The Finite-Volume- Particle Method for Conservation Laws

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1 D. Hietel, M. Junk, R. Keck, D. Teleaga The Finite-Volume- Particle Method for Conservation Laws Berichte des Fraunhofer ITWM, Nr. 22 (2001)

2 Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM 2001 ISSN Bericht 22 (2001) Alle Rechte vorbehalten. Ohne ausdrückliche, schriftliche Genehmigung des Herausgebers ist es nicht gestattet, das Buch oder Teile daraus in irgendeiner Form durch Fotokopie, Mikrofilm oder andere Verfahren zu reproduzieren oder in eine für Maschinen, insbesondere Datenverarbeitungsanlagen, verwendbare Sprache zu übertragen. Dasselbe gilt für das Recht der öffentlichen Wiedergabe. Warennamen werden ohne Gewährleistung der freien Verwendbarkeit benutzt. Die Veröffentlichungen in der Berichtsreihe des Fraunhofer ITWM können bezogen werden über: Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM Gottlieb-Daimler-Straße, Geb Kaiserslautern Telefon: +49 (0) 6 31/ Telefax: +49 (0) 6 31/ Internet:

3 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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14 Bisher erschienene Berichte des Fraunhofer ITWM Die PDF-Files der folgenden Berichte finden Sie unter: 1. D. Hietel, K. Steiner, J. Struckmeier A Finite - Volume Particle Method for Compressible Flows We derive a new class of particle methods for conservation laws, which are based on numerical flux functions to model the interactions 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 b so-called mass packets of particles. 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 S., 1998) 2. M. Feldmann, S. Seibold Damage Diagnosis of Rotors: Application of Hilbert Transform and Multi-Hpothesis Testing In this paper, a combined approach to damage diagnosis of rotors is proposed. The intention is to emplo signalbased 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 frequenc, so that various tpes of nonlinearities due to a damage ma be identified and classified based on measured response data. In a second step, a multi-hpothesis bank of Kalman Filters is emploed for the detection of the size and location of the damage based on the information of the tpe of damage provided b the results of the Hilbert transform. Kewords: Hilbert transform, damage diagnosis, Kalman filtering, non-linear dnamics (23 S., 1998) 3. Y. Ben-Haim, S. Seibold Robust Reliabilit of Diagnostic Multi- Hpothesis Algorithms: Application to Rotating Machiner Damage diagnosis based on a bank of Kalman filters, each one conditioned on a specific hpothesized sstem condition, is a well recognized and powerful diagnostic tool. This multi-hpothesis approach can be applied to a wide range of damage conditions. In this paper, we will focus on the diagnosis of cracks in rotating machiner. The question we address is: how to optimize the multihpothesis algorithm with respect to the uncertaint of the spatial form and location of cracks and their resulting dnamic effects. First, we formulate a measure of the reliabilit of the diagnostic algorithm, and then we discuss modifications of the diagnostic algorithm for the maximization of the reliabilit. The reliabilit of a diagnostic algorithm is measured b the amount of uncertaint consistent with no-failure of the diagnosis. Uncertaint is quantitativel represented with convex models. Kewords: Robust reliabilit, convex models, Kalman filtering, multihpothesis diagnosis, rotating machiner, crack diagnosis (24 S., 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, practicall 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 accurac. The following approaches are studied: 3D diffusion approximations and 3D ra-tracing methods. (23 S., 1998) 5. A. Klar, R. Wegener A hierarch 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 dnamic model is derived. The macroscopic coefficients are deduced from the underling kinetic model. Numerical simulations are presented for all three levels of description in [10]. Moreover, a comparison of the results is given there. (23 S., 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 S., 1998) 6. A. Klar, N. Siedow Boundar Laers and Domain Decomposition for Radiative Heat Transfer and Diffusion Equations: Applications to Glass Manufacturing 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 asmptotic analsis 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 clearl show the advantages of a domain decomposition approach. Accurac equivalent to the solution of the global radiative transfer solution is achieved, whereas computation time is strongl reduced. (24 S., 1998) 7. I. Choquet Heterogeneous catalsis modelling and numerical simulation in rarified gas flows Part I: Coverage locall at equilibrium A new approach is proposed to model and simulate numericall heterogeneous catalsis in rarefied gas flows. It is developed to satisf 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 onl crstalline but also amorphous surfaces, 3) reproduce on average macroscopic laws correlated with experimental results and 4) derive analtic models in a sstematic and exact wa. The problem is stated in the general framework of a non static flow in the vicinit of a cataltic and non porous surface (without aging). It is shown that the exact and sstematic resolution method based on the Laplace transform, introduced previousl b 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 Ele-Rideal and Langmuir-Hinshelwood recombinations, assuming that the coverage is locall 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 wa reproduce with accurac behaviors observed experimentall. (24 S., 1998) 8. J. Ohser, B. Steinbach, C. Lang Efficient Texture Analsis of Binar Images A new method of determining some characteristics of binar 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 directional resolution for texture analsis while keeping lateral resolution as high as possible. (17 S., 1998) 9. J. Orlik Homogenization for viscoelasticit of the integral tpe with aging and shrinkage A multi-phase composite with periodic distributed inclusions with a smooth boundar is considered in this contribution. The composite component materials are supposed to be linear viscoelastic and aging (of the non-convolution integral tpe, for which the Laplace transform with respect to time is not effectivel 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 wa to determine average (effective homogenized) viscoelastic and shrinkage (temperature) composite properties and the homogenized stress-field from known properties of the

15 components. This is done b the extension of the asmptotic homogenization technique known for pure elastic non-homogeneous bodies to the non-homogeneous thermo-viscoelasticit of the integral non-convolution tpe. Up to now, the homogenization theor has not covered viscoelasticit of the integral tpe. Sanchez-Palencia (1980), Francfort & Suquet (1987) (see [2], [9]) have considered homogenization for viscoelasticit of the differential form and onl up to the first derivative order. The integral-modeled viscoelasticit is more general then the differential one and includes almost all known differential models. The homogenization procedure is based on the construction of an asmptotic solution with respect to a period of the composite structure. This reduces the original problem to some auxiliar boundar value problems of elasticit and viscoelasticit on the unit periodic cell, of the same tpe as the original non-homogeneous problem. The existence and uniqueness results for such problems were obtained for kernels satisfing some constrain conditions. This is done b the extension of the Volterra integral operator theor to the Volterra operators with respect to the time, whose 1 kernels are space linear operators for an fixed time variables. Some ideas of such approach were proposed in [11] and [12], where the Volterra operators with kernels depending additionall on parameter were considered. This manuscript delivers results of the same nature for the case of the space-operator kernels. (20 S., 1998) 10. J. Mohring Helmholtz Resonators with Large Aperture The lowest resonant frequenc of a cavit resonator is usuall approximated b 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 cavit. In addition to the high accurac it allows to estimate the damping due to radiation. The result is found b appling the method of matched asmptotic expansions. The correction contains form factors describing the shapes of opening and cavit. The are computed for a number of standard geometries. Results are compared with numerical computations. (21 S., 1998) 11. H. W. Hamacher, A. Schöbel On Center Ccles in Grid Graphs Finding "good" ccles in graphs is a problem of great interest in graph theor as well as in locational analsis. We show that the center and median problems are NP hard in general graphs. This result holds both for the variable cardinalit case (i.e. all ccles of the graph are considered) and the fixed cardinalit case (i.e. onl ccles with a given cardinalit p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polnomial time. In grid graphs, the variable cardinalit case is, for instance, triviall solvable if the shape of the ccle can be chosen freel. If the shape is fixed to be a rectangle one can analze rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinalit, and variable cardinalit. In all cases a complete characterization of the optimal ccles and closed form expressions of the optimal objective values are given, ielding polnomial time algorithms for all cases of center rectangle problems. Finall, it is shown that center ccles can be chosen as rectangles for small cardinalities such that the center ccle problem in grid graphs is in these cases completel solved. (15 S., 1998) 12. H. W. Hamacher, K.-H. Küfer Inverse radiation therap planning - a multiple objective optimisation approach For some decades radiation therap 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 absolutel necessar to keep in the tissue outside the tumor, particularl 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 basicall contradictor 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 currentl used time consuming iterative approach, which measures deviation from an ideal (non-achievable) treatment plan using recursivel trialand-error weights for the organs of interest, we go a new wa tring to avoid a priori weight choices and consider the treatment planning problem as a multiple objective 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 relativel few efficient solutions representing and approximating the variet of Pareto solutions of the multiple objective linear programming problem. This data base can be easil scanned b phsicians looking for an adequate treatment plan with the aid of an appropriate online tool. (14 S., 1999) 13. C. Lang, J. Ohser, R. Hilfer On the Analsis of Spatial Binar Images This paper deals with the characterization of microscopicall heterogeneous, but macroscopicall homogeneous spatial structures. A new method is presented which is strictl 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 analsis of spatial digital images produced b means of Computer Assisted Tomograph. (20 S., 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 functions 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 S., 1999) 15. M. Junk, S. V. Raghurame Rao A new discrete velocit method for Navier- Stokes equations The relation between the Lattice Boltzmann Method, which has recentl become popular, and the Kinetic Schemes, which are routinel used in Computational Fluid Dnamics, is explored. A new discrete velocit model for the numerical solution of Navier-Stokes equations for incompressible fluid flow is presented b combining both the approaches. The new scheme can be interpreted as a pseudo-compressibilit method and, for a particular choice of parameters, this interpretation carries over to the Lattice Boltzmann Method. (20 S., 1999) 16. H. Neunzert Mathematics as a Ke to Ke Technologies The main part of this paper will consist of examples, how mathematics reall helps to solve industrial problems; these examples are taken from our Institute for Industrial Mathematics, from research in the Technomathematics group at m universit, but also from ECMI groups and a compan called TecMath, which originated 10 ears ago from m universit group and has alread a ver successful histor. (39 S. (vier 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 originar formulated for particle sstems 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 accurac 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 S., 1999)

16 18. E. Carrizosa, H. W. Hamacher, R. Klein, S. Nickel Solving nonconvex planar location problems b finite dominating sets It is well-known that some of the classical location problems with polhedral gauges can be solved in polnomial time b 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 currentl 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 b polhedral ones in the objective function of our general model can be analzed with regard to the subsequent error in the optimal objective value. For the approximation problem two different approaches are described, the sandwich procedure and the greed algorithm. Both of these approaches lead - for fixed epsilon - to polnomial approximation algorithms with accurac epsilon for solving the general model considered in this paper. Kewords: Continuous Location, Polhedral Gauges, Finite Dominating Sets, Approximation, Sandwich Algorithm, Greed Algorithm (19 S., 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 qualit 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 phsiolog. We will consider representative examples of different kinds of distortion measures and are going to discuss them. Kewords: Distortion measure, human visual sstem (26 S., 2000) 20. H. W. Hamacher, M. Labbé, S. Nickel, T. Sonneborn Polhedral Properties of the Uncapacitated Multiple Allocation Hub Location Problem We examine the feasibilit polhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal deliver services. In particular we determine the dimension and derive some classes of facets of this polhedron. We develop some general rules about lifting facets from the uncapacitated facilit location (UFL) for UHL and projecting facets from UHL to UFL. B appling 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 performance b benchmarking it on a well known data set. Kewords: integer programming, hub location, facilit location, valid inequalities, facets, branch and cut (21 S., 2000) 21. H. W. Hamacher, A. Schöbel Design of Zone Tariff Sstems in Public Transportation Given a public transportation sstem represented b 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 alread aggregated to zones and good tariffs have to be found in the existing zone sstem. 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 ver successful in the redesign of one of German s transportation sstems. (30 S., 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 hperbolic conservation law is discretized b 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 unit with smooth and overlapping partition functions (the particles), which can even move along prescribed velocit fields. The information exchange between particles is based on standard numerical flux functions. 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 coefficients in the scheme. (16 S., 2001) 23. T. Bender, H. Hennes, J. Kalcsics, M. T. Melo, S. Nickel Location Software and Interface with GIS and Suppl Chain Management The objective of this paper is to bridge the gap between location theor 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 ver active communit on location theor encompassing man research fields such as operations research, computer science, mathematics, engineering, geograph, economics and marketing. As a result, people working on facilit location problems have a ver diverse background and also different needs regarding the software to solve these problems. For those interested in non-commercial applications (e. g. students and researchers), the librar of location algorithms (LoLA can be of considerable assistance. LoLA contains a collection of efficient algorithms for solving planar, network and discrete facilit location problems. In this paper, a detailed description of the functionalit of LoLA is presented. In the fields of geograph and marketing, for instance, solving facilit 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 sstem (GIS) and the details of the combined functionalit are described in the paper. Finall, there is a wide group of practitioners who need to solve large problems and require special purpose software with a good data interface. Man of such users can be found, for example, in the area of suppl chain management (SCM). Logistics activities involved in strategic SCM include, among others, facilit location planning. In this paper, the development of a commercial location software tool is also described. The too is embedded in the Advanced Planner and Optimizer SCM software developed b SAP AG, Walldorf, German. The paper ends with some conclusions and an outlook to future activities. Kewords: facilit location, software development, geographical information sstems, suppl chain management. (48 S., 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 man 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 mainl used to produce good lower bounds for the evacuation time and do not consider an individual behavior during the emergenc situation. These bounds can be used to analze existing buildings or help in the design phase of planning a building. Macroscopic approaches which are based on dnamic network flow models (minimum cost dnamic flow, maximum dnamic 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 ma be densit dependent. Using multicriteria optimization priorit regions and blockage due to fire or smoke ma 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 characteristics 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 onl 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 S., 2001) Stand: Juni 2001

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