TY - THES A1 - Yousefi, Hassan T1 - Discontinuous propagating fronts: linear and nonlinear systems N2 - The aim of this study is controlling of spurious oscillations developing around discontinuous solutions of both linear and non-linear wave equations or hyperbolic partial differential equations (PDEs). The equations include both first-order and second-order (wave) hyperbolic systems. In these systems even smooth initial conditions, or smoothly varying source (load) terms could lead to discontinuous propagating solutions (fronts). For the first order hyperbolic PDEs, the concept of central high resolution schemes is integrated with the multiresolution-based adaptation to capture properly both discontinuous propagating fronts and effects of fine-scale responses on those of larger scales in the multiscale manner. This integration leads to using central high resolution schemes on non-uniform grids; however, such simulation is unstable, as the central schemes are originally developed to work properly on uniform cells/grids. Hence, the main concern is stable collaboration of central schemes and multiresoltion-based cell adapters. Regarding central schemes, the considered approaches are: 1) Second order central and central-upwind schemes; 2) Third order central schemes; 3) Third and fourth order central weighted non-oscillatory schemes (central-WENO or CWENO); 4) Piece-wise parabolic methods (PPMs) obtained with two different local stencils. For these methods, corresponding (nonlinear) stability conditions are studied and modified, as well. Based on these stability conditions several limiters are modified/developed as follows: 1) Several second-order limiters with total variation diminishing (TVD) feature, 2) Second-order uniformly high order accurate non-oscillatory (UNO) limiters, 3) Two third-order nonlinear scaling limiters, 4) Two new limiters for PPMs. Numerical results show that adaptive solvers lead to cost-effective computations (e.g., in some 1-D problems, number of adapted grid points are less than 200 points during simulations, while in the uniform-grid case, to have the same accuracy, using of 2049 points is essential). Also, in some cases, it is confirmed that fine scale responses have considerable effects on higher scales. In numerical simulation of nonlinear first order hyperbolic systems, the two main concerns are: convergence and uniqueness. The former is important due to developing of the spurious oscillations, the numerical dispersion and the numerical dissipation. Convergence in a numerical solution does not guarantee that it is the physical/real one (the uniqueness feature). Indeed, a nonlinear systems can converge to several numerical results (which mathematically all of them are true). In this work, the convergence and uniqueness are directly studied on non-uniform grids/cells by the concepts of local numerical truncation error and numerical entropy production, respectively. Also, both of these concepts have been used for cell/grid adaptations. So, the performance of these concepts is also compared by the multiresolution-based method. Several 1-D and 2-D numerical examples are examined to confirm the efficiency of the adaptive solver. Examples involve problems with convex and non-convex fluxes. In the latter case, due to developing of complex waves, proper capturing of real answers needs more attention. For this purpose, using of method-adaptation seems to be essential (in parallel to the cell/grid adaptation). This new type of adaptation is also performed in the framework of the multiresolution analysis. Regarding second order hyperbolic PDEs (mechanical waves), the regularization concept is used to cure artificial (numerical) oscillation effects, especially for high-gradient or discontinuous solutions. There, oscillations are removed by the regularization concept acting as a post-processor. Simulations will be performed directly on the second-order form of wave equations. It should be mentioned that it is possible to rewrite second order wave equations as a system of first-order waves, and then simulated the new system by high resolution schemes. However, this approach ends to increasing of variable numbers (especially for 3D problems). The numerical discretization is performed by the compact finite difference (FD) formulation with desire feature; e.g., methods with spectral-like or optimized-error properties. These FD methods are developed to handle high frequency waves (such as waves near earthquake sources). The performance of several regularization approaches is studied (both theoretically and numerically); at last, a proper regularization approach controlling the Gibbs phenomenon is recommended. At the end, some numerical results are provided to confirm efficiency of numerical solvers enhanced by the regularization concept. In this part, shock-like responses due to local and abrupt changing of physical properties, and also stress wave propagation in stochastic-like domains are studied. KW - Partielle Differentialgleichung KW - Adaptives System KW - Wavelet KW - Tichonov-Regularisierung KW - Hyperbolic PDEs KW - Adaptive central high resolution schemes KW - Wavelet based adaptation KW - Tikhonov regularization Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220922-47178 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 -