Orthogonale Dünngitter-Teilraumzerlegungen

In der Simulation treten Häufg hochdimensionale partielle Differentialgleichungen auf. Das Lösen dieser wird für volle Gitter sehr schnell zu teuer. In dieser Arbeit wird ein Verfahren für das Lösen partieller Differentialgleichungen mit Hilfe von Dünnen Gittern, welche für mehrdimensionale Probleme besser skalieren, sowie dessen Implementierung in das Programmpaket SG++ vorgestellt. Durch Funktionsdarstellung in einem Erzeugendensystem wird die Verwendung einer L2-orthogonalen Teilraumzerlegung ermöglicht. Projektionsoperatoren ersetzen hierbei die explizite Transformation in eine Prewavelet-Basis. Diese Zerlegung erlaubt das Lumping der Steifgkeitsmatrix, also das Weglassen von großen Blöcken der Matrix. Hiermit wird ein Algorithmus zur Matrixmultiplikation, welcher dem von Schwab und Todor ähnelt implementiert. Dieser wird in einem konjugierten Gradienten-Verfahren verwendet und auch auf krummberandete Gebieten angewendet. Des Weiteren wird die Teilraumzerlegung durch L2-Projektion mit anderen Zerlegungen in Bezug auf Laufzeit und Fehlerentwicklung verglichen

A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification

A fully-implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2D is presented. In addition we combine a second-order fully-implicit time discretisation scheme with variable steps size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, is shown. Due to the high non-linearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour

A Scheme to Numerically Evolve Data for the Conformal Einstein Equation

This is the second paper in a series describing a numerical implementation of the conformal Einstein equation. This paper deals with the technical details of the numerical code used to perform numerical time evolutions from a "minimal" set of data. We outline the numerical construction of a complete set of data for our equations from a minimal set of data. The second and the fourth order discretisations, which are used for the construction of the complete data set and for the numerical integration of the time evolution equations, are described and their efficiencies are compared. By using the fourth order scheme we reduce our computer resource requirements --- with respect to memory as well as computation time --- by at least two orders of magnitude as compared to the second order scheme.Comment: 20 pages, 12 figure

Modeling Warp in Corrugated Cardboard Based on Homogenization Techniques for In-Process Measurement Applications

A model for describing warp—characterized as a systematic, large-scale deviation from the intended flat shape—in corrugated board based on Kirchhoff plate theory is proposed. It is based on established homogenization techniques and only a minimum of model assumptions. This yields general results applicable to any kind of corrugated cardboard. Since the model is intended to be used with industrial data, basic material properties which are usually not measured in practice are summarized to a few parameters. Those parameters can easily be fitted to the measurement data, allowing the user to systematically identify ways to reduce warp in a given situation in practice. In particular, the model can be used both as a filter to separate the warp from other surface effects such as washboarding, and to interpolate between discrete sample points scattered across the surface of a corrugated board sheet. Applying the model only requires height measurements of the corrugated board at several known (not necessarily exactly predetermined) locations across the corrugated board and acts as an interpolation or regression method between those points. These data can be acquired during production in a cost-efficient way and do not require any destructive testing of the board. The principle of an algorithm for fitting measured data to the model is presented and illustrated with examples taken from ongoing measurements. Additionally, the case of warp-free board is analyzed in more detail to deduce additional theoretical conditions necessary to reach this state

Discontinuous Galerkin discretisation with embedded boundary conditions

The purpose of this paper is to introduce discretisation methods of discontinuous Galerkin type for solving second order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretisation of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DG-discretisation is adapted in the cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection dominated boundary value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of a cubic polynomial, the boundary condition treatment appears quite effective in handling such complex situations

Discontinuous Galerkin discretisation with embedded boundary conditions

The purpose of this paper is to introduce discretisation methods of discontinuous Galerkin type for solving second order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretisation of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DG-discretisation is adapted in the cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection dominated boundary value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of a cubic polynomial, the boundary condition treatment appears quite effective in handling such complex situations

Convergence of infinite element methods for scalar waveguide problems

We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the Perfectly Matched Layer (PML) method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments