In the simulation of deformations of plates it is well known that we have to use a special treatment of the thickness dependence. Therewith we achieve a reduction of dimension from 3D to 2D. For linear elasticity and small deformations several techniques are well established to handle the reduction of dimension and achieve acceptable numerical results. In the case of large deformations of plates with non-linear material behaviour there exist different problems. For example the analytical integration over the thickness of the plate is not possible due to the non-linearities arising from the material law and the large deformations themselves. There are several possibilities to introduce a hypothesis for the treatment of the plate thickness from the strong Kirchhoff assumption on one hand up to some hierarchical approaches on the other hand.

We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

We give an alternate description of the usual shell equation that does not depend on the special mid surface coordinates, but uses differential operators defined on the mid surface.

Beschreibung der Finite Elemente Software-Familie SPC-PM3-AdH-XX für: (S)cientific (P)arallel (C)omputing - (P)rogramm-(M)odul (3)D (ad)aptiv (H)exaederelemente. Für XX stehen die einzelnen Spezialvarianten, die in Teil 2 detailliert geschildert werden. Stand: Ende 2013

Reduced Hsieh-Clough-Tocher elements are triangular C1-elements with only nine degrees of freedom. Simple formulas for the basis functions of reduced Hsieh-Clough-Tocher elements based on the edge vectors of the triangle have been given recently for a barycentric splitting. We generalise these formulas to the case of an arbitrary splitting point.

In this preprint, we look onto the theory of linear thermoelasticity. At the beginning, this theory is shortly repeated and afterwards applied to transversely isotropic materials. Then, the corresponding weak formulation is derived, which is the starting point for a FE-discretisation. In the last part, we explain how we added this material behaviour to an adaptive Finite-Element-code and show some numerical results.

We are investigating a kernel-based autoconvolution problem, which has its origin in the physics of ultra short laser pulses. The task in this problem is to reconstruct a complex-valued function $x$ on a finite interval from measurements of its absolute value and a kernel-based autoconvolution of the form [[F(x)](s)=int k(s,t)x(s-t)x(t)de t.] This problem has not been studied in the literature. One reason might be that one has more information than in the classical autoconvolution case, where only the right hand side is available. Nevertheless we show that ill posedness phenomena may occur. We also propose an algorithm to solve the problem numerically and demonstrate its performance with artificial data. Since the algorithm fails to produce good results with real data and we suspect that the data for $|F(x)|$ are not dependable we also consider the whole problem with only $arg(F(x))$ given instead of $F(x)$.

In this paper we study the error behavior of the well known fast Fourier transform for nonequispaced data (NFFT) with respect to the L2-norm. We compare the arising errors for different window functions and show that the accuracy of the algorithm can be significantly improved by modifying the shape of the window function. Based on the considered error estimates for different window functions we are able to state an easy and efficient method to tune the involved parameters automatically. The numerical examples show that the optimal parameters depend on the given Fourier coefficients, which are assumed not to be of a random structure or roughly of the same magnitude but rather subject to a certain decrease.

Beschreibung der Finite Elemente Software-Familie SPC-PM3-AdH-XX für: (S)cientific (P)arallel (C)omputing - (P)rogramm-(M)odul (3)D (ad)aptiv (H)exaederelemente. Für XX stehen die einzelnen Spezialvarianten, die in Teil 2 detailliert geschildert werden. Stand: Ende 2013

In this thesis a theory for large deformation of plates is presented. Herein aspects of the common 3D-theory for large deformation with the Kirchhoff hypothesis for reducing the dimension from 3D to 2D is combined. Even though the Kirchhoff assumption was developed for small strain and linear material laws, the deformation of thin plates made of isotropic non-linear material was investigated in a numerical experiment. Finally a heavily deformed shell without any change in thickness arises. This way of modeling leads to a two-dimensional strain tensor essentially depending on the first two fundamental forms of the deformed mid surface. Minimizing the resulting deformation energy one ends up with a nonlinear equation system defining the unknown displacement vector U. The aim of this thesis was to apply the incremental Newton technique with a conformal, C^1-continuous finite element discretization. For this the computation of the second derivative of the energy functional is the key difficulty and the most time consuming part of the algorithm. The practicability and fast convergence are demonstrated by different numerical experiments.:1 Introduction 2 The deformation problem in the three-dimensional space 2.1 General differential geometry of deformation in the three-dimensional space 2.2 Equilibrium of forces 2.3 Material laws 2.4 The weak formulation 3 Newton’s method 3.1 The modified Newton algorithm 3.2 Second linearization of the energy functional 4 Differential geometry of shells 4.1 The initial mid surface 4.2 The initial shell 4.3 The plate as an exception of a shell 4.4 Kirchhoff assumption and the deformed shell 4.4.1 Differential geometry of the deformed shell 4.4.2 The Lagrangian strain tensor of the deformed plate 5 Shell energy and boundary conditions 5.1 The resulting Kirchhoff deformation energy 5.2 Boundary conditions 5.3 The resulting weak formulation 6 Newton’s method and implementation 6.1 Newton algorithm 6.2 Finite Element Method (FEM) 6.2.1 Bogner-Fox-Schmidt (BFS) elements 6.2.2 Hsiegh-Clough-Tocher (HCT) elements 6.3 Efficient solution of the linear systems of equation 6.4 Implementation 6.5 Newton’s method and mesh refinement 7 Numerical examples 7.1 Plate deflection 7.1.1 Approximation with FEM using BFS-elements 7.1.2 Approximation with FEM using reduced HCT-elements 7.2 Bending-dominated deformation 7.2.1 Approximation with FEM using BFS-elements 7.2.1.1 1st example: Cylinder 7.2.1.2 2nd example: Cylinder with further rotated edge normals 7.2.1.3 3rd example: Möbiusstrip 7.2.1.4 4th example: Plate with twisted edge 7.2.2 Approximation with FEM using reduced HCT-elements 7.2.2.1 1st example: Partly divided annular octagonal plate 7.2.2.2 2nd example: Divided annulus with rotated edge normals 8 Outlook and open questions Bibliography Notation Theses List of Figures List of Tables