Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Differential Geometry

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field

Advances in Discrete Differential Geometry

Differential Geometr

Implicit neural representations of sheet stamping geometries with small-scale features

Geometric deep learning models, like Convolutional Neural Networks (CNNs), show promise as surrogate models for predicting sheet stamping manufacturability but lack design variables essential for inverse problems like geometric optimisation. Recent developments in deep learning have enabled geometry generation from compact latent spaces that are suitable for optimisation. However, current methods do not accurately model small-scale geometric features that are crucial for stamping performance. This study proposes a new deep learning-based method to address this limitation and generate detailed stamping geometries for optimisation. Specifically, neural networks are trained to generate Signed Distance Fields (SDFs) for stamping geometries, where the zero-level-set of each SDF implicitly represents the generated geometry. A new training approach is proposed for generating SDFs of stamping geometries, which involves supervising geometric properties of the SDFs. A novel loss function is introduced that directly acts on the zero-level-set and places high emphasis on learning small-scale features. This approach is compared with the state-of-the-art approach DeepSDF by Park et al. (2019), which explicitly supervises SDF values using ground truth data. The geometry generation performance of networks trained using both approaches is evaluated quantitatively and qualitatively. The results demonstrate significantly greater geometric accuracy with the proposed approach, which can faithfully generate small-scale features. Further analysis of the new approach reveals an organised learned latent space and varying the network input generates high-quality geometries from this space. By integrating with CNN-based manufacturability surrogate models by Attar et al. (2021), this work could enable the first-ever manufacturability-constrained optimisation of arbitrary sheet stamping geometries, potentially reducing geometry design time and cost

Advances in Discrete Differential Geometry

Differential Geometr

Planar panels and planar supporting beams in architectural structures

In this article, we investigate geometric properties and modeling capabilities of quad meshes with planar faces whose mesh polylines enjoy the additional property of being contained in a single plane. This planarity is a major benefit in architectural design and building construction: If a structural element is contained in a plane, it can be manufactured on the ground without scaffolding and put into place as a whole. Further, the plane it is contained in serves as part of a so-called support structure. We discuss design of meshes under the requirement that one half of mesh polylines are planar (“P meshes”), and we also investigate the geometry and design of meshes where all polylines enjoy this property (“PP meshes”). We work in the space of planes and with appropriate transformations of that space. We also incorporate further properties relevant for architectural design, such as near-rectangular panels and repetitive nodes. We provide geometric insights, give explicit constructions, and show an approach to geometric modeling of both P meshes and PP meshes, in particular, the case of nearly rectangular panels