## Fachbereich Mathematik

Over the last decades, mathematical modeling has reached nearly all fields of natural science. The abstraction and reduction to a mathematical model has proven to be a powerful tool to gain a deeper insight into physical and technical processes. The increasing computing power has made numerical simulations available for many industrial applications. In recent years, mathematicians and engineers have turned there attention to model solid materials. New challenges have been found in the simulation of solids and fluid-structure interactions. In this context, it is indispensable to study the dynamics of elastic solids. Elasticity is a main feature of solid bodies while demanding a great deal of the numerical treatment. There exists a multitude of commercial tools to simulate the behavior of elastic solids. Anyhow, the majority of these software packages consider quasi-stationary problems. In the present work, we are interested in highly dynamical problems, e.g. the rotation of a solid. The applicability to free-boundary problems is a further emphasis of our considerations. In the last years, meshless or particle methods have attracted more and more attention. In many fields of numerical simulation these methods are on a par with classical methods or superior to them. In this work, we present the Finite Pointset Method (FPM) which uses a moving least squares particle approximation operator. The application of this method to various industrial problems at the Fraunhofer ITWM has shown that FPM is particularly suitable for highly dynamical problems with free surfaces and strongly changing geometries. Thereby, FPM offers exactly the features that we require for the analysis of the dynamics of solid bodies. In the present work, we provide a numerical scheme capable to simulate the behavior of elastic solids. We present the system of partial differential equations describing the dynamics of elastic solids and show its hyperbolic character. In particular, we focus our attention to the constitutive law for the stress tensor and provide evolution equations for the deviatoric part of the stress tensor in order to circumvent limitations of the classical Hooke's law. Furthermore, we present the basic principle of the Finite Pointset Method. In particular, we provide the concept of upwinding in a given direction as a key ingredient for stabilizing hyperbolic systems. The main part of this work describes the design of a numerical scheme based on FPM and an operator splitting to take the different processes within a solid body into account. Each resulting subsystem is treated separately in an adequate way. Hereby, we introduce the notion of system-inherent directions and dimensional upwinding. Finally, a coupling strategy for the subsystems and results are presented. We close this work with some final conclusions and an outlook on future work.

A Multi-Phase Flow Model Incorporated with Population Balance Equation in a Meshfree Framework
(2011)

This study deals with the numerical solution of a meshfree coupled model of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) for liquid-liquid extraction columns. In modeling the coupled hydrodynamics and mass transfer in liquid extraction columns one encounters multidimensional population balance equation that could not be fully resolved numerically within a reasonable time necessary for steady state or dynamic simulations. For this reason, there is an obvious need for a new liquid extraction model that captures all the essential physical phenomena and still tractable from computational point of view. This thesis discusses a new model which focuses on discretization of the external (spatial) and internal coordinates such that the computational time is drastically reduced. For the internal coordinates, the concept of the multi-primary particle method; as a special case of the Sectional Quadrature Method of Moments (SQMOM) is used to represent the droplet internal properties. This model is capable of conserving the most important integral properties of the distribution; namely: the total number, solute and volume concentrations and reduces the computational time when compared to the classical finite difference methods, which require many grid points to conserve the desired physical quantities. On the other hand, due to the discrete nature of the dispersed phase, a meshfree Lagrangian particle method is used to discretize the spatial domain (extraction column height) using the Finite Pointset Method (FPM). This method avoids the extremely difficult convective term discretization using the classical finite volume methods, which require a lot of grid points to capture the moving fronts propagating along column height.