TY - THES A1 - Schorling, York T1 - Beitrag zur Stabilitätsuntersuchung von Strukturen mit räumlich korrelierten geometrischen Imperfektionen N2 - Für geometrisch imperfekte Strukturen wird die Versagenswahrscheinlichkeit bezüglich Stabilitätskriterien bestimmt. Eine probabilistische Beschreibung der geometrischen Imperfektionen erfolgt mit skalaren ortsdiskretisierten Zufallsfeldern. Die Stabilitätsberechnungen werden mit der Finite Elemente Methode durchgeführt. Ausgangspunkt der Berechnung ist eine systematische Formulierung probabilistisch gewichteter Imperfektionsformen durch eine Eigenwertzerlegung der Kovarianzmatrix. Wenn mit einer strukturmechanisch orientierten Sensitivitätsanalyse ein Unterraum zur näherungsweisen Beschreibung des probabilistischen Strukturverhaltens gefunden wird, kann die Versagenswahrscheinlichkeit numerisch sehr effizient durch ein Interaktionsmodell bestimmt werden. Es zeigte sich, daß dies genau dann möglich ist, wenn die Beulform merklich im Imperfektionsfeld enthalten ist. Die Imperfektionsform am Bemessungspunkt entspricht dann, unabhängig vom Lastniveau, gerade der Beulform. Wenn die Beulform im Imperfektionsfeld einen untergeordneten Beitrag liefert, erscheint eine Reduktion des stochastischen Problems auf wenige Zufallsvariablen dagegen nicht möglich. N2 - The thesis presents a concept for reliability analysis of geometrical imperfect structures with respect to static stability criteria. The geometrical imperfections are modeled as Gaussian random fields. The structural analysis is based on the Finite Element Method. A spectral decomposition of the covariance matrix, enables to formulate independent probabilistically weighted imperfections shapes, which may be analyzed by means of structural mechanics. Reliability calculations with procedures such as the response surface method require the reduction of the random variable space. Examples proved that a suitable definition of a subspace of the random variable space is possible, if the buckling shapes are sufficiently included in the random field. In this case the imperfection shape is-independent of the load level-identical to the buckling shape. In contrast if the buckling shapes are not included in the random field, the structure shows a wide banded behavior. Consequently a reduction of the variable space and the application of an interaction models is then not feasible for the determination of the failure probabilty. KW - Tragwerk KW - Formabweichung KW - Stabilität KW - Beulung KW - Zuverlässigkeit KW - Finite-Elemente-Methode KW - Imperfektion KW - Berechnung KW - Stochastik KW - Zufallsfeld Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20040216-317 ER - TY - THES A1 - Will, Johannes T1 - Beitrag zur Standsicherheitsberechnung im geklüfteten Fels in der Kontinuums- und Diskontinuumsmechanik unter Verwendung impliziter und expliziter Berechnungsstrategien T1 - Structural safety analysis for jointed rock with continuum and discontinuum mechanics in implizit and explizit codes KW - Staumauer KW - Standsicherheit KW - Klüftung KW - Finite-Elemente-Methode KW - Diskrete-Elemente-Methode KW - Kontinuumsmechanik KW - Diskontinuumsmechanik KW - jointed rock KW - continuum mechanics KW - diskontinuum mechanics Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20040310-613 ER - TY - THES A1 - Roos, Dirk T1 - Approximation und Interpolation von Grenzzustandsfunktionen zur Sicherheitsbewertung nichtlinearer Finite-Elemente-Strukturen T1 - Reliability analysis using approximation and interpolation of nonlinear finite element limit state functions N2 - Die vorliegende Arbeit beschäftigt sich mit der Berechnung der Sicherheit von Strukturen mit sowohl geometrisch als auch physikalisch nichtlinearem Verhalten. Die Berechnung der Versagenswahrscheinlichkeit einer Struktur mit Hilfe von Monte-Carlo-Simulationsmethoden erfordert, dass die Funktion der Strukturantwort implizit berechnet wird, zum Beispiel durch nichtlineare Strukturanalysen für jede Realisation der Zufallsvariablen. Die Strukturanalysen bilden jedoch den Hauptanteil am Berechnungsaufwand der Zuverlässigkeitsanalyse, so dass die Analyse von realistischen Strukturen mit nichtlinearem Verhalten durch die begrenzten Computer-Ressourcen stark eingeschränkt ist. Die klassischen Antwortflächenverfahren approximieren die Funktion der Strukturantwort oder aber die Grenzzustandsfunktion durch Polynome niedriger Ordnung. Dadurch ist für die Auswertung des Versagens-Kriteriums nur noch von Interesse, ob eine Realisation der Basisvariablen innerhalb oder außerhalb des von der Antwortflächenfunktion gebildeten Raumes liegt - die Strukturanalyse kann dann entfallen. Bei stark nichtlinearen Grenzzustandsfunktionen versagt die polynomiale Approximation. Das directional sampling neigt bei Problemen mit vielen Zufallsvariablen zu einem systematischen Fehler. Das adaptive importance directional sampling dagegen beseitigt diesen Fehler, verschenkt jedoch Informationen über den Verlauf der Grenzzustandsfunktion, da die aufgefundenen Stützstellen aus den vorangegangenen Simulationsläufen nicht berücksichtigt werden können. Aus diesem Grund erscheint eine Kombination beider Simulationsverfahren und eine Interpolation mittels einer Antwortfläche geeignet, diese Probleme zu lösen. Dies war die Motivation für die Entwicklung eines Verfahren der adaptiven Simulation der Einheitsvektoren und anschließender Interpolation der Grenzzustandsfunktion durch eine Antwortflächenfunktion. Dieses Vorgehen stellt besondere Anforderungen an die Antwortflächenfunktion. Diese muss flexibel genug sein, um stark nichtlineare Grenzzustandsfunktionen beliebig genau annähern zu können. Außerdem sollte die Anzahl der verarbeitbaren Stützstellen nicht begrenzt sein. Auch ist zu berücksichtigen, dass die Ermittlung der Stützstellen auf der Grenzzustandsfunktion nicht regelmäßig erfolgt. Die in dieser Arbeit entwickelten Methoden der lokalen Interpolation der Grenzzustandsfunktion durch Normalen-Hyperebenen bzw. sekantialen Hyperebenen und der sowohl lokalen als auch globalen Interpolation durch gewichtete Radien erfüllen diese Anforderungen. ungen. dieser Arbeit entwickelten Methoden der lokalen Interpolation der Grenzzustandsfunktion durch Normalen-Hyperebenen bzw. sekantialen Hyperebenen und der sowohl lokalen als auch globalen Interpolation durch gewichtete Radien erfüllen diese Anforderungen. N2 - The objective is the analysis of the structural reliability of nonlinear finite element systems. Normally, the limit state of a structural system is described implicitly, e.g. through an algorithmic procedure within finite element analysis. Alternatively, the original limit state function can be approximated by a response surface function. One of the major advantages of the response surface method lies in its potential to selectively determine the number of structural analyses of the support points. By such means the computational effort can be substantially reduced. On the other hand, the global approximation schemes widely used in the application of the response surface method can be quite misleading due to the lack of information in certain regions of the random variable space. It is therefore required to avoid such undesirable interpolation errors at reasonable computational effort. The polynomial approximations are not quite flexible. They always need a predefined number of limit state check points in unimportant directions in order to avoid any approximation problems. On this account the maximum number of limit state check points is limited, too. The study presents an approach to polyhedral and weighted modeling of convex and concave failure surfaces intended to provide reasonably accurate estimates of failure probabilities while maintaining computational efficiency. In particular, these response surfaces can be adaptively refined to consistently increase the accuracy of the estimated failure probability. This is achieved by a combination of random search strategies (based on the adaptive directional sampling approach) as well as deterministic search refinement together with local and global interpolation schemes. The main advantage of these methods is their flexibility for the approximation of highly nonlinear limit state functions. In this sense, the proposed methods are very robust and efficient. KW - Tragwerk KW - Nichtlineares System KW - Tragfähigkeit KW - Mechanisches Versagen KW - Zuverlässigkeitstheorie KW - Finite-Elemente-Methode KW - Sicherheit KW - Antwortflächenverfahren KW - structural reliability KW - response surface method KW - finite elements Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20040311-745 ER - TY - THES A1 - Lehmkuhl, Hansjörg T1 - Zur praktischen Anwendung numerischer Analysemethoden für Stabilitätsprobleme T1 - The application of numerical methods to stability problems in practice N2 - In der täglichen Ingenieurpraxis werden in zunehmenden Maße numerische Analysen im Rahmen der Finite-Elemente-Methode auch zur Untersuchung stabilitätsgefährdeter Strukturen eingesetzt. Für die aktuelle Praxis, insbesondere im konstruktiven Stahlbau, ist jedoch festzustellen, dass zwischen der fortgeschrittenen Theorie und dem Niveau der praktischen Anwendung numerischer Stabilitätsanalysen eine große Kluft besteht. Aus praktischer Sicht erscheint es unumgänglich, die weiter wachsende Diskrepanz zwischen den umfangreichen theoretischen Möglichkeiten und der gegenwärtigen Praxis abzubauen. Damit steht der praktisch tätige Ingenieur vor der Aufgabe, sein Wissen auf dem Gebiet numerischer Stabilitätsanalysen zu vertiefen und bereits vorhandene FE-Programme um Berechnungsalgorithmen für umfassende numerische Stabilitätsanalysen zu erweitern. Dafür werden in der Arbeit die Grundlagen einer FEM- orientierten modernen Stabilitätstheorie einheitlich und aus Sicht einer praktischen Anwendung aufbereitet. Die Darstellung von realisierten programmtechnischen Umsetzungen für erweiterte Analysenmethoden wie Nachbeulanalysen, Pfadwechsel und Approximationen imperfekter Pfade ermöglicht eine Erweiterung des Methodenvorrates. Die innerhalb der Arbeit untersuchten Beispiele zeigen, dass durch die Anwendung der behandelten Verfahren das Tragverhalten einer stabilitätsgefährdeten Struktur wesentlich besser eingeschätzt werden kann als bei Beschränkung auf die herkömmlichen Analysemethoden. N2 - The daily engineering practice is characterized by an increasing use of numerical analyses also for structures influenced by stability problems. However it must be noticed that in practice, especially in steel construction, exists a large gap between the advanced stability theory and the level of practical applications of numerical stability analyses. From a practical point of view it seems be absolutely necessary to break up the continual growing discrepancy between comprehensive theoretical possibilities and current practice. Therefore practical engineers have to deepen their knowledge on the field of numerical stability analyses and have to expand existing finite element programs with special algorithms. For this purpose basics of a finite element orientated modern stability theory are presented from the view of practical application. Furthermore steps are shown for programming special methods like post-buckling analyses, path switching and approximations for load-deformation pathes including imperfections. The importance of the additional methods for a better understanding of the structures stability behaviour clearly comes out by the presentation of different examples of practical relevance. KW - Nichtlineare Stabilitätstheorie KW - Finite-Elemente-Methode KW - Stabilität KW - Stahlbau KW - Imperfektion KW - Nachbeulverhalten KW - Schale KW - Non-linear buckling KW - Finite Element Method KW - stability theory KW - post-buckling analysis KW - koiter approximation Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20051013-7102 ER - TY - THES A1 - Häfner, Stefan T1 - Grid-based procedures for the mechanical analysis of heterogeneous solids N2 - The importance of modern simulation methods in the mechanical analysis of heterogeneous solids is presented in detail. Thereby the problem is noted that even for small bodies the required high-resolution analysis reaches the limits of today's computational power, in terms of memory demand as well as acceptable computational effort. A further problem is that frequently the accuracy of geometrical modelling of heterogeneous bodies is inadequate. The present work introduces a systematic combination and adaption of grid-based methods for achieving an essentially higher resolution in the numerical analysis of heterogeneous solids. Grid-based methods are as well primely suited for developing efficient and numerically stable algorithms for flexible geometrical modeling. A key aspect is the uniform data management for a grid, which can be utilized to reduce the effort and complexity of almost all concerned methods. A new finite element program, called Mulgrido, was just developed to realize this concept consistently and to test the proposed methods. Several disadvantages which generally result from grid discretizations are selectively corrected by modified methods. The present work is structured into a geometrical model, a mechanical model and a numerical model. The geometrical model includes digital image-based modeling and in particular several methods for the theory-based generation of inclusion-matrix models. Essential contributions refer to variable shape, size distribution, separation checks and placement procedures of inclusions. The mechanical model prepares the fundamentals of continuum mechanics, homogenization and damage modeling for the following numerical methods. The first topic of the numerical model introduces to a special version of B-spline finite elements. These finite elements are entirely variable in the order k of B-splines. For homogeneous bodies this means that the approximation quality can arbitrarily be scaled. In addition, the multiphase finite element concept in combination with transition zones along material interfaces yields a valuable solution for heterogeneous bodies. As the formulation is element-based, the storage of a global stiffness matrix is superseded such that the memory demand can essentially be reduced. This is possible in combination with iterative solver methods which represent the second topic of the numerical model. Here, the focus lies on multigrid methods where the number of required operations to solve a linear equation system only increases linearly with problem size. Moreover, for badly conditioned problems quite an essential improvement is achieved by preconditioning. The third part of the numerical model discusses certain aspects of damage simulation which are closely related to the proposed grid discretization. The strong efficiency of the linear analysis can be maintained for damage simulation. This is achieved by a damage-controlled sequentially linear iteration scheme. Finally a study on the effective material behavior of heterogeneous bodies is presented. Especially the influence of inclusion shapes is examined. By means of altogether more than one hundred thousand random geometrical arrangements, the effective material behavior is statistically analyzed and assessed. N2 - Die wichtige Bedeutung moderner Simulationsverfahren in der mechanischen Analyse heterogener Festkörper wird eingangs ausführlich dargestellt. Dabei wird als Problem festgestellt, dass die erforderliche hochauflösende Analyse bereits für relativ kleine Körper an die Grenzen heutiger Rechenleistung stößt, sowohl bezüglich Speicherbedarf als auch akzeptablen Rechenaufwands. Ein weiteres Problem stellt die häufig unzureichend genaue geometrische Modellierung der Zusammensetzung heterogener Körper dar. Die vorliegende Arbeit führt eine systematische Kombination und Anpassung von gitterbasierten Methoden ein, um dadurch eine wesentlich höhere Auflösung in der numerischen Analyse heterogener Körper zu erzielen. Gitterverfahren eignen sich ebenfalls ausgezeichnet, um effiziente und numerisch stabile Algorithmen zur flexiblen geometrischen Modellierung zu entwickeln. Ein Schlüsselaspekt stellt ein gleichmäßiges Datenmanagement für Gitter dar, welches dafür eingesetzt werden kann, um den Aufwand und die Komplexität von nahezu allen beteiligten Methoden zu reduzieren. Ein neues Finite-Elemente Programm, namens Mulgrido, wurde eigens dafür entwickelt, um das vorgeschlagene Konzept konsistent zu realisieren und zu untersuchen. Einige Nachteile, die sich klassischerweise aus Gitterdiskretisierungen ergeben, werden gezielt durch modifizierte Verfahren korrigiert. Die gegenwärtige Arbeit gliedert sich in ein geometrisches Modell, ein mechanisches Modell und ein numerisches Modell. Das geometrische Modell beinhaltet neben Methoden der digitalen Bildverarbeitung, insbesondere sämtliche Verfahren zur künstlichen Generierung von Einschluss-Matrix Geometrien. Wesentliche Beiträge werden bezüglich variabler Form, Größenverteilung, Überschneidungsabfragen und Platzierung von Einschlüssen geleistet. Das mechanische Modell bereitet durch Grundlagen der Kontinuumsmechanik, der Homogenisierung und der Schädigungsmodellierung auf eine numerische Umsetzung vor. Als erstes Thema des numerischen Modells wird eine besondere Umsetzung von B-Spline Finiten Elementen vorgestellt. Diese Finite Elemente können generisch für eine beliebige Ordnung k der B-Splines erzeugt werden. Für homogene Körper verfügen diese somit über beliebig skalierbare Approximationseigenschaften. Mittels des Konzepts mehrphasiger Finite Elemente in Kombination mit Übergangszonen entlang von Materialgrenzen gelingt eine hochwertige Erweiterung für heterogene Körper. Durch die Formulierung auf Elementebene, kann die Speicherung der globalen Steifigkeitsmatrix und somit wesentlicher Speicherplatz eingespart werden. Dies ist möglich in Kombination mit iterativen Lösungsverfahren, die das zweite Thema des numerischen Modells darstellen. Dabei liegt der Fokus auf Mehrgitterverfahren. Diese zeichnen sich dadurch aus, dass die Anzahl der erforderlichen Operationen um ein lineares Gleichungssystem zu lösen, nur linear mit der Problemgröße ansteigt. Durch Vorkonditionierung wird für schlecht konditionierte Probleme eine ganz wesentliche Verbesserung erreicht. Als drittes Thema des numerischen Modells werden Aspekte der Schädigungssimulation diskutiert, die in engem Zusammenhang mit der Gitterdiskretisierung stehen. Die hohe Effizienz der linearen Analyse kann durch ein schädigungskontrolliertes, schrittweise lineares Iterationsschema für die Schädigungsanalyse aufrecht erhalten werden. Abschließend wird eine Studie über das effektive Materialverhalten heterogener Körper vorgestellt. Insbesondere wird der Einfluss der Form von Einschlüssen untersucht. Mittels insgesamt weit über hunderttausend zufälliger geometrischer Anordnungen wird das effektive Materialverhalten statistisch analysiert und bewertet. T2 - Gitterbasierte Verfahren zur mechanischen Analyse heterogener Festkörper KW - B-Spline KW - Finite-Elemente-Methode KW - Mehrgitterverfahren KW - Homogenisieren KW - Schädigung KW - Festkörpermechanik KW - Numerische Mathematik KW - B-Spline Finite Elemente KW - Homogenisierung KW - mehrphasig KW - Lösungsverfahren KW - Modellierung KW - B-spline KW - finite element KW - multigrid KW - multiphase KW - effective properties Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20070830-9185 ER - TY - THES A1 - Eckardt, Stefan T1 - Adaptive heterogeneous multiscale models for the nonlinear simulation of concrete N2 - The nonlinear behavior of concrete can be attributed to the propagation of microcracks within the heterogeneous internal material structure. In this thesis, a mesoscale model is developed which allows for the explicit simulation of these microcracks. Consequently, the actual physical phenomena causing the complex nonlinear macroscopic behavior of concrete can be represented using rather simple material formulations. On the mesoscale, the numerical model explicitly resolves the components of the internal material structure. For concrete, a three-phase model consisting of aggregates, mortar matrix and interfacial transition zone is proposed. Based on prescribed grading curves, an efficient algorithm for the generation of three-dimensional aggregate distributions using ellipsoids is presented. In the numerical model, tensile failure of the mortar matrix is described using a continuum damage approach. In order to reduce spurious mesh sensitivities, introduced by the softening behavior of the matrix material, nonlocal integral-type material formulations are applied. The propagation of cracks at the interface between aggregates and mortar matrix is represented in a discrete way using a cohesive crack approach. The iterative solution procedure is stabilized using a new path following constraint within the framework of load-displacement-constraint methods which allows for an efficient representation of snap-back phenomena. In several examples, the influence of the randomly generated heterogeneous material structure on the stochastic scatter of the results is analyzed. Furthermore, the ability of mesoscale models to represent size effects is investigated. Mesoscale simulations require the discretization of the internal material structure. Compared to simulations on the macroscale, the numerical effort and the memory demand increases dramatically. Due to the complexity of the numerical model, mesoscale simulations are, in general, limited to small specimens. In this thesis, an adaptive heterogeneous multiscale approach is presented which allows for the incorporation of mesoscale models within nonlinear simulations of concrete structures. In heterogeneous multiscale models, only critical regions, i.e. regions in which damage develops, are resolved on the mesoscale, whereas undamaged or sparsely damage regions are modeled on the macroscale. A crucial point in simulations with heterogeneous multiscale models is the coupling of sub-domains discretized on different length scales. The sub-domains differ not only in the size of the finite elements but also in the constitutive description. In this thesis, different methods for the coupling of non-matching discretizations - constraint equations, the mortar method and the arlequin method - are investigated and the application to heterogeneous multiscale models is presented. Another important point is the detection of critical regions. An adaptive solution procedure allowing the transfer of macroscale sub-domains to the mesoscale is proposed. In this context, several indicators which trigger the model adaptation are introduced. Finally, the application of the proposed adaptive heterogeneous multiscale approach in nonlinear simulations of concrete structures is presented. N2 - Das nichtlineare Materialverhalten von Beton ist durch die Entwicklung von Mikrorissen innerhalb der heterogenen Materialstruktur gekennzeichnet. In dieser Arbeit wird ein Mesoskalenmodell entwickelt, welches die einzelnen Bestandteile der Materialstruktur explizit auflöst und somit die Simulation dieser Mikrorisse erlaubt. Dadurch können die wirklichen physikalischen Vorgänge, welche das komplexe nichtlineare Verhalten von Beton verursachen, durch relativ einfache Materialformulierungen abgebildet werden. Für Beton wird auf der Mesoskala ein 3-Phasenmodell vorgeschlagen, bestehend aus groben Zuschlägen, Mörtelmatrix und Übergangszone zwischen Zuschlag und Matrix. In diesem Zusammenhang wird ein effizienter Algorithmus vorgestellt, welcher ausgehend von einer gegebenen Sieblinie dreidimensionale Kornstrukturen mittels Ellipsoiden simuliert. Im Mesoskalenmodell wird das Zugversagen der Mörtelmatrix durch einen Kontinuumsansatz beschrieben. Um Netzabhängigkeiten, welche durch das Entfestigungsverhalten des Materials hervorgerufen werden, zu reduzieren, kommen nichtlokale Materialformulierungen zum Einsatz. Risse innerhalb der Übergangszone zwischen Zuschlag und Matrix werden, basierend auf einem kohäsiven Modell, mittels eines diskreten Rissansatzes abgebildet. Die Verwendung einer neuen Nebenbedingung innerhalb der Last-Verschiebungs-Zwangsmethode führt zu einer Stabilisierung des iterativen Lösungverfahrens, so dass eine effiziente Simulation von Snap-back Phänomenen möglich wird. Anhand von Beispielen wird gezeigt, dass Mesoskalenmodelle die stochastische Streuung von Ergebnissen und Maßstabseffekte abbilden können. Da auf der Mesoskala die Diskretisierung der inneren Materialstruktur erforderlich ist, steigt im Vergleich zu Simulationen auf der Makroskala der numerische Aufwand erheblich. Aufgrund der Komplexität des numerischen Modells sind Mesoskalensimulationen in der Regel auf kleine Probekörper beschränkt. In dieser Arbeit wird ein adaptiver heterogener Mehrskalenansatz vorgestellt, welcher die Verwendung von Mesoskalenmodellen in nichtlinearen Simulationen von Betonstrukturen erlaubt. In heterogenen Mehrskalenmodellen werden nur kritische Bereiche auf der Mesoskala aufgelöst, während ungeschädigte Bereiche auf der Makroskala abgebildet werden. Ein wichtiger Aspekt in Simulationen mit heterogenen Mehrskalenmodellen ist die Kopplung der auf unterschiedlichen Längenskalen diskretisierten Teilgebiete. Diese unterscheiden sich nicht nur in der Größe der finiten Elemente sondern auch in der Beschreibung des Materials. Verschiedene Methoden zur Kopplung nicht übereinstimmender Vernetzungen - Kopplungsgleichungen, die Mortar-Methode und die Arlequin-Methode - werden untersucht und ihre Anwendung in heterogenen Mehrskalenmodellen wird gezeigt. Ein weiterer wichtiger Aspekt ist die Bestimmung kritischer Regionen. Eine adaptive Lösungsstrategie wird entwickelt, welche die Umwandlung von Makroskalengebieten auf die Mesoskala erlaubt. In diesem Zusammenhang werden Indikatoren vorgestellt, die eine Modellanpassung auslösen. Anhand nichtlinearer Simulationen von Betonstrukturen wird die Anwendung des vorgestellten adaptiven heterogenen Mehrskalenansatzes demonstriert. T2 - Adaptive heterogene Mehrskalenmodelle zur nichtlinearen Simulation von Beton T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2010,1 KW - Beton KW - Mehrskalenanalyse KW - Finite-Elemente-Methode KW - Nichtlineare Finite-Elemente-Methode KW - Schadensmechanik KW - Mehrskalenmodell KW - Adaptives Verfahren KW - concrete KW - multiscale method KW - finite element method KW - continuum damage mechanics KW - adaptive simulation Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20100317-15023 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 - TY - THES A1 - Nanthakumar, S.S. T1 - Inverse and optimization problems in piezoelectric materials using Extended Finite Element Method and Level sets T1 - Inverse und Optimierungsprobleme für piezoelektrische Materialien mit der Extended Finite Elemente Methode und Level sets N2 - Piezoelectric materials are used in several applications as sensors and actuators where they experience high stress and electric field concentrations as a result of which they may fail due to fracture. Though there are many analytical and experimental works on piezoelectric fracture mechanics. There are very few studies about damage detection, which is an interesting way to prevent the failure of these ceramics. An iterative method to treat the inverse problem of detecting cracks and voids in piezoelectric structures is proposed. Extended finite element method (XFEM) is employed for solving the inverse problem as it allows the use of a single regular mesh for large number of iterations with different flaw geometries. Firstly, minimization of cost function is performed by Multilevel Coordinate Search (MCS) method. The XFEM-MCS methodology is applied to two dimensional electromechanical problems where flaws considered are straight cracks and elliptical voids. Then a numerical method based on combination of classical shape derivative and level set method for front propagation used in structural optimization is utilized to minimize the cost function. The results obtained show that the XFEM-level set methodology is effectively able to determine the number of voids in a piezoelectric structure and its corresponding locations. The XFEM-level set methodology is improved to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure. The material interfaces are implicitly represented by level sets which are identified by applying regularisation using total variation penalty terms. The formulation is presented for three dimensional structures and inclusions made of different materials are detected by using multiple level sets. The results obtained prove that the iterative procedure proposed can determine the location and approximate shape of material subdomains in the presence of higher noise levels. Piezoelectric nanostructures exhibit size dependent properties because of surface elasticity and surface piezoelectricity. Initially a study to understand the influence of surface elasticity on optimization of nano elastic beams is performed. The boundary of the nano structure is implicitly represented by a level set function, which is considered as the design variable in the optimization process. Two objective functions, minimizing the total potential energy of a nanostructure subjected to a material volume constraint and minimizing the least square error compared to a target displacement, are chosen for the numerical examples. The numerical examples demonstrate the importance of size and aspect ratio in determining how surface effects impact the optimized topology of nanobeams. Finally a conventional cantilever energy harvester with a piezoelectric nano layer is analysed. The presence of surface piezoelectricity in nano beams and nano plates leads to increase in electromechanical coupling coefficient. Topology optimization of these piezoelectric structures in an energy harvesting device to further increase energy conversion using appropriately modified XFEM-level set algorithm is performed . KW - Finite-Elemente-Methode KW - Piezoelectricity KW - Inverse problems KW - Optimization problems KW - Nanostructures KW - XFEM KW - level set method KW - Surface effects Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20161128-27095 ER - TY - THES A1 - Ghasemi, Hamid T1 - Stochastic optimization of fiber reinforced composites considering uncertainties N2 - Briefly, the two basic questions that this research is supposed to answer are: 1. Howmuch fiber is needed and how fibers should be distributed through a fiber reinforced composite (FRC) structure in order to obtain the optimal and reliable structural response? 2. How do uncertainties influence the optimization results and reliability of the structure? Giving answer to the above questions a double stage sequential optimization algorithm for finding the optimal content of short fiber reinforcements and their distribution in the composite structure, considering uncertain design parameters, is presented. In the first stage, the optimal amount of short fibers in a FRC structure with uniformly distributed fibers is conducted in the framework of a Reliability Based Design Optimization (RBDO) problem. Presented model considers material, structural and modeling uncertainties. In the second stage, the fiber distribution optimization (with the aim to further increase in structural reliability) is performed by defining a fiber distribution function through a Non-Uniform Rational BSpline (NURBS) surface. The advantages of using the NURBS surface as a fiber distribution function include: using the same data set for the optimization and analysis; high convergence rate due to the smoothness of the NURBS; mesh independency of the optimal layout; no need for any post processing technique and its non-heuristic nature. The output of stage 1 (the optimal fiber content for homogeneously distributed fibers) is considered as the input of stage 2. The output of stage 2 is the Reliability Index (b ) of the structure with the optimal fiber content and distribution. First order reliability method (in order to approximate the limit state function) as well as different material models including Rule of Mixtures, Mori-Tanaka, energy-based approach and stochastic multi-scales are implemented in different examples. The proposed combined model is able to capture the role of available uncertainties in FRC structures through a computationally efficient algorithm using all sequential, NURBS and sensitivity based techniques. The methodology is successfully implemented for interfacial shear stress optimization in sandwich beams and also for optimization of the internal cooling channels in a ceramic matrix composite. Finally, after some changes and modifications by combining Isogeometric Analysis, level set and point wise density mapping techniques, the computational framework is extended for topology optimization of piezoelectric / flexoelectric materials. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2016,1 KW - Optimization KW - Fiber Reinforced Composite KW - Finite Element Method KW - Isogeometric Analysis KW - Flexoelectricity KW - Finite-Elemente-Methode KW - Optimierung Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20161117-27042 ER - TY - THES A1 - Schwedler, Michael T1 - Integrated structural analysis using isogeometric finite element methods N2 - The gradual digitization in the architecture, engineering, and construction industry over the past fifty years led to an extremely heterogeneous software environment, which today is embodied by the multitude of different digital tools and proprietary data formats used by the many specialists contributing to the design process in a construction project. Though these projects become increasingly complex, the demands on financial efficiency and the completion within a tight schedule grow at the same time. The digital collaboration of project partners has been identified as one key issue in successfully dealing with these challenges. Yet currently, the numerous software applications and their respective individual views on the design process severely impede that collaboration. An approach to establish a unified basis for the digital collaboration, regardless of the existing software heterogeneity, is a comprehensive digital building model contributed to by all projects partners. This type of data management known as building information modeling (BIM) has many benefits, yet its adoption is associated with many difficulties and thus, proceeds only slowly. One aspect in the field of conflicting requirements on such a digital model is the cooperation of architects and structural engineers. Traditionally, these two disciplines use different abstractions of reality for their models that in consequence lead to incompatible digital representations thereof. The onset of isogeometric analysis (IGA) promised to ease the discrepancy in design and analysis model representations. Yet, that initial focus quickly shifted towards using these methods as a more powerful basis for numerical simulations. Furthermore, the isogeometric representation alone is not capable of solving the model abstraction problem. It is thus the intention of this work to contribute to an improved digital collaboration of architects and engineers by exploring an integrated analysis approach on the basis of an unified digital model and solid geometry expressed by splines. In the course of this work, an analysis framework is developed that utilizes such models to automatically conduct numerical simulations commonly required in construction projects. In essence, this allows to retrieve structural analysis results from BIM models in a fast and simple manner, thereby facilitating rapid design iterations and profound design feedback. The BIM implementation Industry Foundation Classes (IFC) is reviewed with regard to its capabilities of representing the unified model. The current IFC schema strongly supports the use of redundant model data, a major pitfall in digital collaboration. Additionally, it does not allow to describe the geometry by volumetric splines. As the pursued approach builds upon a unique model for both, architectural and structural design, and furthermore requires solid geometry, necessary schema modifications are suggested. Structural entities are modeled by volumetric NURBS patches, each of which constitutes an individual subdomain that, with regard to the analysis, is incompatible with the remaining full model. The resulting consequences for numerical simulation are elaborated in this work. The individual subdomains have to be weakly coupled, for which the mortar method is used. Different approaches to discretize the interface traction fields are implemented and their respective impact on the analysis results is evaluated. All necessary coupling conditions are automatically derived from the related geometry model. The weak coupling procedure leads to a linear system of equations in saddle point form, which, owed to the volumetric modeling, is large in size and, the associated coefficient matrix has, due to the use of higher degree basis functions, a high bandwidth. The peculiarities of the system require adapted solution methods that generally cause higher numerical costs than the standard procedures for symmetric, positive-definite systems do. Different methods to solve the specific system are investigated and an efficient parallel algorithm is finally proposed. When the structural analysis model is derived from the unified model in the BIM data, it does in general initially not meet the requirements on the discretization that are necessary to obtain sufficiently accurate analysis results. The consequently necessary patch refinements must be controlled automatically to allowfor an entirely automatic analysis procedure. For that purpose, an empirical refinement scheme based on the geometrical and possibly mechanical properties of the specific entities is proposed. The level of refinement may be selectively manipulated by the structural engineer in charge. Furthermore, a Zienkiewicz-Zhu type error estimator is adapted for the use with isogeometric analysis results. It is shown that also this estimator can be used to steer an adaptive refinement procedure. T3 - ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar - 2016,2 KW - Finite-Elemente-Methode KW - NURBS KW - Isogeometrische Analyse KW - finite element method KW - isogeometric analysis KW - mortar method KW - building information modelling Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170130-27372 ER -