Optimization in geometric materials

This thesis aims to investigate particular geometric materials with algebro-geometric constraints by combining physically meaningful geometric models with coarse discretizations and robust optimization routines. Ultimately, this makes the investigation of previously inaccessible, increasingly complex structures feasible. It is a promising endeavor to explore robust numerical optimization techniques in conjunction with geometric models since the main objective in many physical systems is the minimization of energy. Energies occur in various real-world scenarios, be it facet tension in soap film, intermolecular pair potentials, entropy, and the elastic energy occurring in a material's deformation. All these different types of energy are discussed in some form in this thesis. With primary applications in civil engineering and architecture, framework materials are used to investigate both a structure's rigidity and flexibility. A large class of geometric constraint systems can be modeled by framework materials. These structures are given by embedded graphs with their edges equipped with physical distance constraints. Via the Euclidean distance between points, their deformations provide a compelling connection between materials science and algebraic geometry. When adding elastic elements carrying intrinsic energy to the framework, a tensegrity framework is created, paradoxically stabilizing an otherwise flexible structure. This work considers two distinct applications of rigidity theory. Firstly, we generate random point processes in the square to experimentally verify asymptotic bounds for the local and global rigidity of the corresponding Gilbert frameworks. This leads to an enhanced understanding of what typically prevents a graph's unique realizability in the Euclidean plane. Secondly, the regular dodecahedron with edge length and coplanarity constraints is examined. This polytope is rigid and we can locally describe the deformation space induced by edge-contractions. To study material deformations effectively, a reliable and robust method for traversing the configuration space is essential. For this reason, the metric projection onto the constraint set is considered, matching geodesics up to second order. By using homotopy continuation to solve the implicit closest point problem, we can prove theoretical convergence guarantees. This method relies on a predictor-corrector scheme based on Newton's method with adaptive step sizes. As such, it prevents the jumping between connected components. The retraction map is applied in Riemannian optimization routines after constructing appropriate step sizes and directions of descent. These algorithms provide an invaluable tool for the optimization over smooth constraint sets, which has led to successful applications in dynamical systems, robotics, and computer vision. Analyzing the algorithm's performance on multiple examples, Newton's method is more efficient than the predictor-corrector scheme. Still, the theoretical guarantees justify the computational overhead of the predictor step in many cases. Lipid bilayers minimize surface tension, suggesting that they can be modeled as minimal surfaces using Riemannian geometry. These membranes occur in many organic settings. For ordered phases with crystalline symmetry, periodic bicontinuous minimal surfaces provide established models. Conversely, a minimal surface model for amorphous phases does not exist. To sample the space of minimal surfaces, we propose two models for generating disordered interfaces in combination with a topology-stabilized curvature optimization routine. We provide experimental evidence for why disordered surfaces can form by investigating two homogeneity measures on all generated surfaces. In addition, our work demonstrates the degree to which the curvature homogeneity of the cubic Gyroid surface is superior to its entropy-favored amorphous counterparts. It thereby contributes a general geometric result of relevance to bicontinuous structure formation in soft matter and biology across all lengthscales, relevant to both ordered and amorphous phases. Triply periodic cylinder packings constitute another geometric system that we investigate in this thesis. Some of these packings exhibit a cooperative unwinding mechanism, reminiscent of auxeticity. This property describes an expansive lateral response upon stretching a material in a fixed direction, making such structures a prime target for the design of metamaterials with targeted functionality, such as impact protection and filtration. We create a robust geometric model based on tensegrity frameworks for the configuration space of two filaments in tight, orthogonal contact. The corresponding elastic elements are constructed by assuming that the incompressible cylinders want to minimize their length. It is used to compute the deformation paths of two particular cylinder packings via Riemannian optimization and homotopy-based path-tracking. As a result, we show that both structures are auxetic by multiple definitions. Finally, we generalize the theory of weavings to surfaces. Constructing a robust model based on tensegrity frameworks to study weavings in curved space results in a nonlinear optimization problem. This tensegrity model is suitable for analyzing tangled weaves and complex contact geometries. Our model is most robust on surfaces where all points are geometrically identical. It accurately simulates the behavior of weavings on the sphere and the flat torus. In order to find the weave that energetically prefers the sphere the most, we deform the underlying surface into a spheroid. The energy landscape resulting from this deformation reveals that the more regular the weave, the more it prefers the sphere.