Published in: Bernd Hofmann, Antonio Leitao and Jorge P. Zubelli (Eds.). New Trends in Parameter Identification for Mathematical Models, Birkhäuser Basel 2018
Abstract:
This study investigates a simplified discretized EIT model with eight electrodes distributed equally spaced at the boundary of a disc covered with a small number of material stripes of varying conductivity. The goal of this paper is to evaluate the chances of identifying the conductivity values of each stripe from rotating measurements of potential differences. This setting comes from an engineering background, where the used EIT model is exploited for the detection of conductivities in carbon nanotubes (CNT) and carbon nanofibers (CNF). Connections between electrical conductivity and mechanical strain have been of major interest within the engineering community and has motivated the investigation of such a stripe structure. Up to five conductivity values can be recovered from noisy 8x8 data matrices in a stable manner by a least squares approach. Hence, this is a version of regularization by discretization and additional tools for stabilizing the recovery seem to be superfluous. To our astonishment, no local minima of the squared misfit functional were observed, which seems to indicate uniqueness of the recovery if the number of stripes is quite small.
Published in the Preprint Series of Department of Mathematics as PREPRINT 2017-03 at the Technische Universität Chemnitz, 2017.
Download in pdf-format as Glaenzel-Unger--Clustering_by_optimal_subsets_to_describe_environment_interdependencies.pdf
Published in the 1st Conference on Thermal Issues in Machine Tools. Dresden, 21.03.2018 - 23.03.2018
Abstract:
The paper copes with the problem of finding an optimal subset of interpolation points out of a given large set of computed values, arising from a finite element simulation. This simulation computes environment data, which are on their part input data for finite element simulations of machine tools. For machine tool manufacturers it is still a seriously problem that the machine works imprecisely and products wastrel if environment values like temperature changes. The change of the environment boundary conditions contribute to the phenomenon through sunlight or cold draught owing to open doors of the machine hall or factory. Resulting thermo-elastic effects on the tool center point are one of the major reasons for positioning errors in machine tools. A genetic search algorithm for clustering relevant heat transfer coefficient values over the geometric surface through computational fluid dynamics (CFD) simulations will be described. These values are the input data for a developed thermo-elastic correction algorithm.
Published in the The International Journal of Advanced Manufacturing Technology 88:1279-1288, 2017.
Abstract:
A new interdisciplinary approach is discribed to identifying unknown parameters using an extended version
of the known interval bisection method. This developed method is based on the use of finite elements for
calibrating the simulation calculation. The resulting thermo-elastic deformations which occur in drilling
processes with impaired cooling lubrication are to be used as correction values for tool positioning in the NC control.
Based on the strong impact on workpiece temperature of machining, a simulation approach is presented for
calculating the temperature fields and their thermo-elastic consequences. In addition, methods are
presented to correct these effects. This paper particularly deals with the temperature fields of drilling operations.
Special attention is paid to the technique employed for iterative numerical determination of the unknown heat flux
and heat transfer coefficient values.
Download at http://dx.doi.org/10.1016/j.procir.2016.03.178
Abstract:
Based on the method of adaptive finite elements (FEM) a correction approach has been considered to identify
the influence of thermo-elastic workpiece deformations during the production process milling.
The paper presents a simulation-based tolerance variation calculation of cutting paths, which is caused by the
heat input of the machine tool. Therefore a mathematical method is developed to numerically depict the progress
of the miller with different curves A(t), B(t) and C(t). These curves are used to map the state of the milling
path during the production process as well as to compare the current workpiece contour and the target workpiece contour.
The tool center point (TCP) correction results from mapping of time-dependent deformation fields from the FE simulation.
The aim is, on the one hand, to make statements before the production about keeping the tolerance, and on the other hand,
to derive other correction approaches for the adaption of the cutting path coordinates.
Abstract:
Nonparametrical copula density estimation is a meaningful tool for analyzing the dependence structure of a random vector
from given samples. Usually kernel estimators or penalized maximum likelihood estimators are considered.
We propose solving the Volterra integral equation
to find the copula density c(u1,...,ud) of the given copula C.
In the statistical framework, the copula C is not available and we replace it by the empirical
copula of the pseudo samples, which converges to the unobservable copula C for large samples.
Hence, we can treat the copula density estimation from given samples as an inverse problem and consider
the instability of the inverse operator, which has an important impact if the input data of the operator equation are noisy.
The well-known curse of high dimensions usually results in huge nonsparse linear equations after discretizing the
operator equation. We present a Petrov–Galerkin projection for the numerical computation of the linear integral equation.
A special choice of test and ansatz functions leads to a very special structure of the linear equations,
such that we are able to estimate the copula density also in higher dimensions.
Published in the Chemnitz Scientific Computing Preprints as csc14-03.pdf at the Technische Universität Chemnitz, 2014.
Download in pdf-format as Glaenzel-Unger--High Quality_FEM-Postprocessing_and_Visualization_Using_a_Gnuplot_Based_Toolchain.pdf
Abstract:
In the paper a toolchain for postprocessing and visualization of simulation datas,
arising in finite element computations is described.
This toolchain is based on gnuplot and free software.
It is developed on the simulation problem of thermoelasticity, but
it is not restricted on this problem class and adaptable on all other simulation problems,
dealing with scalar- and vector fields.
The most involved programs are gnuplot, perl, make and ffmpeg.
Some examples, files, scripts e.g. are downloadable at CSC14-03
Published in the Preprint Series of Department of Mathematics as PREPRINT 2014-03 at the Technische Universität Chemnitz, 2014.
Download in pdf-format as Uhlig-Unger--The_Petrov_Galerkin_projection_for_copula_density_estimation_isnt_counting.pdf
Abstract:
Non-parametric copula density estimation in the d-dimensional
case is a big challenge in particular if the dimension d of the problem
increases. In
Preprint 2013-07 we proposed to solve the d-dimensional Volterra
integral equation \int_0^u c(s)ds = C(u) for a given copula C.
In the statistical framework the copula C is unobservable and hence we solved the
linear integral equation for the empirical copula. For the numerical
computation we used a Petrov-Galerkin projection for the approximated
piecewise constant function c_h(u)=Sum_{i=1}^N c_i \phi_i(u).
Other than might be expected, the coefficient vector c does not
count the number of samples in the elements of the discretized grid,
even the approximated solution c_h is a piecewise constant function
on the elements.
We will establish that solving the Volterra integral
equation by a Petrov-Galerkin projection is not simple counting.
Published in the Preprint Series of Department of Mathematics as PREPRINT 2013-07 at the Technische Universität Chemnitz, 2013.
Download in pdf-format as Uhlig-Unger--A_Petrov_Galerkin_projection_for_copula_density_estimation.pdf
Abstract:
The reconstruction of the dependence structure of two or more random variables (d > 1) is a big issue in finance and many other applications.
Looking at samples of the random vector, neither the common distribution nor the copula itself are observable. So the identification
of the copula C or the copula density can be treated as an inverse problem.
In the statistical literature usually kernel estimators or penalized maximum likelihood estimators are considered for the non-parametric estimation of the copula
density c from given samples of the random vector.
Even though the copula C itself is unobservable we can treat the empirical copula as a noisy representation, since it is well known that the
empirical copula converges for large samples to the copula and solve the d-dimensional linear integral equation for determining the copula
density c.
We present a Petrov-Galerkin projection for the numerical computation of the linear integral equation and discuss the assembling algorithm of the
non-sparce matrices and vectors. Furthermore we analyze the stability of the discretized linear equation.
Published in the Preprint Series of Department of Mathematics as PREPRINT 2010-16 at the Technische Universität Chemnitz, 2010.
Download in pdf-format as Kandler-Unger--Population_dispersal_via_diffusion-reaction_equations.pdf
Abstract:
Diffusion-reaction systems are well-established in different life-science disciplines.
When applied to 'human questions' they are used to estimate the demographic processes involved in
major human (or animal) dispersal episodes and to estimate the general spread pattern of new
ideas or technologies through cultures.
This manuscript gives an introduction to diffusion-reaction systems for a
non-mathematical audience. We focus on describing dispersal processes and start with
modelling and analysing the spread dynamic of a single population under different
dispersal and growth hypotheses. Further, we focus on the impacts of population
interactions on spread behaviour of a particular population.
Lastly we introduce an open software package 'CultDiff' which provides
a solution tool for diffusion reaction systems.
Published in the Philosophical Transactions of the Royal Socienty B: Biological Sciences (B 2010 365, 3855-3864)
Abstract:
Language shift is the process whereby members of a community in which more than one
language is spoken abandon their original vernacular language in favour of another. The historical
shifts to English by Celtic language speakers of Britain and Ireland are particularly well-studied
examples, for which good census data exist for the most recent 100-120 years in many areas where
Celtic languages were once the prevailing vernaculars. We model the dynamics of language shift as a
competition process in which the numbers of speakers of each language (both monolingual and
bilingual) vary as a function both of internal recruitment (as the net outcome of birth, death,
immigration and emigration rates of native speakers), and of gains and losses due to language shift.
We examine two models: a basic model in which bilingualism is simply the transitional state for
households moving between alternative monolingual states, and a diglossia model in which there is an
additional demand for the endangered language as the preferred medium of communication in some
restricted sociolinguistic domain, superimposed on the basic shift dynamic. Fitting our models to
census data we successfully reproduce the demographic trajectories of both languages over the past
century. We estimate the rates of recruitment of new Scottish Gaelic speakers that would be required
each year (for instance, through school education) to counteract the 'natural wastage' as households
with one or more Gaelic speaker fail to transmit the language to the next generation informally, for
different rates of loss during informal intergenerational transmission.
to appear
to appear
Published in the Chemnitz Scientific Computing Preprints as csc09-01.pdf at the Technische Universität Chemnitz, 2009.
Download in pdf-format as Unger--Obstacle_Description_with_Radial_Basis_Functions_for_Contact_Problems_in_Elasticity.pdf
Abstract:
In this paper the obstacle description with Radial Basis
Functions for contact problems in three dimensional elasticity
will be done.
A short Introduction of the idea of Radial Basis
Functions will be followed by the usage of Radial Basis Functions
for approximation of isosurfaces.
Then this isosurfaces are used for the obstacle-description
in three dimensional elasticity contact problems.
In the last part some computational examples will be shown.
Published in IEEE Transactions on Automation Science and Engineering Vol. 5 Number 1 01/2008.
Based on my Diploma Thesis
Numerische Simulation von Zugfahrten unter Realbedingungen
respectively the preprint
Numerical simulation of train traffic in large networks via time-optimal control
Download as PDF from http://ieeexplore.ieee.org
Abstract:
From a system-theoretic standpoint, a constrained state-space model for train traffic
in a large railway network is developed.
The novelty of the work is the transformation or rather reduction of the directed graph of the
network to some parallel lists.
Mathematization of this sophisticated problem is thus circumvented.
All the aspects of a real network (such as that of the German Rail) are completely captured by this model.
Some degrees of freedom, as well as some robustness can be injected into the operation of the system.
The problem of time-optimal train traffic in large networks is then defined and
solved using the maximum principle. The solution is obtained by reducing the boundary value
problem arising from the time-optimality criterion to an initial value problem for an
ordinary differential equation. A taxonomy of all possible switching points of the
control actions is presented.
The proposed approach is expected to result in faster-than-real-time simulation
of time-optimal traffic in large networks and, thus, facilitation of real-time
control of the network by dispatchers. This expectation is quantitatively justified by
analysis of simulation results of some small parts of the German Rail Network.
Dissertationsschrift, veröffentlicht im MONARCH der TU-Chemnitz unter http://archiv.tu-chemnitz.de/pub/2007/0027
Download als PDF-Version: dissertation_roman_unger.pdf
Abstract:
Der Gegenstand dieser Arbeit ist die Untersuchung spezieller Lösungsmethoden
zum Problem des Kontaktes eines elastischen Körpers mit einem festen Hindernis
sowie des Kontaktes zweier elastischer Körper miteinander.
Grundlage der Betrachtungen ist dabei ein Lösungsverfahren,
das auf Unterraum-CG-Techniken beruht.
Die zu Grunde liegende partielle Differentialgleichung zur Modellierung der Verformung eines elastischen Körpers ist die Lame-Gleichung. Aufbauend auf dieser Gleichung wird das Problem des Kontaktes in einer neuen Formulierung, die auch große Verformungen zuläßt, betrachtet.
Um diese Probleme mit Hilfe der Finiten-Elemente-Methode numerisch lösen zu können, erfolgt die Betrachtung der üblichen Variationsformulierung mit Hilfe von Variationsungleichungen sowie die Angabe einer alternativen Formulierung, die auf einer Variationsgleichung beruht.
Zur Konstruktion eines effektiven Lösungsalgorithmus werden die Problematiken der a-posteriori Fehlerschätzung, Voraussetzungen an Vernetzungen sowie moderner Lösungsmethoden zum Auflösen des Finiten-Elemente-Gleichungssystems betrachtet.
Um die aus dem Kontaktproblem resultierenden Restriktionen zu erfüllen, wird die Klasse der Unterraum-CG-Verfahren einführend betrachtet und es wird die Anpassung dieser Verfahren auf die betrachteten Probleme vorgestellt. Die für derartige Lösungsmethoden verwendeten Projektoren werden formuliert und es werden verschiedene Formulierungen dieser Projektoren in Bezug auf Effektivität der Implementierung sowie Speicheraufwand miteinander verglichen.
Es wird auf einige verschiedene Möglichkeiten der Beschreibung von Hindernissen sowie des Kontaktproblems zweier elastischer Körper miteinander eingegangen und es werden Referenzimplementierungen zu diesen Problemen angegeben.
Zu den implementierten Projektoren werden Beispielrechnungen am Ende der jeweiligen Abschnitte vorgestellt sowie die Rechenzeiten und Konvergenzverhalten restringierter und unrestringierter Elastizitätsprobleme verglichen. Es zeigt sich dabei der Vorteil der entwickelten Verfahren in einem vergleichbaren numerischen Aufwand zwischen restringierten und unrestringierten Problemen bei einer übersichtlichen Implementierbarkeit und guter Stabilität.
Die Problemklasse von Restriktionen im Inneren des betrachteten Gebietes wird anhand des Clinch-Problems formuliert, und die zur Lösung derartiger Probleme verwendeten Projektoren betrachtet.
Die Referenzimplementierung aller vorgestellen Algorithmen und Projektoren erfolgt dabei in einem adaptiven 2D-FEM-Programm, welches innerhalb des DFG-Sonderforschungsbereichs 393 "Parallele Numerische Simulation für Physik und Kontinuumsmechanik" entstanden ist.
Published in the preprint series of the SFB393 "Parallele Numerische Simulation für Physik und Kontinuumsmechanik" at the Technische Universität Chemnitz, 2005.
Download as Postscript or PDF Version.
Abstract:
Subspace-cg-techniques with projection methods are useful for an easy
extension of an arbitrary finite element code with
error estimation and adaptive strategies to an algorithm for solving
contact problems with additional restrictions such as contact
problems.
In this paper we use the method to apply restrictions in the interior of the domain of an elastic body, not on the boundary.
Published in the preprint series of the SFB393 "Parallele Numerische Simulation für Physik und Kontinuumsmechanik" at the Technische Universität Chemnitz, 2004.
Download as Postscript or PDF Version.
Abstract:
The aim of the paper is showing, how projection methods can be used for
computing contact-problems in elasticity for different classes of obstacles.
Starting with the projection idea for handling hanging nodes in finite element
discretizations the extension of the method for handling penetrated nodes
in contact problems will be described for some obstacle classes.
Veröffentlicht im Tagungsband zum Workshop Stochastische Analysis vom 29.09.2003 - 01.10.2003.
Abstract:
In dieser Arbeit werden stochastische Charakteristiken der
Lösung parabolischer Differentialgleichungen mit zufälligen
Neumann-Randbedingungen mit Hilfe der Finite-Elemente-Methode angegeben.
Dabei wird der Berechnung der Korrelations-bzw. Varianzfunktion
besondere Bedeutung beigemessen.
Das stochastische Randanfangswertproblem wird durch Anwendung von
FEM-Techniken durch ein System gewöhnlicher Differentialgleichungen
mit stochastischen inhomogenen Termen approximiert.
Die Modellierung der stochastischen Eingangsparameter durch
epsilon-korrelierte Felder gestattet Entwicklungen der
Lösungscharakteristiken nach der Korrelationslänge.
Numerische Beispiele enthalten den Vergleich zwischen
analytischen Ergebnissen und Simulationsresultaten.
Published in the preprint series at the Institute of Mathematics, Technische Universität Berlin, 2001. Technical Report 722-2001
Download as Postscript or PDF Version.
Abstract:
We discuss the mathematical modelling of schedule based rail traffic.
The model is used to develop efficient numerical simulation methods for the time optimal control of a large number
of interacting trains in a large network.
The time optimal control is used to model the realistic behaviour of driving in a network like that of Deutsche Bahn.
It is used to allow a real time simulation with incomplete information on real train velocities.
We present numerical examples that demonstrate the efficiency of the model and the simulation method.
Y.Bavafa-Toosi,
Ch.Blendinger,
V.Mehrmann,
H.Ohmori,
A.Steinbrecher and
R.Unger.
Time-Optimal Train Traffic in Large Networks based on a New Model.
Preprints of the 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems:
Theory and Applications, Osaka, Japan, 2004, pages 729-734, 2004.
Y.Bavafa-Toosi,
Ch.Blendinger,
V.Mehrmann,
A.Steinbrecher and
R.Unger.
Modeling, analysis, synthesis, and simulation of time-optimal train traffic in large networks.
Technical Report 27-2006
Institute of Mathematics, Technische Universit\E4t Berlin, 2006.
Download as
PDF Version.
Y.Bavafa-Toosi,
Ch.Blendinger,
V.Mehrmann,
A.Steinbrecher and
R.Unger.
A New Methodology for Modeling, Analysis, Synthesis, and simulation of Time-Optimal Train
Traffic in Large Networks.
IEEE Trans. Autom. Sci. Eng., 2007 (to appear)
Numerische Simulation von Zugfahrten unter Realbedingungen
Autor:
R. Unger.
Diplomarbeit
Download als PDF-Version: diplomarbeit_roman_unger.pdf
Abstract:
Die in dieser Arbeit entwickelten Algorithmen sollen als schnelles Simulationswerkzeug zur
Fahrzeitrechnung zum Einsatz kommen.
Hauptzielgruppe sind die Betriebszentralen der Deutschen Bahn.
Ziel war es, Dispositionshilfen für die Bearbeitung eines einzelnen Zuges und eines Systems aus
mehreren, sich gegenseitig beeinflussenden Zügen zu schaffen.
Schwerpunkt der Entwicklung der Algorithmen war neben der obligatorischen sinnvollen Genauigkeit der
Rechnung eine schnelle Laufzeit der Programme, um Testreihen für Optimierungsstrategien in effektiven
Rechenzeiten absolvieren zu können.
Es wurden sowohl die theoretischen Grundlagen der Simulation sowie der optimalen Steuerung
als auch die algorithmische Umsetzung in ein zeitgemäßes C++ Programm betrachtet.
Auf hohe Portabilität und ISO-Konformität des Quellcodes wurde besonderer Wert gelegt,
da die spätere Einsatzplattform noch nicht feststeht.