J. Kallrath, M. C. Müller, S. Nickel
Batch Presorting Problems:
Models and Complexity Results
Berichte des Fraunhofer ITWM, Nr. 40 (2002)
© Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM 2002
ISSN 1434-9973
Bericht 40 (2002)
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Vorwort
Das Tätigkeitsfeld des Fraunhofer Instituts für Techno- und Wirtschaftsmathematik
ITWM umfasst anwendungsnahe Grundlagenforschung, angewandte Forschung
sowie Beratung und kundenspezifische Lösungen auf allen Gebieten, die für Techno- und Wirtschaftsmathematik bedeutsam sind.
In der Reihe »Berichte des Fraunhofer ITWM« soll die Arbeit des Instituts kontinuierlich einer interessierten Öffentlichkeit in Industrie, Wirtschaft und Wissenschaft vorgestellt werden. Durch die enge Verzahnung mit dem Fachbereich Mathematik der Universität Kaiserslautern sowie durch zahlreiche Kooperationen mit
internationalen Institutionen und Hochschulen in den Bereichen Ausbildung und
Forschung ist ein großes Potenzial für Forschungsberichte vorhanden. In die Berichtreihe sollen sowohl hervorragende Diplom- und Projektarbeiten und Dissertationen als auch Forschungsberichte der Institutsmitarbeiter und Institutsgäste zu
aktuellen Fragen der Techno- und Wirtschaftsmathematik aufgenommen werden.
Darüberhinaus bietet die Reihe ein Forum für die Berichterstattung über die zahlreichen Kooperationsprojekte des Instituts mit Partnern aus Industrie und Wirtschaft.
Berichterstattung heißt hier Dokumentation darüber, wie aktuelle Ergebnisse aus
mathematischer Forschungs- und Entwicklungsarbeit in industrielle Anwendungen
und Softwareprodukte transferiert werden, und wie umgekehrt Probleme der Praxis neue interessante mathematische Fragestellungen generieren.
Prof. Dr. Dieter Prätzel-Wolters
Institutsleiter
Kaiserslautern, im Juni 2001
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Published reports of the
Fraunhofer ITWM
The PDF-files of the following reports
are available under:
www.itwm.fraunhofer.de/zentral/
berichte.html
1. D. Hietel, K. Steiner, J. Struckmeier
A Finite - Volume Particle Method for
Compressible Flows
We derive a new class of particle methods for conser vation laws, which are based on numerical flux
functions to model the interac tions between moving
particles. The derivation is similar to that of classical
Finite-Volume methods; except that the fixed grid
structure in the Finite-Volume method is substituted
by so-called mass packets of par ticles. We give some
numerical results on a shock wave solution for Burgers
equation as well as the well-known one-dimensional
shock tube problem.
(19 pages, 1998)
2. M. Feldmann, S. Seibold
Damage Diagnosis of Rotors: Application
of Hilbert Transform and Multi-Hypothesis Testing
In this paper, a combined approach to damage diagnosis of rotors is proposed. The intention is to employ
signal-based as well as model-based procedures for an
improved detection of size and location of the damage.
In a first step, Hilbert transform signal processing techniques allow for a computation of the signal envelope
and the instantaneous frequency, so that various types
of non-linearities due to a damage may be identified
and classified based on measured response data. In a
second step, a multi-hypothesis bank of Kalman Filters
is employed for the detection of the size and location
of the damage based on the information of the type
of damage provided by the results of the Hilbert transform.
Keywords: Hilbert transform, damage diagnosis, Kalman filtering, non-linear dynamics
(23 pages, 1998)
3. Y. Ben-Haim, S. Seibold
Robust Reliability of Diagnostic MultiHypothesis Algorithms: Application to
Rotating Machinery
Damage diagnosis based on a bank of Kalman filters, each one conditioned on a specific hypothesized
system condition, is a well recognized and powerful
diagnostic tool. This multi-hypothesis approach can
be applied to a wide range of damage conditions. In
this paper, we will focus on the diagnosis of cracks in
rotating machinery. The question we address is: how to
optimize the multi-hypothesis algorithm with respect
to the uncertainty of the spatial form and location of
cracks and their resulting dynamic effects. First, we
formulate a measure of the reliability of the diagnostic algorithm, and then we discuss modifications of
the diagnostic algorithm for the maximization of the
reliability. The reliability of a diagnostic algorithm is
measured by the amount of uncer tainty consistent with
no-failure of the diagnosis. Uncer tainty is quantitatively
represented with convex models.
Keywords: Robust reliability, convex models, Kalman
filtering, multi-hypothesis diagnosis, rotating machinery,
crack diagnosis
(24 pages, 1998)
4. F.-Th. Lentes, N. Siedow
Three-dimensional Radiative Heat Transfer
in Glass Cooling Processes
For the numerical simulation of 3D radiative heat transfer in glasses and glass melts, practically applicable
mathematical methods are needed to handle such
problems optimal using workstation class computers.
Since the exact solution would require super-computer
capabilities we concentrate on approximate solutions with a high degree of accuracy. The following
approaches are studied: 3D diffusion approximations
and 3D ray-tracing methods.
(23 pages, 1998)
5. A. Klar, R. Wegener
A hierarchy of models for multilane
vehicular traffic
Part I: Modeling
In the present paper multilane models for vehicular
traffic are considered. A microscopic multilane model
based on reaction thresholds is developed. Based on
this model an Enskog like kinetic model is developed.
In particular, care is taken to incorporate the correlations between the vehicles. From the kinetic model a
fluid dynamic model is derived. The macroscopic coefficients are deduced from the underlying kinetic model.
Numerical simulations are presented for all three levels
of description in [10]. Moreover, a comparison of the
results is given there.
(23 pages, 1998)
Part II: Numerical and stochastic
investigations
In this paper the work presented in [6] is continued.
The present paper contains detailed numerical investigations of the models developed there. A numerical
method to treat the kinetic equations obtained in [6]
are presented and results of the simulations are shown.
Moreover, the stochastic correlation model used in [6]
is described and investigated in more detail.
(17 pages, 1998)
6. A. Klar, N. Siedow
Boundary Layers and Domain Decomposition for Radiative Heat Transfer and Dif fusion Equations: Applications to Glass Manufac turing Processes
In this paper domain decomposition methods for
radiative transfer problems including conductive heat
transfer are treated. The paper focuses on semi-transparent materials, like glass, and the associated conditions at the interface between the materials. Using
asymptotic analysis we derive conditions for the coupling of the radiative transfer equations and a diffusion
approximation. Several test cases are treated and a
problem appearing in glass manufacturing processes is
computed. The results clearly show the advantages of a
domain decomposition approach. Accuracy equivalent
to the solution of the global radiative transfer solution is achieved, whereas computation time is strongly
reduced.
(24 pages, 1998)
7. I. Choquet
Heterogeneous catalysis modelling and
numerical simulation in rarified gas flows
Part I: Coverage locally at equilibrium
A new approach is proposed to model and simulate
numerically heterogeneous catalysis in rarefied gas
flows. It is developed to satisfy all together the following points:
1) describe the gas phase at the microscopic scale, as
required in rarefied flows,
2) describe the wall at the macroscopic scale, to avoid
prohibitive computational costs and consider not only
crystalline but also amorphous surfaces,
3) reproduce on average macroscopic laws correlated
with experimental results and
4) derive analytic models in a systematic and exact
way. The problem is stated in the general framework
of a non static flow in the vicinity of a catalytic and
non porous surface (without aging). It is shown that
the exact and systematic resolution method based
on the Laplace transform, introduced previously by
the author to model collisions in the gas phase, can
be extended to the present problem. The proposed
approach is applied to the modelling of the EleyRideal
and LangmuirHinshelwood recombinations, assuming
that the coverage is locally at equilibrium. The models
are developed considering one atomic species and
extended to the general case of several atomic species.
Numerical calculations show that the models derived in
this way reproduce with accuracy behaviors observed
experimentally.
(24 pages, 1998)
8. J. Ohser, B. Steinbach, C. Lang
Efficient Texture Analysis of Binary Images
A new method of determining some characteristics
of binary images is proposed based on a special linear
filtering. This technique enables the estimation of the
area fraction, the specific line length, and the specific
integral of curvature. Furthermore, the specific length
of the total projection is obtained, which gives detailed
information about the texture of the image. The
influence of lateral and directional resolution depending on the size of the applied filter mask is discussed in
detail. The technique includes a method of increasing
direc tional resolution for texture analysis while keeping
lateral resolution as high as possible.
(17 pages, 1998)
9. J. Orlik
Homogenization for viscoelasticity of the
integral type with aging and shrinkage
A multiphase composite with periodic distributed
inclusions with a smooth boundary is considered in this
contribution. The composite component materials are
supposed to be linear viscoelastic and aging (of the
nonconvolution integral type, for which the Laplace
transform with respect to time is not effectively applicable) and are subjected to isotropic shrinkage. The
free shrinkage deformation can be considered as a fictitious temperature deformation in the behavior law. The
procedure presented in this paper proposes a way to
determine average (effective homogenized) viscoelastic
and shrinkage (temperature) composite properties and
the homogenized stress field from known properties
of the components. This is done by the extension of
the as ymptotic homogenization technique known for
pure elastic nonhomogeneous bodies to the nonhomogeneous thermoviscoelasticity of the integral noncon-
volution type. Up to now, the homogenization theory
has not covered viscoelasticity of the integral type.
SanchezPalencia (1980), Francfort & Suquet (1987) (see
[2], [9]) have considered homogenization for viscoelasticity of the differential form and only up to the first
derivative order. The integralmodeled viscoelasticity
is more general then the differential one and includes
almost all known differential models. The homogenization procedure is based on the construction of an
asymptotic solution with respect to a period of the
composite struc ture. This reduces the original problem
to some auxiliary boundary value problems of elasticity and viscoelasticity on the unit periodic cell, of the
same type as the original non-homogeneous problem.
The existence and uniqueness results for such problems
were obtained for kernels satisfying some constrain
conditions. This is done by the extension of the Volterra
integral operator theory to the Volterra operators with
respect to the time, whose 1 kernels are space linear
operators for any fixed time variables. Some ideas of
such approach were proposed in [11] and [12], where
the Volterra operators with kernels depending additionally on parameter were considered. This manuscript
delivers results of the same nature for the case of the
spaceoperator kernels.
(20 pages, 1998)
10. J. Mohring
Helmholtz Resonators with Large Aperture
The lowest resonant frequency of a cavity resonator is usually approximated by the classical Helmholtz
formula. However, if the opening is rather large and
the front wall is narrow this formula is no longer valid.
Here we present a correction which is of third order
in the ratio of the diameters of aperture and cavity. In
addition to the high accuracy it allows to estimate the
damping due to radiation. The result is found by applying the method of matched asymptotic expansions. The
correction contains form factors describing the shapes
of opening and cavity. They are computed for a number of standard geometries. Results are compared with
numerical computations.
(21 pages, 1998)
11. H. W. Hamacher, A. Schöbel
On Center Cycles in Grid Graphs
Finding “good” cycles in graphs is a problem of great
interest in graph theory as well as in locational analysis. We show that the center and median problems are
NP hard in general graphs. This result holds both for
the variable cardinality case (i.e. all cycles of the graph
are considered) and the fixed cardinality case (i.e. only
cycles with a given cardinality p are feasible). Hence
it is of interest to investigate special cases where the
problem is solvable in polynomial time. In grid graphs,
the variable cardinality case is, for instance, trivially
solvable if the shape of the cycle can be chosen freely.
If the shape is fixed to be a rectangle one can analyze rectangles in grid graphs with, in sequence, fixed
dimension, fixed cardinality, and variable cardinality.
In all cases a complete charac terization of the optimal
cycles and closed form expressions of the optimal
objec tive values are given, yielding polynomial time
algorithms for all cases of center rectangle problems.
Finally, it is shown that center cycles can be chosen as
rectangles for small cardinalities such that the center
cycle problem in grid graphs is in these cases completely solved.
(15 pages, 1998)
12. H. W. Hamacher, K.-H. Küfer
Inverse radiation therapy planning a multiple objective optimisation approach
15. M. Junk, S. V. Raghurame Rao
A new discrete velocity method for NavierStokes equations
For some decades radiation therapy has been proved
successful in cancer treatment. It is the major task of
clinical radiation treatment planning to realize on the
one hand a high level dose of radiation in the cancer
tissue in order to obtain maximum tumor control. On
the other hand it is obvious that it is absolutely necessary to keep in the tissue outside the tumor, particularly
in organs at risk, the unavoidable radiation as low as
possible.
No doubt, these two objectives of treatment planning
- high level dose in the tumor, low radiation outside the
tumor - have a basically contradictory nature. Therefore,
it is no surprise that inverse mathematical models with
dose distribution bounds tend to be infeasible in most
cases. Thus, there is need for approximations compromising between overdosing the organs at risk and
underdosing the target volume.
Differing from the currently used time consuming
iterative approach, which measures deviation from an
ideal (non-achievable) treatment plan using recursively
trial-and-error weights for the organs of interest, we
go a new way trying to avoid a priori weight choices
and consider the treatment planning problem as a multiple objec tive linear programming problem: with each
organ of interest, target tissue as well as organs at risk,
we associate an objective function measuring the maximal deviation from the prescribed doses.
We build up a data base of relatively few efficient
solutions representing and approximating the variety
of Pareto solutions of the multiple objective linear
programming problem. This data base can be easily
scanned by physicians looking for an adequate treatment plan with the aid of an appropriate online tool.
(14 pages, 1999)
The relation between the Lattice Boltzmann Method,
which has recently become popular, and the Kinetic
Schemes, which are routinely used in Computational
Fluid Dynamics, is explored. A new discrete velocity model for the numerical solution of Navier-Stokes
equations for incompressible fluid flow is presented by
combining both the approaches. The new scheme can
be interpreted as a pseudo-compressibility method and,
for a particular choice of parameters, this interpretation
carries over to the Lattice Boltzmann Method.
(20 pages, 1999)
13. C. Lang, J. Ohser, R. Hilfer
On the Analysis of Spatial Binary Images
This paper deals with the characterization of micro scopically heterogeneous, but macroscopically homogeneous spatial structures. A new method is presented
which is strictly based on integral-geometric formulae
such as Crofton’s intersection formulae and Hadwiger’s
recursive definition of the Euler number. The corresponding algorithms have clear advantages over other
techniques. As an example of application we consider
the analysis of spatial digital images produced by
means of Computer Assisted Tomography.
(20 pages, 1999)
14. M. Junk
On the Construction of Discrete Equilibrium
Distributions for Kinetic Schemes
A general approach to the construction of discrete
equilibrium distributions is presented. Such distribution
func tions can be used to set up Kinetic Schemes as
well as Lattice Boltzmann methods. The general principles are also applied to the construction of Chapman
Enskog distributions which are used in Kinetic Schemes
for compressible Navier-Stokes equations.
(24 pages, 1999)
16. H. Neunzert
Mathematics as a Key to Key Technologies
The main part of this paper will consist of examples,
how mathematics really helps to solve industrial problems; these examples are taken from our Institute for
Industrial Mathematics, from research in the Technomathematics group at my university, but also from
ECMI groups and a company called TecMath, which
originated 10 years ago from my university group and
has already a very successful history.
(39 pages (4 PDF-Files), 1999)
17. J. Ohser, K. Sandau
Considerations about the Estimation of the
Size Distribution in Wicksell’s Corpuscle
Problem
Wicksell’s corpuscle problem deals with the estimation of the size distribution of a population of particles,
all having the same shape, using a lower dimensional
sampling probe. This problem was originary formulated
for particle systems occurring in life sciences but its
solution is of actual and increasing interest in materials
science. From a mathematical point of view, Wicksell’s
problem is an inverse problem where the interesting size distribution is the unknown part of a Volterra
equation. The problem is often regarded ill-posed,
because the structure of the integrand implies unstable
numerical solutions. The accuracy of the numerical
solutions is considered here using the condition number, which allows to compare different numerical methods with different (equidistant) class sizes and which
indicates, as one result, that a finite section thickness
of the probe reduces the numerical problems. Furthermore, the relative error of estimation is computed
which can be split into two parts. One part consists
of the relative discretization error that increases for
increasing class size, and the second part is related
to the relative statistical error which increases with
decreasing class size. For both parts, upper bounds
can be given and the sum of them indicates an optimal
class width depending on some specific constants.
(18 pages, 1999)
18. E. Carrizosa, H. W. Hamacher, R. Klein,
S. Nickel
Solving nonconvex planar location problems by finite dominating sets
It is well-known that some of the classical location
problems with polyhedral gauges can be solved in
polynomial time by finding a finite dominating set, i. e.
a finite set of candidates guaranteed to contain at least
one optimal location.
In this paper it is first established that this result holds
for a much larger class of problems than currently considered in the literature. The model for which this result
can be proven includes, for instance, location problems
with attraction and repulsion, and location-allocation
problems.
Next, it is shown that the approximation of general
gauges by polyhedral ones in the objective function of
our general model can be analyzed with regard to the
subsequent error in the optimal objec tive value. For
the approximation problem two different approaches
are described, the sandwich procedure and the greedy
algorithm. Both of these approaches lead - for fixed
epsilon - to polynomial approximation algorithms with
accuracy epsilon for solving the general model considered in this paper.
Keywords: Continuous Location, Polyhedral Gauges,
Finite Dominating Sets, Approximation, Sandwich Algo rithm, Greedy Algorithm
(19 pages, 2000)
19. A. Becker
A Review on Image Distortion Measures
Within this paper we review image distortion measures.
A distortion measure is a criterion that assigns a “quality number” to an image. We distinguish between
mathematical distortion measures and those distortion
measures in-cooperating a priori knowledge about
the imaging devices ( e. g. satellite images), image processing algorithms or the human physiology. We will
consider representative examples of different kinds of
distortion measures and are going to discuss them.
Keywords: Distortion measure, human visual system
(26 pages, 2000)
20. H. W. Hamacher, M. Labbé, S. Nickel,
T. Sonneborn
Polyhedral Properties of the Uncapacitated
Multiple Allocation Hub Location Problem
We examine the feasibility polyhedron of the unca pacitated hub location problem (UHL) with multiple
allocation, which has applications in the fields of air
passenger and cargo transportation, telecommunication and postal delivery services. In particular we
determine the dimension and derive some classes of
facets of this polyhedron. We develop some general
rules about lifting facets from the uncapacitated facility
location (UFL) for UHL and projecting facets from UHL
to UFL. By applying these rules we get a new class of
facets for UHL which dominates the inequalities in the
original formulation. Thus we get a new formulation of
UHL whose constraints are all facet–defining. We show
its superior computational per formance by benchmarking it on a well known data set.
Keywords: integer programming, hub location, facility
location, valid inequalities, facets, branch and cut
(21 pages, 2000)
21. H. W. Hamacher, A. Schöbel
Design of Zone Tariff Systems in Public
Transpor tation
Given a public transportation system represented by its
stops and direct connections between stops, we consider two problems dealing with the prices for the customers: The fare problem in which subsets of stops are
already aggregated to zones and “good” tariffs have
to be found in the existing zone system. Closed form
solutions for the fare problem are presented for three
objective functions. In the zone problem the design
of the zones is part of the problem. This problem is NP
hard and we therefore propose three heuristics which
prove to be very successful in the redesign of one of
Germany’s transpor tation systems.
(30 pages, 2001)
22. D. Hietel, M. Junk, R. Keck, D. Teleaga:
The Finite-Volume-Particle Method for
Conservation Laws
In the Finite-Volume-Particle Method (FVPM), the weak
formulation of a hyperbolic conservation law is discretized
by restricting it to a discrete set of test functions. In
contrast to the usual Finite-Volume approach, the test
functions are not taken as characteristic functions of the
control volumes in a spatial grid, but are chosen from a
partition of unity with smooth and overlapping partition
functions (the particles), which can even move along
pre- scribed velocity fields. The information exchange
between particles is based on standard numerical flux
func tions. Geometrical information, similar to the surface area of the cell faces in the Finite-Volume Method
and the corresponding normal directions are given as
integral quantities of the partition functions. After a
brief derivation of the Finite-Volume-Particle Method,
this work focuses on the role of the geometric coeffi cients in the scheme.
(16 pages, 2001)
23. T. Bender, H. Hennes, J. Kalcsics,
M. T. Melo, S. Nickel
Location Software and Interface with GIS
and Supply Chain Management
The objective of this paper is to bridge the gap
between location theory and practice. To meet this
objective focus is given to the development of software capable of addressing the different needs of a
wide group of users. There is a very active community on location theory encompassing many research
fields such as operations research, computer science,
mathematics, engineering, geography, economics and
marketing. As a result, people working on facility location problems have a very diverse background and also
different needs regarding the soft ware to solve these
problems. For those interested in non-commercial
applications (e. g. students and researchers), the library
of location algorithms (LoLA can be of considerable
assistance. LoLA contains a collection of efficient algorithms for solving planar, network and discrete facility
location problems. In this paper, a detailed description
of the func tionality of LoLA is presented. In the fields
of geography and marketing, for instance, solving facility location problems requires using large amounts of
demographic data. Hence, members of these groups
(e. g. urban planners and sales managers) often work
with geographical information too s. To address the
specific needs of these users, LoLA was inked to a
geographical information system (GIS) and the details
of the combined functionality are described in the
paper. Finally, there is a wide group of prac titioners
who need to solve large problems and require special
purpose soft ware with a good data inter face. Many of
such users can be found, for example, in the area of
supply chain management (SCM). Logistics activities
involved in strategic SCM include, among others, facility location planning. In this paper, the development of
a commercial location soft ware tool is also described.
The too is embedded in the Advanced Planner and
Optimizer SCM software developed by SAP AG, Walldorf, Germany. The paper ends with some conclusions
and an outlook to future ac tivities.
Keywords: facility location, software development,
geographical information systems, supply chain management.
(48 pages, 2001)
24. H. W. Hamacher, S. A. Tjandra
Mathematical Modelling of Evacuation
Problems: A State of Art
This paper details models and algorithms which can
be applied to evacuation problems. While it concentrates on building evacuation many of the results are
applicable also to regional evacuation. All models
consider the time as main parameter, where the travel
time between components of the building is part of the
input and the overall evacuation time is the output. The
paper distinguishes between macroscopic and microscopic evacuation models both of which are able to
capture the evacuees’ movement over time.
Macroscopic models are mainly used to produce good
lower bounds for the evacuation time and do not consider any individual behavior during the emergency
situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a
building. Macroscopic approaches which are based on
dynamic network flow models (minimum cost dynamic
flow, maximum dynamic flow, universal maximum
flow, quickest path and quickest flow) are described. A
special feature of the presented approach is the fact,
that travel times of evacuees are not restricted to be
constant, but may be density dependent. Using multicriteria optimization priority regions and blockage due
to fire or smoke may be considered. It is shown how
the modelling can be done using time parameter either
as discrete or continuous parameter.
Microscopic models are able to model the individual
evacuee’s charac teristics and the interaction among
evacuees which influence their movement. Due to the
corresponding huge amount of data one uses simulation approaches. Some probabilistic laws for individual evacuee’s movement are presented. Moreover
ideas to model the evacuee’s movement using cellular
automata (CA) and resulting software are presented.
In this paper we will focus on macroscopic models and
only summarize some of the results of the microscopic
approach. While most of the results are applicable to
general evacuation situations, we concentrate on building evacuation.
(44 pages, 2001)
25. J. Kuhnert, S. Tiwari
Grid free method for solving the Poisson
equation
A Grid free method for solving the Poisson equation
is presented. This is an iterative method. The method
is based on the weighted least squares approximation
in which the Poisson equation is enforced to be satisfied in every iterations. The boundary conditions can
also be enforced in the iteration process. This is a local
approximation procedure. The Dirichlet, Neumann and
mixed boundary value problems on a unit square are
presented and the analytical solutions are compared
with the exact solutions. Both solutions matched perfectly.
Keywords: Poisson equation, Least squares method,
Grid free method
(19 pages, 2001)
26. T. Götz, H. Rave, D. Reinel-Bitzer,
K. Steiner, H. Tiemeier
Simulation of the fiber spinning process
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
Bolt zmann, CFD
(19 pages, 2001)
27. A. Zemitis
On interaction of a liquid film with an
obstacle
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.
Keywords: impinging jets, liquid film, models, numerical solution, shape
(22 pages, 2001)
28. I. Ginzburg, K. Steiner
Free surface lattice-Boltzmann method to
model the filling of expanding cavities by
Bingham Fluids
The filling process of viscoplastic metal alloys and plastics in expanding cavities is modelled using the lattice
Bolt zmann method in two and three dimensions. These
models combine the regularized Bingham model for
viscoplastic with a free-interface algorithm. The latter
is based on a modified immiscible lattice Boltzmann
model in which one species is the fluid and the other
one is considered as vacuum. The boundary conditions
at the curved liquid-vacuum interface are met without
any geometrical front reconstruc tion from a first-order
Chapman-Enskog expansion. The numerical results
obtained with these models are found in good agreement with available theoretical and numerical analysis.
Keywords: Generalized LBE, free-surface phenomena,
interface boundary conditions, filling processes, Bingham viscoplastic model, regularized models
(22 pages, 2001)
29. H. Neunzert
»Denn nichts ist für den Menschen als Menschen etwas wert, was er nicht mit Leidenschaft tun kann«
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 Graduier tenkolleg) 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 Sozialwissenschaf ten
hat.
Keywords: Lehre, Forschung, angewandte Mathematik,
Mehr skalenanalyse, Strömungsmechanik
(18 pages, 2001)
30. J. Kuhnert, S. Tiwari
Finite pointset method based on the projection method for simulations of the incompressible Navier-Stokes equations
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.
Keywords: Incompressible Navier-Stokes equations,
Meshfree method, Projection method, Particle scheme,
Least squares approximation
AMS subject classification: 76D05, 76M28
(25 pages, 2001)
31. R. Korn, M. Krekel
Optimal Portfolios with Fixed Consumption
or Income Streams
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.
Keywords: Portfolio optimisation, stochastic control,
HJB equation, discretisation of control problems.
(23 pages, 2002)
32. M. Krekel
Optimal portfolios with a loan dependent
credit spread
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.
Keywords: Portfolio optimisation, stochastic control,
HJB equation, credit spread, log utility, power utility,
non-linear wealth dynamics
(25 pages, 2002)
33. J. Ohser, W. Nagel, K. Schladitz
The Euler number of discretized sets - on
the choice of adjacency in homogeneous
lattices
Two approaches for determining the Euler-Poincaré
charac teristic 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
microstruc tures 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.
Keywords: image analysis, Euler number, neighborhod
relationships, cuboidal lattice
(32 pages, 2002)
34. I. Ginzburg, K. Steiner
Lattice Boltzmann Model for Free-Surface
flow and Its Application to Filling Process in
Casting
A generalized lattice Boltzmann model to simulate freesurface is constructed in both two and three dimensions. The proposed model satisfies the interfacial
boundary conditions accurately. A distinctive feature
of the model is that the collision processes is carried
out only on the points occupied partially or fully by the
fluid. To maintain a sharp interfacial front, the method
includes an anti-diffusion algorithm. The unknown
distribution functions at the interfacial region are constructed according to the first order Chapman-Enskog
analysis. The interfacial boundary conditions are satis-
fied exactly by the coef ficients in the Chapman-Enskog
expansion. The distribution functions are naturally
expressed in the local inter facial coordinates. The macroscopic quantities at the inter face are extracted from
the least-square solutions of a locally linearized system
obtained from the known distribution functions. The
proposed method does not require any geometric front
construction and is robust for any interfacial topology.
Simulation results of realistic filling process are presented: rectangular cavity in two dimensions and Hammer box, Campbell box, Shef field box, and Motorblock
in three dimensions. To enhance the stability at high
Reynolds numbers, various upwind-type schemes are
developed. Free-slip and no-slip boundary conditions
are also discussed.
Keywords: Lattice Bolt zmann models; free-surface
phenomena; interface boundary conditions; filling
processes; injection molding; volume of fluid method;
interface boundary conditions; advec tion-schemes;
upwind-schemes
(54 pages, 2002)
35. M. Günther, A. Klar, T. Materne,
R. Wegener
Multivalued fundamental diagrams and
stop and go waves for continuum traffic
equations
In the present paper a kinetic model for vehicular traffic leading to multivalued fundamental diagrams is
developed and investigated in detail. For this model
phase transitions can appear depending on the local
density and velocity of the flow. A derivation of associated macroscopic traffic equations from the kinetic
equation is given. Moreover, numerical experiments
show the appearance of stop and go waves for highway traffic with a bottleneck.
Keywords: traffic flow, macroscopic equations, kinetic
derivation, multivalued fundamental diagram, stop and
go waves, phase transitions
(25 pages, 2002)
36. S. Feldmann, P. Lang, D. Prätzel-Wolters
Parameter influence on the zeros of network determinants
To a network N(q) with determinant D(s;q) depending on a parameter vector q Î Rr via identification of
some of its vertices, a network N^ (q) is assigned. The
paper deals with procedures to find N ^ (q), such that
its determinant D^ (s;q) admits a factorization in the
determinants of appropriate subnetworks, and with
the estimation of the deviation of the zeros of D^ from
the zeros of D. To solve the estimation problem state
space methods are applied.
Keywords: Networks, Equicofactor matrix polynomials,
Realization theory, Matrix perturbation theory
(30 pages, 2002)
37. K. Koch, J. Ohser, K. Schladitz
Spectral theory for random closed sets and
estimating the covariance via frequency
space
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 spec trum 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 charac teristics in frequency space than from
their real world counterparts.
Keywords: Random set, Bartlett spectrum, fast Fourier
transform, power spectrum
(28 pages, 2002)
38. D. d’Humières, I. Ginzburg
Multi-reflection boundary conditions for
lattice Boltzmann models
We present a unified approach of several boundary
conditions for lattice Boltzmann models. Its general
framework is a generalization of previously introduced
schemes such as the bounce-back rule, linear or quadratic interpolations, etc. The objectives are two fold:
first to give theoretical tools to study the existing
boundary conditions and their corresponding accuracy; secondly to design formally third- order accurate
boundary conditions for general flows. Using these
boundary conditions, Couette and Poiseuille flows are
exact solution of the lattice Boltzmann models for a
Reynolds number Re = 0 (Stokes limit).
Numerical comparisons are given for Stokes flows in
periodic arrays of spheres and cylinders, linear periodic array of cylinders between moving plates and for
Navier-Stokes flows in periodic arrays of cylinders for
Re < 200. These results show a significant improvement of the overall accuracy when using the linear
interpolations instead of the bounce-back reflection
(up to an order of magnitude on the hydrodynamics fields). Further improvement is achieved with the
new multi-reflection boundary conditions, reaching a
level of accuracy close to the quasi-analytical reference
solutions, even for rather modest grid resolutions and
few points in the narrowest channels. More important,
the pressure and velocity fields in the vicinity of the
obstacles are much smoother with multi-reflection
than with the other boundary conditions.
Finally the good stability of these schemes is highlighted by some simulations of moving obstacles: a cylinder between flat walls and a sphere in a cylinder.
Keywords: lattice Boltzmann equation, boudary condistions, bounce-back rule, Navier-Stokes equation
(72 pages, 2002)
39. R. Korn
Elementare Finanzmathematik
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
ver wendet werden, doch sollen einzelne Prinzipien so
heraus gearbeitet werden, dass sie auch an einfachen
Beispielen verstanden werden können.
Keywords: Finanzmathematik, Aktien, Optionen, Portfolio-Optimierung, Börse, Lehrerweiterbildung, Mathe matikunterricht
(98 pages, 2002)
40. J. Kallrath, M. C. Müller, S. Nickel
Batch Presorting Problems:
Models and Complexity Results
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.
Keywords: Complexity theory, Integer programming,
Assigment, Logistics
(19 pages, 2002)
Status quo: November 2002