Polygonal Grids Research Articles

Structure Preserving Polytopal Discontinuous Galerkin Methods for the Numerical Modeling of Neurodegenerative Diseases

Many neurodegenerative diseases are connected to the spreading of misfolded prionic proteins. In this paper, we analyse the process of misfolding and spreading of both α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-synuclein and Amyloid-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}, related to Parkinson’s and Alzheimer’s diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on ϑ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vartheta -$$\\end{document}method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guarantees that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson’s and Alzheimer’s diseases.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

DEVELOPMENT OF THE CALCULATION PROCEDURE ALGORITHM FOR THE CALCULATION OF LIGHTNING RODS PROTECTION ZONES ACCORDING TO THE INTERNATIONAL STANDARD IEC 62305

A series of publications on the problem of lightning protection in power stations and substations, especially in the conditions of decreasing stability of the functioning of the energy system of Ukraine is continued. In addition, the relevance of the work is substantiated by the increase in thunderstorm activity on the planet as a whole and by the data of the national energy company in Ukraine, which leads to emergency shutdowns on power transmission lines. The system of algorithms for the computational procedure for calculating the protection zones of the system, which consists of numerous rod and cable lightning conductors and protective nets, using the rolling sphere method is developed. It is noted, that the majority of strategic energy facilities in Ukraine have lightning protection systems built according to outdated standards, therefore, their inspection and modernization is necessary taking into account current regulatory documents. A generalized algorithm for constructing a protection zone using the Delaunay algorithm and previously obtained mathematical expressions has been developed, which in turn consists of sub-algorithms: constructing a polygonal grid based on the points of the coordinates of the lightning conductors, finding the areas of contact with the sphere of the vertices of the polygonal grid, constructing a grid of nodes of the analysis points, finding the height coordinates for of each analysis point, construction of the protection surface based on the grid coordinates of the analysis point nodes. The software implementation was carried out and an example of the application of algorithms was shown when calculating the protection of an operating substation with a voltage class of 150 kV. The general features of software implementation in the LiGro software complex are given. The developed software product found its application in the modernization of existing lightning protection systems of operating power facilities of Ukraine with a voltage class of 35-750 kV.

A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability.

Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers

Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of Machine Learning (ML) strategies, that can naturally exploit geometrical information about the mesh in order to preserve the grid quality, enhancing performance of numerical methods and reducing the overall computational cost. In particular, we employ the k-means clustering algorithm and Graph Neural Networks (GNNs) to partition the connectivity graph of a computational mesh. Moreover, GNNs have high online inference speed and the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information, such as the areas of the elements or their barycentric coordinates. These techniques are compared with METIS, a standard algorithm for graph partitioning, which is meant to process only the graph information of the mesh. We demonstrate that performance in terms of quality metrics is enhanced for ML strategies. Such models also show a good degree of generalization when applied to more complex geometries, such as brain MRI scans, and the capability of preserving the quality of the grid. The effectiveness of these strategies is demonstrated also when applied to MultiGrid (MG) solvers in a Polygonal Discontinuous Galerkin (PolyDG) framework. In the considered experiments, GNNs show overall the best performance in terms of inference speed, accuracy and flexibility of the approach.

THE USE OF POLYGONS IN VARIOUS FILE FORMATS IN THE DEVELOPING DESIGN OF 3D MODELS

The article describes the basic concepts of topology, which are fundamental for specialists in the field of three-dimensional design. All polygons can be divided into triangles, squares and n-gons, and of these polygons are the most undesirable. To consider this statement, a number of models of varying complex- ity have been developed and transferred to the UNITY game engine in two forms obj and fbx. Both formats are the most commonly used in game design, character design, locations, or object modeling, but they con- vert the polygonal grid of an object in different ways, which is why the shape of the model itself will change in the presence of n-gons.

Discontinuous Galerkin methods for Fisher–Kolmogorov equation with application to [formula omitted]-synuclein spreading in Parkinson’s disease

This spreading of prion proteins is at the basis of brain neurodegeneration. This paper deals with the numerical modelling of the misfolding process of α-synuclein in Parkinson’s disease. We introduce and analyse a discontinuous Galerkin method for the semi-discrete approximation of the Fisher–Kolmogorov (FK) equation that can be employed to model the process. We employ a discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) for space discretization, to accurately simulate the wavefronts typically observed in the prionic spreading and we prove stability and a priori error estimates. Next, we use a Crank–Nicolson scheme to advance in time. For the numerical verification of our numerical model, we first consider a manufactured solution, and then we consider a case with wavefront propagation in two-dimensional polygonal grids. Next, we carry out a simulation of α-synuclein spreading in a two-dimensional brain slice in the sagittal plane with a polygonal agglomerated grid that takes full advantage of the flexibility of PolyDG approximation. Finally, we present a simulation in a three-dimensional geometry reconstructed from magnetic resonance images of a patient’s brain.

A Local Grid-Refined Numerical Groundwater Model Based on the Vertex-centred Finite-Volume Method

Many algorithms have been proposed to cope with groundwater numerical simulation associated with local grid refinements typically for subsurface flow driven by sources acting on diverse scales. In this context, we focus on establishing an efficient local grid refinement method with nonmatching grids for groundwater flow modelling. The new model is based on the vertex-centred finite-volume method (VCFVM). The core idea of the algorithm resting on set all unknowns on vertices, and the flux between two vertices (the numerical flux) is expressed as a function of the hydraulic heads at the vertices of the element containing the lateral surface. The total outflow of the control volume of a given vertex is expressed as the sum of numerical fluxes. Since the algorithm sets all unknowns on vertices and a control volume can be defined for each vertex, our scheme readily embeds treatment of arbitrary polygonal grids, including nonmatching grids, in the presence of local grid refinement, without additional treatment at the nonmatching nodes. Six test cases, including five assumed ones and a real-world case, were adopted to evaluate the accuracy and efficiency of the new algorithm. The hydraulic heads calculated by the new algorithm were compared with those by a widely used and tested numerical groundwater model, called MODFLOW, and the analytic solutions. The mass balance error and the CPU time were compared with those of the MODFLOW 6 model, which owns ability to cope with nonmatching grids. The results show that the new algorithm yields high accuracy and efficiency in simulating groundwater flows with local grid refinements.

Numerical simulation of oil-water displacements in naturally fractured reservoirs using a hybrid grid MPFA-streamline (SLS-HyG) formulation

The numerical simulation of fluid displacements in naturally fractured reservoirs is rather expensive in terms of CPU requirements and challenging, due to the random spatial distribution of fractures of different sizes and shapes. In this paper, we present a numerical formulation to simulate the 2-D two-phase flow of oil and water in naturally fractured oil reservoirs using unstructured quadrilateral meshes. In our Hybrid-Grid MPFA-Streamline (SLS-HyG) formulation, the elliptic pressure equation is discretised by a finite volume method with a Multipoint Flux Approximation using a Diamond stencil. This formulation is able to handle a full permeability tensor representation in an arbitrary polygonal grid. For the advective saturation problem, we used the streamline-based method, which decouples the transport equations into several 1-D problems solved along each streamline. These formulations were implemented in the framework of the Hybrid-Grid (HyG) representation of fractures, where the fractures in the physical domain are represented by 1-D objects in 2-D domains and extended to the same dimension of the matrix problem. The coupling between the flow and the transport problem is done by a modified implicit pressure explicit saturation procedure. We have solved some reference problems from the literature, obtaining promising results when compared to other classical formulations.

DEVELOPMENT OF A POSTGIS-BASED METHOD FOR CREATING RISK MAPS OF NATURAL DISASTERS USING THE EXAMPLE OF GEORGIA

This article discusses a method for creating a risk map of natural disasters for a specific area using PostGIS analysis. This method, which is described in detail, is based entirely on the use of open-source programs and spatial SQL directives. It is designed to use widely available GIS programs to create a flexible method that performs operations on OGC geometry. The area to be analyzed is covered with a polygonal grid. The size of the polygon is determined by the size of the study area. For each element of the grid (polygon), attribute values are determined for the traits characteristic of the area: seismic, landslide, mudflow, rockfall, population density, and number of buildings. The attribution values for geometry are assigned based on PostGIS analysis. The process of creating a risk map involves the summation of different hazard maps. Our method’s originality lies in the fact that the initial data is transformed so that it is possible to carry out map algebra operations with vector data. The method is designed to create risk maps based on OGC vector data analysis, which will help experts and stakeholders working on risk assessment and decision-making to develop a risk management strategy.

Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods

We propose new strategies to handle polygonal grids refinement based on Convolutional Neural Networks (CNNs). We show that CNNs can be successfully employed to identify correctly the “shape” of a polygonal element so as to design suitable refinement criteria to be possibly employed within adaptive refinement strategies. We propose two refinement strategies that exploit the use of CNNs to classify elements' shape, at a low computational cost. We test the proposed idea considering two families of finite element methods that support arbitrarily shaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG) methods and Virtual Element Methods (VEMs). We demonstrate that the proposed algorithms can greatly improve the performance of the discretization schemes both in terms of accuracy and quality of the underlying grids. Moreover, since the training phase is performed off-line and is independent of the differential model the overall computational costs are kept low.

Arbitrary-Order Bernstein–Bézier Functions for DGFEM Transport on 3D Polygonal Grids

In this paper, we present an arbitrary-order discontinuous Galerkin finite element discretization of the SN transport equation on 3D extruded polygonal prisms. Basis functions are formed by the tensor product of 2D polygonal Bernstein–Bézier functions and 1D Lagrange polynomials. For a polynomial degree p, these functions span {xayb}(a+b)≤p⊗{zc}c∈(0,p) with a dimension of np(p+1)+(p+1)(p−1)(p−2)/2 on an extruded n-gon. Numerical tests confirm that the functions capture exactly monomial solutions, achieve expected convergence rates, and provide full resolution in the thick diffusion limit.

Discontinuous Galerkin solutions for elliptic problems on polygonal grids using arbitrary-order Bernstein-Bézier functions

In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their interpolation points. This can enable the use of efficient DGFEM physics-based diffusion preconditioners for the first-order linear transport equation on arbitrary polygons in the computationally-challenging asymptotic diffusion limit. On a polygonal element with n vertices and polynomial degree p, there are n vertex functions, (p−1) functions per edge, and (p−1)(p−2)/2 interior functions that span the {xayb}(a+b)≤p space of bivariate monomials. Numerical experiments highlighting the performance of the proposed method are presented through uniform and adaptive h, p, and hp refinement strategies. Appropriate error rates are observed including hp adaptivity yielding exponential convergence in the presence of singularities in the solution.

A generalised complete flux scheme for anisotropic advection-diffusion equations

In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilise the advantages of each of these methods. The usage of HMM allows us to handle anisotropic diffusion tensors on generic polygonal (polytopal) grids, whereas the CF method provides a framework for the construction of a uniformly second-order method, even when the problem is advection dominated.

Staggered DG Method for Coupling of the Stokes and Darcy--Forchheimer Problems

In this paper we develop a staggered discontinuous Galerkin method for the Stokes and Darcy--Forchheimer problems coupled with the Beavers--Joseph--Saffman conditions. The method is defined by imposing staggered continuity for all the variables involved and the interface conditions are enforced by switching the roles of the variables met on the interface, which eliminate the hassle of introducing additional variables. This method can be flexibly applied to rough grids such as the highly distorted grids and the polygonal grids. In addition, the method allows nonmatching grids on the interface thanks to the special inclusion of the interface conditions, which is highly appreciated from a practical point of view. A new discrete trace inequality and a generalized Poincaré--Friedrichs inequality are proved, which enables us to prove the optimal convergence estimates under reasonable regularity assumptions. Finally, several numerical experiments are given to illustrate the performances of the proposed method, and the numerical results indicate that the proposed method is accurate and efficient, and in addition, it is a good candidate for practical applications.

Probabilistic sampling and estimation for large-scale assessment of poplar plantations in Northern Italy

In the recent decades, growing demand for wood products, combined with efforts to conserve natural forests, has supported a steady increase in the global extent of planted forests. In this paper, a two-phase sampling strategy for large-scale assessment of hybrid poplar plantations in Northern Italy was implemented. The first phase was performed by means of tessellation stratified sampling on high-resolution remotely sensed imagery, covering the survey area by a grid of regular polygons of equal size and randomly and independently selecting one point per quadrat. All the plantations spotted by at least one sample point were selected. In the second phase, we randomly chosen a subset of plantations by stratified sampling that were visited on the ground to collect qualitative and quantitative attributes. The resulting estimates were reliable, and the survey demonstrated relatively easy to be implemented and replicated. These considerations support the use of the proposed sampling strategy to frequently update information on fast-growing forest plantations within agricultural farms, like hybrid poplar crops. Moreover, the results of the case study here presented highlight the relevance of hybrid poplar plantations in Italy, in the context of sustainable development strategies under a green economy perspective.

On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order $$C^1$$ conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete $$H^2$$ norm. Under a full elliptic regularity $$H^4$$ assumption, the $$L^2$$ convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.

A linearity-preserving vertex interpolation algorithm for cell-centered finite volume approximations of anisotropic diffusion problems

PurposeA finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl., 1(4)(1980)217–224 (in Chinese)], which is the so-called nine point scheme on structured quadrilateral meshes. The scheme has both cell-centered unknowns and vertex unknowns which are usually expressed as a linear weighted interpolation of the cell-centered unknowns. The critical factor to obtain the optimal accuracy for the scheme is the reconstruction of vertex unknowns. However, when the mesh deformation is severe or the diffusion tensor is discontinuous, the accuracy of the scheme is not satisfactory, and the author hope to improve this scheme.Design/methodology/approachThe authors propose an explicit weighted vertex interpolation algorithm which allows arbitrary diffusion tensors and does not depend on the location of discontinuity. Both the derivation of the scheme and that of vertex reconstruction algorithm satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The vertex interpolation algorithm can be easily extended to 3 D case.FindingsNumerical results show that it maintain optimal convergence rates for the solution and flux on 2 D and 3 D meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous.Originality/valueThis paper proposes a linearity preserving and explicit weighted vertex interpolation algorithm for cell-centered finite volume approximations of diffusion equations on general grids. The proposed finite volume scheme with the new interpolation algorithm allows arbitrary continuous or discontinuous diffusion tensors; the final scheme is applicable to arbitrary polygonal grids, which may have concave cells or degenerate ones with hanging nodes. The final scheme has second-order convergence rate for the approximate solution and higher than first-order accuracy for the flux on 2 D and 3 D meshes. The explicit weighted interpolation algorithm is easy to implement in three dimensions in case that the diffusion tensor is continuous or discontinuous.

A hybrid subcell-remapping algorithm for staggered multi-material arbitrary Lagrangian-Eulerian methods

A new flux-based hybrid subcell-remapping algorithm for staggered multimaterial arbitrary Lagrangian-Eulerian (MMALE) methods is presented. This new method is an effective generalization of the original subcell-remapping method to the multi-material regime (LOUBERE, R. and SHASHKOV, M. A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. Journal of Computational Physics, 209, 105–138 (2005)). A complete remapping procedure of all fluid quantities is described detailedly in this paper. In the pure material regions, remapping of mass and internal energy is performed by using the original subcell-remapping method. In the regions near the material interfaces, remapping of mass and internal energy is performed with the intersection-based fluxes where intersections are performed between the swept regions and pure material polygons in the Lagrangian mesh, and an approximate approach is then introduced for constructing the subcell mass fluxes. In remapping of the subcell momentum, the mass fluxes are used to construct the momentum fluxes by multiplying a reconstructed velocity in the swept region. The nodal velocity is then conservatively recovered. Some numerical examples simulated in the full MMALE regime and several purely cyclic remapping examples are presented to prove the properties of the remapping method.

Use of exochorion characters for the systematics of Hamadryas Hübner andEctima Doubleday (Nymphalidae: Biblidinae: Ageroniini).

We studied the chorionic morphology of six species of Hamadryas, and together with previous studies, we compared our results with previously published phylogenies for the genus. Samples were obtained from 19 females collected between 2013 and 2017 whose abdomens were sectioned and preserved for later dissection. Eggs were extracted from those dissections and used for the descriptions and illustrations of the chorion. The Hamadryas egg is of the globose type; it is quasi-spheroidal and has multiple polygonal grids with differentiation in specific zones/regions, and knolls with macrocells in their summits that arise in the apical third. These characteristics are very different from those found in the majority of Biblidinae and for those reported in the literature for Batesia and Panacea, which belong to the same subtribe as Hamadryas (Ageroniina, now Ageroniini). Chorionic characters support a previously suggested division of the genus (februa, feronia and laodamia groups) and they agree with the phylogenetic proposal based on morphological characters. Our study expands previous morphological work focused on this genus and compiles all the information available to date about the exochorion of Hamadryas, which now includes data for 10 species and that of Ectima thecla thecla, the putative sister group of Hamadryas.