Geometric Entities Research Articles

Fast character modeling with sketch-based PDE surfaces

Virtual characters are 3D geometric models of characters. They have a lot of applications in multimedia. In this paper, we propose a new physics-based deformation method and efficient character modelling framework for creation of detailed 3D virtual character models. Our proposed physics-based deformation method uses PDE surfaces. Here PDE is the abbreviation of Partial Differential Equation, and PDE surfaces are defined as sculpting force-driven shape representations of interpolation surfaces. Interpolation surfaces are obtained by interpolating key cross-section profile curves and the sculpting force-driven shape representation uses an analytical solution to a vector-valued partial differential equation involving sculpting forces to quickly obtain deformed shapes. Our proposed character modelling framework consists of global modeling and local modeling. The global modeling is also called model building, which is a process of creating a whole character model quickly with sketch-guided and template-based modeling techniques. The local modeling produces local details efficiently to improve the realism of the created character model with four shape manipulation techniques. The sketch-guided global modeling generates a character model from three different levels of sketched profile curves called primary, secondary and key cross-section curves in three orthographic views. The template-based global modeling obtains a new character model by deforming a template model to match the three different levels of profile curves. Four shape manipulation techniques for local modeling are investigated and integrated into the new modelling framework. They include: partial differential equation-based shape manipulation, generalized elliptic curve-driven shape manipulation, sketch assisted shape manipulation, and template-based shape manipulation. These new local modeling techniques have both global and local shape control functions and are efficient in local shape manipulation. The final character models are represented with a collection of surfaces, which are modeled with two types of geometric entities: generalized elliptic curves (GECs) and partial differential equation-based surfaces. Our experiments indicate that the proposed modeling approach can build detailed and realistic character models easily and quickly.

Indoor Semantic Mapping with Efficient Convolutional Neural Networks for Resource-constrained SLAM System

Scene parsing and robotic manipulation using visual sensing are of significant importance for mobile and robotics related applications. And increasing efforts are being made to associate semantic concepts to geometric entities in robot’s surroundings. However, they are not well suitable for resource-constrained and mobile platforms. Toward an efficient semantic mapping solution, we propose a specifically light-weight segmentation network for building a dense 3D semantic map in real-time. The propose architecture is based on depth-wise convolution and channel shuffle to reduce computation cost. An improved Atrous Spatial Pyramid Pooling (ASPP) and encoder-decoder structures are presented and multi-scale features are merged for better prediction. Experiments are compared with other methods to prove the effectiveness and superiority.

Steady-state response of annular wedge-shaped aquifers to arbitrarily-located multiwells with regional flow

Wedge-shaped aquifers are sometimes cut by crossing streams so that the areal geometry can be delineated by an annular wedge of infinite radial extent. Despite their worldwide spread, the problem of groundwater flow in such geometrical entities has not received much attention in analytical studies. This paper presents a closed-form analytical solution for the aquifer head distribution in response to a system of multiwells superimposed on a background regional flow. The aquifer is isotropic, homogeneous and in prefect hydraulic connection with the surrounding streams. A steady-state Green’s function is constructed for this laterally bounded aquifer system. The formulation is general in the sense that the aquifer vertex angle as well as number of wells and their positions can be arbitrary chosen. Closed-form expression for the stream function is obtained by integration of the Cauchy-Riemann equations. Three types of hydrogeological boundary conditions are considered, combining streams of constant and linearly varying head and no-flow barrier. The results are applicable to both confined and unconfined flows. Extremely simple expressions are derived to quantify the amount of water exchange across stream-aquifer interfaces. Flow nets are given for a number of hypothetical test cases, demonstrating different aspects of the generated flow field. Sensitivity maps are prepared for the stream depletion rates due to a single pumping well and the role of aquifer vertex angle on the model response is investigated. Multiple stagnation points in the flow field are identified on a semi-analytical basis. This helps to assess how the position of a single stagnation point is affected by changes in the gradient of regional flow. For relatively high gradients, it is observed that the radial position of the stagnation point coincides with that of the well itself. Finally, the formulation is extended to account for rainfall recharge over an unconfined aquifer having the same areal extent. The aquifer response is found to agree well with other analytical and numerical results.

A multipoint flux approximation with diamond stencil finite volume scheme for the two‐dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid‐grid method

SummaryTwo‐phase flows of oil and water in naturally fractured reservoirs can be described by a system of nonlinear partial differential equations that comprises of an elliptic pressure equation and hyperbolic saturation equation coupled through the total velocity field. Modeling this problem is a great challenge, due to the complexity of the depositional environments, including inclined layers and fractures with different sizes and shapes, and random spatial distribution. In this work, to solve the pressure equation, we adopted a cell‐centered finite‐volume method with a multipoint flux approximation that uses the “diamond stencil” (MPFA‐D) coupled with a hybrid‐grid method (HyG) to deal with the fractures. The classical first‐order upwind method was used to solve the saturation equation, in its explicit and implicit versions. The MPFA‐D is a very robust and flexible formulation that is capable of handling highly heterogeneous and anisotropic domains using general polygonal meshes. In the strategy developed in this work, the mesh that discretize the domain must fit the spatial position of the fractures, so that they are associated to the control surfaces—as (n − 1)D cells—therefore, the calculation of the fluxes in these control surfaces is dependent on the pressures on fractures and on the adjacent volumes. In HyG, the fractures are expanded to nD in the computational domain. The proposed formulation presented quite remarkable results when compared with similar formulations using classical full pressure support and triangle pressure support methods, or even the with MPFA‐D itself when the fractures are treated as nD geometric entities.

TbGAL: A Tensor-Based Library for Geometric Algebra

Geometric algebra is a powerful mathematical framework that allows us to use geometric entities (encoded by blades) and orthogonal transformations (encoded by versors) as primitives and operate on them directly. In this work, we present a high-level C++ library for geometric algebra. By manipulating blades and versors decomposed as vectors under a tensor structure, our library achieves high performance even in high-dimensional spaces ($$\bigwedge \mathbb {R}^{n}$$ with $$n > 256$$) assuming (p, q, r) metric signatures with $$r = 0$$. Additionally, to keep the simplicity of use of our library, the implementation is ready to be used both as a C++ pure library and as a back-end to a Python environment. Such flexibility allows easy manipulation accordingly to the user’s experience, without impact on the performance.

A features subtraction system for rotational parts based on manufacturing and metal removing concepts

Automatic feature recognition (AFR) techniques represent an essential tool that links computer-aided design (CAD) and computer-aided process planning (CAPP) systems. This is by converting low-level geometric entities of part design models created in CAD systems to manufacturing features which are used in CAPP. Whilst different AFR systems have been developed for the purpose of recognising final manufacturing features of a part, most of them do not solve features intersecting issues. The features intersecting is an unavoidable matter that occurs during the rough cutting cycle and forms intermediate manufacturing features. This paper presents a “features subtraction” methodology for solving manufacturing features intersection issues of rotational parts. The system takes the recognised external and internal manufacturing features from a smart interactive AFR (SI-AFR) system as an input. After seven steps of processing, it provides geometrical and topological information of intermediate and final features that can be used to automatically generate CAPP outputs, such as selecting machining process, sequence of operations, cutting tools and cutting conditions, and generating the G-code for machining the part. This has been written using C# coding, and several examples have been processed for this paper to validate the system.

A conservative, consistent, and scalable meshfree mimetic method

Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.

Constrained fitting with free-form curves and surfaces

The accurate reconstruction of engineering parts from measured data is a challenging problem, in particular, when various geometric constraints need to be imposed to meet requirements in downstream CAD/CAM applications. There is a wide range of constraints including incidence, tangency, orthogonality, parallelism, symmetry and many others. In the majority of cases, only regular surfaces (planes, natural quadrics, sweeps) are involved; however, imposing constraints on objects with free-form curves and surfaces is also necessary in engineering practice; this motivated our work.We numerically optimize complex systems of non-linear equations, containing unknown control points and auxiliary geometric entities. We show that auxiliaries are indispensable; their use simplifies the algebra of the constraint equations and speeds up computations. We do not require the auxiliaries to be fixed in advance, rather propose incorporating them into the constraint system. As a result, the procedure will automatically substantiate various entities that have been unknown beforehand, for example, points of intersection or constrained trim curves on B-spline surfaces.After formulating equations for a representative set of free-form constraints, we will discuss how this technique can be applied in practical reverse engineering. We will present case studies and analyze how surfaces get improved by means of constrained fitting.

Morphable components topology optimization for additive manufacturing

This paper addresses the issue of minimizing support material in additive manufacturing (AM) during topology optimization (TO) in order to reduce material and post-processing costs. The TO method developed in this paper utilizes the moving morphable components (MMC) approach, where a structure is composed of several building blocks. This work introduces minimum build angle constraints to eliminate overhanging edges, supplementing these with penalty functions to ensure connectivity between building blocks, such that the TO output is printable. The MMC approach uses explicit geometric entities for the morphable components that are controlled by geometric parameters, such as length, thickness, and angle. These parameters are the design variables. Using this approach enables the formulation of geometric manufacturing constraints and the construction of CAD models, which are important advantages of the MMC method. Examples of a short cantilever beam and an MBB beam demonstrate the capabilities of the TO methods.

A Minimal Stabilization Procedure for Isogeometric Methods on Trimmed Geometries

Trimming is a common operation in CAD, and, in its simplest formulation, consists in removing superfluous parts from a geometric entity described via splines (a spline patch). After trimming the geometric description of the patch remains unchanged, but the underlying mesh is unfitted with the physical object. We discuss the main problems arising when solving elliptic PDEs on a trimmed domain. First we prove that, even when Dirichlet boundary conditions are weakly enforced using Nitsche's method, the resulting method suffers lack of stability. Then, we develop novel stabilization techniques based on a modification of the variational formulation, which allow us to recover well-posedness and guarantee accuracy. Optimal a priori error estimates are proven, and numerical examples confirming the theoretical results are provided.

Reverse modeling of the human mandible 3D geometric model

Background/Aim. The geometry of each bone in the human skeletal system is unique. The aim of this research was to present application of a new method, method of anatomical features (MAF), for the creation of the geometrical model (surface and solid) of the human mandible. Methods. The method was based on Referential Geometrical Entities (RGEs) which have been defined on mandible polygonal model in accordance with anatomical properties of the mandible. Polygonal model was created over the input data (anatomical landmarks of the mandible) acquired from computed tomography scans. For the creation of computer-aided design (CAD) models in CATIA software, referential geometrical entities were defined according to the bone geometry and morphology features. Results. Definition of B-spline curves was performed on the body and on the ramus of the mandible. In this way, it was possible to create the geometrically accurate and anatomically correct three-dimensional geometric (surface and solid) models. The accuracy of the obtained surface model was tested through comparison with the geometry of the original bone model. Conclusion. Compared to the previously applied methods for creating geometric models, MAF provides more satisfactory results, and in some cases even better.

A Multiscale Control Volume framework using the Multiscale Restriction Smooth Basis and a non-orthodox Multi-Point Flux Approximation for the simulation of two-phase flows on truly unstructured grids

In these days, the advances in the fields of geology have led to geocellular models containing up to hundreds millions cells. Meanwhile, the standard petroleum reservoir simulations are limited to models with only a fraction of this amount. Thus, the multiples simulation of multiphase flow in heterogeneous and anisotropic media on the geological scale are simply impractical. To overcome this problem Multiscale Finite Volume (MsFV) uses operators to project the fine-scale system of equations onto a coarse-scale space originated from a lower-resolution grid. The resulting coarse system is solved, and by using the multiscale operator projected back onto the higher-resolution grid. Notwithstanding, the MsFV is not capable of capturing geologic properties on unstructured grids, as in its classical form, it relies on a Two Point Flux Approximation (TPFA) for fluxes, which is only consistent for k-orthogonal grids. Furthermore, the algorithm used by the MsFV to generate the geometric entities, such as, coarse-scale and dual meshes and its centers are not valid for general unstructured grids. The Multiscale Restricted Smoothed Basis (MsRSB) method expands this framework by creating the algorithms capable of generating these structures and by proposing an iterative multiscale operator capable of handling simulation on unstructured coarse grids. However, it fails to produce consistent solutions on fine-scale unstructured grids and for arbitrary permeability tensors, as it uses the TPFA method. In this article, we couple a Multi-Point Flux Approximation with a Diamond stencil to the MsRSB, to extend its use to general unstructured grids. We call our framework the “Multiscale Control-Volume” (MsCV) method. Moreover, we bring the iterative Dirichlet pressure correction scheme and the upscaling flux correction to the unstructured multiscale context. On the first examples, we study the impact of these corrections on the simulation of one-phase flow using the SPE Model 2 benchmark. In addition, we show the behavior of the MsCV framework on the simulation of two-phase flows on general unstructured grids and highly heterogeneous and isotropic permeability fields.

CAD-integrated Parametric Design Cycle for Structural Membranes

The design cycle of membrane structures consists of three interactive and highly non-linear disciplines. Formfinding is performed in order to find a geometry that allows the prestressed structure to carry loads through tension only. Once a formfound geometry is set, structural analysis needs to assess the structure's safety and usability. The most challenging step is the cutting pattern generation, which aims at finding the planar pieces which can be elevated to build spatial membrane structures with a minimum deviation from the desired shape and prestress state.<br/> Isogeometric B-Rep Analysis (IBRA) allows the designer to perform analyses on the original CAD model without leaving the CAD environment. High quality is ensured for the geometry and the mechanical approximation by using Non-Uniform Rational B-Splines (NURBS). Additionally, the topology information of multipatches can be transferred to the analysis in order to enrich the design space. Trimmed and coupled surfaces can thus be included in the analysis. Parametric models allow the designer to examine a large variety of geometrical and mechanical entities with one model.<br/> The advantages of the CAD-integration with IBRA for the highly interactive design of structural membranes are shown in this contribution.

A comparative study about some application packages used in Photogrammetry

Photogrammetry is the technique used to convert a real geometry of a part into a 3D computer file starting from images. The paper aims to compare some free and commercial application packages used in Photogrammetry, based on the same images taken from an axial turbine blade. The blade geometry can be included in a bounding box with the following dimensions: 233 x 125 x 82 mm. The geometrical entities generated with these applications will be compared through Cloud Compare software.

Mixed projection- and density-based topology optimization with applications to structural assemblies

In this paper we present a mixed projection- and density-based topology optimization approach. The aim is to combine the benefits of both parametrizations: the explicit geometric representation provides specific controls on certain design regions while the implicit density representation provides the ultimate design freedom elsewhere. This approach is particularly suited for structural assemblies, where the optimization of the structural topology is coupled with the optimization of the shape of the interface between the sub-components in a unified formulation. The interface between the assemblies is defined by a segmented profile made of linear geometric entities. The geometric coordinates of the nodes connecting the profile segments are used as shape variables in the problem, together with density variables as in conventional topology optimization. The variable profile is used to locally impose specific geometric constraints or to project particular material properties. Examples of the properties considered herein are a local volume constraint, a local maximum length scale control, a variable Young's modulus for the distributed solid material, and spatially variable minimum and maximum length scale. The resulting optimization approach is general and various geometric entities can be used. The potential for complex design manipulations is demonstrated through several numerical examples.

Modeling of High-Tc Superconducting Bulk using Different Jc–T Relationships over Dynamic Permanent Magnet Guideway

The linear temperature dependence of critical current density Jc∝((Tc-T)/(Tc-T0)) and the nonlinear functions of Jc∝(1-(T/Tc)2)α with the exponent α equal to 1, 3/2, and 2 are used to calculate the dynamic levitation force, the temperature distribution, and the current density distribution of the high-temperature superconducting (HTS) YBaCuO bulk over a permanent magnetic guideway (PMG). The calculations were based on the H-formulation and E–J power law. The model of the HTS bulk and the PMG has been built as a geometric entity by finite element software. To simulate the magnetic field fluctuation caused by the PMG arrangement irregularity, a small amplitude vibration in the vertical direction is applied to the PMG during the calculations. Both the low vibration frequency of 2 Hz and the high vibration frequency of 60 Hz are analyzed as the representative converted linear speeds of 34 km/h and 1018 km/h for magnetic levitation (Maglev) application. We compared the electromagnetic-thermo-force modeling with the experiments and the previous model without considering the thermal effect. The levitation force computed by the Jc–T relationship, in which Jc is proportional to (1-(T/Tc)2)2, is found to be in best agreement with the experimental data under quasi-static conditions. This work can provide a reference for the HTS electromagnetic-thermal-force coupling reproduction method of HTS Maglev at high speed.

Complex and Hypercomplex-Valued Support Vector Machines: A Survey

In recent years, the field of complex, hypercomplex-valued and geometric Support Vector Machines (SVM) has undergone immense progress due to the compatibility of complex and hypercomplex number representations with analytic signals, as well as the power of description that geometric entities provide to object descriptors. Thus, several interesting applications can be developed using these types of data and algorithms, such as signal processing, pattern recognition, classification of electromagnetic signals, light, sonic/ultrasonic and quantum waves, chaos in the complex domain, phase and phase-sensitive signal processing and nonlinear filtering, frequency, time-frequency and spatiotemporal domain processing, quantum computation, robotics, control, time series prediction, and visual servoing, among others. This paper presents and discusses the importance, recent progress, prospective applications, and future directions of complex, hypercomplex-valued and geometric Support Vector Machines.

Topology of configuration space of the mean-field ϕ^{4} model by Morse theory.

In this paper we present the study of the topology of the equipotential hypersurfaces of configuration space of the mean-field ϕ^{4} model with a Z_{2} symmetry. Our purpose is discovering, if any, the relation between the second-order Z_{2}-symmetry-breaking phase transition and the geometric entities mentioned above. The mean-field interaction allows us to solve analytically either the thermodynamic in the canonical ensemble or the topology by means of Morse theory. We have analyzed the results in the light of two theorems on sufficiency conditions for symmetry-breaking phase transitions recently proven. This study is part of a research line based on the general framework of geometric-topological approach to Hamiltonian chaos and critical phenomena.

Tuning of conic parameters using Tikhonov regularization and L-Curve simulation

In this article, a method to infer the parameters of a conic given a set of rectangular coordinates that belong to the geometric entity is shown. The methodology consists of solving a Tikhonov regulation problem where the unregulated term introduces the non-linear nature of the conical body and the regulated the restriction associated to the discriminant of the quadratic equation, then the solution is computed minimizing the resulting cost function where the Regularization parameter is tuned using the L-Curve technique. The model was validated with synthetic and real data from digital images, as well as subject to comparison against other state of the art alternatives. The results show that the method is robust against atypical values and the phenomenon of occlusion present in the data.

Design of Broadband Dielectric Collimators With the Phase-Field Design Method for Applications in the X-Band Range

An electromagnetic (EM) collimator is a device to narrow EM wave propagation in a desired way. Recalling the device has been widely used in several EM applications such as radar detectors, it is encouraged to maximize its beam-focusing capacity over a wide operating band. This study aims to design mono-scale all-dielectric collimators, which exhibit a broadband functionality in the X-band range. The phase-field design method is employed to systematically derive topological configurations of the broadband collimators. Two types of collimators for steering beams into different focusing zones are under consideration. In order to iteratively improve the frequency response characteristics during design update process, the statistical moments of the frequency response representing the collimator performance are directly incorporated into a multiobjective optimization formulation. All-dielectric prototypes were fabricated by additive manufacturing, accommodating complicated geometrical entities. The improvement over frequency dependence is validated through numerical simulations and experimental confirmation with the printed structures.