The model is imbedded in a principal concept of Reverse Engineering, a general optimization paradigm for design problems. Relevant parts of the Pareto-set are approximated by using extreme compromises as cornerstone solutions, a concept that is always feasible if box constraints for objective funtions are available. A major practical drawback of generic multicriteria concepts trying to compute or approximate parts of the Pareto-set is the high computational effort. This problem can be overcome by exploitation of an inherent asymmetry of the IMRT planning problem and an adaptive approximation scheme for optimal solutions based on an adaptive clustering preprocessing technique. Finally, a coherent approach for calculating and selecting solutions in a real-timeinteractive decision-making process is presented. The paper is concluded with clinical examples and a discussion of ongoing research topics.

To simulate the influence of process parameters to the melt spinning process a fiber model is used and coupled with CFD calculations of the quench air flow. In the fiber model energy, momentum and mass balance are solved for the polymer mass flow. To calculate the quench air the Lattice Boltzmann method is used. Simulations and experiments for different process parameters and hole configurations are compared and show a good agreement. Keywords: Melt spinning, fiber model, Lattice Boltzmann, CFD.

A Lagrangian particle scheme is applied to the projection method for the incompressible Navier-Stokes equations. The approximation of spatial derivatives is obtained by the weighted least squares method. The pressure Poisson equation is solved by a local iterative procedure with the help of the least squares method. Numerical tests are performed for two dimensional cases. The Couette flow, Poiseuelle flow, decaying shear flow and the driven cavity flow are presented. The numerical solutions are obtained for stationary as well as instationary cases and are compared with the analytical solutions for channel flows. Finally, the driven cavity in a unit square is considered and the stationary solution obtained from this scheme is compared with that from the finite element method.

In nancial mathematics stock prices are usually modelled directly as a result of supply and demand and under the assumption that dividends are paid continuously. In contrast economic theory gives us the dividend discount model assuming that the stock price equals the present value of its future dividends. These two models need not to contradict each other - in their paper Korn and Rogers (2005) introduce a general dividend model preserving the stock price to follow a stochastic process and to be equal to the sum of all its discounted dividends. In this paper we specify the model of Korn and Rogers in a Black-Scholes framework in order to derive a closed-form solution for the pricing of American Call options under the assumption of a known next dividend followed by several stochastic dividend payments during the option's time to maturity.

We consider the problem of pricing European forward starting options in the presence of stochastic volatility. By performing a change of measure using the asset price at the time of strike determination as a numeraire, we derive a closed-form solution based on Heston’s model of stochastic volatility.

A unified approach to Credit Default Swaption and Constant Maturity Credit Default Swap valuation
(2006)

In this paper we examine the pricing of arbitrary credit derivatives with the Libor Market Model with Default Risk. We show, how to setup the Monte Carlo-Simulation efficiently and investigate the accuracy of closed-form solutions for Credit Default Swaps, Credit Default Swaptions and Constant Maturity Credit Default Swaps. In addition we derive a new closed-form solution for Credit Default Swaptions which allows for time-dependent volatility and abitrary correlation structure of default intensities.1

If an investor borrows money he generally has to pay higher interest rates than he would have received, if he had put his funds on a savings account. The classical model of continuous time portfolio optimisation ignores this effect. Since there is obviously a connection between the default probability and the total percentage of wealth, which the investor is in debt, we study portfolio optimisation with a control dependent interest rate. Assuming a logarithmic and a power utility function, respectively, we prove explicit formulae of the optimal control.

In this paper we deal with dierent statistical modeling of real world accident data in order to quantify the eectiveness of a safety function or a safety conguration (meaning a specic combination of safety functions) in vehicles. It is shown that the eectiveness can be estimated along the so-called relative risk, even if the eectiveness does depend on a confounding variable which may be categorical or continuous. For doing so a concrete statistical modeling is not necessary, that is the resulting estimate is of nonparametric nature. In a second step the quite usual and from a statistical point of view classical logistic regression modeling is investigated. Main emphasis has been laid on the understanding of the model and the interpretation of the occurring parameters. It is shown that the eectiveness of the safety function also can be detected via such a logistic approach and that relevant confounding variables can and should be taken into account. The interpretation of the parameters related to the confounder and the quantication of the in uence of the confounder is shown to be rather problematic. All the theoretical results are illuminated by numerical data examples.

Modeling and formulation of optimization problems in IMRT planning comprises the choice of various values such as function-specific parameters or constraint bounds. These values also affect the characteristics of the optimization problem and thus the form of the resulting optimal plans. This publication utilizes concepts of sensitivity analysis and elasticity in convex optimization to analyze the dependence of optimal plans on the modeling parameters. It also derives general rules of thumb how to choose and modify the parameters in order to obtain the desired IMRT plan. These rules are numerically validated for an exemplary IMRT planning problems.

We consider some portfolio optimisation problems where either the investor has a desire for an a priori specified consumption stream or/and follows a deterministic pay in scheme while also trying to maximize expected utility from final wealth. We derive explicit closed form solutions for continuous and discrete monetary streams. The mathematical method used is classical stochastic control theory.

Elementare Finanzmathematik
(2002)

Im Rahmen dieser Arbeit soll eine elementar gehaltene Einführung in die Aufgabenstellungen und Prinzipien der modernen Finanzmathematik gegeben werden. Insbesondere werden die Grundlagen der Modellierung von Aktienkursen, der Bewertung von Optionen und der Portfolio-Optimierung vorgestellt. Natürlich können die verwendeten Methoden und die entwickelte Theorie nicht in voller Allgemeinheit für den Schuluntericht verwendet werden, doch sollen einzelne Prinzipien so heraus gearbeitet werden, dass sie auch an einfachen Beispielen verstanden werden können.

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. Definition and proof of existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khintchine theorem for the power spectrum are used to two ends: First, well known second order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second order characteristics in frequency space. Examples show, that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real world counterparts.

An efficient mathematical model to virtually generate woven metal wire meshes is
presented. The accuracy of this model is verified by the comparison of virtual structures with three-dimensional
images of real meshes, which are produced via computer tomography. Virtual structures
are generated for three types of metal wire meshes using only easy to measure parameters. For these
geometries the velocity-dependent pressure drop is simulated and compared with measurements
performed by the GKD - Gebr. Kufferath AG. The simulation results lie within the tolerances of
the measurements. The generation of the structures and the numerical simulations were done at
GKD using the Fraunhofer GeoDict software.

The objective of the present article is to give an overview of an application of Fuzzy Logic in Regulation Thermography, a method of medical diagnosis support. An introduction to this method of the complementary medical science based on temperature measurements – so-called thermograms – is provided. The process of modelling the physician’s thermogram evaluation rules using the calculus of Fuzzy Logic is explained.

In the article the application of kernel functions – the so-called »kernel trick« – in the context of Fisher’s approach to linear discriminant analysis is described for data sets subdivided into two groups and having real attributes. The relevant facts about functional Hilbert spaces and kernel functions including their proofs are presented. The approximative algorithm published in [Mik3] to compute a discriminant function given the data and a kernel function is briefly reviewed. As an illustration of the technique an artificial data set is analysed using the algorithm just mentioned.

In this paper a three dimensional stochastic model for the lay-down of fibers on a moving conveyor belt in the production process of nonwoven materials is derived. The model is based on stochastic diferential equations describing the resulting position of the fiber on the belt under the influence of turbulent air ows. The model presented here is an extension of an existing surrogate model, see [6, 3].

In this work we establish a hierarchy of mathematical models for the numerical simulation of the production process of technical textiles. The models range from highly complex three-dimensional fluid-solid interactions to one-dimensional fiber dynamics with stochastic aerodynamic drag and further to efficiently handable stochastic surrogate models for fiber lay-down. They are theoretically and numerically analyzed and coupled via asymptotic analysis, similarity estimates and parameter identification. Themodel hierarchy is applicable to a wide range of industrially relevant production processes and enables the optimization, control and design of technical textiles.

After a short introduction to the basic ideas of lattice Boltzmann methods and a brief description of a modern parallel computer, it is shown how lattice Boltzmann schemes are successfully applied for simulating fluid flow in microstructures and calculating material properties of porous media. It is explained how lattice Boltzmann schemes compute the gradient of the velocity field without numerical differentiation. This feature is then utilised for the simulation of pseudo-plastic fluids, and numerical results are presented for a simple benchmark problem as well as for the simulation of liquid composite moulding.

In this paper we consider short term storage systems. We analyze presorting strategies to improve the effiency of these storage systems. The presorting task is called Batch PreSorting Problem (BPSP). The BPSP is a variation of an assigment problem, i.e., it has an assigment problem kernel and some additional constraints. We present different types of these presorting problems, introduce mathematical programming formulations and prove the NP-completeness for one type of the BPSP. Experiments are carried out in order to compare the different model formulations and to investigate the behavior of these models.

Territory design and districting may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. The availability of GIS on computers and the growing interest in Geo-Marketing leads to an increasing importance of this area. Despite the wide range of applications for territory design problems, when taking a closer look at the models proposed in the literature, a lot of similarities can be noticed. Indeed, the models are many times very similar and can often be, more or less directly, carried over to other applications. Therefore, our aim is to provide a generic application-independent model and present efficient solution techniques. We introduce a basic model that covers aspects common to most applications. Moreover, we present a method for solving the general model which is based on ideas from the field of computational geometry. Theoretical as well as computational results underlining the efficiency of the new approach will be given. Finally, we show how to extend the model and solution algorithm to make it applicable for a broader range of applications and how to integrate the presented techniques into a GIS.