Particle-Reinforced Aluminium Solid Self-piercing Rivets for Joining Aluminium Alloy Sheets - Koch, Steffen; Weber, Joshua; Meschut, Gerson; Stadelmann, Claudia; Böhm, Wolfgang; Merklein, Marion (Proceedings in Engineering Mechanics)
The use of aluminium materials in the structural and bodywork areas of assemblies has proven to be a targeted option for lightweight design. How-ever, the reliable and cost-efficient joining of aluminium components remains a challenge. Mechanical joining methods, such as riveting, are frequently used in the automotive and aerospace construction industries. Rivets are made from heat-treated steels. Compared to steel, the use of aluminium materials for fasten-ers offers several advantages in terms of joining properties, particularly in terms of recyclability, corrosion resistance and reduced weight of the joined structure. Additionally, the manufacturing process is shorter since aluminium fasteners do not require coating. However, aluminium rivets can often not be used due to the insufficient mechanical strength of the fastener material in relation to the joining component materials. This study systematically investigates the requirements for using solid alu-minium self-piercing rivets. The influence of rivet geometry adjustments on the joint quality is analysed using numerical simulation. The results are used to derive and evaluate an optimised rivet geometry for joining pure aluminium sheets. On this basis, solid self-drilling rivets with optimised geometry are manufactured from particle-reinforced aluminium produced in a continuous extrusion process by machining. The integration of particles increases the material’s strength. Exper-imental tests are conducted to evaluate the use of optimised solid self-piercing rivets. The quality-relevant parameters are determined and evaluated based on macrographs of the joints.
Keywords: Solid self-piercing riveting cdot particle-reinforced aluminium cdot continuous powder extrusion cdot Joining technology cdot Rivet geometry cdot lightweight design

Which categories do prospective teachers apply when selecting tasks for the scalar product? - Herrmann, Janine (Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education)
Selecting tasks is a typical job in everyday teaching that requires teachers to have content-specific expertise, including categories to think about and perceive conceptual understanding. This contribution investigates the activation of prospective teachers’ categories when solving tasks on the scalar product and evaluating their relevance for teaching in upper secondary school. To further investigate, whether input on content-related aspects could be helpful to promote a professional justification for the selection of tasks, two groups with and without input are to be differentiated. Different ways of dealing with underlying concept elements and evaluating the relevance of tasks are discussed by examples.
Keywords: Teacher expertise; noticing; analytical vectorial geometry

Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification - Offen, Christian
We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schrödinger equation. The article constitutes an extension of our previous article arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes.
Keywords: System identification; inverse problem of variational calculus; Gaussian process; Lagrangian learning; physics informed machine learning; geometry aware learning

Student Teachers ’ Knowledge of Congruence before a University Course on Geometry - Hoffmann, Max; Biehler, Rolf (Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022))
Keywords: Teaching and learning of specific topics in university mathematics; Transition to; across and from university mathematics; Student Teachers; Geometry; Congruence; Double Discontinuity.

Poisson transforms for trees of bounded degree - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (Journal of Spectral Theory)
Keywords: Geometry and Topology; Mathematical Physics; Statistical and Nonlinear Physics

Symmetries of analytic curves - Hanusch, Maximilian (Differential Geometry and its Applications)
Keywords: Geometry and Topology; Analysis

Strength of self-piercing riveted Joints with conventional Rivets and Rivets made of High Nitrogen Steel - Uhe, Benedikt; Kuball, Clara-Maria; Merklein, Marion; Meschut, Gerson
The use of high-strength steel and aluminium is rising due to the intensified efforts being made in lightweight design, and self-piercing riveting is becoming increasingly important. Conventional rivets for self-piercing riveting differ in their geometry, the material used, the condition of the material and the coating. To shorten the manufacturing process, the use of stainless steel with high strain hardening as the rivet material represents a promising approach. This allows the coating of the rivets to be omitted due to the corrosion resistance of the material and, since the strength of the stainless steel is achieved by cold forming, heat treatment is no longer required. In addition, it is possible to adjust the local strength within the rivet. Because of that, the authors have elaborated a concept for using high nitrogen steel 1.3815 as the rivet material. The present investigation focusses on the joint strength in order to evaluate the capability of rivets in high nitrogen steel by comparison to conventional rivets made of treatable steel. Due to certain challenges in the forming process of the high nitrogen steel rivets, deviations result from the targeted rivet geometry. Mainly these deviations cause a lower joint strength with these rivets, which is, however, adequate. All in all, the capability of the new rivet is proven by the results of this investigation.
Keywords: Self-piercing Riveting; Joining Technology; Rivet Geometry; Rivet Material; High Nitrogen Steel; Joint Strength

On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry - Jean, Frédéric; Maslovskaya, Sofya; Zelenko, Igor (Geometriae Dedicata)
Keywords: Geometry and Topology

Improvement of a rivet geometry for the self-piercing riveting of high-strength steel and multi-material joints - Uhe, Benedikt; Kuball, Clara-Maria; Merklein, Marion; Meschut, Gerson (Production Engineering)
As a result of lightweight design, increased use is being made of high-strength steel and aluminium in car bodies. Self-piercing riveting is an established technique for joining these materials. The dissimilar properties of the two materials have led to a number of different rivet geometries in the past. Each rivet geometry fulfils the requirements of the materials within a limited range. In the present investigation, an improved rivet geometry is developed, which permits the reliable joining of two material combinations that could only be joined by two different rivet geometries up until now. Material combination 1 consists of high-strength steel on both sides, while material combination 2 comprises aluminium on the punch side and high-strength steel on the die side. The material flow and the stress and strain conditions prevailing during the joining process are analysed by means of numerical simulation. The rivet geometry is then improved step-by-step on the basis of this analysis. Finally, the improved rivet geometry is manufactured and the findings of the investigation are verified in experimental joining tests.
Keywords: Self-piercing riveting; Joining technology; Rivet geometry; Multi-material design; High-strength steel; Aluminium

On Efficiently Solvable Cases of Quantum k-SAT - Aldi, Marco; de Beaudrap, Niel; Gharibian, Sevag; Saeedi, Seyran (43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018))
The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere.
Keywords: search complexity; local Hamiltonian; Quantum SAT; algebraic geometry

Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci - McLachlan, Robert I; Offen, Christian (New Zealand Journal of Mathematics)
In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.
Keywords: Hamiltonian boundary value problems; singularities; conformal symplectic geometry; catastrophe theory; conjugate loci

On the semiclassical functional calculus for h-dependent functions - Küster, Benjamin (Annals of Global Analysis and Geometry)
Keywords: Geometry and Topology; Analysis

Shape-Dependent Transmittable Tangential Force of Wire Bond Tools - Althoff, Simon; Meyer, Tobias; Unger, Andreas; Sextro, Walter; Eacock, Florian (IEEE 66th Electronic Components and Technology Conference)
Wire bonding has been an established packaging technology for decades. When introducing copper as wire material for high power applications, adaptations to the bonding process and to machines became necessary. Here, challenges occur due to the stiffer wire material and changing oxide layers on the contact partners. To achieve sufficient process stability, a clean bond area is required, which can only be achieved with high shear stresses in the contact partners surfaces. These necessitate high normal forces to plastically deform the wire and substrate. To achieve such high stresses in the contact area, the bonding tool needs to be able to transmit the needed tangential forces to the top side of the wire. The wire itself performs a shear movement and transmits the force into the contact area to clean the contaminant and oxide layers and to level the desired bond surfaces. The main function of the tool is to transmit these forces. If the bond tool can only transmit low forces in the direction of excitation, the parameter space for a stable bond process is severely restricted. Here, a modeling approach to estimate how well different tool shapes meet the demand of transmitting high tangential forces is presented. The model depends on wire deformation and thus on the ultrasonic softening effect.
Keywords: finite element simulation; wire bonding; tool geometry

Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions - Borthwick, David; Weich, Tobias (Journal of Spectral Theory)
Keywords: Geometry and Topology; Mathematical Physics; Statistical and Nonlinear Physics

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian - Rösler, Margit; Voit, Michael (Symmetry, Integrability and Geometry: Methods and Applications)
Keywords: Geometry and Topology; Mathematical Physics; Analysis

DoA-Based Microphone Array Position Self-Calibration Using Circular Statistic - Jacob, Florian; Schmalenstroeer, Joerg; Haeb-Umbach, Reinhold (38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013))
In this paper we propose an approach to retrieve the absolute geometry of an acoustic sensor network, consisting of spatially distributed microphone arrays, from reverberant speech input. The calibration relies on direction of arrival measurements of the individual arrays. The proposed calibration algorithm is derived from a maximum-likelihood approach employing circular statistics. Since a sensor node consists of a microphone array with known intra-array geometry, we are able to obtain an absolute geometry estimate, including angles and distances. Simulation results demonstrate the effectiveness of the approach.
Keywords: Geometry calibration; microphone arrays; position self-calibration

Microphone Array Position Self-Calibration from Reverberant Speech Input - Jacob, Florian; Schmalenstroeer, Joerg; Haeb-Umbach, Reinhold (International Workshop on Acoustic Signal Enhancement (IWAENC 2012))
In this paper we propose an approach to retrieve the geometry of an acoustic sensor network consisting of spatially distributed microphone arrays from unconstrained speech input. The calibration relies on Direction of Arrival (DoA) measurements which do not require a clock synchronization among the sensor nodes. The calibration problem is formulated as a cost function optimization task, which minimizes the squared differences between measured and predicted observations and additionally avoids the existence of minima that correspond to mirrored versions of the actual sensor orientations. Further, outlier measurements caused by reverberation are mitigated by a Random Sample Consensus (RANSAC) approach. The experimental results show a mean positioning error of at most 25 cm even in highly reverberant environments.
Keywords: Unsupervised; geometry calibration; microphone arrays; position self-calibration

A hierarchical approach to unsupervised shape calibration of microphone array networks - Hennecke, Marius; Ploetz, Thomas; Fink, Gernot A.; Schmalenstroeer, Joerg; Haeb-Umbach, Reinhold (IEEE/SP 15th Workshop on Statistical Signal Processing (SSP 2009))
Microphone arrays represent the basis for many challenging acoustic sensing tasks. The accuracy of techniques like beamforming directly depends on a precise knowledge of the relative positions of the sensors used. Unfortunately, for certain use cases manually measuring the geometry of an array is not feasible due to practical constraints. In this paper we present an approach to unsupervised shape calibration of microphone array networks. We developed a hierarchical procedure that first performs local shape calibration based on coherence analysis and then employs SRP-PHAT in a network calibration method. Practical experiments demonstrate the effectiveness of our approach especially for highly reverberant acoustic environments.
Keywords: acoustic sensing tasks; array geometry; calibration; coherence analysis; hierarchical procedure; local shape calibration; microphone array networks; microphone arrays; network calibration method; sensor arrays; SRP-PHAT; unsupervised shape calibration

Sampling methods for shortest vectors, closest vectors and successive minima - Blömer, Johannes; Naewe, Stefanie (Theoretical Computer Science)
Keywords: Geometry of numbers; Lattices; Shortest vectors

A Limit Relation for Dunkl-Bessel Functions of Type A and B - Rösler, Margit; Voit, Michael (Symmetry, Integrability and Geometry: Methods and Applications)
Keywords: Geometry and Topology; Mathematical Physics; Analysis