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- Finite-Elemente-Methode (3)
- Isogeometric Analysis (3)
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- Augmented Reality Displays (1)
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Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDEs), which was introduced with the aim of integrating finite element analysis with computer-aided design systems. The main idea of the method is to use the same spline basis functions which describe the geometry in CAD systems for the approximation of solution fields in the finite element method (FEM). Originally, NURBS which is a standard technology employed in CAD systems was adopted as basis functions in IGA but there were several variants of IGA using other technologies such as T-splines, PHT splines, and subdivision surfaces as basis functions. In general, IGA offers two key advantages over classical FEM: (i) by describing the CAD geometry exactly using smooth, high-order spline functions, the mesh generation process is simplified and the interoperability between CAD and FEM is improved, (ii) IGA can be viewed as a high-order finite element method which offers basis functions with high inter-element continuity and therefore can provide a primal variational formulation of high-order PDEs in a straightforward fashion. The main goal of this thesis is to further advance isogeometric analysis by exploiting these major advantages, namely precise geometric modeling and the use of smooth high-order splines as basis functions, and develop robust computational methods for problems with complex geometry and/or complex multi-physics.
As the first contribution of this thesis, we leverage the precise geometric modeling of isogeometric analysis and propose a new method for its coupling with meshfree discretizations. We exploit the strengths of both methods by using IGA to provide a smooth, geometrically-exact surface discretization of the problem domain boundary, while the Reproducing Kernel Particle Method (RKPM) discretization is used to provide the volumetric discretization of the domain interior. The coupling strategy is based upon the higher-order consistency or reproducing conditions that are directly imposed in the physical domain. The resulting coupled method enjoys several favorable features: (i) it preserves the geometric exactness of IGA, (ii) it circumvents the need for global volumetric parameterization of the problem domain, (iii) it achieves arbitrary-order approximation accuracy while preserving higher-order smoothness of the discretization. Several numerical examples are solved to show the optimal convergence properties of the coupled IGA–RKPM formulation, and to demonstrate its effectiveness in constructing volumetric discretizations for complex-geometry objects.
As for the next contribution, we exploit the use of smooth, high-order spline basis functions in IGA to solve high-order surface PDEs governing the morphological evolution of vesicles. These governing equations are often consisted of geometric PDEs, high-order PDEs on stationary or evolving surfaces, or a combination of them. We propose an isogeometric formulation for solving these PDEs. In the context of geometric PDEs, we consider phase-field approximations of mean curvature flow and Willmore flow problems and numerically study the convergence behavior of isogeometric analysis for these problems. As a model problem for high-order PDEs on stationary surfaces, we consider the Cahn–Hilliard equation on a sphere, where the surface is modeled using a phase-field approach. As for the high-order PDEs on evolving surfaces, a phase-field model of a deforming multi-component vesicle, which consists of two fourth-order nonlinear PDEs, is solved using the isogeometric analysis in a primal variational framework. Through several numerical examples in 2D, 3D and axisymmetric 3D settings, we show the robustness of IGA for solving the considered phase-field models.
Finally, we present a monolithic, implicit formulation based on isogeometric analysis and generalized-alpha time integration for simulating hydrodynamics of vesicles according to a phase-field model. Compared to earlier works, the number of equations of the phase-field model which need to be solved is reduced by leveraging high continuity of NURBS functions, and the algorithm is extended to 3D settings. We use residual-based variational multi-scale method (RBVMS) for solving Navier–Stokes equations, while the rest of PDEs in the phase-field model are treated using a standard Galerkin-based IGA. We introduce the resistive immersed surface (RIS) method into the formulation which can be employed for an implicit description of complex geometries using a diffuse-interface approach. The implementation highlights the robustness of the RBVMS method for Navier–Stokes equations of incompressible flows with non-trivial localized forcing terms including bending and tension forces of the vesicle. The potential of the phase-field model and isogeometric analysis for accurate simulation of a variety of fluid-vesicle interaction problems in 2D and 3D is demonstrated.
Im Rahmen der Dissertation wurde ein Toolboxmodell für transdisziplinäres Wasserressourcenmanagement entwickelt. Das Modell liefert den methodischen Rahmen Wasserressourcen nachhaltig und transdisziplinär zu bewirtschaften.
Der Begriff der Nachhaltigkeit und eine Konkretisierung der nachhaltigen Bewirtschaftung globaler Wasserressourcen scheinen unüberschaubar und suggerieren die Forderung nach einer neuen Weltformel. Die globale Bedeutung der Wasserressourcen, die für Regionen spezifischen Besonderheiten des natürlichen Wasserhaushalts und der anthropogenen Nutzung, die Zeitskala und die Kontextualisierung in alle betroffenen und benachbarten Disziplinen deuten auf die Komplexität der Thematik hin. Es wird eine Systematisierung des Planungsprozesses von Wasserressourcen notwendig, anhand derer eine holistische Herangehensweise mit einer Strategieentwicklung für Regionen spezifischer Schwerpunktprobleme erfolgt. Ziel der Arbeit ist die Erarbeitung einer Strategie zur Systematisierung nach diesen Forderungen und die Bereitstellung eines Toolboxmodelles als Planungswerkzeug für das transdisziplinäre Wasserressourcenmanagement.
Das Toolboxmodell stellt den konzeptionellen Rahmen für die Bewirtschaftung von Wasserressourcen mit der Anwendung transdisziplinärer Forschungsmethoden bereit. Wesentliche Herausforderung bei der Anwendung der transdisziplinären Methode sind die Implementierung verschiedener Skalenbereiche, der Umgang mit der Komplexität von Daten, das Bewahren von Transparenz und Objektivität sowie die Ermöglichung eines auf andere Regionen übertragbaren Planungsprozesses.
Die theoretischen Grundlagen naturwissenschaftlicher Forschung zur Nachhaltigkeit haben ihren Ursprung in den biologischen und geographischen Disziplinen. Das Ineinandergreifen naturräumlicher Zusammenhänge und der Einfluss anthropogener Nutzung und technischer Innovationen auf den Naturhaushalt sind Kern der Kausalität übergreifenden Denkens und Verstehens. Mit dem Ansatz des integrierten Wasserressourcenmanagements (IWRM) erfolgt die Berücksichtigung wirtschaftlicher und sozioökonomischer Ziele in den Planungsprozess für ökologisch nachhaltige Wasserwirtschaft. Das Instrument der Wasserrahmenrichtlinie (EU-WRRL) ist auf eine Gewässerökologie ausgerichtete Richtlinie, welche die Integration verschiedener Interessenvertreter in den Planungsprozess vorsieht. Das Konzept der neuartigen Sanitärsysteme basiert auf Stoffflüssen zwischen konkurrierenden Handlungsbereichen, wie Abfall-, Ressourcen- und Landwirtschaft.
Den integrierten Ansätzen fehlt eine übergeordnete gemeinsame Zielstrategie – eine sogenannte Phase Null. Diese Phase Null – das Lernen aller 7 Zusammenfassung 157 relevanten, konkurrierenden und harmonisierenden Handlungsfelder eines Planungshorizontes wird durch eine transdisziplinäre Perspektive ermöglicht. Während bei der integralen Perspektive eine disziplinorientierte Kooperation im Vordergrund steht, verlangt die transdisziplinäre Perspektive nach einer problemorientierten Kooperation zwischen den Interessenvertretern (Werlen 2015). Die bestehenden Konzepte und Richtlinien für das nachhaltige Management von Wasserressourcen sind etabliert und evaluiert. Der Literatur zur Folge ist eine Weiterentwicklung nach der Perspektive der Transdisziplinarität erforderlich. Das Toolboxmodell für integrales Wasserressourcenmanagement entspricht einem Planungstool bestehend aus Werkzeugen für die Anwendung wissenschaftlicher Methoden. Die Zusammenstellung der Methoden/Werkzeuge erfüllt im Rahmen die Methode transdisziplinärer Forschung. Das Werkzeug zum Aufstellen der relevanten Handlungsfelder umfasst die Charakterisierung eines Untersuchungsgebietes und Planungsrahmens, die kausale Verknüpfung des Bewirtschaftungskonzeptes und konkurrierender sowie sich unterstützender Stakeholder. Mit dem Werkzeug der Kontextualisierung und Indikatorenaufstellung wird eine Methode der stufenweisen und von einer Skala unabhängigen Bewertung des Umweltzustandes für die Zielpriorisierung vorgenommen. Damit wird das Toolboxmodell dem Problem der Komplexität und Datenverfügbarkeit gerecht. Anhand der eingesetzten ABC Methode, werden die Bewertungsgrößen differenziert strukturiert auf verschiedene Skalen und Datenressourcen (A=Ersterkennung,B=Zeigerwerte, C=Modell/Index). Die ABC-Methode ermöglicht die Planung bereits mit unsicherer und lückenhafter Datengrundlage, ist jederzeit erweiterbar und bietet somit eine operative Wissensgenerierung während des Gestaltungsprozesses.
Für das Werkzeug zur Bewertung und Priorisierung wird der Algorithmus der Composite Programmierung angewandt. Diese Methode der Mehrfachzielplanung erfüllt den Anspruch der permanenten Erweiterbarkeit und der transparenten und objektiven Entscheidungsfindung. Die Komplexität des transdisziplinären Wasserressourcenmanagements kann durch die Methode der Composite Programmierung systematisiert werden. Das wesentliche Ergebnis der Arbeit stellt die erfolgreiche Erarbeitung und Anwendung des Tool-boxmodells für das transdisziplinäre Wasserressourcenmanagement im Untersuchungsgebiet Stadt Darkhan in der Mongolei dar. Auf Grund seiner besonderen hydrologischen und strukturellen Situa-tion wird die Relevanz eines nachhaltigen Bewirtschaftungskonzeptes deutlich. Im Rahmen des Querschnittsmoduls des MoMo-Projektes wurde eine für das Toolboxmodell geeignete Datengrundlage erarbeitet. Planungsrelevante Handlungsfelder wurden im Rahmen eines Workshops mit verschiedenen Interessenvertretern erarbeitet. Im Ergebnis dessen wurde die Systematik eines Zielbaumes mit Hauptzielen und untergeordneten Teilzielen als Grundlage der Priorisierung nach den holistischen Anspruch der transdisziplinären Forschung aufgestellt. Für die Messbarkeit, in-wieweit Teilziele erreicht sind oder Handlungsbedarf besteht, wurden Indikatoren erarbeitet. Die Indikatoren-Aufstellung erfolgte exemplarisch für das Handlungsfeld Siedlungswasserwirtschaft in allen Skalen des ABC-Systems. Die im BMBF-MoMo Projekt generierte umfassende Datengrundlage ermöglichte die Anwendung und Evaluierung des Toolboxmodells mit unterschiedlichem quantitativem und qualitativem Dateninput. Verschiedene Kombination von A (Ersterkennung), B (Zeigerwerte) und C (Modell/Index) als Grundlage der Priorisierung mit der Compostite Programmierung ermöglichten die Durchführung und Bewertung des transdisziplinären Planungstools. Die er-mittelten Rangfolgen von Teilzielen mit unterschiedlichen Bewertungsvarianten ergaben ähnliche
Tendenzen. Das ist ein Hinweis dafür, dass für die zukünftige Anwendung des Toolboxmodells die operative Wissensgenerierung, d.h. das schrittweise Hinzufügen neu ermittelter, gesicherterer Daten, funktioniert. Eine schwierige Datenverfügbarkeit oder eine noch im Prozess befindliche wissenschaftliche Analyse sollen keine Hindernisse für eine schrittweise und erweiterbare Zielpriorisierung und Maßnahmenplanung sein. Trotz der Komplexität des transdisziplinären Ansatzes wird durch die Anwendung des Toolboxmodells eine effiziente und zielorientierte Handlungspriorisierung ermöglicht. Die Effizienz wird erreicht durch ressourcenschonende und flexible, Ziel fokussierte Datenermittlung. Zeit und Kosten im Planungsprozess können eingespart werden. Die erzielte Priorisierung von letztlich Handlungsempfehlungen erfolgt individuell auf die Eigenart des Untersuchungsgebietes angepasst, was hinsichtlich seiner Wirkung als erfolgsversprechend gilt.
Landwirtschaftliche Biogasanlagen leisten mit ca. 9.300 Anlagen und einem Anteil von 5,3% an der Stromerzeugung, einen Beitrag zur Erzeugung Erneuer-barer Energien in Deutschland. Die Optimierung dieser Anlagen fördert die nachhaltige Bereitstellung von Strom, Wärme und BioErdgas.
Das Ergebnis dieser Forschungsarbeit ist die Entwicklung eines mehrmethodi-schen Bewertungsansatzes zur Beschreibung der Qualität der Eingangs-substrate als Teil einer ganzheitlichen Prozessoptimierung. Dies gelingt durch die kombinierte Nutzung klassischer Analysesätze, der Nutzung organolepti-scher Parameter – der humansensorischen Sinnenprüfung – und der Integration von prozess- und substratspezifischem Erfahrungswissen. Anhand von halbtechnischen Versuchen werden Korrelationen und Kausalitäten zwi-schen chemisch-physikalischen, biologischen, organoleptischen und erfahrungsbezogenen Parametern erforscht. Die Entwicklung einer Fallbasis mit Hilfe des Fallbasierten Schließens, einer Form Künstlicher Intelligenz, zeigt das Entwicklungs- und Integrationspotenzial der Automatisierung auf, insbesondere auch im Hinblick auf neue Ansätze z.B. Industrie 4.0. Erste Lösungen zur Bewältigung der identifizierten Herausforderungen der mehrmethodischen Prozessbewertung werden vorgestellt.
Abschließend wird ein Ausblick auf den weiteren Forschungsbedarf gegeben und die Übertragbarkeit des mehrmethodischen Bewertungsansatzes auf andere Anwendungsfelder z.B. Bioabfallbehandlung, Kläranlagen angeregt.
Gaze based human-computer-interaction has been a research topic for over a quarter century. Since then, the main scenario for gaze interaction has been helping handicapped people to communicate an interact with their environment. With the rapid development of mobile and wearable display technologies, a new application field for gaze interaction has appeared, opening new research questions.
This thesis investigates the feasibility of mobile gaze based interaction, studying deeply the use of pie menus as a generic and robust widget for gaze interaction as well as visual and perceptual issues on head mounted (wearable) optical see-through displays.
It reviews conventional gaze-based selection methods and investigates in detail the use of pie menus for gaze control. It studies and discusses layout issues, selection methods and applications. Results show that pie menus can allocate up to six items in width and multiple depth layers, allowing a fast and accurate navigation through hierarchical levels by using or combining multiple selection methods. Based on these results, several text entry methods based on pie menus are proposed. Character-by-character text entry, text entry with bigrams and with text entry with bigrams derived by word prediction, as well as possible selection methods, are examined in a longitudinal study. Data showed large advantages of the bigram entry methods over single character text entry in speed and accuracy. Participants preferred the novel selection method based on saccades (selecting by borders) over the conventional and well established dwell time method.
On the one hand, pie menus showed to be a feasible and robust widget, which may enable the efficient use of mobile eye tracking systems that may not be accurate enough for controlling elements on conventional interface. On the other hand, visual perception on mobile displays technologies need to be examined in order to deduce if the mentioned results can be transported to mobile devices.
Optical see-through devices enable observers to see additional information embedded in real environments. There is already some evidence of increasing visual load on the respective systems. We investigated visual performance on participants with a visual search tasks and dual tasks presenting visual stimuli on the optical see-through device, only on a computer screen, and simultaneously on both devices. Results showed that switching between the presentation devices (i.e. perceiving information simultaneously from both devices) produced costs in visual performance. The implications of these costs and of further perceptual and technical factors for mobile gaze-based interaction are discussed and solutions are proposed.
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.
The thesis investigates at the computer aided simulation process for operational vibration analysis of complex coupled systems. As part of the internal methods project “Absolute Values” of the BMW Group, the thesis deals with the analysis of the structural dynamic interactions and excitation interactions. The overarching aim of the methods project is to predict the operational vibrations of engines.
Simulations are usually used to analyze technical aspects (e. g. operational vibrations, strength, ...) of single components in the industrial development. The boundary conditions of submodels are mostly based on experiences. So the interactions with neighboring components and systems are neglected. To get physically more realistic results but still efficient simulations, this work wants to support the engineer during the preprocessing phase by useful criteria.
At first suitable abstraction levels based on the existing literature are defined to identify structural dynamic interactions and excitation interactions of coupled systems. So it is possible to separate different effects of the coupled subsystems. On this basis, criteria are derived to assess the influence of interactions between the considered systems. These criteria can be used during the preprocessing phase and help the engineer to build up efficient models with respect to the interactions with neighboring systems. The method was developed by using several models with different complexity levels. Furthermore, the method is proved for the application in the industrial environment by using the example of a current combustion engine.
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 .
In this thesis, a new approach is developed for applications of shape optimization on the time harmonic wave propagation (Helmholtz equation) for acoustic problems. This approach is introduced for different dimensional problems: 2D, 3D axi-symmetric and fully 3D problems. The boundary element method (BEM) is coupled with the isogeometric analysis (IGA) forming the so-called (IGABEM) which speeds up meshing and gives higher accuracy in comparison with standard BEM. BEM is superior for handling unbounded domains by modeling only the inner boundaries and avoiding the truncation error, present in the finite element method (FEM) since BEM solutions satisfy the Sommerfeld radiation condition automatically. Moreover, BEM reduces the space dimension by one from a volumetric three-dimensional problem to a surface two-dimensional problem, or from a surface two-dimensional problem to a perimeter one-dimensional problem. Non-uniform rational B-splines basis functions (NURBS) are used in an isogeometric setting to describe both the CAD geometries and the physical fields.
IGABEM is coupled with one of the gradient-free optimization methods, the Particle Swarm Optimization (PSO) for structural shape optimization problems. PSO is a straightforward method since it does not require any sensitivity analysis but it has some trade-offs with regard to the computational cost. Coupling IGA with optimization problems enables the NURBS basis functions to represent the three models: shape design, analysis and optimization models, by a definition of a set of control points to be the control variables and the optimization parameters as well which enables an easy transition between the three models.
Acoustic shape optimization for various frequencies in different mediums is performed with PSO and the results are compared with the benchmark solutions from the literature for different dimensional problems proving the efficiency of the proposed approach with the following remarks:
- In 2D problems, two BEM methods are used: the conventional isogeometric boundary element method (IGABEM) and the eXtended IGABEM (XIBEM) enriched with the partition-of-unity expansion using a set of plane waves, where the results are generally in good agreement with the linterature with some computation advantage to XIBEM which allows coarser meshes.
-In 3D axi-symmetric problems, the three-dimensional problem is simplified in BEM from a surface integral to a combination of two 1D integrals. The first is the line integral similar to a two-dimensional BEM problem. The second integral is performed over the angle of revolution. The discretization is applied only to the former integration. This leads to significant computational savings and, consequently, better treatment for higher frequencies over the full three-dimensional models.
- In fully 3D problems, a detailed comparison between two BEM methods: the conventional boundary integral equation (CBIE) and Burton-Miller (BM) is provided including the computational cost. The proposed models are enhanced with a modified collocation scheme with offsets to Greville abscissae to avoid placing collocation points at the corners. Placing collocation points on smooth surface enables accurate evaluation of normals for BM formulation in addition to straightforward prediction of jump-terms and avoids singularities in $\mathcal{O} (1/r)$ integrals eliminating the need for polar integration. Furthermore, no additional special treatment is required for the hyper-singular integral while collocating on highly distorted elements, such as those containing sphere poles. The obtained results indicate that, CBIE with PSO is a feasible alternative (except for a small number of fictitious frequencies) which is easier to implement. Furthermore, BM presents an outstanding treatment of the complicated geometry of mufflers with internal extended inlet/outlet tube as an interior 3D Helmholtz acoustic problem instead of using mixed or dual BEM.
Finite Element Simulations of dynamically excited structures are mainly influenced by the mass, stiffness, and damping properties of the system, as well as external loads. The prediction quality of dynamic simulations of vibration-sensitive components depends significantly on the use of appropriate damping models. Damping phenomena have a decisive influence on the vibration amplitude and the frequencies of the vibrating structure. However, developing realistic damping models is challenging due to the multiple sources that cause energy dissipation, such as material damping, different types of friction, or various interactions with the environment.
This thesis focuses on thermoelastic damping, which is the main cause of material damping in homogeneous materials. The effect is caused by temperature changes due to mechanical strains. In vibrating structures, temperature gradients arise in adjacent tension and compression areas. Depending on the vibration frequency, they result in heat flows, leading to increased entropy and the irreversible transformation of mechanical energy into thermal energy.
The central objective of this thesis is the development of efficient simulation methods to incorporate thermoelastic damping in finite element analyses based on modal superposition. The thermoelastic loss factor is derived from the structure's mechanical mode shapes and eigenfrequencies. In subsequent analyses that are performed in the time and frequency domain, it is applied as modal damping.
Two approaches are developed to determine the thermoelastic loss in thin-walled plate structures, as well as three-dimensional solid structures. The realistic representation of the dissipation effects is verified by comparing the simulation results with experimentally determined data. Therefore, an experimental setup is developed to measure material damping, excluding other sources of energy dissipation.
The three-dimensional solid approach is based on the determination of the generated entropy and therefore the generated heat per vibration cycle, which is a measure for thermoelastic loss in relation to the total strain energy. For thin plate structures, the amount of bending energy in a modal deformation is calculated and summarized in the so-called Modal Bending Factor (MBF). The highest amount of thermoelastic loss occurs in the state of pure bending. Therefore, the MBF enables a quantitative classification of the mode shapes concerning the thermoelastic damping potential.
The results of the developed simulations are in good agreement with the experimental results and are appropriate to predict thermoelastic loss factors. Both approaches are based on modal superposition with the advantage of a high computational efficiency. Overall, the modeling of thermoelastic damping represents an important component in a comprehensive damping model, which is necessary to perform realistic simulations of vibration processes.
This thesis presents the advances and applications of phase field modeling in fracture analysis. In this approach, the sharp crack surface topology in a solid is approximated by a diffusive crack zone governed by a scalar auxiliary variable. The uniqueness of phase field modeling is that the crack paths are automatically determined as part of the solution and no interface tracking is required. The damage parameter varies continuously over the domain. But this flexibility comes with associated difficulties: (1) a very fine spatial discretization is required to represent sharp local gradients correctly; (2) fine discretization results in high computational cost; (3) computation of higher-order derivatives for improved convergence rates and (4) curse of dimensionality in conventional numerical integration techniques. As a consequence, the practical applicability of phase field models is severely limited.
The research presented in this thesis addresses the difficulties of the conventional numerical integration techniques for phase field modeling in quasi-static brittle fracture analysis. The first method relies on polynomial splines over hierarchical T-meshes (PHT-splines) in the framework of isogeometric analysis (IGA). An adaptive h-refinement scheme is developed based on the variational energy formulation of phase field modeling. The fourth-order phase field model provides increased regularity in the exact solution of the phase field equation and improved convergence rates for numerical solutions on a coarser discretization, compared to the second-order model. However, second-order derivatives of the phase field are required in the fourth-order model. Hence, at least a minimum of C1 continuous basis functions are essential, which is achieved using hierarchical cubic B-splines in IGA. PHT-splines enable the refinement to remain local at singularities and high gradients, consequently reducing the computational cost greatly. Unfortunately, when modeling complex geometries, multiple parameter spaces (patches) are joined together to describe the physical domain and there is typically a loss of continuity at the patch boundaries. This decrease of smoothness is dictated by the geometry description, where C0 parameterizations are normally used to deal with kinks and corners in the domain. Hence, the application of the fourth-order model is severely restricted. To overcome the high computational cost for the second-order model, we develop a dual-mesh adaptive h-refinement approach. This approach uses a coarser discretization for the elastic field and a finer discretization for the phase field. Independent refinement strategies have been used for each field.
The next contribution is based on physics informed deep neural networks. The network is trained based on the minimization of the variational energy of the system described by general non-linear partial differential equations while respecting any given law of physics, hence the name physics informed neural network (PINN). The developed approach needs only a set of points to define the geometry, contrary to the conventional mesh-based discretization techniques. The concept of `transfer learning' is integrated with the developed PINN approach to improve the computational efficiency of the network at each displacement step. This approach allows a numerically stable crack growth even with larger displacement steps. An adaptive h-refinement scheme based on the generation of more quadrature points in the damage zone is developed in this framework. For all the developed methods, displacement-controlled loading is considered. The accuracy and the efficiency of both methods are studied numerically showing that the developed methods are powerful and computationally efficient tools for accurately predicting fractures.