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This report discusses two approaches for a posteriori error indication in the linear elasticity solver DDFEM: An indicator based on the Richardson extrapolation and Zienkiewicz-Zhu-type indicator. The solver handles 3D linear elasticity steady-state problems. It uses own input language to describe the mesh and the boundary conditions. Finite element discretization over tetrahedral meshes with first or second order shape functions (hierarchical basis) has been used to resolve the model. The parallelization of the numerical method is based on the domain decomposition approach. DDFEM is highly portable over a set of parallel computer architectures supporting the MPI-standard.

Within the last decades, a remarkable development in materials science took place -- nowadays, materials are not only constructed for the use of inert structures but rather designed for certain predefined functions. This innovation was accompanied with the appearance of smart materials with reliable recognition, discrimination and capability of action as well as reaction. Even though ferroelectric materials serve smartly in real applications, they also possess several restrictions at high performance usage. The behavior of these materials is almost linear under the action of low electric fields or low mechanical stresses, but exhibits strong non-linear response under high electric fields or mechanical stresses. High electromechanical loading conditions result in a change of the spontaneous polarization direction with respect to individual domains, which is commonly referred to as domain switching. The aim of the present work is to develop a three-dimensional coupled finite element model, to study the rate-independent and rate-dependent behavior of piezoelectric materials including domain switching based on a micromechanical approach. The proposed model is first elaborated within a two-dimensional finite element setting for piezoelectric materials. Subsequently, the developed two-dimensional model is extended to the three-dimensional case. This work starts with developing a micromechanical model for ferroelectric materials. Ferroelectric materials exhibit ferroelectric domain switching, which refers to the reorientation of domains and occurs under purely electrical loading. For the simulation, a bulk piezoceramic material is considered and each grain is represented by one finite element. In reality, the grains in the bulk ceramics material are randomly oriented. This property is taken into account by applying random orientation as well as uniform distribution for individual elements. Poly-crystalline ferroelectric materials at un-poled virgin state can consequently be characterized by randomly oriented polarization vectors. Energy reduction of individual domains is adopted as a criterion for the initiation of domain switching processes. The macroscopic response of the bulk material is predicted by classical volume-averaging techniques. In general, domain switching does not only depend on external loads but also on neighboring grains, which is commonly denoted as the grain boundary effect. These effects are incorporated into the developed framework via a phenomenologically motivated probabilistic approach by relating the actual energy level to a critical energy level. Subsequently, the order of the chosen polynomial function is optimized so that simulations nicely match measured data. A rate-dependent polarization framework is proposed, which is applied to cyclic electrical loading at various frequencies. The reduction in free energy of a grain is used as a criterion for the onset of the domain switching processes. Nucleation in new grains and propagation of the domain walls during domain switching is modeled by a linear kinetics theory. The simulated results show that for increasing loading frequency the macroscopic coercive field is also increasing and the remanent polarization increases at lower loading amplitudes. The second part of this work is focused on ferroelastic domain switching, which refers to the reorientation of domains under purely mechanical loading. Under sufficiently high mechanical loading, however, the strain directions within single domains reorient with respect to the applied loading direction. The reduction in free energy of a grain is used as a criterion for the domain switching process. The macroscopic response of the bulk material is computed for the hysteresis curve (stress vs strain) whereby uni-axial and quasi-static loading conditions are applied on the bulk material specimen. Grain boundary effects are addressed by incorporating the developed probabilistic approach into this framework and the order of the polynomial function is optimized so that simulations match measured data. Rate dependent domain switching effects are captured for various frequencies and mechanical loading amplitudes by means of the developed volume fraction concept which relates the particular time interval to the switching portion. The final part of this work deals with ferroelectric and ferroelastic domain switching and refers to the reorientation of domains under coupled electromechanical loading. If this free energy for combined electromechanical loading exceeds the critical energy barrier elements are allowed to switch. Firstly, hysteresis and butterfly curves under purely electrical loading are discussed. Secondly, additional mechanical loads in axial and lateral directions are applied to the specimen. The simulated results show that an increasing compressive stress results in enlarged domain switching ranges and that the hysteresis and butterfly curves flatten at higher mechanical loading levels.

In this thesis viscoelastic material models are established to investigate the nature of continuous calving processes at Antarctic ice shelves. Physics-based descriptions of calving require appropriate fracture criteria to separate icebergs from the remaining ice shelf. Hence, criteria of the stress, the strain, and the self-similarity criterion are considered within finite-element computations. Crucial parameters in the models to determine the position of calving are the accurate knowledge of the geometry, especially the freeboard height, while the material parameters mainly influence the time span between two successive calving events. The extension to nonlinear material models is necessary to properly analyze the internal forces also for large deformations that occur for longer times of the viscous ice flow.

This work describes the development of a continuum phase field model that can describe static as well as dynamic wetting scenarios on the nano- and microscale.
The model reaches this goal by a direct integration of an equation of state as well as a direct integration of the dissipative properties of a specific fluid, which are both obtained from molecular simulations. The presented approach leads to good agreement between the predictions of the phase field model and the physical properties of the regarded fluid.
The implementation of the model employs a mixed finite element formulation, a newly developed semi-implicit time integration scheme, as well as the concept of hyper-dual numbers. This ensures a straightforward and robust exchangeability of the constitutive equation for the regarded fluid.
The presented simulations show good agreement between the results of the present phase field model and results from molecular dynamics simulations. Furthermore, the results show that the model enables the investigation of wetting scenarios on the microscale. The continuum phase field model of this work bridges the gap between the molecular models on the nanoscale and the phenomenologically motivated continuum models on the macroscale.

This thesis investigates the electromechanic coupling of dielectric elastomers for the static and dynamic case by numerical simulations. To this end, the fundamental equations of the coupled field problem are introduced and the discretisation procedure for the numerical implementation is described. Furthermore, a three field formulation is proposed and implemented to treat the nearly incompressible behaviour of the elastomer. Because of the reduced electric permittivity of the material, very high electric fields are required for actuation purposes. To improve the electromechanic coupling a heterogeneous microstructure consisting of an elastomer matrix with barium titanate inclusions is proposed and studied.

A general framework for the thermodynamics of open systems is developed in the spatial and the material setting. Special emphasis is placed on the balance of mass which is enhanced by additional source and flux terms. Different solution strategies within the finite element technique are derived and compared. A number of numerical examples illustrates the features of the proposed approach.

This thesis is concerned with a phase field model for brittle fracture.
The high potential of phase field modeling in computational fracture mechanics lies in the generality of the approach and the straightforward numerical implementation, combined with a good accuracy of the results in the sense of continuum fracture mechanics.
However, despite the convenient numerical application of phase field fracture models, a detailed understanding of the physical properties is crucial for a correct interpretation of the numerical results. Therefore, the driving mechanisms of crack propagation and nucleation in the proposed phase field fracture model are explored by a thorough numerical and analytical investigation in this work.

This thesis treats the application of configurational forces for the evaluation of fracture processes in Antarctic ice shelves. FE simulations are used to analyze the influence of geometric scales, material parameters and boundary conditions on single surface cracks. A break-up event at the Wilkins Ice Shelf that coincided with a major temperature drop motivates the consideration of frost wedging as a mechanism for ice shelf disintegration. An algorithm for the evaluation of the crack propagation direction is used to analyze the horizontal growth of rifts. Using equilibrium considerations for a viscoelastic fluid, a method is introduced to compute viscous volume forces from measured velocity fields as loads for a linear elastic fracture mechanical analysis.

The phase field approach is a powerful tool that can handle even complicated fracture phenomena within an apparently simple framework. Nonetheless, a profound understanding of the model is required in order to be able to interpret the obtained results correctly. Furthermore, in the dynamic case the phase field model needs to be verified in comparison to experimental data and analytical results in order to increase the trust in this new approach. In this thesis, a phase field model for dynamic brittle fracture is investigated with regard to these aspects by analytical and numerical methods

In this thesis we present a new method for nonlinear frequency response analysis of mechanical vibrations.
For an efficient spatial discretization of nonlinear partial differential equations of continuum mechanics we employ the concept of isogeometric analysis. Isogeometric finite element methods have already been shown to possess advantages over classical finite element discretizations in terms of exact geometry representation and higher accuracy of numerical approximations using spline functions.
For computing nonlinear frequency response to periodic external excitations, we rely on the well-established harmonic balance method. It expands the solution of the nonlinear ordinary differential equation system resulting from spatial discretization as a truncated Fourier series in the frequency domain.
A fundamental aspect for enabling large-scale and industrial application of the method is model order reduction of the spatial discretization of the equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. We investigate the concept of modal derivatives theoretically and using computational examples we demonstrate the applicability and accuracy of the reduction method for nonlinear static computations and vibration analysis.
Furthermore, we extend nonlinear vibration analysis to incompressible elasticity using isogeometric mixed finite element methods.