Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Straining the Identity of Majorana Fermions

We propose an experimental setup of an interferometer for the observation of neutral Majorana fermions on topological insulator - superconductor - ferromagnet junctions. We show that the extended lattice defects naturally present in materials, dislocations, induce spin currents on the edges while keeping the bulk time-reversal symmetry intact. We propose a simple two terminal conductance measurement in an interferometer formed by two edge point contacts, which reveals the nature of Majorana states through the effect of dislocations. The zero temperature magneto-conductance changes from even oscillations with period phi/2 (phi is the flux quantum hc/e) to odd oscillations with period phi, when non-trivial dislocations are present and the Majorana states are sufficiently strongly coupled. Additionally, the conductance acquires a notable asymmetry as a function of the incident electron energy, due to the topological influence of the dislocations, while resonances appear at the coupling energy of Majorana states.Comment: 5 pages, 3 figures, three-point bending setup with Hg(Cd)Te analyze

TiGL - An Open Source Computational Geometry Library for Parametric Aircraft Design

This paper introduces the software TiGL: TiGL is an open source high-fidelity geometry modeler that is used in the conceptual and preliminary aircraft and helicopter design phase. It creates full three-dimensional models of aircraft from their parametric CPACS description. Due to its parametric nature, it is typically used for aircraft design analysis and optimization. First, we present the use-case and architecture of TiGL. Then, we discuss it's geometry module, which is used to generate the B-spline based surfaces of the aircraft. The backbone of TiGL is its surface generator for curve network interpolation, based on Gordon surfaces. One major part of this paper explains the mathematical foundation of Gordon surfaces on B-splines and how we achieve the required curve network compatibility. Finally, TiGL's aircraft component module is introduced, which is used to create the external and internal parts of aircraft, such as wings, flaps, fuselages, engines or structural elements

Parallel implementation of the non-smooth contact dynamics method for large particle systems

In numerous industrial applications there is the need to realistically model granular material. For instance, simulating the interaction of vehicles and tools with soil is of great importance for the design of earth moving machinery. The Discrete Element Method (DEM) has been successfully applied to this task [1, 2]. Large scale problems require a lot of computational resources. Hence, for the application in the industrial engineering process, the computational effort is an issue. In DEM parallelization is straight forward, since each contact between adjacent particles is resolved locally without regard of the other contacts. However, modelling a contact as a stiff spring imposes strong limitations on the time step size to maintain a stable simulation. The Non–Smooth Contact Dynamics Method (NSCD), on the other hand, models contacts globally as a set of inequality constraints on a system of perfectly rigid bodies [3]. At the end of every time step, all inequality constraints must be satisfied simultaneously, which can be achieved by solving a complementarity problem. This leads to a numerically stable method that is robust with respect to much larger time steps in comparison to DEM. Since a global problem must be solved, parallelization now strongly depends on the numerical solver that is used for the complementarity problem. We present our first massively parallel implementation of NSCD based on the projected Gauß-Jacobi (PGJ) iterative scheme presented in [4]. Focusing on one-sided asynchronous communication patterns with double buffering for data exchange, global synchronizations can be avoided. Only weak synchronization due to data dependencies of neighboring domains remains. The implementation is based on the Global address space Programming Interface (GPI), supplemented by the Multi Core Threading Package (MCTP) [5] on the processor level. This allows to efficiently overlap calculation and communication between processors

Type-II Bose-Mott insulators

The Mott insulating state formed from bosons is ubiquitous in solid He-4, cold atom systems, Josephson junction networks and perhaps underdoped high-Tc superconductors. We predict that close to the quantum phase transition to the superconducting state the Mott insulator is not at all as featureless as is commonly believed. In three dimensions there is a phase transition to a low temperature state where, under influence of an external current, a superconducting state consisting of a regular array of 'wires' that each carry a quantized flux of supercurrent is realized. This prediction of the "type-II Mott insulator" follows from a field theoretical weak-strong duality, showing that this 'current lattice' is the dual of the famous Abrikosov lattice of magnetic fluxes in normal superconductors. We argue that this can be exploited to investigate experimentally whether preformed Cooper pairs exist in high-Tc superconductors.Comment: RevTeX, 17 pages, 6 figures, published versio

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development

Universal probes of two-dimensional topological insulators: Dislocation and pi-flux

We show that the pi-flux and the dislocation represent topological observables that probe two-dimensional topological order through binding of the zero-energy modes. We analytically demonstrate that pi-flux hosts a Kramers pair of zero modes in the topological Gamma (Berry phase skyrmion at the zero momentum) and M (Berry phase skyrmion at a finite momentum) phases of the M-B model introduced for the HgTe quantum spin Hall insulator. Furthermore, we analytically show that the dislocation acts as a pi-flux, but only so in the M phase. Our numerical analysis confirms this through a Kramers pair of zero modes bound to a dislocation appearing in the M phase only, and further demonstrates the robustness of the modes to disorder and the Rashba coupling. Finally, we conjecture that by studying the zero modes bound to dislocations all translationally distinguishable two-dimensional topological band insulators can be classified.Comment: 5 pages, 2 figures; version accepted in Physical Review Letter

Mixed-integer programming techniques for the minimum sum-of-squares clustering problem

The minimum sum-of-squares clustering problem is a very important problem in data mining and machine learning with very many applications in, e.g., medicine or social sciences. However, it is known to be NP-hard in all relevant cases and to be notoriously hard to be solved to global optimality in practice. In this paper, we develop and test different tailored mixed-integer programming techniques to improve the performance of state-of-the-art MINLP solvers when applied to the problem—among them are cutting planes, propagation techniques, branching rules, or primal heuristics. Our extensive numerical study shows that our techniques significantly improve the performance of the open-source MINLP solver SCIP. Consequently, using our novel techniques, we can solve many instances that are not solvable with SCIP without our techniques and we obtain much smaller gaps for those instances that can still not be solved to global optimality

Constructing a Volume Geometry Map For Hexahedra With Curved Boundary Geometries (Presentation)

In (dynamic) adaptive mesh refinement (AMR), a given input mesh is refined and coarsened during the computation to optimally adapt the resolution of the computational mesh to specific requirements. The input mesh is often the output of a mesh generator and provides information on the geometry of the domain. It is desired to keep its resolution as coarse as possible in order to benefit from the AMR mesh hierarchy and efficient mesh indexing algorithms. We present a novel approach to equip the coarse mesh with high-order geometry data and evaluate this geometry on the fine mesh elements in order to ensure geometric accuracy of the refined mesh elements, even for coarse input meshes. To this end, we construct a volume geometry map for hexahedral cells from given curved boundary geometry data and discuss our implementation in a state-of-the-art AMR library