@phdthesis{AlYasiri2017, author = {Al-Yasiri, Zainab Riyadh Shaker}, title = {Function Theoretic Methods for the Analytical and Numerical Solution of Some Non-linear Boundary Value Problems with Singularities}, doi = {10.25643/bauhaus-universitaet.3898}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20190506-38987}, school = {Bauhaus-Universit{\"a}t Weimar}, pages = {164}, year = {2017}, abstract = {The p-Laplace equation is a nonlinear generalization of the well-known Laplace equation. It is often used as a model problem for special types of nonlinearities, and therefore it can be seen as a bridge between very general nonlinear equations and the linear Laplace equation, too. It appears in many problems for instance in the theory of non-Newtonian fluids and fluid dynamics or in rockfill dam problems, as well as in special problems of image restoration and image processing. The aim of this thesis is to solve the p-Laplace equation for 1 < p < 2, as well as for 2 < p < 3 and to find strong solutions in the framework of Clifford analysis. The idea is to apply a hypercomplex integral operator and special function theoretic methods to transform the p-Laplace equation into a p-Dirac equation. We consider boundary value problems for the p-Laplace equation and transfer them to boundary value problems for a p-Dirac equation. These equations will be solved iteratively by applying Banach's fixed-point principle. Applying operator-theoretical methods for the p-Dirac equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. In addition, using a finite difference approach on a uniform lattice in the plane, the fundamental solution of the Cauchy-Riemann operator and its adjoint based on the fundamental solution of the Laplacian will be calculated. Besides, we define gener- alized discrete Teodorescu transform operators, which are right-inverse to the discrete Cauchy-Riemann operator and its adjoint in the plane. Furthermore, a new formula for generalized discrete boundary operators (analogues of the Cauchy integral operator) will be considered. Based on these operators a new version of discrete Borel-Pompeiu formula is formulated and proved. This is the basis for an operator calculus that will be applied to the numerical solution of the p-Dirac equation. Finally, numerical results will be presented showing advantages and problems of this approach.}, subject = {Finite-Differenzen-Methode}, language = {en} } @article{GuerlebeckLegatiukNilssonetal., author = {G{\"u}rlebeck, Klaus and Legatiuk, Dmitrii and Nilsson, Henrik and Smarsly, Kay}, title = {Conceptual modelling: Towards detecting modelling errors in engineering applications}, series = {Mathematical Methods in Applied Sciences}, journal = {Mathematical Methods in Applied Sciences}, doi = {10.1002/mma.5934}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20200110-40614}, pages = {1 -- 10}, abstract = {Rapid advancements of modern technologies put high demands on mathematical modelling of engineering systems. Typically, systems are no longer "simple" objects, but rather coupled systems involving multiphysics phenomena, the modelling of which involves coupling of models that describe different phenomena. After constructing a mathematical model, it is essential to analyse the correctness of the coupled models and to detect modelling errors compromising the final modelling result. Broadly, there are two classes of modelling errors: (a) errors related to abstract modelling, eg, conceptual errors concerning the coherence of a model as a whole and (b) errors related to concrete modelling or instance modelling, eg, questions of approximation quality and implementation. Instance modelling errors, on the one hand, are relatively well understood. Abstract modelling errors, on the other, are not appropriately addressed by modern modelling methodologies. The aim of this paper is to initiate a discussion on abstract approaches and their usability for mathematical modelling of engineering systems with the goal of making it possible to catch conceptual modelling errors early and automatically by computer assistant tools. To that end, we argue that it is necessary to identify and employ suitable mathematical abstractions to capture an accurate conceptual description of the process of modelling engineering systems.}, subject = {Angewandte Mathematik}, language = {en} } @inproceedings{KerstenRodehorst, author = {Kersten, Jens and Rodehorst, Volker}, title = {TOWARDS STEREO VISION- AND LASER SCANNER-BASED UAS POSE ESTIMATION}, series = {Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar}, booktitle = {Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar}, editor = {G{\"u}rlebeck, Klaus and Lahmer, Tom}, organization = {Bauhaus-Universit{\"a}t Weimar}, issn = {1611-4086}, doi = {10.25643/bauhaus-universitaet.2807}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170314-28072}, pages = {7}, abstract = {A central issue for the autonomous navigation of mobile robots is to map unknown environments while simultaneously estimating its position within this map. This chicken-eggproblem is known as simultaneous localization and mapping (SLAM). Asctec's quadrotor Pelican is a powerful and flexible research UAS (unmanned aircraft system) which enables the development of new real-time on-board algorithms for SLAM as well as autonomous navigation. The relative UAS pose estimation for SLAM, usually based on low-cost sensors like inertial measurement units (IMU) and barometers, is known to be affected by high drift rates. In order to significantly reduce these effects, we incorporate additional independent pose estimation techniques using exteroceptive sensors. In this article we present first pose estimation results using a stereo camera setup as well as a laser range finder, individually. Even though these methods fail in few certain configurations we demonstrate their effectiveness and value for the reduction of IMU drift rates and give an outlook for further works towards SLAM.}, subject = {Angewandte Informatik}, language = {en} } @inproceedings{AlaladeKafleWuttkeetal., author = {Alalade, Muyiwa and Kafle, Binod and Wuttke, Frank and Lahmer, Tom}, title = {CALIBRATION OF CYCLIC CONSTITUTIVE MODELS FOR SOILS BY OSCILLATING FUNCTIONS}, series = {Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar}, booktitle = {Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar}, editor = {G{\"u}rlebeck, Klaus and Lahmer, Tom}, organization = {Bauhaus-Universit{\"a}t Weimar}, issn = {1611-4086}, doi = {10.25643/bauhaus-universitaet.2793}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170314-27932}, pages = {6}, abstract = {In order to minimize the probability of foundation failure resulting from cyclic action on structures, researchers have developed various constitutive models to simulate the foundation response and soil interaction as a result of these complex cyclic loads. The efficiency and effectiveness of these model is majorly influenced by the cyclic constitutive parameters. Although a lot of research is being carried out on these relatively new models, little or no details exist in literature about the model based identification of the cyclic constitutive parameters. This could be attributed to the difficulties and complexities of the inverse modeling of such complex phenomena. A variety of optimization strategies are available for the solution of the sum of least-squares problems as usually done in the field of model calibration. However for the back analysis (calibration) of the soil response to oscillatory load functions, this paper gives insight into the model calibration challenges and also puts forward a method for the inverse modeling of cyclic loaded foundation response such that high quality solutions are obtained with minimum computational effort. Therefore model responses are produced which adequately describes what would otherwise be experienced in the laboratory or field.}, subject = {Angewandte Informatik}, language = {en} } @inproceedings{KraussharConstalesGuerlebecketal., author = {Kraußhar, Rolf S{\"o}ren and Constales, Denis and G{\"u}rlebeck, Klaus and Spr{\"o}ßig, Wolfgang}, title = {APPLICATIONS OF QUATERNIONIC ANALYSIS IN ENGINEERING}, editor = {G{\"u}rlebeck, Klaus and K{\"o}nke, Carsten}, organization = {Bauhaus-Universit{\"a}t Weimar}, doi = {10.25643/bauhaus-universitaet.2912}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170327-29128}, pages = {8}, abstract = {The quaternionic operator calculus can be applied very elegantly to solve many important boundary value problems arising in fluid dynamics and electrodynamics in an analytic way. In order to set up fully explicit solutions. In order to apply the quaternionic operator calculus to solve these types of boundary value problems fully explicitly, one has to evaluate two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. Recently the theory of quaternionic holomorphic multiperiodic functions and automorphic forms provided new impulses to set up explicit representation formulas for large classes of hyperbolic polyhedron type domains. These include block shaped domains, wedge shaped domains (with or without additional rectangular restrictions) and circular symmetric finite and infinite cylinders as particular subcases. In this talk we want to give an overview over the recent developments in this direction.}, subject = {Architektur }, language = {en} } @phdthesis{AhmedElSayed2003, author = {Ahmed El-Sayed, Ahmed Mohammed}, title = {One some classes and spaces of holomorphic and hyperholomorphic functions}, doi = {10.25643/bauhaus-universitaet.25}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20040216-271}, school = {Bauhaus-Universit{\"a}t Weimar}, year = {2003}, abstract = {In this Thesis we study some complex and hypercomplex function spaces and classes such as hypercomplex Besov spaces, Bloch space and Op spaces as well as the class of basic sets of polynomials in several complex variables. It is shown that hyperholomorphic Besov spaces can be applied to characterize the hyperholomorphic Bloch space. Moreover, we consider BMOM and VMOM spaces.}, subject = {Funktionenraum}, language = {en} } @article{LegatiukWeiszPatrault, author = {Legatiuk, Dmitrii and Weisz-Patrault, Daniel}, title = {Coupling of Complex Function Theory and Finite Element Method for Crack Propagation Through Energetic Formulation: Conformal Mapping Approach and Reduction to a Riemann-Hilbert Problem}, series = {Computational Methods and Function Theory}, volume = {2021}, journal = {Computational Methods and Function Theory}, publisher = {Springer}, address = {Heidelberg}, doi = {10.1007/s40315-021-00403-7}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20210805-44763}, pages = {1 -- 23}, abstract = {In this paper we present a theoretical background for a coupled analytical-numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical-numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann-Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.}, subject = {Angewandte Mathematik}, language = {en} } @article{LegatiukGuerlebeckHommel, author = {Legatiuk, Anastasiia and G{\"u}rlebeck, Klaus and Hommel, Angela}, title = {Estimates for the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice}, series = {Mathematical Methods in the Applied Sciences}, volume = {2021}, journal = {Mathematical Methods in the Applied Sciences}, publisher = {Wiley}, address = {Chichester}, doi = {10.1002/mma.7747}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220209-45829}, pages = {1 -- 23}, abstract = {This paper presents numerical analysis of the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice. Additionally, to provide estimates in interior and exterior domains, two different regularisations of the discrete fundamental solution are considered. Estimates for the absolute difference and lp-estimates are constructed for both regularisations. Thus, this work extends the classical results in the discrete potential theory to the case of a rectangular lattice and serves as a basis for future convergence analysis of the method of discrete potentials on rectangular lattices.}, subject = {diskrete Fourier-Transformation}, language = {en} } @article{CerejeirasKaehlerLegatiuketal., author = {Cerejeiras, Paula and K{\"a}hler, Uwe and Legatiuk, Anastasiia and Legatiuk, Dmitrii}, title = {Discrete Hardy Spaces for Bounded Domains in Rn}, series = {Complex Analysis and Operator Theory}, volume = {2021}, journal = {Complex Analysis and Operator Theory}, number = {Volume 15, article 4}, publisher = {Springer}, address = {Heidelberg}, doi = {10.1007/s11785-020-01047-6}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20210804-44746}, pages = {1 -- 32}, abstract = {Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in Rn. On this way, discrete Stokes' formula, discrete Borel-Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.}, subject = {Dirac-Operator}, language = {en} } @misc{Hamzah, type = {Master Thesis}, author = {Hamzah, Abdulrazzak}, title = {L{\"o}sung von Randwertaufgaben der Bruchmechanik mit Hilfe einer approximationsbasierten Kopplung zwischen der Finite-Elemente-Methode und Methoden der komplexen Analysis}, doi = {10.25643/bauhaus-universitaet.4093}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20200211-40936}, school = {Bauhaus-Universit{\"a}t Weimar}, abstract = {Das Hauptziel der vorliegenden Arbeit war es, eine stetige Kopplung zwischen der ananlytischen und numerischen L{\"o}sung von Randwertaufgaben mit Singularit{\"a}ten zu realisieren. Durch die inter-polationsbasierte gekoppelte Methode kann eine globale C0 Stetigkeit erzielt werden. F{\"u}r diesen Zweck wird ein spezielle finite Element (Kopplungselement) verwendet, das die Stetigkeit der L{\"o}sung sowohl mit dem analytischen Element als auch mit den normalen CST Elementen gew{\"a}hrleistet. Die interpolationsbasierte gekoppelte Methode ist zwar f{\"u}r beliebige Knotenanzahl auf dem Interface ΓAD anwendbar, aber es konnte durch die Untersuchung von der Interpolationsmatrix und numerische Simulationen festgestellt werden, dass sie schlecht konditioniert ist. Um das Problem mit den numerischen Instabilit{\"a}ten zu bew{\"a}ltigen, wurde eine approximationsbasierte Kopplungsmethode entwickelt und untersucht. Die Stabilit{\"a}t dieser Methode wurde anschließend anhand der Untersuchung von der Gramschen Matrix des verwendeten Basissystems auf zwei Intervallen [-π,π] und [-2π,2π] beurteilt. Die Gramsche Matrix auf dem Intervall [-2π,2π] hat einen g{\"u}nstigeren Konditionszahl in der Abh{\"a}ngigkeit von der Anzahl der Kopplungsknoten auf dem Interface aufgewiesen. Um die dazu geh{\"o}rigen numerischen Instabilit{\"a}ten ausschließen zu k{\"o}nnen wird das Basissystem mit Hilfe vom Gram-Schmidtschen Orthogonalisierungsverfahren auf beiden Intervallen orthogonalisiert. Das orthogonale Basissystem l{\"a}sst sich auf dem Intervall [-2π,2π] mit expliziten Formeln schreiben. Die Methode des konsistentes Sampling, die h{\"a}ufig in der Nachrichtentechnik verwendet wird, wurde zur Realisierung von der approximationsbasierten Kopplung herangezogen. Eine Beschr{\"a}nkung dieser Methode ist es, dass die Anzahl der Sampling-Basisfunktionen muss gleich der Anzahl der Wiederherstellungsbasisfunktionen sein. Das hat dazu gef{\"u}hrt, dass das eingef{\"u}hrt Basissys-tem (mit 2 n Basisfunktionen) nur mit n Basisfunktion verwendet werden kann. Zur L{\"o}sung diese Problems wurde ein alternatives Basissystems (Variante 2) vorgestellt. F{\"u}r die Verwendung dieses Basissystems ist aber eine Transformationsmatrix M n{\"o}tig und bei der Orthogonalisierung des Basissystems auf dem Intervall [-π,π] kann die Herleitung von dieser Matrix kompliziert und aufwendig sein. Die Formfunktionen wurden anschließend f{\"u}r die beiden Varianten hergeleitet und grafisch (f{\"u}r n = 5) dargestellt und wurde gezeigt, dass diese Funktionen die Anforderungen an den Formfunktionen erf{\"u}llen und k{\"o}nnen somit f{\"u}r die FE- Approximation verwendet werden. Anhand numerischer Simulationen, die mit der Variante 1 (mit Orthogonalisierung auf dem Intervall [-2π,2π]) durchgef{\"u}hrt wurden, wurden die grundlegenden Fragen (Beispielsweise: Stetigkeit der Verformungen auf dem Interface ΓAD, Spannungen auf dem analytischen Gebiet) {\"u}ber- pr{\"u}ft.}, subject = {Mathematik}, language = {de} }