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We consider a linearized kinetic BGK equation and the associated acoustic system on a network.
Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network.
This analysis leads to the consideration of a fixpoint problem involving the solutions of kinetic half-space problems.
This work extends the procedure developed in [13], where coupling conditions for a simplified BGK model have been derived.
Numerical comparisons between different coupling conditions
confirm the accuracy of the proposed approximation.
Im Zuge der Bologna-Reform besteht der Anspruch die universitäre Lehre kompetenzorien-tiert zu gestalten. Es werden gemäß der KMK-Rahmenvorgaben in den universitären Mo-dulplänen Kompetenzen formuliert, jedoch realisiert sich die Umsetzung in nur wenigen Fällen und es bleibt oftmals bei der formalen Angabe von Qualifikationszielen. Im Sinne einer adä-quaten Berufsvorbereitung der Studierenden ist eine kompetenzorientiere Lehre aber als ele-mentar anzusehen.
An der Technischen Universität Kaiserslautern wird den Lehramtsstudierenden im Rahmen des Sportstudiums eine breit gefächerte Schneesportausbildung angeboten. Eine praxisorien-tierte Schneesportausbildung bietet eine besondere Form des Erlebens und Erfahrens, wel-che Studierenden in rein theoretischen Veranstaltungen nicht geboten werden kann. Jedes Erleben und Erfahren bewirkt Emotionen; Kompetenzen lassen sich nur in emotionsaktivie-renden Lernarrangements aneignen.
Das Ziel dieses Beitrags besteht darin, die Schneesportausbildung der TU Kaiserslautern un-ter dem Aspekt einer Kompetenzermöglichung vorzustellen. Hierfür wurde zunächst das Ausbildungskonzept erhoben, und untersucht, inwiefern den Studierenden im Rahmen der Schneesportausbildung eine Kompetenzentwicklung ermöglicht wird. Als Grundlage dient hier der Kompetenzatlas nach Heyse und Erpenbeck. Die sich ergebenden Resultate zu Inhalten und Methoden der Schneesportausbildung werden zusammenfassend in einem Kompetenz-profil dargestellt. Darüber hinaus werden die Ausbildungsmaßnahmen beurteilt, Möglichkeiten und Potentiale zur Kompetenzförderung aufgezeigt und beispielhaft im Sinne eines Best Practice Beispiels in einem Wochenplan zusammengefasst.
Simulating the flow of water in district heating networks requires numerical methods which are independent of the CFL condition. We develop a high order scheme for networks of advection equations allowing large time steps. With the MOOD technique unphysical oscillations of non smooth solutions are avoided. In numerical tests the applicability to real networks is shown.
A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.
Spatial regression models provide the opportunity to analyse spatial data and spatial processes. Yet, several model specifications can be used, all assuming different types of spatial dependence. This study summarises the most commonly used spatial regression models and offers a comparison of their performance by using Monte Carlo experiments. In contrast to previous simulations, this study evaluates the bias of the impacts rather than the regression coefficients and additionally provides results for situations with a non-spatial omitted variable bias. Results reveal that the most commonly used spatial autoregressive (SAR) and spatial error (SEM) specifications yield severe drawbacks. In contrast, spatial Durbin specifications (SDM and SDEM) as well as the simple SLX provide accurate estimates of direct impacts even in the case of misspecification. Regarding the indirect `spillover' effects, several - quite realistic - situations exist in which the SLX outperforms the more complex SDM and SDEM specifications.
SDE-driven modeling of phenotypically heterogeneous tumors: The influence of cancer cell stemness
(2018)
We deduce cell population models describing the evolution of a tumor (possibly interacting with its
environment of healthy cells) with the aid of differential equations. Thereby, different subpopulations
of cancer cells allow accounting for the tumor heterogeneity. In our settings these include cancer
stem cells known to be less sensitive to treatment and differentiated cancer cells having a higher
sensitivity towards chemo- and radiotherapy. Our approach relies on stochastic differential equations
in order to account for randomness in the system, arising e.g., by the therapy-induced decreasing
number of clonogens, which renders a pure deterministic model arguable. The equations are deduced
relying on transition probabilities characterizing innovations of the two cancer cell subpopulations,
and similarly extended to also account for the evolution of normal tissue. Several therapy approaches
are introduced and compared by way of tumor control probability (TCP) and uncomplicated tumor
control probability (UTCP). A PDE approach allows to assess the evolution of tumor and normal
tissue with respect to time and to cell population densities which can vary continuously in a given set
of states. Analytical approximations of solutions to the obtained PDE system are provided as well.
Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε.
In this article a new numerical solver for simulations of district heating networks is presented. The numerical method applies the local time stepping introduced in [11] to networks of linear advection equations. In combination with the high order approach of [4] an accurate and very efficient scheme is developed. In several numerical test cases the advantages for simulations of district heating networks are shown.
We continue in this paper the study of k-adaptable robust solutions for combinatorial optimization problems with bounded uncertainty sets. In this concept not a single solution needs to be chosen to hedge against the uncertainty. Instead one is allowed to choose a set of k different solutions from which one can be chosen after the uncertain scenario has been revealed. We first show how the problem can be decomposed into polynomially many subproblems if k is fixed. In the remaining part of the paper we consider the special case where k=2, i.e., one is allowed to choose two different solutions to hedge against the uncertainty. We decompose this problem into so called coordination problems. The study of these coordination problems turns out to be interesting on its own. We prove positive results for the unconstrained combinatorial optimization problem, the matroid maximization problem, the selection problem, and the shortest path problem on series parallel graphs. The shortest path problem on general graphs turns out to be NP-complete. Further, we present for minimization problems how to transform approximation algorithms for the coordination problem to approximation algorithms for the original problem. We study the knapsack problem to show that this relation does not hold for maximization problems in general. We present a PTAS for the corresponding coordination problem and prove that the 2-adaptable knapsack problem is not at all approximable.
This paper presents a case study of duty rostering for physicians at a department of orthopedics and trauma surgery. We provide a detailed description of the rostering problem faced and present an integer programming model that has been used in practice for creating duty rosters at the department for more than a year. Using real world data, we compare the model output to a manually generated roster as used previously by the department and analyze the quality of the rosters generated by the model over a longer time span. Moreover, we demonstrate how unforeseen events such as absences of scheduled physicians are handled.
We extend the standard concept of robust optimization by the introduction of an alternative solution. In contrast to the classic concept, one is allowed to chose two solutions from which the best can be picked after the uncertain scenario has been revealed. We focus in this paper on the resulting robust problem for combinatorial problems with bounded uncertainty sets. We present a reformulation of the robust problem which decomposes it into polynomially many subproblems. In each subproblem one needs to find two solutions which are connected by a cost function which penalizes if the same element is part of both solutions. Using this reformulation, we show how the robust problem can be solved efficiently for the unconstrained combinatorial problem, the selection problem, and the minimum spanning tree problem. The robust problem corresponding to the shortest path problem turns out to be NP-complete on general graphs. However, for series-parallel graphs, the robust shortest path problem can be solved efficiently. Further, we show how approximation algorithms for the subproblem can be used to compute approximate solutions for the original problem.
We propose a multiscale model for tumor cell migration in a tissue network. The system of equations involves a structured population model for the tumor cell density, which besides time and
position depends on a further variable characterizing the cellular state with respect to the amount
of receptors bound to soluble and insoluble ligands. Moreover, this equation features pH-taxis and
adhesion, along with an integral term describing proliferation conditioned by receptor binding. The
interaction of tumor cells with their surroundings calls for two more equations for the evolution of
tissue fibers and acidity (expressed via concentration of extracellular protons), respectively. The
resulting ODE-PDE system is highly nonlinear. We prove the global existence of a solution and
perform numerical simulations to illustrate its behavior, paying particular attention to the influence
of the supplementary structure and of the adhesion.
We propose and analyze a multiscale model for acid-mediated tumor invasion
accounting for stochastic effects on the subcellular level.
The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density,
the movement being directed towards pH gradients in the local microenvironment,
which is coupled to a PDE-SDE system characterizing the
dynamics of extracellular and intracellular proton concentrations, respectively.
The global well-posedness of the model is shown and
numerical simulations are performed in order to illustrate the solution behavior.
This work presents a framework for the computation of complex geometries containing intersections of multiple patches with Reissner-Mindlin shell elements. The main objective is to provide an isogeometric finite element implementation which neither requires drilling rotation stabilization, nor user interaction to quantify the number of rotational degrees of freedom for every node. For this purpose, the following set of methods is presented. Control points with corresponding physical location are assigned to one common node for the finite element solution. A nodal basis system in every control point is defined, which ensures an exact interpolation of the director vector throughout the whole domain. A distinction criterion for the automatic quantification of rotational degrees of freedom for every node is presented. An isogeometric Reissner-Mindlin shell formulation is enhanced to handle geometries with kinks and allowing for arbitrary intersections of patches. The parametrization of adjacent patches along the interface has to be conforming. The shell formulation is derived from the continuum theory and uses a rotational update scheme for the current director vector. The nonlinear kinematic allows the computation of large deformations and large rotations. Two concepts for the description of rotations are presented. The first one uses an interpolation which is commonly used in standard Lagrange-based shell element formulations. The second scheme uses a more elaborate concept proposed by the authors in prior work, which increases the accuracy for arbitrary curved geometries. Numerical examples show the high accuracy and robustness of both concepts. The applicability of the proposed framework is demonstrated.
An isogeometric Reissner-Mindlin shell derived from the continuum theory is presented. The geometry is described by NURBS surfaces. The kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system. Hence, the definition of nodal basis systems at the control points is necessary for the proposed formulation. The control points are in general not located on the shell reference surface and thus, several choices for the nodal values are possible. The proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof. Thus, the initial director vector coincides with the normal vector even for the coarsest mesh. In addition to that a more accurate interpolation of the current director and its variation is proposed. Instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations. Account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation. The exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes. The convergence behavior is shown to be correct for k-refinement allowing the use of coarse meshes with high orders of NURBS basis functions. This is potentially advantageous for applications with high numerical effort per integration point. The geometrically nonlinear formulation accounts for large rotations. The consistent tangent matrix is derived. Various standard benchmark examples show the superior accuracy of the presented shell formulation. A new benchmark designed to test the convergence behavior for free form surfaces is presented. Despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound.
In this contribution a mortar-type method for the coupling of non-conforming NURBS surface patches is proposed. The connection of non-conforming patches with shared degrees of freedom requires mutual refinement, which propagates throughout the whole patch due to the tensor-product structure of NURBS surfaces. Thus, methods to handle non-conforming meshes are essential in NURBS-based isogeometric analysis. The main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation. The deformations of the interface control points of adjacent patches are interrelated with a master-slave relation. This relation is established numerically using the weak form of the equality of mutual deformations along the interface. With the help of this relation the interface degrees of freedom of the slave patch can be condensated out of the system. A natural connection of the patches is attained without additional terms in the weak form. The proposed method is also applicable for nonlinear computations without further measures. Linear and geometrical nonlinear examples show the high accuracy and robustness of the new method. A comparison to reference results and to computations with the Lagrange multiplier method is given.
We propose and study a strongly coupled PDE-ODE-ODE system modeling cancer cell invasion through a tissue network
under the go-or-grow hypothesis asserting that cancer cells can either move or proliferate. Hence our setting features
two interacting cell populations with their mutual transitions and involves tissue-dependent degenerate diffusion and
haptotaxis for the moving subpopulation. The proliferating cells and the tissue evolution are characterized by way of ODEs
for the respective densities. We prove the global existence of weak solutions and illustrate the model behaviour by
numerical simulations in a two-dimensional setting.
In this paper, we discuss the problem of approximating ellipsoid uncertainty sets with bounded (gamma) uncertainty sets. Robust linear programs with ellipsoid uncertainty lead to quadratically constrained programs, whereas robust linear programs with bounded uncertainty sets remain linear programs which are generally easier to solve.
We call a bounded uncertainty set an inner approximation of an ellipsoid if it is contained in it. We consider two different inner approximation problems. The first problem is to find a bounded uncertainty set which sticks close to the ellipsoid such that a shrank version of the ellipsoid is contained in it. The approximation is optimal if the required shrinking is minimal. In the second problem, we search for a bounded uncertainty set within the ellipsoid with maximum volume. We present how both problems can be solved analytically by stating explicit formulas for the optimal solutions of these problems.
Further, we present in a computational experiment how the derived approximation techniques can be used to approximate shortest path and network flow problems which are affected by ellipsoidal uncertainty.
For the prediction of digging forces from a granular material simulation, the
Nonsmooth Contact Dynamics Method is examined. First, the equations of motion
for nonsmooth mechanical systems are laid out. They are a differential
variational inequality that has the same structure as classical discrete algebraic equations. Using a Galerkin projection in time, it becomes possible to derive
nonsmooth versions of the classical SHAK and RATTLE integrators.
A matrix-free Interior Point Method is used for the complementarity
problems that need to be solved in every time step. It is shown that this method
outperforms the Projected Gauss-Jacobi method by several orders of magnitude
and produces the same digging force result as the Discrete Element Method in comparable computing time.
We consider the problem to evacuate several regions due to river flooding, where sufficient time is given to plan ahead. To ensure a smooth evacuation procedure, our model includes the decision which regions to assign to which shelter, and when evacuation orders should be issued, such that roads do not become congested.
Due to uncertainty in weather forecast, several possible scenarios are simultaneously considered in a robust optimization framework. To solve the resulting integer program, we apply a Tabu search algorithm based on decomposing the problem into better tractable subproblems. Computational experiments on random instances and an instance based on Kulmbach, Germany, data show considerable improvement compared to an MIP solver provided with a strong starting solution.