Global Geometry Research Articles

Wess-Zumino-Witten terms of Sp QCD by bordism theory

We investigate the four-dimensional Wess-Zumino-Witten (WZW) terms within the framework of Sp quantum chromodynamics (QCD) using invertible field theory through bordism theory. We present a novel approach aimed at circumventing both perturbative and non-perturbative gauge anomalies on spacetime manifolds endowed with spin structures. We study both ungauged and gauged WZW terms including the problems of the topological consistency of gauged WZW terms.

The effect of productive and quality deposition strategies on residual stress for Directed Energy Deposition (DED) process

In the Directed Energy Deposition (DED) realm, challenges persist concerning process stability and mechanical properties in as-built components. Facts like inconsistent laser head feed rate and accumulated heat during multi-layer deposition result in variations in melt pool energy, causing variant clad widths and heights. Moreover, significant residual stress in as-built DED parts can induce excessive distortions and cracking. To tackle these challenges, offline energy control systems that adjust laser power or feed rate have been implemented to manage melt pool energy and improve productivity. Additionally, an innovative online clad height compensation system has been developed to achieve global height geometry, further boosting productivity. However, the impact of these productive and quality control systems on the mechanical properties of austenitic stainless-steel parts, specifically residual stress, remains unexplored. In this research, residual stress measurements in as-built DED thin wall parts fabricated with energy and/or height control strategies using neutron diffraction have been investigated. The control of three key DED parameters influencing residual stress distribution and magnitude has been identified and discussed. Measurements indicate that neither offline energy control nor online height control adversely affects the quality of the thin wall parts and offline LP-based energy control can reduce residual stress magnitude and its variation. Additionally, the height control results in compressive residual stress with elevated magnitudes. Furthermore, aggressive DED parameters hold the potential for further improvement by pairing with an energy control strategy. This study establishes a foundation for further developing comprehensive control strategies designed to enhance the productivity and quality of DED systems.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.

Space–time topology optimization for anisotropic materials in wire and arc additive manufacturing

Wire and Arc Additive Manufacturing (WAAM) has great potential for efficiently producing large metallic components. However, like other additive manufacturing techniques, materials processed by WAAM exhibit anisotropic properties. Assuming isotropic material properties in design optimization thus leads to less efficient material utilization. Instead of viewing WAAM-induced material anisotropy as a limitation, we consider it an opportunity to improve structural performance. This requires the integration of process planning into structural design. In this paper, we propose a novel method to utilize material anisotropy to enhance the performance of structures both during fabrication and in their use. Our approach is based on space–time topology optimization, which simultaneously optimizes the structural layout and the fabrication sequence. To model material anisotropy in space–time topology optimization, we derive the material deposition direction from the gradient of the pseudo-time field, which encodes the fabrication sequence. Numerical results demonstrate that leveraging material anisotropy effectively improves the performance of intermediate structures during fabrication as well as the overall structure.

Toroidal modified Miller-Turner CME model in EUHFORIA: Validation and comparison with flux rope and spheromak

Rising concerns about the impact of space-weather-related disruptions demand modelling and reliable forecasting of coronal mass ejection (CME) impacts. In this study, we demonstrate the application of the modified Miller-Turner (mMT) model implemented within EUropean Heliospheric FORecasting Information Asset (EUHFORIA) in forecasting the geo-effectiveness of observed coronal mass ejection (CME) events in the heliosphere. Our goal is to develop a model that not only has a global geometry, in order to improve overall forecasting, but is also fast enough for operational space-weather forecasting. We test the original full torus implementation and introduce a new three-fourths Torus version called the Horseshoe CME model. This new model has a more realistic CME geometry, and overcomes the inaccuracies of the full torus geometry. We constrain the torus geometrical and magnetic field parameters using observed signatures of the CMEs before, during, and after the eruption. We perform EUHFORIA simulations for two validation cases -- the isolated CME event of 12 July 2012 and the CME--CME interaction event of 8-10 September 2014. We performed an assessment of the model's capability to predict the most important $B_z$ component using the advanced dynamic time-warping (DTW) technique. The Horseshoe model predictions of CME arrival time and geo-effectiveness for both validation events compare well with the observations and are weighed against the results obtained with the spheromak and FRi3D models, which were already available in EUHFORIA. The runtime of the Horseshoe model simulations is close to that of the spheromak model, which is suitable for operational space weather forecasting. However, the capability of the magnetic field prediction at 1 AU of the Horseshoe model is close to that of the FRi3D model. In addition, we demonstrate that the Horseshoe CME model can be used for simulating successive CMEs in EUHFORIA, overcoming a limitation of the FRi3D model.

Static perfect fluid spacetime on Riemannian manifolds admitting concurrent-recurrent vector field with Bach tensor

Abstract In this paper, we first consider the 𝕊 ⁢ ℙ ⁢ 𝔽 {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M. Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a 𝕊 ⁢ ℙ ⁢ 𝔽 {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold.

Energetics and spectroscopic studies of clusters and the temperature dependencies of the isomers: An approach based on a combined recipe of parallel tempering and quantum chemical methods.

A system associated with several number of weak interactions supports numerous number of stable structures within a narrow range of energy. Often, a deterministic search method fails to locate the global minimum geometry as well as important local minimum isomers for such systems. Therefore, in this work, the stochastic search technique, namely parallel tempering, has been executed on the quantum chemical surface of the system for -8 to generate global minimum as well as several number of local minimum isomers. IR spectrum can act as the fingerprint property for such system to be identified. Thus, IR spectroscopic features have also been included in this work. Vertical detachment energy has also been calculated to obtain clear information about number of water molecules in several spheres around the central anion. In addition, in a real experimental scenario, not only the global but also the local minimum isomers play an important role in determining the average value of a particular physically observable property. Therefore, the relative conformational populations have been determined for all the evaluated structures for the temperature range between 20K and 400K. Further to understand the phase change behavior, the configurational heat capacities have also been calculated for different sizes.

Regularization in space–time topology optimization for additive manufacturing

In additive manufacturing, the fabrication sequence has a large influence on the quality of manufactured components. While planning of the fabrication sequence is typically performed after the component has been designed, recent developments have demonstrated the possibility and benefits of simultaneous optimization of both the structural layout and the corresponding fabrication sequence. This is particularly relevant in multi-axis additive manufacturing, where rotational motion offers enhanced flexibility compared to planar fabrication. The simultaneous optimization approach, called space–time topology optimization, introduces a pseudo-time field to encode the manufacturing process order, alongside a pseudo-density field representing the structural layout. To comply with manufacturing principles, the pseudo-time field needs to be monotonic, i.e., free of local minima. However, explicitly formulated constraints proposed in prior work are not always effective, particularly for complex structural layouts that commonly result from topology optimization.In this paper, we introduce a novel method to regularize the pseudo-time field in space–time topology optimization. We conceptualize the monotonic additive manufacturing process as a virtual heat conduction process starting from the surface upon which a component is constructed layer by layer. The virtual temperature field, which shall not be confused with the actual temperature field during manufacturing, serves as an analogy for encoding the fabrication sequence. In this new formulation, we use local virtual heat conductivity coefficients as optimization variables to steer the temperature field and, consequently, the fabrication sequence. The virtual temperature field is inherently free of local minima due to the physics it resembles. We numerically validate the effectiveness of this regularization in space–time topology optimization under process-dependent loads, including gravity and thermomechanical loads.

Binary segmentation of relief patterns on point clouds

Analysis of 3D textures, also known as relief patterns is a challenging task that requires separating repetitive surface patterns from the underlying global geometry. Existing works classify entire surfaces based on one or a few patterns by extracting ad-hoc statistical properties. Unfortunately, these methods are not suitable for objects with multiple geometric textures and perform poorly on more complex shapes. In this paper, we propose a neural network for binary segmentation to infer per-point labels based on the presence of surface relief patterns. We evaluated the proposed architecture on a high resolution point cloud dataset, surpassing the state-of-the-art, while maintaining memory and computation efficiency.

Poisson electrodynamics with charged matter fields

Poisson electrodynamics is the low-energy limit of a rank-one noncommutative gauge theory. It admits a closed formulation in terms of a Poisson structure on the space-time manifold and reproduces ordinary classical electrodynamics in the commutative limit. In this paper, we address and solve the problem of minimal coupling to charged matter fields with a proper commutative limit. Our construction relies essentially on the geometry of symplectic groupoids and works for all integrable Poisson manifolds. An additional advantage of our approach is that the corresponding Lagrangians can be defined on an arbitrary metric background.

Transferable, Living Data Sets for Predicting Global Minimum Energy Nanocluster Geometries.

Modeling of nanocluster geometries is essential for studying the dependence of catalytic activity on the available active sites. In heterogeneous catalysis, the interfacial interaction of the support with the metal can result in modification of the structural and electronic properties of the clusters. To tackle the study of a diverse array of cluster shapes, data-driven methodologies are essential to circumvent prohibitive computational costs. At their core, these methods require large data sets in order to achieve the necessary accuracy to drive structural exploration. Given the similarity in binding character of the transition metals, cluster shapes encountered for various systems show a large amount of overlap. This overlap has been utilized to construct a living data set which may be carried over across multiple studies. Iterative refinement of this data set provides a low-cost pathway for initialization of cluster studies. It is shown that utilization of transferable structural information can reduce model construction costs by more than 90%. The benefits of this approach are particularly notable for alloy systems, which possess significantly larger configurational spaces compared to the pure-phase counterparts.

Analysis of Planar Double-Layer Timber Spatial Frames by Using Parametric Tools

It is in the preliminary design phase of a project that the designer makes decisions concerning the global geometry of the structure. When working with space frames, the choice of the frame topology is key for the structural behavior. It is difficult to find manuals that provide guidance on which of the most common topologies is the right one for the project, let alone in wood construction. In response to this shortcoming, the use of parametric software is proposed (Grasshopper build1.0.0007 and Karamba 3D 2.2.0.16-220828). The aim is to create a dynamic catalog that responds instantaneously to changes in the parameters to provide information on structural behavior, pre-dimensioning and metrics. With the display of all this information, the architect will have enough technical argumentation to choose or reject options. The proposal is developed through a case study: the early design and analysis stages of flat double-layer timber spatial frames as for rectangular medium-span roofs.

Reductive quotients of klt singularities

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} endowed with the action of a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document} and admitting a quasi-projective good quotient X→X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X\\rightarrow X/\\!/G$\\end{document}, we can find a boundary B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$B$\\end{document} on X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/G$\\end{document} so that the pair (X//G,B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(X/\\!/G,B)$\\end{document} is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document}-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-dimensional K-polystable smooth Fano varieties of volume v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

Time-reversed information flow through a wormhole in scalar–tensor gravity

This Letter aims to advance unexplored properties of a new class of Closed Timelike Curves recently discovered in scalar–tensor gravity, reported in Nguyen and Lobo (2023) [31] and Nguyen and Azreg-Aïnou (2023) [30]. Therein, it was shown that when the Weak Energy Condition is violated, the topology of spacetime in scalar–tensor gravity is altered and directly enables the formation of two-way traversable wormholes. Furthermore, each of these wormholes acts a gateway between two time-mirrored worlds, where the two asymptotically flat sheets in the Kruskal–Szekeres diagram are glued antipodally along three directions—time t and the polar and azimuth angles (θ,φ) of the 2-sphere—to form a wormhole throat. This contrasts with the standard embedding diagram which typically glues the sheets only along the θ and φ directions. Crucially, due to the ‘gluing’ along the t direction, the wormhole becomes a portal connecting the two spacetime sheets with opposite physical time flows, enabling the emergence of closed timelike loops which straddle the throat. In this Letter, we shall point out that this portal mathematically permits the possibility of backward propagation of information against time. This feature is ubiquitous for wormholes in scalar–tensor theories. In addition, we formulate the Feynman sum for transition amplitudes of microscopic particles in the proximity of a wormhole throat in which we account for timelike paths that experience time reversal.

Space–time isogeometric analysis of tire aerodynamics with complex tread pattern, road contact, and tire deformation

AbstractThe space–time (ST) computational method “ST-SI-TC-IGA” and recently-introduced complex-geometry isogeometric analysis (IGA) mesh generation methods have enabled high-fidelity computational analysis of tire aerodynamics with near-actual tire geometry, road contact, tire deformation, and aerodynamic influence of the car body. The tire geometries used in the computations so far included the longitudinal and transverse grooves. Here, we bring the tire geometry much closer to an actual tire geometry by using a complex, asymmetric tread pattern. The complexity of the tread pattern required an updated version of the NURBS Surface-to-Volume Guided Mesh Generation (NSVGMG) method, which was introduced recently and is robust even in mesh generation for complex shapes with distorted boundaries. The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). They all play a key role. The ST-TC, uniquely offered by the ST framework, enables moving-mesh computation even with the topology change created by the contact between the tire and the road. It deals with the contact while maintaining high-resolution flow representation near the tire.The computational analysis we present is the first of its kind and shows the effectiveness of the ST-SI-TC-IGA and NSVGMG in tire aerodynamic analysis with complex tread pattern, road contact, and tire deformation.

Constraints on the topology of Type IIB string theory

We discuss some topological constraints on Type IIB string theory that cannot be described by elementary characteristic classes. Anomaly cancellation of the worldvolume theory of a D3-brane requires a shifted Dirac quantization condition of the Ramond-Ramond 5-form flux. However, the required shift is sometimes inconsistent depending on the topology of spacetime backgrounds. The obstruction to the existence of a shifted quantization is described by a degree-6 cohomology class whose definition involves spin structure of spacetime manifolds. The cohomology class is interpreted as a background D3-charge, and the Gauss law constraint requires inclusion of additional D3-branes to cancel it.

Hyper-Kähler instantons, symmetries, and flat spaces

We find all hyper-Kähler 4-manifolds admitting conformal Kähler structures with respect to either orientation, and we show that these structures can be expressed as a combination of twistor elementary states (and possibly a self-dual dyon) in locally flat spaces. The complex structures of different flat pieces are not compatible however, reflecting that the global geometry is not a linear superposition. For either orientation, the space must be Gibbons–Hawking (thus excluding the Atiyah–Hitchin metric), and, if the orientations are opposite, it must also be toric and have an irreducible Killing tensor. We also show that the only hyper-Kähler 4-metric with a non-constant Killing–Yano tensor is the half-flat Taub–NUT instanton.

The Einstein–Hilbert action for entropically dominant causal sets

In the path integral formulation of causal set quantum gravity, the quantum partition function is a phase-weighted sum over locally finite partially ordered sets, which are viewed as discrete quantum spacetimes. It is known, however, that the number of ‘layered’ sets—a class of causal sets that look nothing like spacetime manifolds—grows superexponentially with the cardinality n, giving an entropic contribution that can potentially dominate that of the action. We show here that in any dimension, the discrete Einstein–Hilbert action for a typical K-layered causal set reduces to the simple link action to leading order in n. Combined with earlier work, this completes the proof that the layered sets, although entropically dominant, are very strongly suppressed in the path sum of causal set quantum gravity whenever the discreteness scale is greater than or equal to a (mildly dimension-dependent) order one multiple of the Planck scale.

Bayesian Uncertainty Estimation by Hamiltonian Monte Carlo: Applications to Cardiac MRI Segmentation

Deep learning (DL)-based methods have achieved state-of-the-art performance for many medical image segmentation tasks. Nevertheless, recent studies show that deep neural net- works (DNNs) can be miscalibrated and overconfident, leading to ”silent failures” that are risky for clinical applications. Bayesian DL provides an intuitive approach to DL failure de- tection, based on posterior probability estimation. However, the posterior is intractable for large medical image segmentation DNNs. To tackle this challenge, we propose a Bayesian learning framework using Hamiltonian Monte Carlo (HMC), tempered by cold posterior (CP) to accommodate medical data augmentation, named HMC-CP. For HMC compu- tation, we further propose a cyclical annealing strategy, capturing both local and global geometries of the posterior distribution, enabling highly efficient Bayesian DNN training with the same computational budget as training a single DNN. The resulting Bayesian DNN outputs an ensemble segmentation along with the segmentation uncertainty. We evaluate the proposed HMC-CP extensively on cardiac magnetic resonance image (MRI) segmentation, using in-domain steady-state free precession (SSFP) cine images as well as out-of-domain datasets of quantitative T1 and T2 mapping. Our results show that the proposed method improves both segmentation accuracy and uncertainty estimation for in- and out-of-domain data, compared with well-established baseline methods such as Monte Carlo Dropout and Deep Ensembles. Additionally, we establish a conceptual link between HMC and the commonly known stochastic gradient descent (SGD) and provide general insight into the uncertainty of DL. This uncertainty is implicitly encoded in the training dynamics but often overlooked. With reliable uncertainty estimation, our method provides a promising direction toward trustworthy DL in clinical applications. We release our code in <a href='https://gitlab.tudelft.nl/yidongzhao/hmc_uncertainty'>https://gitlab.tudelft.nl/yidongzhao/hmc_uncertainty</a>

Experimental investigation of the characteristics of flame geometry and heat transfer from wind-driven pool fires

Wind-driven pool fire is a frequently encountered thermal hazard in cases of liquid fuel leakage accidents. Given the intricate interplay between wind flow and thermal buoyancy, understanding the flame geometry and external heat transfer behavior of pool fires is vital for effective liquid fuel security management. This paper presents a study to investigate these characteristics under various square pool sizes (side length, D = 0.31, 0.38, 0.44, 0.49, and 0.54 m) and wind speeds (u = 0, 0.5, 1, 2, 4, and 6 m/s). Five parameters pertaining to global flame geometry and two parameters concerning external heat transfer were measured. The findings indicate that the change rate of overall flame geometry parameters accelerates at first and then decelerates with wind speed. The ratio of the net flame horizontal length dominated by inertia force (Li) and the net flame horizontal length dominated by buoyancy force (Lb) to the total horizontal flame length (Lh) (Li / Lh and Lb / Lh) present a non-monotonic trend with increasing Fr number. To further comprehend the global flame geometry, a corresponding physical model is developed based on the ratio of Grs number to the 5/2 power of Rem number (Grs / Rem5/2). Due to the effects of flame thickness and flame attachment length, the variation in radiant heat transfer with wind speed can be bifurcated into two categories. The convective heat coefficient increases with wind speed, enhancing the local convection. Finally, the new models of radiant and convective heat transfers are established using dimensionless wind speed (u*, u∗=u(ṁ″g/ρ0)1/3) and Ri number. To ascertain the suitability of these new models, they have been validated against two classic flame models.