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Wed, 21 Jul 2010 14:10:41 +0200Wed, 21 Jul 2010 14:10:41 +0200Identification of Temperature Dependent Parameters in Radiative Heat Transfer
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2216
Laser-induced thermotherapy (LITT) is an established minimally invasive percutaneous technique of tumor ablation. Nevertheless, there is a need to predict the effect of laser applications and optimizing irradiation planning in LITT. Optical attributes (absorption, scattering) change due to thermal denaturation. The work presents the possibility to identify these temperature dependent parameters from given temperature measurements via an optimal control problem. The solvability of the optimal control problem is analyzed and results of successful implementations are shown.Oliver Tse; René Pinnau; Norbert Siedowpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2216Wed, 21 Jul 2010 14:10:41 +0200Optimal Control of Melt Spinning Processes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1928
An optimal control problem for a mathematical model of a melt spinning process is considered. Newtonian and non--Newtonian models are used to describe the rheology of the polymeric material, the fiber is made of. The extrusion velocity of the polymer at the spinneret as well as the velocity and temperature of the quench air serve as control variables. A constrained optimization problem is derived and the first--order optimality system is set up to obtain the adjoint equations. Numerical solutions are carried out using a steepest descent algorithm.Thomas Götz; Shyam Pererapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1928Sun, 20 Jan 2008 12:37:46 +0100Optimal Control of Film Casting Processes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1927
We present an optimal control approach for the isothermal film casting process with free surfaces described by averaged Navier-Stokes equations. We control the thickness of the film at the take-up point using the shape of the nozzle. The control goal consists in finding an even thickness profile. To achieve this goal, we minimize an appropriate cost functional. The resulting minimization problem is solved numerically by a steepest descent method. The gradient of the cost functional is approximated using the adjoint variables of the problem with fixed film width. Numerical simulations show the applicability of the proposed method.Thomas Götz; K. Selvanayagampreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1927Sun, 20 Jan 2008 12:37:37 +0100A piecewise analytical solution for Jiangs model of elastoplasticity
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1898
In this article, we present an analytic solution for Jiang's constitutive model of elastoplasticity. It is considered in its stress controlled form for proportional stress loading under the assumptions that the one-to-one coupling of the yield surface radius and the memory surface radius is switched off, that the transient hardening is neglected and that the ratchetting exponents are constant.Holger Lang; Klaus Dressler; Rene Pinnaureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1898Tue, 02 Oct 2007 12:27:59 +0200A condition that a continuously deformed, simply connected body does not penetrate itself
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1874
In this article we give a sufficient condition that a simply connected flexible body does not penetrate itself, if it is subjected to a continuous deformation. It is shown that the deformation map is automatically injective, if it is just locally injective and injective on the boundary of the body. Thereby, it is very remarkable that no higher regularity assumption than continuity for the deformation map is required. The proof exclusively relies on homotopy methods and the Jordan-Brouwer separation theorem.Holger Langreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1874Sat, 23 Jun 2007 13:31:37 +0200Convergent Finite Element Discretizations of the Density Gradient Equation for Quantum Semiconductors
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1864
We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. Especially, we give a finite element description of the so--called nonlinear scheme introduced by {it Ancona}. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, especially with respect to the influence of approximate vacuum boundary conditions.Rene Pinnau; Jorge Mauricio Ruizpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1864Fri, 11 May 2007 20:46:05 +0200The Semiconductor Model Hierarchy in Optimal Dopant Profiling
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1781
We consider optimal design problems for semiconductor devices which are simulated using the energy transport model. We develop a descent algorithm based on the adjoint calculus and present numerical results for a ballistic diode. Further, we compare the optimal doping profile with results computed on basis of the drift diffusion model. Finally, we exploit the model hierarchy and test the space mapping approach, especially the aggressive space mapping algorithm, for the design problem. This yields a significant reduction of numerical costs and programming effort.Concetta Drago; Rene Pinnaupreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1781Tue, 26 Sep 2006 19:06:17 +0200A homotopy between the solutions of the elastic and elastoplastic boundary value problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1776
In this article, we give an explicit homotopy between the solutions (i.e. stress, strain, displacement) of the quasistatic linear elastic and nonlinear elastoplastic boundary value problem, where we assume a linear kinematic hardening material law. We give error estimates with respect to the homotopy parameter.Holger Lang; Klaus Dressler; Rene Pinnau; Gerd Bitschreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1776Fri, 22 Sep 2006 16:05:14 +0200Error estimates for quasistatic global elastic correction and linear kinematic hardening material
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1775
We consider in this paper the quasistatic boundary value problems of linear elasticity and nonlinear elastoplasticity with linear kinematic hardening material. We derive expressions and estimates for the difference of solutions (i.e. stress, strain and displacement) of both models. Further, we study the error between the elastoplastic solution and the solution of a postprocessing method, that corrects the solution of the linear elastic problem in order to approximate the elastoplastic model.Holger Lang; Klaus Dressler; Rene Pinnau; Michael Speckertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1775Thu, 21 Sep 2006 20:54:30 +0200Lipschitz estimates for the stop and the play operator
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1773
In this article, we give some generalisations of existing Lipschitz estimates for the stop and the play operator with respect to an arbitrary convex and closed characteristic a separable Hilbert space. We are especially concerned with the dependency of their outputs with respect to different scalar products.Holger Lang; Klaus Dressler; Rene Pinnaureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1773Thu, 21 Sep 2006 12:05:33 +0200Model Reduction Techniques for Frequency Averaging in Radiative Heat
Transfer
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1738
We study model reduction techniques for frequency averaging in radiative heat transfer. Especially, we employ proper orthogonal decomposition in combination with the method of snapshots to devise an automated a posteriori algorithm, which helps to reduce significantly the dimensionality for further simulations. The reliability of the surrogate models is tested and we compare the results with two other reduced models, which are given by the approximation using the weighted sum of gray gases and by an frequency averaged version of the so-called \(\mathrm{SP}_n\) model. We present several numerical results underlining the feasibility of our approach.Rene Pinnau; Alexander Schulzepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1738Tue, 16 May 2006 14:12:20 +0200Parameter optimization for a stress-strain correction scheme
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1702
A gradient based algorithm for parameter identification (least-squares) is applied to a multiaxial correction method for elastic stresses and strains at notches. The correction scheme, which is numerically cheap, is based on Jiang's model of elastoplasticity. Both mathematical stress-strain computations (nonlinear PDE with Jiang's constitutive material law) and physical strain measurements have been approximized. The gradient evaluation with respect to the parameters, which is large-scale, is realized by the automatic forward differentiation technique.Holger Lang; Rene Pinnau; Klaus Dreßlerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1702Sat, 21 Jan 2006 15:51:35 +0100A multiaxial stress-strain correction scheme
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1701
A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.Holger Lang; Rene Pinnau; Klaus Dreßlerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1701Sat, 21 Jan 2006 15:51:16 +0100Regularized Fixed-Point Iterations for Nonlinear Inverse Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1653
In this paper we introduce a derivative-free, iterative method for solving nonlinear ill-posed problems \(Fx=y\), where instead of \(y\) noisy data \(y_\delta\) with \(|| y-y_\delta ||\leq \delta\) are given and \(F:D(F)\subseteq X \rightarrow Y\) is a nonlinear operator between Hilbert spaces \(X\) and \(Y\). This method is defined by splitting the operator \(F\) into a linear part \(A\) and a nonlinear part \(G\), such that \(F=A+G\). Then iterations are organized as \(A u_{k+1}=y_\delta-Gu_k\). In the context of ill-posed problems we consider the situation when \(A\) does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators \(A\) and \(G\) we study the behavior of the iteration error. We obtain its stability with respect to the iteration number \(k\) as well as the optimal convergence rate with respect to the noise level \(\delta\), provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behavior. The theoretical assumptions are illustrated by a computational experiment.S.S. Pereverzyev; R. Pinnau; N. Siedowpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1653Wed, 10 Aug 2005 17:10:20 +0200Initial Temperature Reconstruction for a Nonlinear Heat Equation: Application to Radiative Heat Transfer
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1607
Consider a cooling process described by a nonlinear heat equation. We are interested to recover the initial temperature from temperature measurements which are available on a part of the boundary for some time. Up to now even for the linear heat equation such a problem has been usually studied as a nonlinear ill-posed operator equation, and regularization methods involving Frechet derivatives have been applied. We propose a fast derivative-free iterative method. Numerical results are presented for the glass cooling process, where nonlinearity appears due to radiation.Sergiy Pereverzyev; Rene Pinnau; Norbert Siedowpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1607Fri, 04 Feb 2005 10:33:11 +0100On an asymptotic expansion for porous media flow of Carreau fluids
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1568
Porous media flow of polymers with Carreau law viscosities and their application to enhanced oil recovery (EOR) is considered. Applying the homogenization method leads to a nonlinear two-scale problem. In case of a small difference between the Carreau and the Newtonian case an asymptotic expansion based on the small deviation of the viscosity from the Newtonian case is introduced. For uni-directional pressure gradients, which is a reasonable assumption in applications like EOR, auxiliary problems to decouple the micro- from the macrovariables are derived. The microscopic flow field obtained by the proposed approach is compared to the solution of the two-scale problem. Finite element calculations for an isotropic and an anisotropic pore cell geometries are used to validate the accuracy and speed-up of the proposed approach. The order of accuracy has been studied by performing the simulations up to the third order expansion for the isotropic geometry.Thomas Götz; Hanna Parhusippreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1568Thu, 09 Sep 2004 19:06:00 +0200Algebraic Systems Theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1567
Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).Eva Zerzreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1567Wed, 25 Aug 2004 09:34:44 +0200Wave Reflection and Refraction in Triclinic Crystalline media
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1414
In this paper, the reflection and refraction of a plane wave at an interface between .two half-spaces composed of triclinic crystalline material is considered. It is shown that due to incidence of a plane wave three types of waves namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) will be generated governed by the propagation condition involving the acoustic tensor. A simple procedure has been presented for the calculation of all the three phase velocities of the quasi waves. It has been considered that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. Relations are established between directions of motion and propagation, respectively. The expressions for reflection and refraction coefficients of qP, qSV and qSH waves are obtained. Numerical results of reflection and refraction coefficients are presented for different types of anisotropic media and for different types of incident waves. Graphical representation have been made for incident qP waves and for incident qSV and qSH waves numerical data are presented in two tables.Amares Chattopadhyaypreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1414Tue, 15 Jul 2003 13:10:48 +0200Regularized Multiresolution Recovery of the Mass Density Distribution from Satellite Data of the Earth's Gravitational Field
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1413
The inverse problem of recovering the Earth's density distribution from satellite data of the first or second derivative of the gravitational potential at orbit height is discussed. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust.Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1413Tue, 15 Jul 2003 13:04:31 +0200A Model for Spherical SH-Wave Propagation in Self-reinforced Linearly Elastic Media
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1412
The original publication is available at www.springerlink.com. This original publication also contains further results. We study a spherical wave propagating in radius- and latitude-direction and oscillating in latitude-direction in case of fibre-reinforced linearly elastic material. A function system solving Euler's equation of motion in this case and depending on certain Bessel and associated Legendre functions is derived.Amares Chattopadhyay; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1412Tue, 15 Jul 2003 10:45:21 +0200Smoothing Splines in Multiscale Geopotential Determination from Satellite Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1396
SST (satellite-to-satellite tracking) and SGG (satellite gravity gradiometry) provide data that allows the determination of the first and second order radial derivative of the earth's gravitational potential on the satellite orbit, respectively. The modeling of the gravitational potential from such data is an exponentially ill-posed problem that demands regularization. In this paper, we present the numerical studies of an approach, investigated in [24] and [25], that reconstructs the potential with spline smoothing. In this case, spline smoothing is not just an approximation procedure but it solves the underlying compact operator equation of the SST-problem and the SGG-problem. The numerical studies in this paper are performed for a simplified geometrical scenario with simulated data, but the approach is designed to handle first or second order radial derivative data on a real satellite orbit.Kerstin Hesse; Martin Guttingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1396Mon, 26 May 2003 10:49:20 +0200SST-Regularisierung durch Multiresolutionstechniken auf der Basis von CHAMP-Daten
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1377
Die Bestimmung des Erdgravitationspotentials aus den Meßdaten des Forschungssatelliten CHAMP lässt sich als Operatorgleichung formulieren (SST-Problem). Dieser Ansatz geht davon aus, dass ein geometrischer Orbit des Satelliten CHAMP vorliegt. Mittels numerischer Differentiation unter Einsatz eines geeigneten Denoising Verfahrens kann dann aus dem geometrischen Orbit der Gradient des Potentials längs der Bahn bestimmt werden. Damit sind insbesondere die Radialableitung (und der Flächengradient) auf einem Punktgitter auf der Bahn bekannt. In einem erdfesten System stellt sich dies als eine nahezu vollständige Überdeckung der Erde (bis auf Polar Gaps) mit einem ziemlich dichten Datengitter auf Flughöhe des Satelliten dar. Die Lösung der SST-Operatorgleichung (Bestimmung des Potentials auf der Erdoberfläche aus Kenntnis der Radialableitung auf einem Datengitter auf Flughöhe) ist ein schlecht gestelltes inverses Problem, das mit einer geeigneten Regularisierungstechnik gelöst werden muß. Im vorliegenden Fall wurde eine solche Regularisierung mit Hilfe von nicht-bandlimitierten Regularisierungsskalierungsfunktionen und Regularisierungswavelets umgesetzt. Diese sind stark ortslokalisierend und führen daher auf ein Potentialmodell, welches eine Linearkombination stark ortslokalisierender Funktionen ist. Ein solches Modell kann als Lokalmodell auch aus nur lokalen Daten berechnet werden und bietet daher gegenüber Kugelfunktionsmodellen wie EGM96 erhebliche Vorteile für die moderne Geopotentialbestimmung. Die Diskretisierung und numerische Umsetzung der Berechnung eines solchen Modells erfolgt mit Splines, die hier ebenfalls Linearkombinationen stark ortslokalisierender Funktionen sind. Die großen linearen Gleichungssysteme, die zur Berechnung der glättenden oder interpolierenden Splines gelöst werden müssen, können auf schnelle und effiziente Weise mit dem Schwarzschen alternierenden Algorithmus in Verbindung mit schnellen Summationsverfahren (Fast Multipole Methods) gelöst werden. Eine Kombination des Schwarzschen alternierenden Algorithmus mit solchen schnellen Summationsverfahren ermöglicht eine weitere erhebliche Beschleunigung beim Lösen dieser Gleichungssysteme. Zur Bestimmung von Glättungsparametern (Spline-Smoothing) und Regularisierungsparametern kann die L-Curve Method zum Einsatz kommen.Willi Freeden; Thorsten Maierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1377Mon, 24 Feb 2003 14:15:34 +0100Multiresolution Data Analysis - Numerical Realization by use of Domain Decomposition Methods and Fast Multipole Techniques / Multiscale Solutions of Oblique Boundary-Value Problems by Layer Potentials
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1344
This survey paper deals with multiresolution analysis from geodetically relevant data and its numerical realization for functions harmonic outside a (Bjerhammar) sphere inside the Earth. Harmonic wavelets are introduced within a suit- able framework of a Sobolev-like Hilbert space. Scaling functions and wavelets are defined by means of convolutions. A pyramid scheme provides efficient implementation und economical computation. Essential tools are the multiplicative Schwarz alternating algorithm (providing domain decomposition procedures) and fast multipole techniques (accelerating iterative solvers of linear systems).W. Freeden; C. Mayerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1344Thu, 12 Sep 2002 10:37:46 +0200New Numerical Schemes based on Relaxation Systems for Conservation Laws
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1337
S. V. Rao Raghuramapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1337Fri, 19 Jul 2002 00:00:00 +0200Spline Modelling of Geostrophic Flow: Theoretical and Algorithmic Aspects
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1338
Spline functions that approximate (geostrophic) wind field or ocean circulation data are developed in a weighted Sobolev space setting on the (unit) sphere. Two problems are discussed in more detail: the modelling of the (geostrophic) wind field from (i)discrete scalar air pressure data and (ii) discrete vectorial velocity data. Domain decomposition methods based on the Schwarz alternating algorithm for positive definite symmetric matrices are described for solving large linear systems occuring in vectorial spline interpolation or smoothing of geostrophic flow.W. Freeden; K. Hessepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1338Fri, 19 Jul 2002 00:00:00 +0200