arXiv subjects

Lukas Schnelle

Publications and source records attributed to Lukas Schnelle.

4 recordsLinked to original sources

arXiv2026

An algorithmic approach for computing fundamental domains of crystallographic groups

A crystallographic group is a discrete subgroup of the Euclidean group $\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $Γ$ is infinite, computing fundamental domains of $Γ$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $Γ$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $Γ$ acting on $\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on these results, we design an algorithm for the computation of fundamental domains of crystallographic groups and exploit it to study the construction of topological interlocking assemblies.

math.MG↗

arXiv2025

On the impact of geometric variance on the performance of formed parts: A probabilistic approach on the example of airbag pressure bins

Scatter in properties resulting from manufacturing is a great challenge in lightweight design, requiring consideration of not only the average mechanical performance but also the variance which is done e.g., by conservative safety factors. One contributor to this variance is the inherent geometric variability in the formed part. To isolate and quantify this effect, we present a probabilistic numerical study, aiming to assess the impact of geometric variance on the resulting part performance. By modelling geometric deviations stochastically, we aim to establish a correlation between the variance in geometry with the resulting variance in performance. The study is done on the example of an airbag pressure bin, where a better understanding of this correlation is crucial, as it allows for the design of a lighter part without changing the manufacturing process. Instead, we aim to implement more targeted and effective quality assurance, informed by the performance impact of geometric deviations.

cs.CE↗

arXiv2025

Influence of a generative parameter on the mechanical performance of topological interlocking assemblies of a hexagonal block

A topological interlocking assembly is an arrangement of blocks, where all blocks are kinematically constrained by their neighboring blocks and a fixed frame. This concept has been known for a long time, attracting recent interest due to its advantageous mechanical properties, such as reusability, redundancy and limited crack propagation. New mathematical methods enable the generation of vast numbers of new topologically interlocking blocks. A natural next question is the quantification of the mechanical performance of these new blocks. We conduct a numerical study of topological interlocking assemblies whose blocks are constructed based on the hexagonal grid. By varying a design parameter used in the generation of these blocks, we study its influence on the structural performance of the entire assembly. The results improve our understanding of the link between the block parameters and the mechanical performance. This enhances the ability to custom design blocks for certain mechanical requirements of the topological interlocking assemblies.

cond-mat.mtrl-sci↗

arXiv2023

Mechanical Comparison of Arrangement Strategies for Topological Interlocking Assemblies

Topological Interlocking assemblies are arrangements of blocks kinematically constrained by a fixed frame, such that all rigid body motions of each block are constrained only by its permanent contact with other blocks and the frame. In the literature several blocks are introduced that can be arranged into different interlocking assemblies. In this study we investigate the influence of arrangement on the overall structural behaviour of the resulting interlocking assemblies. This is performed using the Versatile Block, as it can be arranged in three different doubly periodic ways given by wallpaper symmetries. Our focus lies on the load transfer mechanisms from the assembly onto the frame. For fast a priori evaluation of the assemblies we introduce a combinatorial model called Interlocking Flows. To investigate our assemblies from a mechanical point of view we conduct several finite element studies. These reveal a strong influence of arrangement on the structural behaviour, for instance, an impact on both the point and amount of maximum deflection. The results of the finite element analysis are in very good agreement with the predictions of the Interlocking Flow model. Our source code, data and examples are available under https://doi.org/10.5281/zenodo.10246034.

cs.CE↗