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We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using Wire-Arc Additive Manufacturing. As the first task, we consider the automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this we employ Lloyd’s algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages.
Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces.
In this thesis we study efficient time integration methods for linear parabolic PDEs to solve practical problems that arise in a variety of real-world applications. The classical construction of numerical methods for solving PDEs is based on the method of lines, which leads to a large sparse semi-discretised system of ODEs to which any numerical method for initial value ODE problems can be applied. When dealing with parabolic-type problems, the underlying ODE systems are known to be stiff. Therefore, in the context of linear model problems, the use of implicit schemes is usually considered to be the best choice in practice.
However, this statement is not correct for some relevant real-world applications. In particular, implicit schemes can cause high computational costs that are equipped with certain model conditions. The model problems considered here are coupled with various settings, ranging from many different initial conditions over long-term simulation with relatively frequent model updates, to dealing with very large-scale problems for which the matrix size can exceed several millions. For this reason, we are interested in sophisticated and computationally efficient numerical methods that bring the aspects of approximation accuracy as well as computational and storage complexity into balance.
Even nowadays it is still a challenging task to devise a numerical method that combines high accuracy, robustness and computational efficiency for the model problem to be solved. Therefore, the main objective is to find an easy and efficient ODE integration scheme for each individual model problem. On this basis, we first give a comprehensive introduction to the state-of-the-art methods that are often used for practical purposes. In this framework, we will investigate very detailed the theoretical and numerical foundations of two popular techniques, namely the fast explicit methods and the model order reduction techniques. This is primarily important in order to fully understand the numerical methods, and also absolutely essential in finding the best numerical method that is specifically suitable for the intended purpose.
Our second goal is then to efficiently solve the practical problems that arise in connection with shape correspondence, geothermal energy storage and image osmosis filtering. For each application we specify a complete setup, and in order to provide an efficient and accurate numerical approximation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. We validate our numerical findings through many experiments using synthetic and real-world data. In addition, the thesis provides a complete and detailed description of the powerful methods that can be very useful for tackling similar problems that are the subject of interest in many applications.