Geometric Relationship Research Articles

GeoExplainer: Interpreting Graph Convolutional Networks with geometric masking

For Graph Convolutional Network (GCN), the inherent geometric topology and relationships within spatio-temporal graphs are critical to factors affect its predictive performance. Understanding the decision-making process of GCN remains a challenging task. Existing perturbation-based explanation methods are primarily tailored for Graph Neural Network (GNN) models and focus on how the information of nodes or subgraphs in static graphs impacts GNN predictions. Due to the geometric properties of spatio-temporal graphs, screw theory serves as crucial mathematical tools for describing the geometric relationships and patterns of motion within the geometric space of spatio-temporal graphs. Therefore,we propose GeoExplainer, which utilizes screw theory to geometric masking for spatio-temporal graphs with strong geometric correlations, generating explanations for GCN. Our method perturbs spatio-temporal graphs using geometric masking, and ultimately generating explanations in the form of geometric masks serve as importance mappings for model prediction results. Firstly, we design a screw motion vector model with geometric constraints to describe the spatio-temporal graph in screw system. Then, we design a geometric screw perturbation to generate smooth geometric masks. Finally, we propose a geometric masking approach to optimize and update the geometric mixture mask. We utilize pointwise mutual information (PMI) to maximize the correlation between the model predictions after geometric perturbation and the original predictions, obtaining the optimal geometric mask as an explanation for GCN. The interpretable experiments of ST-GCN, HD-GCN, GCN and GIN models on four datasets with geometric relationships demonstrate that GeoExplainer outperforms eight competitive interpretation methods in quantitative and visualization analysis.

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

Integrating local energetics into Maxwell-Calladine constraint counting to design mechanical metamaterials.

The Maxwell-Calladine index theorem plays a central role in our current understanding of the mechanical rigidity of discrete materials. By considering the geometric constraints each material component imposes on a set of underlying degrees of freedom, the theorem relates the emergence of rigidity to constraint counting arguments. However, the Maxwell-Calladine paradigm is significantly limited-its exclusive reliance on the geometric relationships between constraints and degrees of freedom completely neglects the actual energetic costs of deforming individual components. To address this limitation, we derive a generalization of the Maxwell-Calladine index theorem based on susceptibilities that naturally incorporate local energetic properties such as stiffness and prestress. Using this extended framework, we investigate how local energetics modify the classical constraint counting picture to capture the relationship between deformations and external forces. We then combine this formalism with group representation theory to design mechanical metamaterials where differences in symmetry between local energy costs and structural geometry are exploited to control responses to external forces.

Effect of Roundness Error of the Grooves on the Inner Ring Runout of Angular Contact Ball Bearings

In this paper, a prediction model of the inner ring runout of angular contact ball bearings is established according to the geometric and kinematic relationships of the bearing, considering factors such as the roundness error of the inner and outer grooves, the dimensional error of the balls, and the change of the contact angle between the balls and the grooves. The correctness of the model is verified through experiments. The effects of the order and amplitude of the roundness error of the inner groove and the order and amplitude of the roundness error of the outer groove on the inner ring runout are analyzed. The coupling effect of the roundness error of the inner and outer grooves on the inner ring runout is further analyzed. The results show that the inner ring runout changes periodically with a change to the roundness error order of the grooves, which increases with an increase in the roundness error amplitude. Under the coupling of the roundness error of the inner and outer grooves, the magnification of the inner ring runout increases as a whole. When there are specific relationships between the roundness error orders of the grooves and the number of balls, the magnification of the axial or radial runout changes significantly.

Development of a Simple Analytical Model to Facilitate Preoperative Surgical Planning in Valve-Sparing Aortic Root Replacement.

Valve-sparing root replacement (VSRR) is attractive for aortic root dilation as it preserves the native aortic valve (AoV). Low effective height (eH) after reconstruction is a risk factor for repair failure and reoperation. We developed and validated a quantitative AoV repair strategy to reliably restore normal valve proportions to promote long-term function. Normal AoV proportions were used to derive geometric relationships for sinotubular junction diameter (DSTJ), free edge length (FEL), free edge angle, and commissure height. These relationships informed two models for predicting eH following VSRR: (1) assuming valve symmetry and (2) accounting for valve asymmetry. Porcine heart (n = 6) ex vivo validation was performed under 4 VSRR scenarios: "Ideal" (tube graft size targeting FEL/DSTJ = 1.28), "Oversized" (one graft size larger than Ideal), "Undersized" (two sizes smaller), and "Undersized + Plicated" (FEL/DSTJ = 1.28 restored with leaflet plication). Our analytical models predicted eH using preoperative measurements and estimated reconstructed dimensions. The Oversized graft exhibited similar eH to Ideal but higher regurgitation in the ex vivo model, whereas the Undersized graft demonstrated lower eH and regurgitation. Plication in the Undersized graft restored valve function (regurgitation & eH) similar to Ideal in the ex vivo model and above Ideal in the analytical models. Both analytical models predicted ex vivo eH well except in the Oversized and Undersized + Plicated conditions. Utilizing measurements from preoperative imaging and simple mathematical models, patient-specific operative plans for VSRR can be created by estimating valve dimensions necessary to achieve favorable valve features post-repair. Clinical application of this approach promises to improve consistency in achieving optimal long-term dimensions and durability.

Partial order on involutive permutations and double Schubert cells

As shown by Melnikov, the orbits of a Borel subgroup acting by conjugation on upper-triangular matrices with square zero are indexed by involutions in the symmetric group. The inclusion relation among the orbit closures defines a partial order on involutions. We observe that the same order on involutive permutations also arises while describing the inclusion order on [Formula: see text]-orbit closures in the direct product of two Grassmannians. We establish a geometric relation between these two settings.

Representation-theoretical characterization of canonical custodial symmetry in NHDM potentials

By considering the basis-covariant constituents of N-Higgs-doublet potentials, we derive necessary and sufficient conditions for canonical SO(4)C Custodial Symmetry (CS) of potentials with N>2 doublets, based on representation-theoretical and geometrical relations. In essence, our characterization relates the presence of canonical CS to basis-covariant vectors corresponding to particular bases of the defining representation of the orthogonal Lie algebras. For N=3,4 and 5, the conditions demand little computational effort to be evaluated, and we provide practical algorithms that may be efficiently implemented in a computer program, for deciding whether or not a potential is custodial-symmetric.

A scaling relationship for the width of secondary deformation around strike-slip faults

Simple mechanical arguments suggest that slip along interlocked, rough faults, damages surrounding rocks. The same arguments require that the scale of secondary damage is proportional to the size of geometric irregularities along the main fault. This relationship could apply at all scales, but has, so far, been difficult to observe at the 10s to 100 s of km scales of large, natural faults, often because large-scale deformation is distributed across wide, complex plate-boundary fault systems, like the San Andreas Fault. The geometry and geology of another large-scale plate-boundary strike slip fault—the Queen Charlotte Fault (QCF)—is, in contrast, especially simple. Here, we show that observations of secondary deformation are well-aligned with predictions of stress variations caused by geometric irregularities along the QCF, suggesting a geometric relationship between primary fault geometry and secondary deformation. The analytic stress solution reveals that the highest stresses and highest likelihood of failure are confined to a zone of influence (ZOI) with a width quantified by ZOI=λ/2π, where λ is the wavelength of geometric variations along the main fault. This simple model is consistent with ∼100-km-scale observations along the QCF and can theoretically be used to predict the width of secondary deformation at all scales.

Are spatial terms rooted in geometry or force-dynamics? Yes.

Theories of spatial term meanings often focus on geometric properties of objects and locations as the key to understanding meaning. For example, in English, "The cat is on the mat" might engage geometric properties characterizing the figure ('cat', a point) and the ground ('mat', a plane) as well as the geometric relationship between the two objects ('on', + vertical, 0 distance from ground object). However, substantial literature suggests that geometric properties are far from sufficient to capture the meanings of many spatial expressions, and that instead, force-dynamic properties of objects that afford containment or support relationships may be crucial to the meanings of those expressions. I will argue that both approaches are needed to understand the variety of spatial terms that appear in language and further, that spatial terms fall into two distinct sets, one represented by geometric properties of figure and ground and their spatial relationships, and the other by the force-dynamic properties of objects and their relationships. This division of labor within spatial terms has many consequences, with the two types differing in the nature of the acquisition problem and likely learning mechanisms, the extent and kind of cross-linguistic variation that has been observed, and the application of pragmatic principles to spatial terms. Speculatively, the two types may also be rooted in different cognitive systems and their neural substrates.

Extended Kalman filter-based robust roll angle estimation method for spinning vehicles

Attitude measurement is a basic technique for monitoring vehicle motion states and safety. The spin motion of a vehicle couples the attitude angles with each other, which has an impact on the navigation and control of the vehicle. Global navigation satellite system (GNSS) signals-based roll angle measurement methods are important for vehicle attitude measurement. Most of existing studies use continuous signal power, but the case of loop lock loss leading to discontinuous power reception has not been considered. A robust estimation method for the roll angle based on the Tukey weight function is proposed to improve the measurement accuracy in cases of discontinuous reception. The characteristics of the GNSS signals, the geometric relationship between the signal power and roll angle of the vehicle are discussed. By installing a GNSS receiver with a single patched antenna on a rotating platform with a controllable rolling speed, the proposed method was verified by experiments. The robust estimation errors of different weight functions are analyzed. According to the characteristics of the gross measurement errors, a robust estimation method of multisatellite power observations is proposed to obtain a high-precision and stable estimation of the vehicle roll angle. The results show that the proposed algorithm can improve the accuracy of roll angle estimation even with gross measurement errors. As a result of the experiments, the estimation errors of the algorithm are 6.57° at a confidence level of 68 % and 15.49°at the confidence level of 95 %. In contrast, they are 11.38° and 37.31° for the traditional LS method. Moreover, the estimation accuracy of the algorithm is not significantly correlated with the vehicle rotational speed. Therefore, the vehicle roll angle can be estimated with high accuracy under a variety of rotational speeds.

Thermal expansion properties of double concave arrow honeycomb structure with double materials

This article proposes a kind of double concave double-arrow cell. The equivalent thermal expansion coefficient is obtained by using the geometric relation of deformation. The theoretical results agree well with the finite element results. The effect of material matching and geometry size on the equivalent thermal expansion coefficient is discussed. Based on the proposed two-material double-arrow cell, a two-material honeycomb structure is constructed. The thermal expansion coefficient, equivalent elastic modulus and Poisson’s ratio of the structure under thermal load are numerically discussed. It is found that the thermal expansion coefficient of the structure increases with the increase of the thermal expansion coefficient of the material, and the thermal expansion coefficient of the up arrow has a greater effect on the thermal expansion performance of the structure than that of the down arrow. Compared with the single material honeycomb structure, the double-material concave honeycomb structure does not produce the deformation mode of surrounding diffusion, and its longitudinal expansion is smaller. It shows that zero expansion can be achieved by adjusting the material and geometric parameters, and this structure provides a new idea for the design of other zero expansion structures.

Research on Multi Beam Sideline Model Based on Geometric Relationships

Research on Multi Beam Sideline Model Based on Geometric Relationships

Key indicators of caprock sealing assessment with consideration of faults in potential CO2 geological storage sites in Subei Basin, China

Good caprock sealing is crucial for ensuring the long-term safety and security of carbon dioxide (CO2) geological storage. However, concealed faults within caprocks and reservoirs can impact the transport and accumulation of CO2 plumes. Due to the uncertainties associated with the geometric structure, physical properties, and combined relationships among reservoir, caprock and fault, accurately evaluating caprock sealing is challenging. To address the challenge of identifying key indicators for caprock sealing considering faults, this study focuses on the one potential CO2 geological storage field in the Subei Basin, China. First, this study establishes a hydraulic-mechanical (HM) coupled finite element model considering a combination of caprock, reservoir and fault. Then, 22 research indicators, including fault offset, fault dip, caprock thickness, burial depth, permeability and mechanical properties, are analysed to evaluate their influence on caprock sealing via the tornado analysis and response surface methods. Finally, key indicators are determined based on evaluation criteria, such as pore pressure increment (ΔPP) and Coulomb failure stress (CFS), in the fault plane. The research results indicate that fault dip, fault offset, fault permeability, burial depth, and reservoir permeability are key indicators. The caprock thickness, caprock permeability, and caprock Young's modulus significantly influence the caprock sealing capacity. The ΔPP and CFS increase with increasing fault dip and offset. When the fault dip exceeds 45°, the ΔPP and CFS reach maximum values at the interface between the reservoir and caprock. Fault permeability has a greater impact than reservoir permeability and caprock permeability. The research results may provide guidance for the evaluation of caprock sealing capacity for CO2 geological storage.

Performance Improvement of Three-Spring Quasi-Zero-Stiffness Isolator by Shape Memory Alloy Damper

Quasi-zero stiffness vibration isolators (QZSI) can effectively resolve the contradiction between the load bearing and the vibration isolation for low-frequency vibration systems. The three-spring QZSI is a classical QZSI. However, constrained by its geometrical relationship, the working stroke of the QZSI is limited, and the isolation performance will deteriorate when the displacement response is significant. To solve this question, the shape memory alloy (SMA) damper, characterized by lower secondary stiffness, partially replaces the positive-stiffness spring of the three-spring QZS isolator to widen its effective working stroke, which forms a novel SMA-QZSI. Based on a piecewise linear constitutive model of SMA, a theoretical model of the force–displacement relationship of SMA-QZSI is established. Subsequently, the approximate analytic amplitude–frequency response equation of the non-smooth nonlinear SMA-QZSI system under harmonic excitation is obtained by the average method and verified numerically. In addition, the effects of mechanical parameters of SMA springs and external excitation parameters on mechanical behaviors and low-frequency isolation performance of the SMA-QZSI isolator are discussed. Finally, the static and dynamic experiments of SMA-QZSI are conducted to verify its force–displacement relationship and isolation performance. The results show that the proposed SMA-QZSI has remarkable low-frequency vibration isolation performance.

The radiative and geometric properties of melting first-year landfast sea ice in the Arctic

Abstract. In polar regions, sea ice is a crucial mediator of the interaction between Earth's atmosphere and oceans. Its formation and breakup is intimately connected with large-scale climatic processes, local weather patterns, and the use of sea ice in coastal Arctic regions by Indigenous people. In order to investigate the physical phenomena at the heart of this process, a set of targeted, intensive observations were made over spring sea ice melt and breakup in Kotzebue Sound, Alaska. These observations were planned and executed through a collaborative effort in which an Indigenous Elder advisory council from Kotzebue and scientists participated in co-production of hypotheses and observational research, including a stronger understanding of the physical properties of sea ice during spring melt. Here we present the results of observations performed using high-endurance, fixed-wing uncrewed aerial vehicles (UAVs) containing custom-built scientific payloads. Repeated flights over the measurement period captured the early stages of the transition from a white, snow-covered state to a broken-up, bare blue-green state. We found that the reflectance of sea ice features depends strongly on their size. Snow patches get darker as they get smaller, an effect owed to the geometric relationship between the bright interior and the darker, melting feature edges. Conversely, bare patches get darker as they get larger. For the largest ice features observed, bare blue-green ice patches were found to be ≈ 20 % less reflective than average across all observational cases, while large snowy white ice patches were found to be ≈ 20 % more reflective than that same average.

Monge Surfaces. Generation, Discretisation and Application in Architecture

AbstractThis paper describes a revision of so-called Monge surfaces through using digital graphic tools, based on the synthetic conception exposed by Gaspard Monge in the 18th and 19th centuries, through their geometric relation with polar surface. As a starting point, we propose a graphical system that integrally solves the generation of these kind of surfaces, their rationalization and discretization, with the objective of their specific application in architecture. The geometric system described allows the generation of a type of surface in which the pair of curves -generatrix and directrix- give rise to the network of principal lines of curvature (LCP) of the final surface. In this way, we propose a process that follows a bottom-up generative system based on Monge surfaces, which offers wide possibilities of formal exploration, while imposing geometrical-constructive properties that are very useful in fabrication and assembly processes.

Trajectory design based on frictional resistance of deflector in a hydraulic radial drilling system

The trajectory design is a key factor affecting the frictional resistance of the deflector, which is crucial for the drilling ability of the hydraulic radial drilling system. In this research, the trajectory design model and geometric constraints model of a deflector are established by analyzing the geometric relationship of the deflector and considering the geometric constraints of the casing. According to the actual contact form between the high-pressure hose and the deflector, the theoretical model of frictional resistance is built based on segmental rigidization and micro-element methods. The theoretical model is verified by the deflector friction experiment. The results demonstrate a good agreement between the experimental results of friction resistance of different sizes of high-pressure hoses and the theoretical results, with an error of less than 7%. The influence of critical parameters of deflector trajectory on the friction resistance is studied by coupling the geometric constraints model and the frictional resistance theoretical model of the deflector. The findings show that the deflector’s frictional resistance increases with the inclination angle β of the inclined-straight section and decreases with the increase in the radius R of the bending segment, the inclination angle ω of the straightening segment, and the channel radius of r deflector. A new trajectory parameters of the deflector are presented by taking a five-inch casing as an example, they are β = 9°, R = 92 mm, ω = 108°, r = 17 mm. The research results provide a theoretical rationale for designing a low-frictional resistance deflector.

Cross-region feature fusion with geometrical relationship for OCR-based image captioning

Automatically generating a readable sentence that describes the text-contained image is a challenging task. Compared to traditional image captioning algorithms, OCR-based image captioning focuses on reading OCR tokens in images and understanding them with the image content to generate descriptions. However, existing research mainly concentrate on improving the quantity and quality of obtaining OCR tokens and exploring their spatial relationships while lacking investigation into how to effectively join OCR tokens with image content. This paper proposes a cross-region feature fusion with a geometrical relationship Transformer(CFGR-Transformer) for OCR-based image captioning. The network first establishes the associations between the OCR and object regions of the image by constructing relative geometric relationships, including width/height difference, distance, IOU(Intersection over Union), inclusion relationships, and angles offset, and then incorporates intra-region and cross-region features to aggregate entities from different modalities by a multi-head attention mechanism based on relative relationships. Benefiting from the guidance of the relative relationship, visual entities like OCR tokens and object regions can consider multiple relative relationships as the attention weight for feature fusion within each subspace. Extensive experiments conducted on the TextCaps dataset demonstrate the effectiveness of the proposed CFGR-Transformer method. In particular, our results on the online testing of TextCaps achieve an improvement in CIDEr score from 93.0% to 98.2%.

Cotunneling Effects in the Geometric Statistics of a Nonequilibrium Spin‐Resolved Junction

AbstractIn the nonequilibrium steady state of electronic transport across a spin‐resolved quantronic junction, the role of cotunneling on the emergent statistics under phase‐different adiabatic modulation of the reservoirs' chemical potentials is investigated. By explicitly identifying the sequential and inelastic cotunneling rates, the geometric or Pancharatnam–Berry contribution to the spin exchange flux between the spin system and the right reservoir is numerically evaluated. The relevant conditions wherein the sequential and cotunneling processes compete and selectively influence the total geometric flux upshot are identified. The Fock space coherences are found to suppress the cotunneling effects when the system reservoir couplings are comparable. The cotunneling contribution to the total geometric flux can be made comparable to the sequential contribution by creating a right‐sided asymmetrically stronger system‐reservoir coupling strength. Using a recently proposed geometric thermodynamic uncertainty relationship, the total rate of minimal entropy production is estimated. The geometric flux and the minimum entropy is found to be nonlinear as a function of the interaction energy of the junctions' spin orbitals.

Semantic segmentation for large-scale point clouds based on hybrid attention and dynamic fusion

This paper investigates the semantic segmentation problem for large-scale point clouds. Recent segmentation methods usually employ an encoder–decoder architecture. However, these methods may not effectively extract neighboring information in the encoder. Additionally, they typically use nearest neighbor interpolation and skip connections in the decoder, overlooking the semantic gap between encoder and decoder features. To resolve these issues, we propose HADF-Net, which consists of a Hybrid Attention Encoder (HAE), an Edge Dynamic Fusion module (EDF), and a Dynamic Cross-attention Decoder (DCD). HAE leverages the distinctive properties of geometric and semantic relations to aggregate local features at different stages. EDF aims to alleviate information loss during decoder upsampling by dynamically integrating the neighboring information. DCD employs an enhanced fusion mechanism with spatial-wise cross-attention to bridge the semantic gap between encoder and decoder features. Experimental results on 4 datasets demonstrate that our HADF-Net achieves superior performance.