Geometric Entities Research Articles

GPU based discrete element modeling for convex polyhedral shape particles: Development and validation

Particle dynamics simulations face a significant challenge in understanding the intricate behaviors of convex polyhedral particles due to their complex geometries and interactions. DEM emerges as a key method, illuminating the concealed intricacies of these geometric entities. Traditional algorithms often require cumbersome processes to check each type of contact individually. However, Gilbert-Johnson-Keerthi’s (GJK) and the expanding polytope algorithm (EPA) provide efficient numerical solutions for polyhedral contact detection and contact resolution. These Minkowski difference-based methods streamline contact detection and overlap computation, paving the way for deeper exploration of three-dimensional contact theory within DEM simulations. By leveraging GPU computational power, this paper outlines key algorithmic steps and verifies the code’s accuracy through comparison with simulated and experimental data, with an average deviation of less than 5%. This study explores the impact of particle shape on the dynamics and mechanical behavior of densely packed systems, particularly in hoppers and tumblers. Spherical particles discharge faster but mix more slowly than polyhedral shapes, with icosahedrons achieving quicker full mixing. These results align with experimental findings, further validating the simulation approach.

2D-Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II

In earlier research, we developed two techniques designed to expand the construction of a periodic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom. Our methodology involved transforming a periodic orbit into a torus or cylinder, thereby elevating it to a higher-dimensional structure within the energy surface (refer to [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). Recently, we introduced two new methods for creating dividing surfaces, which do not rely on periodic orbits. Instead, we used 2D surfaces (geometric entities) or 3D surfaces in a Hamiltonian system with three degrees of freedom (see [Katsanikas & Wiggins, 2024a, 2024b, 2024c]). In these studies, we applied these surfaces within a quadratic normal-form Hamiltonian system with three degrees of freedom. This series of two papers (this paper and [Katsanikas et al., 2024]) extends our results to 2D-generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper focuses on presenting the second method of constructing 2D-generating surfaces.

2D Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – I

In previous studies, we developed two techniques aimed at expanding the scope of constructing a periodic orbit dividing surface within a Hamiltonian system with three or more degrees of freedom. Our approach involved extending a periodic orbit into a torus or cylinder, thereby elevating it into a higher-dimensional entity within the energy surface (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). Recently, we introduced two alternative methods for creating dividing surfaces, distinct from the utilization of periodic orbits, by employing 2D surfaces (geometric entities) or 3D surfaces within a Hamiltonian system with three degrees of freedom (refer to [Katsanikas & Wiggins, 2024a, 2024b, 2024c]). In these studies, we applied these surfaces in a quadratic normal form Hamiltonian system with three degrees of freedom. In this series of two papers, we extend our results to 2D generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper presents the first method of constructing 2D generating surfaces.

Modeling Pressure Effect of Circular Tourniquet Based on Digital Arm

Background: This study aims to investigate displacement deformation of human tissue in the force region subjected to annular pressure. Methods: In this patent, 727 images of a Chinese digital human arm, captured from shoulder to fingertip, were used as the reconstruction data. The geometric entities of tissue structure were obtained after tissue segmentation, three-dimensional modeling, and reverse engineering to establish the working mechanism model of the tourniquet of the human forearm in the finite element simulation software (COMSOL Multiphysics 5.5). By setting different parameter models (tourniquet pressure and width models), we analyzed the force conduction mechanism and the displacement deformation mechanism of the viscoelastic and rigid tissues of the forearm when subjected to annular pressure. Results: Modeling analysis showed that when a pressure of 800 kPa was applied on a width of 40 mm, the annular pressure on the viscoelastic tissues was converted into displacement deformation, thus changing the tissue structure in the body and realizing the hemostatic effect of the tourniquet. In the case of fixed tourniquet width but variable tourniquet pressure, with the gradual increase of the pressure, displacement deformation showed an increasing trend. When the externally applied pressure was fixed and the tourniquet width was different, with the gradual increase of the tourniquet width, the displacement deformation showed a decreasing trend. Conclusion: This patent study demonstrates that both the amount of externally applied pressure and the width of the tourniquet affect the hemostatic effect of the tourniquet. The hemostatic effect on the damaged body will be more obvious under a small tourniquet width and large pressure.

A 3-D extension of the Multiscale Control Volume method for the simulation of the single-phase flow in anisotropic and heterogeneous porous media

The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heterogeneous media with phenomena taking place in different scales. Scale transferring techniques allow for the solution of such problems in a lower resolution scale at reduced computational cost. Among these techniques, the Multiscale Finite Volume (MsFV) method constructs a set of numerical operators in order to map quantities from the fine-scale mesh to a coarser one and vice-versa while maintaining flux conservation on both scales. However, the MsFV formulation, as originally stated, is only consistent on k-orthogonal grids since it uses a linear Two-point Flux Approximation (TPFA) method and may struggle to generate consistent primal-dual coarse grids pairs on unstructured grids. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and the concept of support regions which reduces the method's complexity when applied to unstructured fine and coarse grids. The original version was only consistent on k-orthogonal fine grids due to the TPFA discretization, but filtering methods have been developed to also enable consistent multipoint schemes on the fine scale. Meanwhile, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow unstructured grids on the fine and coarse scales for two-dimensional simulations. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of a background grid, a coarser grid used as a proxy for the primal coarse grid. We modify the original MPFA-D method to use the very robust Global Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We also introduce an enhanced version of the 3-D MsCV with the incorporation of the enhanced MsRSB (E-MsRSB) to enforce M-matrix properties and improve convergence. Finally, we employ the MsCV operators in a two-stage smoother to show how it can be used as a good iterative procedure to recover the fine-scale solution. We show that the 3-D MsCV method produces good results employing unstructured grids on both scales to handle the simulation of the single-phase flow in anisotropic, and heterogeneous porous media and that the smoothing procedure is able to accurately retrieve the fine-scale solution within a few iterations.

Detailed Design of Special-Shaped Steel Structures Based on DfMA: The BIM-FEM Model Conversion Method

(1) This paper, based on the characteristics of complex steel structures as well as difficult points in the process of their detailed design, introduces the product design concept of DfMA (Design for Manufacturing and Assembly) from the manufacturing industry and studies the detailed design method of BIM-FEM model conversion. The BIM software Revit (2020) is used as the basis for the BIM detailed design of the project, which achieves the purpose of rapid modeling and provides a detailed design model basis for finite element analysis. (2) Utilizing the Revit API and C# for secondary development as the technical means, this approach involves converting the geometric entity model described by CSG-Brep into an APDL stream. This creates an interface with the finite element analysis software ANSYS (12.0) to implement the detailed design of BIM-FEM model conversion, optimizing the algorithm for converting complex analysis models that require high precision for special-shaped steel structures. (3) This research addresses issues such as the disconnection between the design, manufacturing, and construction of special-shaped steel structures, providing support for design decisions. Moreover, it enhances the detailed design method by improving the standardization of special-shaped components under the condition of design diversity. (4) These studies provide sustainability for engineering design, manufacturing, and construction projects, enabling the maximization of benefits and product lifecycle management (PLM) through these projects. (5) Finally, a case study analysis was conducted on the Wuhan City New Generation Weather Radar Construction Project, designed by the Central South Architectural Design Institute (CSADI), to verify the detailed design of BIM-FEM model conversion. This proved the scientific validity, practicality, and necessity of this research.

GRAPH FRACTALS WITH THE VARIABILITY OF THE FORMATION PROCESS

Fractals, which are characterized by their self-similarity at different scales, are complex geometric entities created using recursive algorithms. They are widely used in computer graphics to create complex visual effects and to model natural phenomena such as river networks and mountain landscapes. Graph fractals combine the properties of fractals and graph structures and can be used for research in fields such as computer networks or medicine. This work presents the approach of constructive-production modeling, based on formal grammars, for the generation of graph fractals with the variability of the formation process.

High-order kinematics of uniform flexures

In the last decades, many efforts have been made to analyze and model small and large deflections of flexures, considering complex load cases and different solution techniques. However, few investigations focused on the kinematic aspects related to the deflection analysis of the flexible elements, and limited the study to the second-order kinematics. In this paper, an analytical formulation based on the instantaneous geometric invariants is developed to give deep kinematic insight, up to the fourth order, into the motion generated by the deflections. The problem is addressed from a geometrical point of view, defining the fundamental geometric entities that characterize the motion, that are inflection circle, cubic of stationary curvature and its derivative, Ball’s point, and Burmester’s points. Thanks to these special points on the plane, straight and circular paths can be approximated to the third and to the fourth order, respectively. The proposed formulation defines the geometric characteristics of flexures with different curvatures. An application regarding the definition of pseudo-rigid body models is discussed. Finite element simulations are performed to validate the results.

Multi-Objective Optimization of Cyclone Separators Based on Geometrical Parameters for Performance Enhancement

The present study focuses on performing multi-objective optimization of the cyclone separator geometry to lower the pressure losses and enhance the collection efficiency. For this, six geometrical entities, such as the main body diameter of the cyclone, the vortex finder diameter and its insertion length, the cone tip diameter, and the height of the cylindrical and conical segment, have been accounted for optimization, and the Muschelknautz method of modeling has been used as an objective function for genetic algorithms. To date, this is one of the most popular mathematical models that accurately predicts the cyclone performance, such as the pressure drop and cut-off particle size. Three cases have been selected from the Pareto fronts, and the cyclone performance is calculated using advanced closure large-eddy simulation—the results are then compared to the baseline model to evaluate the relative improvement. It has been observed that in one of the models, with merely a 2% reduction in the collection efficiency and an increase of 12% in the cut-off particle size, more than a 43% reduction in pressure drop value was obtained (an energy-efficient model). In another model, a nearly 25% increment in the collection efficiency and a reduction of 42% in the cut-off particle size with a nearly 36% increase in pressure drop value were observed (a high-efficiency model).

Exploring geometry with the help of technological resources in a school in the amazon floodplains

The study dealt with the implementation of a pedagogical project for teaching Plane and Spatial Geometry in the 9th grade of Elementary School II, using the GeoGebra application as an innovative teaching tool. The project aimed to overcome challenges perceived in the traditional teaching of Geometry, seeking to integrate concepts in a contextualized and motivating way. The theoretical framework highlights the importance of innovative approaches, the contextualization of Geometry throughout history and the use of technological resources to enhance learning. The methodology adopted is predominantly qualitative, involving action and participatory research. The project was carried out at the São Jose Municipal School, located in a floodplain area in the municipality of Santarém, west of Pará. GeoGebra was introduced as a practical tool for exploring geometric entities, solids and figures, providing a visual and interactive approach. Analysis of the results reveals a significant improvement in the students' understanding and attitude towards geometry after the GeoGebra intervention. However, important considerations are pointed out, such as the need for adequate technological infrastructure and ongoing teacher training to maximize the impact of these initiatives. The study highlights the effectiveness of GeoGebra as a pedagogical tool for overcoming perceived challenges and promoting a more positive and contextualized view of the subject.

Acoustic Fractional Propagation in Terms of Porous Xerogel and Fractal Parameters.

This article portrays solid xerogel-type materials, based on chitosan, TEGylated phenothiazine, and TEG (tri-ethylene glycol), dotted with a large number of pores, that are effectively represented in their constitutive structure. They were assumed to be fractal geometrical entities and adjudged as such. The acoustic fractional propagation equation in a fractal porous media was successfully applied and solved with the help of Bessel functions. In addition, the fractal character was demonstrated by the produced fractal analysis, and it has been proven on the evaluated scanning electron microscopy (SEM) pictures of porous xerogel compounds. The fractal parameters (more precisely, the fractal dimension), the lacunarity, and the Hurst index were calculated with great accuracy.

FIKA: A Conformal Geometric Algebra Approach to a Fast Inverse Kinematics Algorithm for an Anthropomorphic Robotic Arm

This paper presents a geometric approach to solve the inverse kinematics for an anthropomorphic robotic arm with seven degrees of freedom (DoF). The proposal is based on conformal geometric algebra (CGA), by which many geometric primitives can be operated naturally and directly. CGA allows for the intersection of geometric entities such as two or more spheres or a plane’s projection over a sphere. Rigid transformations of such geometric entities are performed using only one operation through another geometric entity called a motor. CGA imposes geometric restrictions on the inverse kinematics solution, which avoids computation of the forward kinematics or other numerical solutions, unlike traditional approaches. Comparisons with state-of-the-art algorithms are included to prove our algorithm’s superior performance: such as decreased execution time and errors of the end-effector for a series of desired poses.

Parallel and automatic mesh sizing field generation for complicated CAD models

PurposeThe purpose of this paper is to present an automatic approach for mesh sizing field generation of complicated computer-aided design (CAD) models.Design/methodology/approachIn this paper, the authors present an automatic approach for mesh sizing field generation. First, a source point extraction algorithm is applied to capture curvature and proximity features of CAD models. Second, according to the distribution of feature source points, an octree background mesh is constructed for storing element size value. Third, mesh size value on each node of background mesh is calculated by interpolating the local feature size of the nearby source points, and then, an initial mesh sizing field is obtained. Finally, a theoretically guaranteed smoothing algorithm is developed to restrict the gradient of the mesh sizing field.FindingsTo achieve high performance, the proposed approach has been implemented in multithreaded parallel using OpenMP. Numerical results demonstrate that the proposed approach is remarkably efficient to construct reasonable mesh sizing field for complicated CAD models and applicable for generating geometrically adaptive triangle/tetrahedral meshes. Moreover, since the mesh sizing field is defined on an octree background mesh, high-efficiency query of local size value could be achieved in the following mesh generation procedure.Originality/valueHow to determine a reasonable mesh size for complicated CAD models is often a bottleneck of mesh generation. For the complicated models with thousands or even ten thousands of geometric entities, it is time-consuming to construct an appropriate mesh sizing field for generating high-quality mesh. A parallel algorithm of mesh sizing field generation with low computational complexity is presented in this paper, and its usability and efficiency have been verified.

Automatic Fingerprint Data Labeling Using WiFi Signal and Smartphone Camera for Indoor Positioning

WiFi fingerprinting has been one of the most practical approaches for implementing an indoor positioning system. However, the need to measure location labels for fingerprint data has hindered the deployment of WiFi fingerprint‐based positioning systems. To reduce the cost of location labeling, several methods for automatically labeling locations or bypassing location labeling have been proposed. However, these methods still need to measure location labels for partial fingerprint data, or the estimated location labels are inaccurate. In this paper, we propose Heimdall, an automatic fingerprint data labeling method using a smartphone camera and WiFi signal for an indoor positioning system. Since Heimdall utilizes camera geometry for accurate automatic labeling, it aims to find the location labels of unlabeled fingerprint data composed of WiFi received signal strengths (RSSs) and photos collected using smartphones in multiple reference points (RPs). Using structure from motion (SfM) with all photos in the unlabeled fingerprint dataset, the location and orientation of the cameras can be recovered on an arbitrary world coordinate system. However, the world coordinate system derived by SfM is warped. In other words, it is different from the real‐world coordinate system where we express the location labels since SfM has similarity ambiguity, including scale, rotation, and translation ambiguity. Heimdall finds the location labels by estimating and applying an unwarping transformation that removes those ambiguities. We use two types of geometric constraints for estimating the unwarping transformation. First, several geometric entities in the warped‐world coordinates and real‐world coordinates are extracted from the photos and prior information about the indoor layout. The unwarping transformation transforms the geometric entities in warped coordinates to those in real‐world coordinates. Second, we utilize the log‐distance pathloss model with RSSs to derive the constraint about distances from the location label to the access points. We construct the optimization problem subject to those constraints to estimate the unwarping transformation. To evaluate the performance of Heimdall, we performed several experiments. Heimdall yields a mean location labeling error of 0.12 m and a standard deviation of location labeling error of 0.06 m for the best case in the experiment. These results demonstrate that a fingerprint‐based indoor positioning system can use the location labels estimated by Heimdall without performance degradation.

Integration of geometry modelling and behavior simulation based on graph-based design languages and functional mockup units

This paper explores the automated creation of a digital twin, employing a framework based on graph-based design languages (GBDLs) and functional mock-up units (FMUs). Through this framework, all entities that represent a digital twin can be represented geometrically and enriched with behavioral properties. The goal is to create not only geometric entities, but also simulation architectures automatically. Kinematics, pneumatics, hydraulics and many other disciplines can be mapped and linked through this kind of simulation architecture. For this purpose, either a monolithic simulation, which transfers all domains into one simulation, is possible, or the setup of a co-simulation. In this case, each simulation is modeled separately and stored as a FMU. Each interface for input and output parameters of an FMU is defined by the functional mock-up interface (FMI) standard and therefore unique. This uniqueness allows different FMUs to be linked together and built up into a co-simulation. The links are defined via a simulation architecture and present the interface between two domains. This simulation architecture, like the whole digital twin, relates to GBDLs. This paper discusses the approach through the exemplary creation of a simulation architecture for a Festo™ assembly system.

Realism, irrationality, and spinor spaces

Mathematics, as Eugene Wigner noted, is unreasonably effective in physics. The argument of this paper is that the disproportionate attention that philosophers have paid to discrete structures such as the natural numbers, for which a nominalist construction may be possible, has deprived us of the best argument for Platonism, which lies in continuous structures—in fields and their derived algebras, such as Clifford algebras. The argument that Wigner was making is best made with respect to such structures—in a loose sense, with respect to geometry rather than arithmetic. The purpose of the present paper is to make this connection between mathematical realism and geometrical entities. It thus constitutes an argument against formalism, for which mathematics is merely a game with humanly set rules; and nominalism, in which whatever mathematics is used is eliminable in the final analysis, by often insufficiently specified means. The hope is that light may be cast on the stubborn mysteries of the nature of quantum mechanics and its mathematical formulation, with particular reference to spinor representations—as they have been developed by Andrej Trautman. Thus, according to our argument, quantum mechanics (QM) may appear more natural, as we have better reasons to take spinor structures as irreducibly real, a view consonant with the work of Trautman and Penrose in particular.

A topology optimization method for problems with design-dependent loads based on non-uniform rational basis spline entities

In this paper, topology optimization (TO) problems of structures subjected to design-dependent loads are formulated in the context of a special density-based TO approach wherein a Non Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor. Unlike classical density-based TO approaches, the NURBS-density-based method allows representing the pseudo-density field through a purely geometric computer-aided design compatible entity. In this context, the problem is formulated in the most general case of inhomogeneous Neumann-Dirichlet boundary conditions. Moreover, a thorough study of the penalty function of the design-dependent loads is carried out to investigate its effect on the optimized topologies and overcome the singularity effect related to the zones characterized by low values of the pseudo-density field. Finally, a wide campaign of sensitivity analyses is conducted to investigate the influence of the integer parameters of the NURBS entity, of the combination of design-depedent loads and inhomogeneous Neumann-Dirichlet boundary conditions, and of the concentrated load on the optimized topology. The effectiveness of the approach is tested on 2D and 3D benchmark problems.

Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations

This contribution presents data-locality optimizations of the additive Schwarz method (ASM) based on the fast-diagonalization method defined on overlapping cell-centric and vertex-star patches in the context of high-order matrix-free finite-element computations on modern CPU-based hardware. The developments are guided by detailed performance models of the ASM in the context of Chebyshev iterations when used as smoothers for p-multigrid. The proposed efficient implementation of ASM adopts concepts known from cell-loop infrastructures for efficient operator evaluation, in particular, the storage of information per geometric entity and the cache-friendly interleaving of cell loops and vector updates as a means to increase data locality. We use the latter concept for both applying the weights required by ASM and performing the vector updates required by the Chebyshev iteration, which are memory-bound operations with non-negligible costs in comparison to efficient operator evaluation. Furthermore, the solution of a scalar Poisson problem on a highly anisotropic and an unstructured mesh with p-multigrid using the developed smoothers indicates that efficient implementations of the additive Schwarz method can outperform optimized point-Jacobi preconditioners already for simple problems despite being more than twice as expensive per iteration. Even though ASM introduces additional communication steps per smoother application, the reduced number of iterations can lead to improved parallel scalability for intermediate problem sizes. At the scaling limit, the results are inconclusive due to these two opposing effects.

BOOLEAN OPERATION-BASED FAST CALCULATION OF CUTTER-WORKPIECE ENGAGEMENT DURING PERIPHERAL MILLING

The cutter-workpiece engagement (CWE) is an important basis for accurately predicting milling force and vibration of machining system, which can be dramatically affected by the complex tool path, variable part allowance and various tool profiles. The paper presents a fast calculation method of CWE during peripheral milling process based on the Boolean operation. A second-developed simulation environment embedded the kernel program with the aid of commercial CAM is established. In the computing model, the locally past-cut material entity of the workpiece is replaced by a set of neighboring tools, and the geometric entity of CWE is quickly obtained through the Boolean operation between the tool, workpiece blank, and past-cut entities. The instantaneous uncut chip thickness (IUCT) is further obtained after extracting the tooth start and end cutting angles. Thanks to the first-order computation time complexity, the method has a significant advantage of fast computing speed when compared to the traditional method with a globally updated workpiece. In the milling case of S-shaped workpiece, the results indicate that the method can effectively calculate the CWE status along the entire tool path, and achieve the fast prediction of milling force under a long-time machining condition.

AAGNet: A graph neural network towards multi-task machining feature recognition

Machining feature recognition (MFR) is an essential step in computer-aided process planning (CAPP) that infers manufacturing semantics from the geometric entities in CAD models. Traditional rule-based MFR methods struggle to handle intersecting features due to the complexity of representing their variable topological structures. This motivates the development of deep-learning-based methods, which can learn from data and overcome the limitations of rule-based methods. However, some existing deep learning methods compromise geometric and topological information when using certain representations such as voxel or point cloud. To address these challenges, we propose a novel graph neural network, named AAGNet, for automatic feature recognition using a geometric Attributed Adjacency Graph (gAAG) representation that preserves topological, geometric, and extended attributes from neutral boundary representation (B-Rep) models. Furthermore, some existing methods (such as UV-Net, Hierarchical CADNet) lack the capability of machining feature instance segmentation, which is a sub-task of feature recognition that requires the network to identify different machining features and the B-Rep face entities that constitute them, and it is a crucial task for subsequent process planning. AAGNet is designed as a multi-task network that can perform semantic segmentation, instance segmentation, and bottom face segmentation simultaneously for recognizing machining features, the faces associated with those features, and their bottom faces. The AAGNet is evaluated on various open-source datasets, including MFCAD, MFCAD++, and the newly introduced MFInstSeg dataset with over 60,000 STEP files and machining feature instance labels. The experimental results demonstrate that AAGNet outperforms other state-of-the-art methods in terms of accuracy and complexity, showing its potential as a flexible solution for MFR in CAPP.