General Polygons Research Articles

Representing and processing polygonal maps with Sector Lists

In this paper we describe a novel data structure called Sector Lists (SLs) to represent discrete scalar fields equivalent to polygonal maps. The idea is to model scalar fields as a sum of elementary, wedge-shaped constant fields — what we call sectors. Sectors are organized in sorted lists in order to support an efficient processing using algorithms based on the sweeping plane paradigm. As a result, SLs combine properties often associated with rasters with those of general polygons, in the sense the borders of regions mapped to the same value can be represented exactly. Algorithms to convert, evaluate, add, transform and morph SLs are described herein, with proof-of-concept implementations publicly available. We also show how several operations commonly supported by raster and polygon-based representations can be computed with SLs. In particular, we include the results of two experiments that demonstrate the flexibility and efficiency of our prototype implementation.

General polygon mesh discrete element method for arbitrarily shaped particles and complex structures based on an energy-conserving contact model

General polygon mesh discrete element method for arbitrarily shaped particles and complex structures based on an energy-conserving contact model

Quadratic serendipity element shape functions on general planar polygons

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L2-norm and the H1-seminorm.

Crystalline flow starting from a general polygon

<p style='text-indent:20px;'>This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.</p>

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

RBFCUB: A numerical package for near-optimal meshless cubature on general polygons

In this paper we improve the cubature rules discussed in Sommariva and Vianello (2021) for the computation of integrals by radial basis functions (RBFs). More precisely, we introduce in the context of meshless cubature a leave-one-out cross validation criterion for the optimization of the RBF shape parameter. This choice allows us to get highly reliable and accurate results for any kind of both infinity and finite regularity RBF. The efficacy of this approximation scheme is tested by numerical experiments on complicated polygonal regions. The related Matlab software is provided to the scientific community in [1].

An efficient numerical method for dynamic analysis of polygonal plate under moving loads

Plate structures under moving loads are frequently encountered in practical engineering. Over the past decades, the moving-load models for plates are mainly developed for rectangular plates. In this study, an efficient numerical method is developed for triangular and quadrilateral plates, which are the two most representative polygonal plates and also the basic elements to form general polygon plates. To avoid the mismatch between the moving load and nodes that encountered in Finite Element Method (FEM), the entire domain of the plate is firstly mapped into a square domain using the isoparametric shape functions. The domain decomposition method is also introduced to handle the general polygon shaped plates, whereas the penalty function method is adopted to handle arbitrary boundary conditions. The first-order shear deformation theory (FSDT) is used to describe the elastic deflection of the plates, and then the deformation fields in the mapped domain are spatially discretized by using Chebyshev polynomials of the first kind (CPOFK). Afterwards, the governing equations of motion is derived using Lagrange’s equation, in which the inertia force, Coriolis force and centrifugal force due to the interactions between the moving load and supporting plates are taken into account simultaneously. The mode reduction method is employed to reduce the dimension of dynamic equations, and then the first-order generalized-α method is used to solve the dynamic response in time domain. Finally, the correctness and efficiency of the presented method are validated from convergence analysis and comparisons.

A new isosurface extraction method on arbitrary grids

The development of interface-capturing methods (such as level-set, phase-field or volume of fluid (VOF) methods) for arbitrary 3D grids has further highlighted the need for more accurate and efficient interface reconstruction procedures. In this work, we propose a new method for the extraction of isosurfaces on arbitrary polyhedra that can be used with advantage for this purpose. The isosurface is extracted from volume fractions by a general polygon tracing procedure, which is valid for convex or non-convex geometries, even with non-planar faces. The proposed method, which can be considered as an extension of the marching cubes technique, produces consistent results even for ambiguous situations in polyhedra of arbitrary shape. To show the reproducibility of the results presented in this work, we provide the open source library isoap, which has been developed to implement the proposed method and includes test programs to demonstrate the successful extraction of isosurfaces on several grids with polyhedral cells of different types. We present results obtained not only for isosurface extraction from discrete volume fractions resulting from a volume of fluid method, but also from data sets obtained from implicit mathematical functions and signed distances to scanned surfaces. The improvement provided by the proposed method for the extraction of isosurfaces in arbitrary grids will also be very useful in other fields, such as CFD visualization or medical imaging.

A simple and complete discrete exterior calculus on general polygonal meshes

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

An automatic EEG-based sleep staging system with introducing NAoSP and NAoGP as new metrics for sleep staging systems.

Different biological signals are recorded in sleep labs during sleep for the diagnosis and treatment of human sleep problems. Classification of sleep stages with electroencephalography (EEG) is preferred to other biological signals due to its advantages such as providing clinical information, cost-effectiveness, comfort, and ease of use. The evaluation of EEG signals taken during sleep by clinicians is a tiring, time-consuming, and error-prone method. Therefore, it is clinically mandatory to determine sleep stages by using software-supported systems. Like all classification problems, the accuracy rate is used to compare the performance of studies in this domain, but this metric can be accurate when the number of observations is equal in classes. However, since there is not an equal number of observations in sleep stages, this metric is insufficient in the evaluation of such systems. For this purpose, in recent years, Cohen's kappa coefficient and even the sensitivity of NREM1 have been used for comparing the performance of these systems. Still, none of them examine the system from all dimensions. Therefore, in this study, two new metrics based on the polygon area metric, called the normalized area of sensitivity polygon and normalized area of the general polygon, are proposed for the performance evaluation of sleep staging systems. In addition, a new sleep staging system is introduced using the applications offered by the MATLAB program. The existing systems discussed in the literature were examined with the proposed metrics, and the best systems were compared with the proposed sleep staging system. According to the results, the proposed system excels in comparison with the most advanced machine learning methods. The single-channel method introduced based on the proposed metrics can be used for robust and reliable sleep stage classification from all dimensions required for real-time applications.

A mimetic method for polygons

A new method for mimetic interpolation on polygonal meshes is described. This new method is based on harmonic function interpolation. Explicit formulas for harmonic functions on general polygons do not exist, so truncated harmonic polynomial expansions are used for computational efficiency. We show that the naive harmonic polynomial expansion, is not stable for arbitrary polygons. However, higher level truncations of harmonic interpolations are stable and accurate. This new method is shown to be a direct extension of the lowest order Raviart-Thomas finite elements to polygons. This method is also a direct extension of the finite volume MAC method to polygons. The accuracy of the interpolation is shown to be first-order irrespective of the polynomial truncation level. However the accuracy of vector Laplace equation solutions using this inner product is shown to be second-order accurate, in keeping with other lowest order Mimetic methods. The versatility of this numerical method is demonstrated on a multiphase incompressible flow problem with a density jump of 1000, on a moving polygonal mesh.

Analytic gradient for the moment-of-fluid method in axisymmetric and on general polyhedrons in any dimension

The moment-of-fluid method (MOF) is an interface reconstruction method similar to the volume-of-fluid method with piecewise linear interface reconstruction (VOF-PLIC). In the MOF method, the normal to the interface is found by minimizing the distance between the centroid of the polyhedron below the interface and a reference centroid under a volume constraint. To solve this minimization problem, the gradient of the objective function must be evaluated. Analytic formulas have been proposed by many authors to compute the gradient in 2D on general polygons with a polar parametrization and in 3D on convex polyhedrons with a spherical parametrization. In this short note, we propose a more general formula that covers non-convex polyhedrons in any dimension and axisymmetric coordinates. Furthermore, this formula does not depend on the parametrization of the normal to the interface. We also provide some practical way to use the formula in a code.

Polygon Laplacian Made Simple

AbstractThe discrete Laplace‐Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de‐facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non‐convex, and even non‐planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] — while being simpler to compute.

Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique

In this paper, we employ the linear virtual element spaces to discretize the semilinear sine–Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank–Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton’s method. We derive a priori error estimations in L2 and H1 norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine–Gordon equation are demonstrated with academic numerical experiments.

Compressed algebraic cubature over polygons with applications to optical design

In this paper we propose an algorithm to determine cubature rules of algebraic degree of exactness δ on general polygons P, by means of Matlab polyshape objects and near minimal rules on triangles, obtaining by Caratheodory–Tchakaloff subsampling a PI (Positive Interior) final formula with cardinality at most Nδ=(δ+1)(δ+2)∕2. We test our algorithm on polygons with different shape, and we also discuss an application to the computation of the RMSWE (Root Mean Square Wavefront Error) on obscured and vignetted pupils, in the framework of optical design by numerical ray tracing for the LSST (Large Synoptic Survey Telescope).

Atlas refinement with bounded packing efficiency

We present a novel algorithm to refine an input atlas with bounded packing efficiency. Central to this method is the use of the axis-aligned structure that converts the general polygon packing problem to a rectangle packing problem, which is easier to achieve high packing efficiency. Given a parameterized mesh with no flipped triangles, we propose a new angle-driven deformation strategy to transform it into a set of axis-aligned charts, which can be decomposed into rectangles by the motorcycle graph algorithm. Since motorcycle graphs are not unique, we select the one balancing the trade-off between the packing efficiency and chart boundary length, while maintaining bounded packing efficiency. The axis-aligned chart often contains greater distortion than the input, so we try to reduce the distortion while bounding the packing efficiency and retaining bijection. We demonstrate the efficacy of our method on a data set containing over five thousand complex models. For all models, our method is able to produce packed atlases with bounded packing efficiency; for example, when the packing efficiency bound is set to 80%, we elongate the boundary length by an average of 78.7% and increase the distortion by an average of 0.0533%. Compared to state-of-the-art methods, our method is much faster and achieves greater packing efficiency.

Clipping simple polygons with degenerate intersections

Polygon clipping is a frequent operation in many fields, including computer graphics, CAD, and GIS. Thus, efficient and general polygon clipping algorithms are of great importance. Greiner and Hormann (1998) propose a simple and time-efficient algorithm that can clip arbitrary polygons, including concave and self-intersecting polygons with holes. However, the Greiner–Hormann algorithm does not properly handle degenerate intersection cases, without the undesirable need for perturbing vertices. We present an extension of the Greiner–Hormann polygon clipping algorithm that properly deals with such degenerate cases.

Development of an Embedded Discrete Fracture Model for Field-Scale Reservoir Simulation With Complex Corner-Point Grids

Summary Corner-point grids have the capability to model complex geological features, such as faults and irregular reservoir boundaries. As an industry standard, they are widely used to simulate different types of reservoirs, including conventional and unconventional reservoirs. It is necessary to effectively simulate natural or hydraulic fractures in such reservoirs. In this work, a discrete fracture model is developed to conveniently simulate fractures in geologically complex reservoirs represented by corner-point grids. The method is an extension of the embedded discrete fracture model (EDFM). We first present the difficulties in terms of geometrical calculations pertaining to corner-point grids, including the irregularity and degeneracy of block geometry and irregular connections between fracture segments. A general-purpose geometrical algorithm is developed to find the intersections between the matrix and fractures in corner-point grids. This algorithm properly handles the intersection between a general polyhedron and a general polygon, in which both the polyhedron and the polygon can be convex or concave. Transmissibility-factor formulations are also further developed for connections and intersections between fractures. The calculation of effective well indices in different situations is also discussed in detail. Several case studies are presented to illustrate the accuracy and applicability of the developed model in standard black-oil or compositional simulators. The accuracy of the developed model is demonstrated by comparing its simulation results with those of local-grid-refinement (LGR) models. It is also found that the accuracy of the EDFM is not sensitive to matrix gridding when the average size of gridblocks is similar. Field-scale studies using synthetic and realistic reservoir models are presented to illustrate the significance of fractures during secondary recovery. Existing simulators can directly be used in conjunction with the proposed approach with slight modification in simulation input, if the simulators have nonneighboring-connection (NNC) functionality. Through the case studies, the algorithms and methodology developed in this work are shown to be highly effective for the modeling of fractures in field-scale reservoir-simulation studies with complex corner-point grids.

Method for Extracting the Effective Tensor Surface Impedance Function From Nonuniform, Anisotropic, Conductive Patterns

A patterning technique known as the point shifting method has enabled the generation of smoothly varying and highly anisotropic impedance surfaces with a wide range of patch sizes and shapes. Previously, the surface impedances of different shapes of unit cells were assumed by the impedance of similar size rectangle cells. In this paper, we study an approach to calculate the surface impedances for anisotropic polygon unit cells more accurately, based on the area moment of inertia equations. We define an alternative unit cell called the equivalent rectangle, which has the same tensor impedance properties of a general polygon unit cell in the surface impedance pattern. The size of the equivalent rectangle cell is calculated by using the moment of inertia equations between the polygon and the rectangle. The extracted surface impedance from the equivalent rectangle is compared to the surface impedance of polygon in the unit cell simulation, validating our method. We also verify the method by comparing the results between PEC patterns and impedance boundary sheets to which the extracted impedances are applied. Simulations of the patterns are verified by measurements as well.

A multiscale virtual element method for elliptic problems in heterogeneous porous media

In this paper, we propose a Multiscale Virtual Element Method (MsVEM) for elliptic problems in heterogeneous porous media. The use of very general coarse grids has advantages in subsurface simulations since they provide flexibility and can render more accurate coarse model. It is often necessary to use an irregular coarse grid when highly heterogeneous reservoirs are discretized via irregular anisotropic fine grids. The proposed MsVEM aims to build a coarse model on a flexible grid by integrating Generalized Multiscale Finite Element Method (GMsFEM) with Virtual Element Method (VEM). Because of the irregularity merit of VEM, MsVEM is desirable to simulate complex real-world problems. In the framework of MsVEM, both coarse grid and fine grid may consist of more general polygons. To build multiscale basis functions on coarse grid, a few local problems are solved on each coarse block to enrich multiscale information and get a snapshot space. To significantly reduce degree of freedoms of the coarse multiscale model, we construct a low dimensional multiscale basis space by using greedy selection procedure and penalized matrix decomposition. The greedy selection procedure is utilized to select optimal multiscale basis functions. The penalized matrix decomposition is employed to improve approximation accuracy and get the sparse basis functions. To show the efficiency and accuracy, we present a few numerical examples for flows in heterogeneous porous media. In particular, MsVEM is applied to the subsurface flow in high-contrast porous media and fractured media.