Geometric Procedure Research Articles

Analysis of a Dry Friction Force Law for the Covariant Optimal Control of Mechanical Systems with Revolute Joints

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh’s dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh’s potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb’s law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin’s principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control.

A deformation procedure using position-based dynamics to optimize the geometric model of woven fabrics

Generating realistic three-dimensional (3D) yarn-level models for fabrics has become a significant research topic due to the complexity of fabric structures and the requirement for high-quality models with no interpenetration between yarns. The generation process of idealized geometric models leads to inaccurate descriptions of yarn contact, specifically the interpenetrations and spurious voids between yarns. A geometric optimization procedure was developed to address the defects in the idealized models, involving a series of geometry-driven operations, such as the shrinking and expansion of yarn volume and the straightening of the yarn centerline, to obtain accurate and consistent fabric models for Finite Element Analysis. During the geometric optimization, a method for yarn deformation based on position-based dynamics (PBD) was applied to simulate the real deformation during yarn interweaving, ensuring no interpenetrations between yarns while preserving yarn volume. The accuracy of the model is validated by comparison with the real fabric images. Although the methods involved in the procedure are all geometric, their deformation results have realistic physical effects. In addition, the procedure can be applied to various woven fabric structures.

The Influence of Different Solar System Planetary Ephemerides on Pulsar Timing

Pulsar timing offers a comprehensive avenue for exploring diverse topics in physics and astrophysics. High-precision solar system planetary ephemeris is crucial for pulsar timing as it provides the positions and velocities of solar system planets including the Earth. However, it is inevitable that inherent inconsistencies exist in these ephemerides. Differences between various ephemerides can significantly impact pulsar timing and parameter estimations. Currently, pulsar timing highly depends on the JPL DE ephemeris, for instance, the Pulsar Timing Array data analysis predominantly utilizes DE436. In this study, we examine inconsistencies across various ephemeris series, including JPL DE, EPM, and INPOP. Notably, discrepancies emerge particularly between the current ephemeris DE436 and the earliest released ephemeris DE200, as well as the most recent ephemerides, e.g., DE440, INPOP21A, and EPM2021. Further detailed analysis of the effects of ephemeris on geometric correction procedures for the conversion of measured topocentric times of arrival is presented in this study. Our researches reveal that variations in the Roemer delays across different ephemerides lead to distinct differences. The timing residuals and the fact that these discrepancies can be readily incorporated into the subsequent pulsar parameters, leading to inconsistent fitting estimates, suggest that the influence of errors in the ephemeris on the timing process might currently be underappreciated.

Mathematical connections involved in area measurement processes

ABSTRACT The present study seeks to explore the mathematical connections that 13–14-year-old secondary school students establish when solving area tasks. Emphasis is placed on different mathematical objects, and the connections between them, that allow students to successfully solve the tasks. The study follows a mixed methodology using qualitative and quantitative method of analysis. The results show that representations play a key role in solving area tasks, as they condition the use of alternative procedures to the use of formulas. Likewise, the properties involved in area measurement processes may condition the use of geometric procedures, such as surface decomposition and reorganisation. Finally, results show that to accurately carry out area measurement processes, it is necessary to bring different mathematical objects into play simultaneously. If these connections between mathematical objects do not occur, there is a risk of using the formulas in a mechanical way.

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Impulsive gravitational waves are theoretical models of short but violent bursts of gravitational radiation. They are commonly described by two distinct spacetime metrics, one of local Lipschitz regularity and the other one even distributional. These two metrics are thought to be ‘physically equivalent’ since they can be formally related by a ‘discontinuous coordinate transformation’. In this paper we provide a mathematical analysis of this issue for the entire class of nonexpanding impulsive gravitational waves propagating in a background spacetime of constant curvature. We devise a natural geometric regularisation procedure to show that the notorious change of variables arises as the distributional limit of a family of smooth coordinate transformations. In other words, we establish that both spacetimes arise as distributional limits of a smooth sandwich wave taken in different coordinate systems which are diffeomorphically related.

Geometric optimisation of a multiple coiled-up resonator for broadband and octave band acoustic absorption

This study presents a comprehensive approach to designing sound absorption metamaterials, aiming to achieve broadband absorption capabilities while maintaining a compact thickness, starting from an exceptionally low frequency of 100 Hz. By employing a parallel arrangement of resonators with varying lengths, the proposed approach exploits the envelopes of absorption frequencies to realize broad-spectrum sound absorption.An analytical model is utilized in the geometric optimization phase, accounting for inlet losses and thermo-viscous losses near the walls. The absorption for multiple ducts coupled in parallel is derived using the series and parallel impedance method. A geometric parameterization procedure is implemented to maximize normal incidence acoustic absorption, leading to design optimization for both broadband and octave-band absorbers. Design maps are established to assess acoustic performance and physical space requirements, facilitating informed decision-making in metamaterial design.Prototype fabrication and experimental validation involve 3D printing of optimized geometries using polylactic acid, followed by testing within impedance tubes according to ISO 10534–2 standards. The results demonstrate the effectiveness of the proposed method in achieving broadband sound absorption across specified frequency ranges.

A numerical study on a novel demountable cold-formed steel composite beam with profiled steel sheeting

Cold-formed steel composite beams are known for their unique advantages, like being lightweight and ease of installation. The use of profiled steel sheeting in cold-formed composite beams reduces construction time and costs by acting as a permanent formwork in the composite beams. The current study presents a 3D finite element model of cold-formed steel composite beam specimens comprising a cold-formed double-lipped channel section, profiled steel sheeting, concrete slab, and bolted shear connector. Employing bolted shear connectors, structural components can be deconstructed and replaced after their service life expires or if they are damaged. The characteristics of the materials obtained from an experimental program were assigned to the finite element model. Geometric characteristics, material nonlinearities, and loading procedures were attentively simulated, and a dynamic explicit procedure was employed for the numerical analyses. A comparison of the results obtained from the finite element models and the available experimental results validated the precision of the models. Then, numerical studies were conducted to investigate the effects of various parameters, including compressive strength of concrete, thickness of concrete slab, height and grade of cold-formed steel section, thickness of profiled steel sheeting, number and diameter of shear connectors, on the behavior of the composite beam. The results showed that the height and grade of the cold-formed steel section and compressive strength and thickness of the concrete slab have a significant effect on increasing the capacity of the composite beam.

Cell-centered indirect Arbitrary Lagrangian-Eulerian numerical strategy for solving 3D gas dynamics equations

Solving the Euler equations under the Lagrangian formalism enables to simulate various complex engineering applications. However, the use of this formalism can lead to significant mesh deformations as the mesh follows the fluid velocity. The mesh quality may be considerably deteriorated requiring a regularization procedure. In the present document, it is shown that the ideas presented by J. Yao (2013)[68] and J. Yao and D. Stillman (2016) [30] may be used efficiently in a 3D hydrodynamics code to perform reliable regularization steps. The flexibility of the methodology in addition to its simplicity, since it only relies on trivial geometrical considerations, opens the way for various extensions. The remapping step then considered, uses the geometrical splitting procedure of the Lagrangian phase to perform effective projections. This coherence ensures the compactness of the overall algorithm. At last but not least, a 3D Flux-Corrected-Remapping method is presented. This yields a particularly robust remapping algorithm while also leading the way for higher order projection extensions.

Geometrical projection improved multi-objective particle swarm optimization for unsupervised nonlinear hyperspectral unmixing

ABSTRACT In recent years, a series of bilinear mixture models (BMMs) have been well studied, based on which nonlinear unmixing algorithms for hyperspectral images have been proposed. However, the performance of unsupervised nonlinear spectral unmixing can be largely affected by the collinearity and the nonconvex unmixing problem’s inherent complexity. To solve the problems, this paper proposes a geometrical projection improved multi-objective particle swarm optimization (GPMOPSO) method for nonlinear unmixing, which consists of two dynamically updating procedures. First, under the assumption of the BMMs, observed pixels are geometrically projected to their approximate linear mixture components to alleviate the collinearity. Second, the linear mixture components are efficiently decomposed to refined endmembers and abundances in an improved linear unmixing framework using multi-objective particle swarm optimization. Two advanced multi-objective optimization strategies are adopted to balance the influence of an endmember distance constraint and an abundance sparsity constraint on the achieved unsupervised linear unmixing process, respectively. Then, accurate unmixing variables can be fed back to the geometrical projection procedure to produce more reliable linear mixture components which facilitate the execution of the former in turn. Experimental results of both simulated data and real hyperspectral remote sensing data verify the superior nonlinear unmixing performance of the proposed method.

Finite Element modeling and convergence analysis of a new Biomimetic Branching Structure.

Branching structures are gaining popularity in the field of advanced structures and building design; they offer high performance in terms of strength and lightweight design, along with the flexibility and precision enabled by modern processing technologies like Additive Manufacturing. This paper provides a concise overview of a geometric design procedure for a novel ribbed class of structures which was previously developed by the authors as a biomimetic optimal Micro-architected dome. Hereinafter, linear lattice models are suggested to carry out structural calculations using the finite element method (FEM). The objective is to examine discretization thresholding and strain energy convergence criteria. Results show that convergence is reached for numbers of elements per leg, ranging from 2 to 6, depending on the geometrical configuration of the dome being studied. The strain energy balance also exposes the influence of each internal force on the total mechanical response of the structure, pinpointing bending moment and axial force as the main decisive factors. As a perspective, the study will focus on limit state design calculations and Analysis of how the local geometry influences the overall stability and strength of this new design.

Comparative analysis between spectral indices obtained in the Guangüiltagua metropolitan park in Quito – Ecuador, using remote sensing

The article presents an exhaustive comparative analysis of NDVI, SAVI and NDWI spectral indices, extracted from satellite images of the Guangüiltagua Metropolitan Park in Quito. The relevance of remote sensing, using satellites such as Sentinel-2, in the monitoring of biodiversity and environmental conditions is highlighted. The research emphasizes the importance of vegetation indices, especially NDVI, to assess vegetation health and detect changes in vegetation cover. The study discusses in detail vegetation sensitivity, water identification, water stress response, soil effect compensation, and specific applications of NDVI, SAVI, and NDWI. The ability of these indices to provide crucial information on ecosystem health, water presence, and potential water stress conditions in Metropolitan Park is highlighted. The methodology applied involves the use of the Copernicus platform and the Sentinel-2 satellite, with radiometric and geometric correction procedures, as well as atmospheric correction. The results obtained reveal high NDVI values, indicating a dense vegetation cover, while NDWI suggests potential areas of water scarcity. The comparative analysis between the indices offers a deeper understanding of the relationship between vegetation and water in the park.

Examination of Transmission Zeros in the MIMO Sensor-Based Propagation Environment Using a New Geometric Procedure

In this paper, we propose the application of a new geometric procedure in order to calculate a set of transmission zeros of a propagation environment. Since the transmission zeros play a crucial role in modern communication systems, there is a need to apply the efficient solutions characterized by a maximum speed operation. It turns out that the classical method based on the Smith–McMillan factorization is time-consuming, so its contribution to the detection of transmission zeros could be unsatisfactory. Therefore, in order to fill the gap, we present a new algorithm strictly dedicated to the multivariable telecommunications systems described by the transfer-function approach. Consequently, a set of new achievements resulted, particularly in terms of computational efforts. Indeed, the proposed procedure allows us to overcome obstacles derived from technological limitations. The representative simulation examples confirm the great potential of this new method. Finally, it has been pointed out that the newly introduced geometric-originated approach has significantly reduced the computational burden. Indeed, for the randomly selected matrix of the 5×5 dimension describing the sensor-related propagation environment, two representative scenarios were performed in order to manifest the crucial properties. In the first scenario, the sets of multiple transmission zeros were analyzed, ultimately leading to intriguing results. The Smith–McMillan solution took three times longer to discover the mentioned sets. On the other hand, the second instance brought us the same result. Naturally, the discussed difference has increased as a function of the number of matrix elements. For the square matrices involving 100 components, we have observed the respective differences, both over QI=100 and QII=60. It should be emphasized that the finding derived from the Smith–McMillan factorization corresponds to the geometric-related approach in the context of some mechanisms. This is particularly visible when appointing the greatest common divisors.

A layerwise geometric error compensation procedure for additive manufacturing

PurposeThe purpose of this paper is to provide a new procedure for in-plane compensation of geometric errors that often appear in the layers deposited by an additive manufacturing (AM) process when building a part, regardless of the complexity of the layer geometry.Design/methodology/approachThe procedure is based on comparing the real layer contours to the nominal ones extracted from the STL model of the part. Considering alignment and form deviations, the compensation algorithm generates new compensated contours that match the nominal ones as closely as possible. To assess the compensation effectiveness, two case studies were analysed. In the first case, the parts were not manufactured, but the distortions were simulated using a predictive model. In the second example, the test part was actually manufactured, and the distortions were measured on a coordinate measuring machine.FindingsThe geometric deviations detected in both case studies, as evaluated by various quality indicators, reduced significantly after applying the compensation procedure, meaning that the compensated and nominal contours were better matched both in shape and size.Research limitations/implicationsAlthough large contours showed deviations close to zero, dimensional overcompensation was observed when applied to small contours. The compensation procedure could be enhanced if the applied compensation factor took into account the contour size of the analysed layer and other geometric parameters that could have an influence.Originality/valueThe presented method of compensation is applicable to layers of any shape obtained in any AM process.

Ray and caustic structure of Ince-Gauss beams

The Ince-Gauss beams, separable in elliptic coordinates, are studied through a ray-optical approach. Their ray structure can be represented over a Poincaré sphere by generalized Viviani curves (intersections of a cylinder and a sphere). This representation shows two topologically different regimes, in which the curve is composed of one or two loops. The overall beam shape is described by the ray caustics that delimit the beams’ bright regions. These caustics are inferred from the generalized Viviani curve through a geometric procedure that reveals connections with other physical systems and geometrical constructions. Depending on the regime, the caustics are composed either of two confocal ellipses or of segments of an ellipse and a hyperbola that are confocal. The weighting of the rays is shown to follow the two-mode meanfield Gross–Pitaevskii equations, which can be mapped to the equation of a simple pendulum. Finally, it is shown that the wave field can be accurately estimated from the ray description.

Pseudo‐solitonic magnetic flows associated with nonlinear integrable systems in the Minkowski 3‐space

In this paper, we introduce a novel class of magnetic curves and call it pseudo‐solitonic magnetic curves by considering the connection between two significant types of integrable equations, that is, the nonlinear heat system/the nonlinear Schrödinger equation and moving space curves in the Minkowski 3‐space. Later, we construct a very special class of soliton surfaces and call it pseudo‐solitonic magnetic surfaces by considering the effects of the pseudo‐solitonic magnetic flows in the Minkowski 3‐space. These curves and surfaces are very useful since they not only contain geometric and physical features in their inner forms but also their constructions rely on a simple geometric procedure compared to other models.

AI-Based Geometry Recognition for Evaluating the Feasibility of Intensified Reaction and Separation Systems.

Reactive distillation (RD) provides notable advantages over conventional processes, regarding reduced energy requirements and CO2 emissions. However, as the benefits of RD may not be universally applicable, a comprehensive feasibility assessment is necessary. This study introduced an automated feasibility evaluation procedure for an RD column using an AI-based region recognition approach, reducing the reliance on expert knowledge and heuristics in graphical methods. Through k-means clustering-based image segmentation, topological information on the reaction and separation reachable region was extracted from ternary diagram landscapes. Subsequently, the extracted information was integrated into tray-by-tray calculations to automate the evaluation. This geometric calculation procedure was applied to assess the feasibility of RD columns with different types of reactions. The feasibility results were obtained within seconds, demonstrating the efficiency of the proposed approach. Furthermore, case studies validated the feasibility of the evaluation results for three practical examples using rigorous simulations, confirming its reliability and applicability.

First-principle study of shear deformation effect on Mg adsorption by monolayer SnS2.

In this study, the effects of different shear deformations on the structural stability, electronic structure, and optical properties of a Mg atom adsorption system of S vacancy defect SnS2 are systematically investigated based on density functional theory. It is shown that the presence of an S-vacancy defect makes the band gap of the SnS2 system significantly smaller than that of the perfect SnS2 system, and the SnS2 system is changed from a direct band gap semiconductor to an indirect band gap semiconductor. The optimal adsorption position of a Mg atom on the S-vacancy SnS2 system is above the S atom where the adsorption energy is the largest and the system is the most stable. The density of states of the adsorption system is predominantly contributed by the S-3p and Sn-5s orbital electrons. The imposition of shear deformation leads to the introduction of certain impurity energy levels in the adsorption system, and the forbidden bandwidth near the Fermi energy level decreases. As compared to the intrinsic SnS2, the absorption and reflection peaks of adsorption systems under different shear deformation are red-shifted and appear in the ultraviolet region. This improves the utilization of the adsorption system for ultraviolet light to a great extent. The model calculations in this paper are performed using the CASTEP module of the Material Studio (MS) software based on the first principles of Density Functional Theory (DFT) (Wei et al. in Physica B 545:99-106, 2018) for plane wave artifacts. Geometrical optimization and computational procedures are used to calculate the exchange-correlation energy using the Perdew-Burke-Ernzerhof (PBE) generalized function (Perdew et al. in Phys Rev B Condens Matter 48:4978, 1993) of the generalized gradient approximation (GGA). The Monkhorst-Pack method (Monkhorst and Pack in Phys Rev B 13:5188-5192, 1976) was used to rationalize the sampling of the highly symmetric k-points in the Brillouin zone. The grid of k-points is set to be 6 × 6 × 1. The plane-wave truncation energy is set to be 400eV. The energy convergence criterion is 1.0 × 10-5eV. The residual stress of all atoms is 0.01eV/Å. A vacuum layer with a thickness of 15Å is set up in the z-direction, which ensures that the interactions of the system along the z-axis between the top and the bottom layers can be ignored during the whole simulation process. We construct a 3 × 3 × 1 SnS2 system containing 27 atoms as the computational model. The intrinsic SnS2 contains 9 Sn atoms and 18 S atoms.

BIM-based framework of automatic tunnel segment assembly and deviation control

The design of universal segments and deviation control of segment assembly are essential for robust and low-risk tunnel construction. A building information modeling (BIM)-based framework was proposed for parametric modeling, automatic assembly, and deviation control of universal segments. First, segment models of different levels of detail (LoDs) were built based on BIM visual programming language (VPL) for different project life cycles. Then, the geometric constraints, requirements, and procedures for parametric segment assembly were distilled to develop a program that combines a novel typesetting algorithm with a 3D path replanning algorithm. Typesetting is implemented by introducing a point indication matrix, characterizing segments by sides, and manipulating geometries in a VPL. Simultaneously, 3D path replanning, with non-uniform rational B-splines (NURBS) and arcs as basic shapes, was used to resolve unacceptable deviation situations after typesetting. Finally, the proposed framework was validated on a water diversion line and was found to be more effective and accurate than the previous method.

Object-Agnostic Vision Measurement Framework Based on One-Shot Learning and Behavior Tree.

Vision measurement is important for intelligent systems to obtain the precise structural and spatial information of objects. Beyond the object-specific vision measurement developed for fixed object type, it is appealing to explore the object-agnostic vision measurement, which can be efficiently reconfigured and adapted to various novel objects. This article proposes a framework to mimic the human's versatile visual measurement behavior: extract a set of contour primitives of interest (CPIs) from an image, then utilize the CPIs to calculate the key geometric information. First, a deep convolutional neural network (CNN) CPieNet + is proposed under the one-shot learning scheme, aiming to extract the pixel-level object CPI from a raw query image, given an annotated support image. The fine-grained CPI prototypes are formed by sampling multiple points on the feature map of the support image. To leverage the explicit geometric knowledge in the CNN inference, the annotation map is encoded as a shape descriptor to guide the feature channel attention, and the geometric attribute awareness is realized by supervising the model to predict the direction and size of CPI. Second, the measurement behavior tree (BT) is designed to model the hierarchical geometric calculation procedure, which is flexibly configurable for different measurement requirements and is interpretable for nonexpert users. After the execution of the measurement BT, the pixel-level CPIs are converted to the required key geometric data. The effectiveness of the proposed methods is validated by a series of experiments.

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra.