TY - THES A1 - Nikulla, Susanne T1 - Quality assessment of kinematical models by means of global and goal-oriented error estimation techniques T1 - Anwendung globaler und zielorientierter Fehlerschätzer zur Beurteilung der Notwendigkeit einer geometrisch nicht-linearen Berechnung N2 - Methods for model quality assessment are aiming to find the most appropriate model with respect to accuracy and computational effort for a structural system under investigation. Model error estimation techniques can be applied for this purpose when kinematical models are investigated. They are counted among the class of white box models, which means that the model hierarchy and therewith the best model is known. This thesis gives an overview of discretisation error estimators. Deduced from these, methods for model error estimation are presented. Their general goal is to make a prediction of the inaccuracies that are introduced using the simpler model without knowing the solution of a more complex model. This information can be used to steer an adaptive process. Techniques for linear and non-linear problems as well as global and goal-oriented errors are introduced. The estimation of the error in local quantities is realised by solving a dual problem, which serves as a weight for the primal error. So far, such techniques have mainly been applied in material modelling and for dimensional adaptivity. Within the scope of this thesis, available model error estimators are adapted for an application to kinematical models. Their applicability is tested regarding the question of whether a geometrical non-linear calculation is necessary or not. The analysis is limited to non-linear estimators due to the structure of the underlying differential equations. These methods often involve simplification, e.g linearisations. It is investigated to which extent such assumptions lead to meaningful results, when applied to kinematical models. N2 - Die verschiedenen Methoden zur Bewertung der Modellqualität haben ein Ziel: Das passende Modell in Bezug auf Genauigkeit und Berechnungsaufwand für eine konkrete Struktur zu finden. Steht dabei die Untersuchung eines Kinematik-Modells im Vordergrund, können Modellfehlerschätzer zur Modellbewertung verwendet werden. Dieser Zusammenhang gilt, solange es sich um mechanisch motivierte Modelle handelt, bei denen die Modellhierarchie und damit das beste Modell bekannt sind. Die vorliegende Arbeit beschreibt den Weg von den einfachen Fehlerschätzern für Diskretisierungsfehler bis zu den daraus abgeleiteten Modellfehlerschätzern. Das Ziel der letztgenannten besteht in der Vorhersage von Ungenauigkeit, die durch die Verwendung eines vereinfachten anstatt des komplexen Modells entstehen. Aus den gewonnenen Informationen wird eine adaptive Modellanpassung entwickelt. Die Methoden lassen sich dabei nach verschiedenen Kriterien unterscheiden. Diese diffenzieren zwischen den verschiedenen Anwendungsbereichen, zwischen linearen und nicht-linearen Modellen sowie zwischen globalen und ziel-orientierten Fehlern. Die bislang in der Literatur hauptsächlich zu findenden Anwendungsgebiete sind die Materialmodellierung und die Dimensionsadaptivität. Im Rahmen dieser Arbeit werden nun die bekannten Methoden zur Abschätzung des Modellfehlers auf kinematische Modelle erweitert. Zudem wird die Frage, ob eine geometrisch nicht-lineare Berechnung notwendig ist oder nicht, untersucht. Aufgrund der Struktur der zugrunde liegenden Differentialgleichungen beschränken sich die Analysen auf nicht-lineare Fehlerschätzer. Da diese Methoden oft auf Vereinfachungen wie z.B. die Linearisierung der Grundgleichungen zurückgreifen, wird in der vorliegenden Arbeit untersucht, inwieweit diese Annahmen zu verwertbaren Ergebnissen führen. T3 - Schriftenreihe des DFG Graduiertenkollegs 1462 Modellqualitäten // Graduiertenkolleg Modellqualitäten - 4 KW - Model quality, Model error estimation, Kinematical model, Geometric non-linearity, Finite Element method KW - Modellqualität, Modellfehlerschätzer, Geometrisch nicht-lineare Berechnung, Kinematik Modell, Finite Elemente Methode Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20120419-16161 PB - Verlag der Bauhaus-Universität Weimar CY - Weimar ER - TY - THES A1 - Stein, Peter T1 - Procedurally generated models for Isogeometric Analysis N2 - Increasingly powerful hard- and software allows for the numerical simulation of complex physical phenomena with high levels of detail. In light of this development the definition of numerical models for the Finite Element Method (FEM) has become the bottleneck in the simulation process. Characteristic features of the model generation are large manual efforts and a de-coupling of geometric and numerical model. In the highly probable case of design revisions all steps of model preprocessing and mesh generation have to be repeated. This includes the idealization and approximation of a geometric model as well as the definition of boundary conditions and model parameters. Design variants leading to more resource-efficient structures might hence be disregarded due to limited budgets and constrained time frames. A potential solution to above problem is given with the concept of Isogeometric Analysis (IGA). Core idea of this method is to directly employ a geometric model for numerical simulations, which allows to circumvent model transformations and the accompanying data losses. Basis for this method are geometric models described in terms of Non-uniform rational B-Splines (NURBS). This class of piecewise continuous rational polynomial functions is ubiquitous in computer graphics and Computer-Aided Design (CAD). It allows the description of a wide range of geometries using a compact mathematical representation. The shape of an object thereby results from the interpolation of a set of control points by means of the NURBS functions, allowing efficient representations for curves, surfaces and solid bodies alike. Existing software applications, however, only support the modeling and manipulation of the former two. The description of three-dimensional solid bodies consequently requires significant manual effort, thus essentially forbidding the setup of complex models. This thesis proposes a procedural approach for the generation of volumetric NURBS models. That is, a model is not described in terms of its data structures but as a sequence of modeling operations applied to a simple initial shape. In a sense this describes the "evolution" of the geometric model under the sequence of operations. In order to adapt this concept to NURBS geometries, only a compact set of commands is necessary which, in turn, can be adapted from existing algorithms. A model then can be treated in terms of interpretable model parameters. This leads to an abstraction from its data structures and model variants can be set up by variation of the governing parameters. The proposed concept complements existing template modeling approaches: templates can not only be defined in terms of modeling commands but can also serve as input geometry for said operations. Such templates, arranged in a nested hierarchy, provide an elegant model representation. They offer adaptivity on each tier of the model hierarchy and allow to create complex models from only few model parameters. This is demonstrated for volumetric fluid domains used in the simulation of vertical-axis wind turbines. Starting from a template representation of airfoil cross-sections, the complete "negative space" around the rotor blades can be described by a small set of model parameters, and model variants can be set up in a fraction of a second. NURBS models offer a high geometric flexibility, allowing to represent a given shape in different ways. Different model instances can exhibit varying suitability for numerical analyses. For their assessment, Finite Element mesh quality metrics are regarded. The considered metrics are based on purely geometric criteria and allow to identify model degenerations commonly used to achieve certain geometric features. They can be used to decide upon model adaptions and provide a measure for their efficacy. Unfortunately, they do not reveal a relation between mesh distortion and ill-conditioning of the equation systems resulting from the numerical model. T3 - Schriftenreihe des DFG Graduiertenkollegs 1462 Modellqualitäten // Graduiertenkolleg Modellqualitäten - 7 KW - NURBS KW - Isogeometric Analysis KW - Procedural modeling KW - Templates KW - Mesh quality Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20130212-18483 SN - 978-3-86068-488-7 PB - Universitätsverlag CY - Weimar ER - TY - THES A1 - Nguyen-Thanh, Nhon T1 - Isogeometric analysis based on rational splines over hierarchical T-mesh and alpha finite element method for structural analysis N2 - This thesis presents two new methods in finite elements and isogeometric analysis for structural analysis. The first method proposes an alternative alpha finite element method using triangular elements. In this method, the piecewise constant strain field of linear triangular finite element method models is enhanced by additional strain terms with an adjustable parameter a, which results in an effectively softer stiffness formulation compared to a linear triangular element. In order to avoid the transverse shear locking of Reissner-Mindlin plates analysis the alpha finite element method is coupled with a discrete shear gap technique for triangular elements to significantly improve the accuracy of the standard triangular finite elements. The basic idea behind this element formulation is to approximate displacements and rotations as in the standard finite element method, but to construct the bending, geometrical and shear strains using node-based smoothing domains. Several numerical examples are presented and show that the alpha FEM gives a good agreement compared to several other methods in the literature. Second method, isogeometric analysis based on rational splines over hierarchical T-meshes (RHT-splines) is proposed. The RHT-splines are a generalization of Non-Uniform Rational B-splines (NURBS) over hierarchical T-meshes, which is a piecewise bicubic polynomial over a hierarchical T-mesh. The RHT-splines basis functions not only inherit all the properties of NURBS such as non-negativity, local support and partition of unity but also more importantly as the capability of joining geometric objects without gaps, preserving higher order continuity everywhere and allow local refinement and adaptivity. In order to drive the adaptive refinement, an efficient recovery-based error estimator is employed. For this problem an imaginary surface is defined. The imaginary surface is basically constructed by RHT-splines basis functions which is used for approximation and interpolation functions as well as the construction of the recovered stress components. Numerical investigations prove that the proposed method is capable to obtain results with higher accuracy and convergence rate than NURBS results. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2013,4 KW - Isogeometric analysis KW - NURBS KW - FEM KW - RHT-splines KW - Isogeometric analysis KW - NURBS KW - FEM KW - RHT-splines Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20131125-20781 SN - 1610-7381 ER - TY - THES A1 - Schrader, Kai T1 - Hybrid 3D simulation methods for the damage analysis of multiphase composites T1 - Hybride 3D Simulationsmethoden zur Abbildung der Schädigungsvorgänge in Mehrphasen-Verbundwerkstoffen N2 - Modern digital material approaches for the visualization and simulation of heterogeneous materials allow to investigate the behavior of complex multiphase materials with their physical nonlinear material response at various scales. However, these computational techniques require extensive hardware resources with respect to computing power and main memory to solve numerically large-scale discretized models in 3D. Due to a very high number of degrees of freedom, which may rapidly be increased to the two-digit million range, the limited hardware ressources are to be utilized in a most efficient way to enable an execution of the numerical algorithms in minimal computation time. Hence, in the field of computational mechanics, various methods and algorithms can lead to an optimized runtime behavior of nonlinear simulation models, where several approaches are proposed and investigated in this thesis. Today, the numerical simulation of damage effects in heterogeneous materials is performed by the adaption of multiscale methods. A consistent modeling in the three-dimensional space with an appropriate discretization resolution on each scale (based on a hierarchical or concurrent multiscale model), however, still contains computational challenges in respect to the convergence behavior, the scale transition or the solver performance of the weak coupled problems. The computational efficiency and the distribution among available hardware resources (often based on a parallel hardware architecture) can significantly be improved. In the past years, high-performance computing (HPC) and graphics processing unit (GPU) based computation techniques were established for the investigationof scientific objectives. Their application results in the modification of existing and the development of new computational methods for the numerical implementation, which enables to take advantage of massively clustered computer hardware resources. In the field of numerical simulation in material science, e.g. within the investigation of damage effects in multiphase composites, the suitability of such models is often restricted by the number of degrees of freedom (d.o.f.s) in the three-dimensional spatial discretization. This proves to be difficult for the type of implementation method used for the nonlinear simulation procedure and, simultaneously has a great influence on memory demand and computational time. In this thesis, a hybrid discretization technique has been developed for the three-dimensional discretization of a three-phase material, which is respecting the numerical efficiency of nonlinear (damage) simulations of these materials. The increase of the computational efficiency is enabled by the improved scalability of the numerical algorithms. Consequently, substructuring methods for partitioning the hybrid mesh were implemented, tested and adapted to the HPC computing framework using several hundred CPU (central processing units) nodes for building the finite element assembly. A memory-efficient iterative and parallelized equation solver combined with a special preconditioning technique for solving the underlying equation system was modified and adapted to enable combined CPU and GPU based computations. Hence, it is recommended by the author to apply the substructuring method for hybrid meshes, which respects different material phases and their mechanical behavior and which enables to split the structure in elastic and inelastic parts. However, the consideration of the nonlinear material behavior, specified for the corresponding phase, is limited to the inelastic domains only, and by that causes a decreased computing time for the nonlinear procedure. Due to the high numerical effort for such simulations, an alternative approach for the nonlinear finite element analysis, based on the sequential linear analysis, was implemented in respect to scalable HPC. The incremental-iterative procedure in finite element analysis (FEA) during the nonlinear step was then replaced by a sequence of linear FE analysis when damage in critical regions occured, known in literature as saw-tooth approach. As a result, qualitative (smeared) crack initiation in 3D multiphase specimens has efficiently been simulated. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2013,2 KW - high-performance computing KW - finite element method KW - heterogeneous material KW - domain decomposition KW - scalable smeared crack analysis KW - FEM KW - multiphase KW - damage KW - HPC KW - solver Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20131021-20595 ER - TY - THES A1 - Hommel, Angela T1 - Diskret holomorphe Funktionen und deren Bedeutung bei der Lösung von Differenzengleichungen N2 - Auf der Grundlage diskreter Cauchy-Riemann Operatoren werden diskret holomorphe Funktionen definiert und detailliert studiert. Darauf aufbauend wird die Lösung von Differenzengleichungen mit Hilfe der diskret holomorphen Funktionen beschrieben. KW - Differenzengleichung KW - Holomorphe Funktion Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20180827-37846 ER - TY - THES A1 - Hamdia, Khader T1 - On the fracture toughness of polymeric nanocomposites: Comprehensive stochastic and numerical studies N2 - Polymeric nanocomposites (PNCs) are considered for numerous nanotechnology such as: nano-biotechnology, nano-systems, nanoelectronics, and nano-structured materials. Commonly , they are formed by polymer (epoxy) matrix reinforced with a nanosized filler. The addition of rigid nanofillers to the epoxy matrix has offered great improvements in the fracture toughness without sacrificing other important thermo-mechanical properties. The physics of the fracture in PNCs is rather complicated and is influenced by different parameters. The presence of uncertainty in the predicted output is expected as a result of stochastic variance in the factors affecting the fracture mechanism. Consequently, evaluating the improved fracture toughness in PNCs is a challenging problem. Artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) have been employed to predict the fracture energy of polymer/particle nanocomposites. The ANN and ANFIS models were constructed, trained, and tested based on a collection of 115 experimental datasets gathered from the literature. The performance evaluation indices of the developed ANN and ANFIS showed relatively small error, with high coefficients of determination (R2), and low root mean square error and mean absolute percentage error. In the framework for uncertainty quantification of PNCs, a sensitivity analysis (SA) has been conducted to examine the influence of uncertain input parameters on the fracture toughness of polymer/clay nanocomposites (PNCs). The phase-field approach is employed to predict the macroscopic properties of the composite considering six uncertain input parameters. The efficiency, robustness, and repeatability are compared and evaluated comprehensively for five different SA methods. The Bayesian method is applied to develop a methodology in order to evaluate the performance of different analytical models used in predicting the fracture toughness of polymeric particles nanocomposites. The developed method have considered the model and parameters uncertainties based on different reference data (experimental measurements) gained from the literature. Three analytical models differing in theory and assumptions were examined. The coefficients of variation of the model predictions to the measurements are calculated using the approximated optimal parameter sets. Then, the model selection probability is obtained with respect to the different reference data. Stochastic finite element modeling is implemented to predict the fracture toughness of polymer/particle nanocomposites. For this purpose, 2D finite element model containing an epoxy matrix and rigid nanoparticles surrounded by an interphase zone is generated. The crack propagation is simulated by the cohesive segments method and phantom nodes. Considering the uncertainties in the input parameters, a polynomial chaos expansion (PCE) surrogate model is construed followed by a sensitivity analysis. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2018,4 KW - Bruch KW - Unsicherheit KW - Rissausbreitung KW - Bayes KW - Sensitivitätsanalyse KW - Fracture mechanics KW - Uncertainty analysis KW - Polymer nanocomposites KW - Bayesian method KW - Phase-field modeling Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20180712-37652 ER - TY - THES A1 - Habtemariam, Abinet Kifle T1 - Generalized Beam Theory for the analysis of thin-walled circular pipe members N2 - The detailed structural analysis of thin-walled circular pipe members often requires the use of a shell or solid-based finite element method. Although these methods provide a very good approximation of the deformations, they require a higher degree of discretization which causes high computational costs. On the other hand, the analysis of thin-walled circular pipe members based on classical beam theories is easy to implement and needs much less computation time, however, they are limited in their ability to approximate the deformations as they cannot consider the deformation of the cross-section. This dissertation focuses on the study of the Generalized Beam Theory (GBT) which is both accurate and efficient in analyzing thin-walled members. This theory is based on the separation of variables in which the displacement field is expressed as a combination of predetermined deformation modes related to the cross-section, and unknown amplitude functions defined on the beam's longitudinal axis. Although the GBT was initially developed for long straight members, through the consideration of complementary deformation modes, which amend the null transverse and shear membrane strain assumptions of the classical GBT, problems involving short members, pipe bends, and geometrical nonlinearity can also be analyzed using GBT. In this dissertation, the GBT formulation for the analysis of these problems is developed and the application and capabilities of the method are illustrated using several numerical examples. Furthermore, the displacement and stress field results of these examples are verified using an equivalent refined shell-based finite element model. The developed static and dynamic GBT formulations for curved thin-walled circular pipes are based on the linear kinematic description of the curved shell theory. In these formulations, the complex problem in pipe bends due to the strong coupling effect of the longitudinal bending, warping and the cross-sectional ovalization is handled precisely through the derivation of the coupling tensors between the considered GBT deformation modes. Similarly, the geometrically nonlinear GBT analysis is formulated for thin-walled circular pipes based on the nonlinear membrane kinematic equations. Here, the initial linear and quadratic stress and displacement tangent stiffness matrices are built using the third and fourth-order GBT deformation mode coupling tensors. Longitudinally, the formulation of the coupled GBT element stiffness and mass matrices are presented using a beam-based finite element formulation. Furthermore, the formulated GBT elements are tested for shear and membrane locking problems and the limitations of the formulations regarding the membrane locking problem are discussed. N2 - Eine detaillierte Strukturanalyse dünnwandiger, kreisförmiger Rohrelemente erfordert oft die Verwendung von Schalenelementen in der Finite Elemente Methode. Diese Methode ermöglicht eine sehr gute Approximation des Verformungszustandes, erfordert jedoch einen hohen Grad der Diskretisierung, welcher wiederum einen hohen Rechenaufwand verursacht. Eine alternative Methode hierzu basiert auf klassischen Balkentheorien, welche eine einfache Modellierung ermöglichen und wesentlich geringeren Rechenaufwand erfordern. Diese weisen jedoch Einschränkungen bei der Approximation von Verformungen auf, da Querschnittsverformungen nicht berücksichtigt werden können. Schwerpunkt dieser Dissertation ist eine Untersuchung der Verallgemeinerten Technischen Biegetheorie (VTB), die sowohl eine genaue als auch eine effiziente Analyse von dünnwandigen Tragwerkselementen ermöglicht. Diese Theorie basiert auf einer Trennung der Variablen, in der das Verschiebungsfeld als eine Kombination von vorbestimmten Verformungsmoden der Querschnitts und unbekannten Amplitudenfunktionen in Längsrichtung ausgedrückt wird. Obwohl die VTB ursprünglich für lange, gerade Elemente entwickelt wurde, können durch die Berücksichtigung komplementärer Verformungsmoden, welche die Null-Annahmen der klassischen VTB für Quer- und Schubmembrandehnung abändern, Probleme mit kurzen Elementen, Rohrbögen und geometrischer Nichtlinearität analysiert werden. In dieser Dissertation wird die VTB-Formulierung für die Analyse dieser Probleme entwickelt. Die Anwendung und Möglichkeiten der Methode werden anhand mehrerer numerischer Beispiele veranschaulicht, deren Verschiebungs- und Spannungsfeldanalysen anhand eines äquivalenten, verfeinerten, schalenbasierten Finite-Elemente-Modells verifiziert werden. Die entwickelten statischen und dynamischen VTB-Formulierungen für Rohrbogenelemente basieren auf der linearen kinematischen Beschreibung der Theorie gekrümmter Schalen. In diesen Formulierungen wird das komplexe Problem in Rohrbögen aufgrund des starken Kopplungseffekts der Längsbiegung, der Verwölbung und der Querschnittsovalisierung durch die Herleitung der Kopplungstensoren zwischen den betrachteten VTB-Verformungsmoden präzise behandelt. In ähnlicher Weise wird die geometrisch nichtlineare VTB-Analyse für gerade Rohrelemente auf der Grundlage der nichtlinearen kinematischen Membrangleichungen formuliert. Die anfänglichen linearen und quadratischen Spannungs- und Verschiebungs-Tangentensteifigkeitsmatrizen werden dabei unter Verwendung der VTB-Kopplungstensoren dritter und vierter Ordnung aufgebaut. In Längsrichtung wird die Formulierung der gekoppelten VTB-Element-Steifigkeits- und Massenmatrizen unter Verwendung einer balkenbasierten Finite-Elemente Formulierung dargestellt. Weiterhin werden die VTB-Elemente auf Schub- und Membran-Locking-Probleme getestet und die Einschränkungen der Formulierungen bezüglich des Membran-Locking-Problems diskutiert. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2022,2 KW - Finite-Elemente-Methode KW - Dynamische Analyse KW - Generalized Beam Theory (GBT) KW - Finite Element Method KW - Dynamic Analysis KW - Geometrically nonlinear analysis KW - Curved thin-walled circular pipes Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220127-45723 ER - TY - THES A1 - Al-Yasiri, Zainab Riyadh Shaker T1 - Function Theoretic Methods for the Analytical and Numerical Solution of Some Non-linear Boundary Value Problems with Singularities N2 - 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. N2 - Die p-Laplace-Gleichung ist eine nichtlineare Verallgemeinerung der wohlbekannten Laplace-Gleichung Die p-Laplace-Gleichung wird häufig als Referenzbeispiel für spezielle Typen von Nichtlinearitäten benutzt und kann daher auch als Brücke zwischen sehr allgemeinen nichtlinearen partiellen Differentialgleichungen und der linearen Laplace-Gleichung gesehen werden. Sie ist darüber hinaus auch das mathematische Modell für eine Reihe praxisrelevanter Probleme, wie z.B. in der Theorie nicht-newtonscher Flüssigkeiten, der Strömungsmechanik, der Durchfeuchtung von Schütt- dämmen und auch ein wichtiges Werkzeug zur Behandlung spezieller Probleme der Bildrekonstruktion und Bildverarbeitung. Das Ziel dieser Arbeit ist es, die p-Laplace-Gleichung sowohl für 1 < p < 2 als auch ür 2 < p < 3 zu lösen. Strenge Lösungen werden unter Benutzung der Clifford- Analysis konstruiert. Die Idee ist dabei, einen hyperkomplexen Integraloperator und funktionentheoretische Methoden auf die p-Laplace-Gleichung anzuwenden und diese Gleichung dadurch in eine p-Dirac-Gleichung zu transformieren, die dann besser gelöst werden kann. Es werden spezielle Randwertprobleme für die p-Laplace-Gleichung in Dirichlet-Probleme für die p-Dirac-Gleichung transformiert und dabei die Ordnung der Differentialgleichung reduziert. Die Randwertprobleme für die p-Dirac-Gleichung werden mit Hilfe des Banachschen Fixpunktprinzips iterativ analytisch gelöst. Durch Anwendung operator-theoretischer Methoden kann die Existenz und Eindeutigkeit der Lösung in bestimmten Sobolev-Räumen nachgewiesen werden. Darüber hinaus wird eine Finite Differenzenmethode auf einem gleichmäßigen Gitter in der Ebene angewandt, um die Fundamentallösung des diskreten Laplace- Operators numerisch zu berechnen. In der Folge werden daraus Fundamentallösungen des diskreten Cauchy-Riemann-Operators und seines adjungierten Operators erzeugt. Auf dieser Grundlage werden über Faltungen mit den Fundamentallösungen diskrete Teodorescu-Operatoren definiert, die rechtsinvers zum diskreten Cauchy-Riemann- Operator bzw. zum adjungierten diskreten Cauchy-Riemann-Operator sind. Weiterhin werden diskrete Randoperatoren, die analog zum Cauchyschen Integraloperator sind, eingeführt. Alle vorgenannten Operatoren werden in einer neuen Version einer diskreten Borel-Pompeiu-Formel zusammengeführt und bilden die Grundlage für eine diskrete Operatorenrechnung. Diese Untersuchungen erweitern bekannte Resultate auf wesentlich größere Funktionenklassen als bisher möglich waren. Die diskrete Opera- torenrechnung wird benutzt, um die diskretisierten Randwertprobleme für die p-Dirac- Gleichung numerisch zu lösen. Numerische Resultate werden vorgestellt und diskutiert. Dabei wird auf Vor- und Nachteile der entwickelten Methode eingegangen. KW - discrete function theory KW - finite difference methods KW - p-Laplace equation KW - Finite-Differenzen-Methode Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20190506-38987 ER - TY - THES A1 - Keßler, Andrea T1 - Matrix-free voxel-based finite element method for materials with heterogeneous microstructures T1 - Matrixfreie voxelbasierte Finite-Elemente-Methode für Materialien mit komplizierter Mikrostruktur N2 - Modern image detection techniques such as micro computer tomography (μCT), magnetic resonance imaging (MRI) and scanning electron microscopy (SEM) provide us with high resolution images of the microstructure of materials in a non-invasive and convenient way. They form the basis for the geometrical models of high-resolution analysis, so called image-based analysis. However especially in 3D, discretizations of these models reach easily the size of 100 Mill. degrees of freedoms and require extensive hardware resources in terms of main memory and computing power to solve the numerical model. Consequently, the focus of this work is to combine and adapt numerical solution methods to reduce the memory demand first and then the computation time and therewith enable an execution of the image-based analysis on modern computer desktops. Hence, the numerical model is a straightforward grid discretization of the voxel-based (pixels with a third dimension) geometry which omits the boundary detection algorithms and allows reduced storage of the finite element data structure and a matrix-free solution algorithm. This in turn reduce the effort of almost all applied grid-based solution techniques and results in memory efficient and numerically stable algorithms for the microstructural models. Two variants of the matrix-free algorithm are presented. The efficient iterative solution method of conjugate gradients is used with matrix-free applicable preconditioners such as the Jacobi and the especially suited multigrid method. The jagged material boundaries of the voxel-based mesh are smoothed through embedded boundary elements which contain different material information at the integration point and are integrated sub-cell wise though without additional boundary detection. The efficiency of the matrix-free methods can be retained. N2 - Moderne bildgebende Verfahren wie Mikro-Computertomographie (μCT), Magnetresonanztomographie (MRT) und Rasterelektronenmikroskopie (SEM) liefern nicht-invasiv hochauflösende Bilder der Mikrostruktur von Materialien. Sie bilden die Grundlage der geometrischen Modelle der hochauflösenden bildbasierten Analysis. Allerdings erreichen vor allem in 3D die Diskretisierungen dieser Modelle leicht die Größe von 100 Mill. Freiheitsgraden und erfordern umfangreiche Hardware-Ressourcen in Bezug auf Hauptspeicher und Rechenleistung, um das numerische Modell zu lösen. Der Fokus dieser Arbeit liegt daher darin, numerische Lösungsmethoden zu kombinieren und anzupassen, um den Speicherplatzbedarf und die Rechenzeit zu reduzieren und damit eine Ausführung der bildbasierten Analyse auf modernen Computer-Desktops zu ermöglichen. Daher ist als numerisches Modell eine einfache Gitterdiskretisierung der voxelbasierten (Pixel mit der Tiefe als dritten Dimension) Geometrie gewählt, die die Oberflächenerstellung weglässt und eine reduzierte Speicherung der finiten Elementen und einen matrixfreien Lösungsalgorithmus ermöglicht. Dies wiederum verringert den Aufwand von fast allen angewandten gitterbasierten Lösungsverfahren und führt zu Speichereffizienz und numerisch stabilen Algorithmen für die Mikrostrukturmodelle. Es werden zwei Varianten der Anpassung der matrixfreien Lösung präsentiert, die Element-für-Element Methode und eine Knoten-Kanten-Variante. Die Methode der konjugierten Gradienten in Kombination mit dem Mehrgitterverfahren als sehr effizienten Vorkonditionierer wird für den matrixfreien Lösungsalgorithmus adaptiert. Der stufige Verlauf der Materialgrenzen durch die voxelbasierte Diskretisierung wird durch Elemente geglättet, die am Integrationspunkt unterschiedliche Materialinformationen enthalten und über Teilzellen integriert werden (embedded boundary elements). Die Effizienz der matrixfreien Verfahren bleibt erhalten. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2018,7 KW - Dissertation KW - Finite-Elemente-Methode KW - Konjugierte-Gradienten-Methode KW - Mehrgitterverfahren KW - conjugate gradient method KW - multigrid method KW - grid-based KW - finite element method KW - matrix-free Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20190116-38448 ER - TY - THES A1 - Jia, Yue T1 - Methods based on B-splines for model representation, numerical analysis and image registration N2 - The thesis consists of inter-connected parts for modeling and analysis using newly developed isogeometric methods. The main parts are reproducing kernel triangular B-splines, extended isogeometric analysis for solving weakly discontinuous problems, collocation methods using superconvergent points, and B-spline basis in image registration applications. Each topic is oriented towards application of isogeometric analysis basis functions to ease the process of integrating the modeling and analysis phases of simulation. First, we develop reproducing a kernel triangular B-spline-based FEM for solving PDEs. We review the triangular B-splines and their properties. By definition, the triangular basis function is very flexible in modeling complicated domains. However, instability results when it is applied for analysis. We modify the triangular B-spline by a reproducing kernel technique, calculating a correction term for the triangular kernel function from the chosen surrounding basis. The improved triangular basis is capable to obtain the results with higher accuracy and almost optimal convergence rates. Second, we propose an extended isogeometric analysis for dealing with weakly discontinuous problems such as material interfaces. The original IGA is combined with XFEM-like enrichments which are continuous functions themselves but with discontinuous derivatives. Consequently, the resulting solution space can approximate solutions with weak discontinuities. The method is also applied to curved material interfaces, where the inverse mapping and the curved triangular elements are considered. Third, we develop an IGA collocation method using superconvergent points. The collocation methods are efficient because no numerical integration is needed. In particular when higher polynomial basis applied, the method has a lower computational cost than Galerkin methods. However, the positions of the collocation points are crucial for the accuracy of the method, as they affect the convergent rate significantly. The proposed IGA collocation method uses superconvergent points instead of the traditional Greville abscissae points. The numerical results show the proposed method can have better accuracy and optimal convergence rates, while the traditional IGA collocation has optimal convergence only for even polynomial degrees. Lastly, we propose a novel dynamic multilevel technique for handling image registration. It is application of the B-spline functions in image processing. The procedure considered aims to align a target image from a reference image by a spatial transformation. The method starts with an energy function which is the same as a FEM-based image registration. However, we simplify the solving procedure, working on the energy function directly. We dynamically solve for control points which are coefficients of B-spline basis functions. The new approach is more simple and fast. Moreover, it is also enhanced by a multilevel technique in order to prevent instabilities. The numerical testing consists of two artificial images, four real bio-medical MRI brain and CT heart images, and they show our registration method is accurate, fast and efficient, especially for large deformation problems. KW - Finite-Elemente-Methode KW - isogeometric methods Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20151210-24849 ER -