Solution of the Kirchhoff-Plateau problem

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments

Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure

Finite element simulation of nonlinear bending models for thin elastic rods and plates

Nonlinear bending phenomena of thin elastic structures arise in various modern and classical applications. Characterizing low energy states of elastic rods has been investigated by Bernoulli in 1738 and related models are used to determine configurations of DNA strands. The bending of a piece of paper has been described mathematically by Kirchhoff in 1850 and extensions of his model arise in nanotechnological applications such as the development of externally operated microtools. A rigorous mathematical framework that identifies these models as dimensionally reduced limits from three-dimensional hyperelasticity has only recently been established. It provides a solid basis for developing and analyzing numerical approximation schemes. The fourth order character of bending problems and a pointwise isometry constraint for large deformations require appropriate discretization techniques which are discussed in this article. Methods developed for the approximation of harmonic maps are adapted to discretize the isometry constraint and gradient flows are used to decrease the bending energy. For the case of elastic rods, torsion effects and a self-avoidance potential that guarantees injectivity of deformations are incorporated. The devised and rigorously analyzed numerical methods are illustrated by means of experiments related to the relaxation of elastic knots, the formation of singularities in a M\"obius strip, and the simulation of actuated bilayer plates.Comment: To appear in Handbook of Numerical Analysi

A methodological framework for the formulation of geometrically exact beam models

The theme of high flexible beams has received growing attention during the last decades and, in parallel, a number of beam models have been proposed in the last half century based on several modeling strategies. Aim of this dissertation is to describe the mathematical fundamentals and outline a methodological framework for formulating geometrically exact models for the analysis of beams undergoing large displacements. The modeling approach followed is the geometrically exact one. Is is based on a reduction process deriving the beam kinematics from the exact deformation analysis of a solid body. The beam model is derived by constraining the three-dimensional solid with the introduction of specific kinematic assumptions. The formulation leads to conceive the beam in terms of a three-dimensional orthogonal moving frame, with one of its axis remaining orthogonal to the beam cross-section in any configuration. This moving frame is also the reference system at which the resultant force and torque, acting on the typical cross-section, are evaluated. The geometric description of the beam model leads to a characterization of the beam cross-section configuration as an affine transformation within the physical space. Then, the space of the spatial proper rigid motions is assumed as the configuration space and the beam model is formulated in terms of curves of the special Euclidean Lie group, namely SE(3)

Tracing back the source of contamination

From the time a contaminant is detected in an observation well, the question of where and when the contaminant was introduced in the aquifer needs an answer. Many techniques have been proposed to answer this question, but virtually all of them assume that the aquifer and its dynamics are perfectly known. This work discusses a new approach for the simultaneous identification of the contaminant source location and the spatial variability of hydraulic conductivity in an aquifer which has been validated on synthetic and laboratory experiments and which is in the process of being validated on a real aquifer

Numerical Methods in Shape Spaces and Optimal Branching Patterns

The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound