This thesis addresses optimal control problems with elasticity for large deformations. A hyperelastic model with a polyconvex energy density is employed to describe the elastic behavior of a body. The two approaches to derive the nonlinear partial differential equation, a balance of forces and an energy minimization, are compared. Besides the conventional volume and boundary loads, two novel internal loads are presented. Furthermore, curvilinear coordinates and a hierarchical plate model can be incorporated into the formulation of the elastic forward problem. The forward problem can be solved with Newton\\\'s method, though a globalization technique should be used to avoid divergence of Newton\\\'s method. The repeated solution of the Newton system is done by a CG or MinRes method with a multigrid V-cycle as a preconditioner. The optimal control problem consists of the displacement (as the state) and a load (as the control). Besides the standard tracking-type objective, alternative objective functionals are presented for problems where a reasonable desired state cannot be provided. Two methods are proposed to solve the optimal control problem: an all-at-once approach by a Lagrange-Newton method and a reduced formulation by a quasi-Newton method with an inverse limited-memory BFGS update. The algorithms for the solution of the forward problem and the optimal control problem are implemented in the finite-element software FEniCS, with the geometrical multigrid extension FMG. Numerical experiments are performed to demonstrate the mesh independence of the algorithms and both optimization methods.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

In diesem Preprint werden grundlegende Gleichungen zur Behandlung von transversal isotropem Materialverhalten zusammengetragen. Wir betrachten ein transversal isotropes Materialgesetz mit linear elastischem Verhalten. Die angegebenen Materialgleichungen sind zur Beschreibung sowohl kleiner als auch großer Deformationen geeignet. Sie bilden eine wesentliche Grundlage zur Lösung statischer Probleme mit der Methode der finiten Elemente. Es werden Gleichungen für den ebenen Spannungszustand und den ebenen Verzerrungszustand hergeleitet.:1 Einführung 2 Energiefunktional 3 Umrechnung der Materialkonstanten 4 Elastizitätsmatrix 5 Eigenwerte 6 Ebener Verzerrungszustand 7 Ebener Spannungszustand 8 Anhang

By counting 1's in the "right half" of 2w consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w. Then the matrix can be correctly centered and factored into block-diagonal permutation matrices. Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined "at infinity" in general, but from only 2w rows for banded permutations.

We present a MPI based software library for computing the fast Fourier transforms on massively parallel, distributed memory architectures. Similar to established transpose FFT algorithms, we propose a parallel FFT framework that is based on a combination of local FFTs, local data permutations and global data transpositions. This framework can be generalized to arbitrary multi-dimensional data and process meshes. All performance relevant building blocks can be implemented with the help of the FFTW software library. Therefore, our library offers great flexibility and portable performance. Likewise FFTW, we are able to compute FFTs of complex data, real data and even- or odd-symmetric real data. All the transforms can be performed completely in place. Furthermore, we propose an algorithm to calculate pruned FFTs more efficiently on distributed memory architectures. For example, we provide performance measurements of FFTs of size 512^3 and 1024^3 up to 262144 cores on a BlueGene/P architecture.

The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a Bramble-Pasciak conjugate gradient method. With some modifications the simulation can be improved.:1. Introduction 2. Basics 3. Mixed variational formulation 4. Solution method 5. Error estimation 6. LBB conditions 7. Improvement suggestions

Adaptive anisotropic refinement of finite element meshes allows to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.:1 Introduction 2 Extension of FEM ansatz spaces 3 Optimality of the extension 4 Application 1: graded refinement 5 Application 2: anisotropic refinement in 2D 6 Numerical experiments 7 Conclusions and outlook

The topic of this thesis is the numerical simulation of transversely isotropic 3D structures and laminates by means of the adaptive finite element method. To achieve this goal, the theoretical background of elastic deformation problems, transverse isotropy, plate theory, and the classical laminate theory is recapitulated. The classical laminate theory implies a combination of the membrane problem and the plate problem with additional coupling terms. The focus of this work is the adjustment of two integral parts of the adaptive FE algorithm according to the classical laminate theory. One of these parts is the solution of the FE system; a good preconditioner is needed in order to use the conjugate gradient method efficiently. It is shown via a spectral equivalence bound that the combination of existing preconditioners for the membrane and plate problems poses a capable preconditioner for the combined laminate problem. The other part is the error estimation process; the error estimator determines where the current mesh has to be refined for the next step. Existing results on residual error estimators for the elasticity problem, the biharmonic problem, and the plate problem are combined and extended to obtain a posteriori local residual error indicators for the classical laminate theory problem. The effectiveness of both results is demonstrated by numerical examples.:1 Introduction 1.1 Motivation 1.2 Organisation of this work 1.3 Notation and basic definitions 2 Basic theory of 3D simulation 2.1 Differential geometry 2.1.1 Initial and deformed domain 2.1.2 Strain tensor 2.2 Energy functional 2.2.1 Linearly elastic material law 2.2.2 Equilibrium of forces 2.2.3 Large deformations 2.2.4 Small deformations 2.3 Voigt notation and elasticity matrix 3 Transversely isotropic material law 3.1 Elasticity tensor 3.2 Conversion of the material constants 3.3 Elasticity matrix 3.4 Eigenvalues 3.5 State of plane strain 3.6 State of plane stress 4 Plate theory and classical laminate theory 4.1 The Kirchhoff–Love hypothesis 4.2 Constitutive law and bilinear form of the laminated plate 4.3 Definition of resultants 4.4 Boundary conditions 4.5 From the equilibrium conditions to the weak formulation 4.5.1 Membrane equilibrium 4.5.2 Plate equilibrium 4.5.3 Combined weak formulation 4.5.4 The CLT problem in Voigt notation 5 Discretisation 5.1 Short introduction to FEM 5.2 Adaptive FEM 5.3 Finite elements for 3D elasticity problems 5.4 Finite elements for plates 5.4 Finite elements for plates 5.4.1 BFS rectangles 5.4.2 rHCT triangles 5.5 CLT elements 5.5.1 Rectangles 5.5.2 Triangles 6 Solver and preconditioner 6.1 The preconditioned conjugate gradient method 6.2 Hierarchical basis and BPX preconditioners 6.3 Preconditioning of CLT problems 6.3.1 General laminates 6.3.2 Some special cases and examples 7 A posteriori residual error estimation 7.1 Residual error estimator for 3D elements 7.2 Residual error estimator for plate and CLT elements 7.2.1 Auxiliary definitions and assumptions on the mesh 7.2.2 Interpolation operators 7.2.3 Important inequalities 7.2.4 Cut-off functions 7.2.5 Definition of the error 7.2.6 Reliability inequality 7.2.7 Efficiency inequality 8 Some details of the implementation 8.1 The adaptive FE package SPC-PM 8.2 Remarks on some added features 8.2.1 Capability of the current code 8.2.2 Cuntze’s failure mode concept 8.3 Coordinate transformation of higher-order derivatives 8.3.1 Mapping of coordinates 8.3.2 Transformation of derivatives of up to the third-order 8.3.3 Recursive construction of transformation matrices 8.3.4 Simplification for axis-parallel rectangles 9 Numerical examples 9.1 A three-dimensional example from eniPROD 9.2 Example problems for laminates 9.2.1 Rectangular plate under in-plane load 9.2.2 Rectangular plate under vertical load 9.2.3 L-shaped plate with inhomogeneous natural boundary conditions 10 Conclusion and outlook Bibliography Acknowledgements List of main symbols Theses

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. While typically B-splines are applied in the scope of particle mesh methods, we consider for the first time also an approximation by Bessel functions. We show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that if the parameters are tuned appropriately the Bessel window function can keep up with the B-spline window and is in many cases even the better choice with respect to computational costs.

Der Schwerpunkt der vorliegenden Arbeit liegt auf Methoden zur Lösung nichtlinearer Optimierungsprobleme mit der Anforderung, jedes globale Optimum garantiert zu finden und mit einer im Voraus festgesetzten Genauigkeit zu approximieren. Eng verbunden mit dieser deterministischen Optimierung ist die Berechnung von Schranken für den Wertebereich einer Funktion über einem gegebenen Hyperquader. Verschiedene Ansätze, z. B. auf Basis der Intervallarithmetik, werden vorgestellt und analysiert. Im Besonderen werden Methoden zur Schrankengenerierung für multivariate ganz- und gebrochenrationale Polynome mit Hilfe der Darstellung in der Basis der Bernsteinpolynome weiterentwickelt. Weiterhin werden in der Arbeit schrittweise die Bausteine eines deterministischen Optimierungsverfahrens unter Verwendung der berechneten Wertebereichsschranken beschrieben und Besonderheiten für die Optimierung polynomialer Aufgaben näher untersucht. Die Analyse und Bearbeitung einer Aufgabenstellung aus dem Entwicklungsprozess für Fahrzeuggetriebe zeigt, wie die erarbeiteten Ansätze zur Lösung nichtlinearer Optimierungsprobleme die Suche nach energieeffizienten Getrieben mit einer optimalen Struktur unterstützen kann. Kontakt zum Autor: [Nachname] [.] [Vorname] [@] gmx [.] de