Arbitrary Polygon Research Articles

An optimal coverage path plan for an autonomous vehicle based on polygon decomposition and ant colony optimisation

This paper proposes a coverage path planner for the autonomous vehicle used in inspection operation. The proposed coverage path planner is developed based on a back and forth pattern (equivalent to boustrophedon pattern), producing the output path for the inspection operation with the least trajectory length and number of turns covering the entire workspace. Implementing a back and forth pattern on an arbitrary polygon (extracted from workspace boundaries) is not always viable due to the existence of concave vertices in the polygon. So, this work proposes the tree search model based on the polygon width and back and forth pattern’s sweep line, which decomposes the input polygon in numerous ways. The Dijkstra algorithm solves this tree model to obtain back and forth pattern viable polygons with the least number of turns. Later, the vehicle visits these polygons one by one during the inspection operation. The order in which polygon visits are made by the vehicle affects the overall trajectory length. So, this work proposes the ant colony optimisation based trajectory length optimiser, which finds the polygon visiting order that yields minimum trajectory length. The developed coverage path planner is versatile, and it applies to any autonomous vehicle that operates in the planar region.

Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle

Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle

Shrinkable Self-Similar Structure Design

Abstract Origami techniques, as folding and unfolding, can be utilized in shrinkable structures. Especially when the crease pattern is rigid foldable, it can be treated as a mechanical linkage of rigid panels connected by hinges. Since rigid foldable crease patterns have the strong geometrical constraint of the facets not being able to stretch or bend, it is difficult to design new crease patterns, and variations of existing patterns are limited. However, it is known that there are cases where crease patterns can be made rigid foldable by adding some slits. This paper proposes a mechanical linkage that folds into a similar flat shape by adding slits. A method is presented of generating rigid foldable crease patterns in arbitrary polygons that fold smaller, and it is confirmed that structures that have a mechanism for shrinking can be generated from these crease patterns using rigid thick panels and hinges.

Revised SBFEM on arbitrary polygons and faceted polyhedrons with the second order completeness by elimination of bubble functions

By the theoretical analysis on the scaled boundary finite element (SBFEM), some appropriate 2D or 3D bubble functions are necessary in order that the elements can possess the second order completeness. However, the extra 2D bubble function is difficult to be adopted as facet shape function for 3D polyhedral element. In this paper, a revised-scaled-boundary element (denoted by 2DRSBd2) only corresponding to the boundary nodal displacements is constructed by elimination of the 2D bubble function derived from the constant right-hand term in 2D Poisson equation. This polygonal element 2DRSBd2 possesses the second order completeness and the third order convergence in L2-norm. Then we construct two polyhedral elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 based on the polygonal element 2DRSBd2. The difference between the two 3D elements is: the 3DSB-2DRSBd2 element is constructed with three 3D bubble functions derived from the constant body loads in three dimensional elastostatic problem, while the 3D revised-scaled-boundary element 3DRSB-2DRSBd2 is constructed by elimination of those 3D bubble functions. Both elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 possess the second order completeness and the third order convergence in L2-norm. Numerical examples verify that they have good accuracy and insensitive to mesh distortion.

A Cell-Based Linear Smoothed Finite Element Method for Polygonal Topology Optimization

The aim of this work is to employ a modified cell-based smoothed finite element method (S-FEM) for topology optimization with the domain discretized with arbitrary polygons. In the present work, the linear polynomial basis function is used as the weight function instead of the constant weight function used in the standard S-FEM. This improves the accuracy and yields an optimal convergence rate. The gradients are smoothed over each smoothing domain, then used to compute the stiffness matrix. Within the proposed scheme, an optimum topology procedure is conducted over the smoothing domains. Structural materials are distributed over each smoothing domain and the filtering scheme relies on the smoothing domain. Numerical tests are carried out to pursue the performance of the proposed optimization by comparing convergence, efficiency and accuracy.

An improved numerical model for groundwater flow simulation with MPFA method on arbitrary polygon grids

Groundwater models are critical for simulating subsurface hydrological processes and guiding informed policy-making for groundwater management. However, the widely applied groundwater models typically use regular-shaped grids to discretize aquifer systems and require that the directions of the grid edges are aligned with the hydraulic conductivity tensor. Such rigorous requirements for spatial discretization have constrained the models’ application in aquifer systems with anisotropic hydrogeological characteristics. To address such limitations, we develop an improved groundwater flow model based on the multipoint flux approximation (MPFA) method in this study. The new model allows us to use arbitrary-shaped polygon grids to discretize aquifer systems and relaxes the rigorous requirement of the alliance between polygon edges and hydraulic conductivity tensor. The functionality and performance of the new model are demonstrated by comparing the output between our model, MODFLOW, and analytical solution in four case studies with various hydrogeological conditions. In a real-world watershed with complex-shaped boundaries, our model outperforms the conventional groundwater model in boundaries. The modeling results show that our model can yield accurate simulation of subsurface hydrological processes in aquifer systems with complex-shaped boundaries. Furthermore, our model can provide a more flexible discretization solution to couple surface water and groundwater model in integrated hydrological model development.

Distance distribution between two random points in arbitrary polygons

Distance distributions are a key building block in many subfields in mathematics, science and engineering. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random points each uniformly randomly distributed in respective arbitrary polygon regions. The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary (concave or convex) polygons, with or without holes, which may be disjoint or overlap or coincide and determine the distance distribution numerically.

On the Boundary Components of Central Streams

Foliations on the space of $p$-divisible groups were studied by Oort in 2004. In his theory, special leaves called central streams play an important role. In this paper, we give a complete classification of the boundary components of the central streams for an arbitrary Newton polygon in the unpolarized case. Hopefully this classification would help us to know the boundaries of other leaves and more detailed structure of the boundaries of central streams.

PSBFEM-Abaqus: Development of User Element Subroutine (UEL) for Polygonal Scaled Boundary Finite Element Method in Abaqus

The polygonal scaled boundary finite element method (PSBFEM) is a novel method integrating the standard scaled boundary finite element method (SBFEM) and the polygonal mesh technique. This work discusses developing a PSBFEM framework within the commercial finite element software Abaqus. The PSBFEM is implemented by the User Element Subroutine (UEL) feature of the software. The details on the main procedures to interact with Abaqus, defining the UEL element, and solving the stiffness matrix by the eigenvalue decomposition are present. Moreover, we also develop the preprocessing module and the postprocessing module using the Python script to generate meshes automatically and visualize results. Several benchmark problems from two-dimensional linear elastostatics are solved to validate the proposed implementation. The results show that PSBFEM-UEL has significantly better than FEM convergence and accuracy rate with mesh refinement. The implementation of PSBFEM-UEL can conveniently use arbitrary polygon elements by the polygon/quadtree discretizations in the Abaqus. The developed UEL and the associated input files can be downloaded from https://github.com/hhupde/PSBFEM-Abaqus.

General design equations for arbitrary odd-way filtering Bagley polygon power divider with notch band

The general design equations are derived for arbitrary odd-way Bagley polygon power divider (BPPD) with filtering response and notch band. Firstly, conditions are derived to achieve a perfect input port impedance matching at centre frequency. Then, the formulas of introduced transmission zeros (TZs) are obtained for high-frequency selectivity, wide upper stopband and notch band. Based on the deduced equations, wideband BPPD (WBPPD), filtering BPPD (FBPPD) and FBPPD integrating notch band (FBPPD-N) with arbitrary odd-way power splitting are designed. Finally, a FBPPD-N with a wide passband bandwidth is fabricated and measured to verify the presented ideal. The experimental results indicate that the proposed FBPPD-N has compact size, high-frequency selectivity, wide upper stopband and required notch band. Consequently, the presented general design equations can provide effective guidance for the design of arbitrary odd-way power splitting FBPPD-N.

Automated Dynamic Mascon Generation for GRACE and GRACE-FO Harmonic Processing

Commonly used mass-concentration (mascon) solutions estimated from Level-1B Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On data, provided by processing centers such as the Jet Propulsion Laboratory (JPL) or the Goddard Space Flight Center (GSFC), do not give users control over the placement of mascons or inversion assumptions, such as regularization. While a few studies have focused on regional or global mascon optimization from spherical harmonics data, a global optimization based on the geometry of geophysical signal as a standardized product with user-defined points has not been addressed. Finding the optimal configuration with enough coverage to account for far-field leakage is not a trivial task and is often approached in an ad-hoc manner, if at all. Here, we present an automated approach to defining non-uniform, global mascon solutions that focus on a region of interest specified by the user, while maintaining few global degrees of freedom to minimize noise and leakage. We showcase our approach in High Mountain Asia (HMA) and Alaska, and compare the results with global uniform mascon solutions from range-rate data. We show that the custom mascon solutions can lead to improved regional trends due to a more careful sampling of geophysically distinct regions. In addition, the custom mascon solutions exhibit different seasonal variation compared to the regularized solutions. Our open-source pipeline will allow the community to quickly and efficiently develop optimized global mascon solutions for an arbitrary point or polygon anywhere on the surface of the Earth.

Mechanical characteristics and deformation calculation of steel diagrid structures in high-rise buildings

In order to propose a simple and practical methodology for calculating the deformation of diagrid structure with polygon plane in arbitrary direction, the diagrid structure is considered to be mainly composed of “element” by geometric method. The vertical stiffness and lateral stiffness of diagrid structure with arbitrary polygon plane under vertical load and horizontal load are derived respectively. The corresponding laboratory test and numerical simulation analysis are carried out through the scale models of steel diagrid structures. The analysis results show that the calculated results of each deformation equation are in good agreement with the test results and numerical simulation results. In addition, the influence of the parameters on the stiffness of the structure is revealed theoretically. The results demonstrate that when the diagonal angle is about 55°, the variation of the diagonal angle has the greatest effect on the vertical stiffness of the diagrid structure. Moreover, the vertical stiffness of steel diagrid structure is smaller than that of traditional structure, and the vertical displacement is obviously larger. It is necessary to limit the inter-story drift ratio of vertical modules. Furthermore, current paper theoretically explains that the more edges of diagrid structure, the greater the horizontal shear stiffness of the structure.

Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

Groundwater solute transport models are critical for simulating subsurface contaminant processes, guiding policy making for groundwater management, and pollution remediation. In this study, we develop an improved groundwater solute transport model which uses arbitrary-shaped polygons for spatial discretization. This allows for the efficient simulation of solutes in complex boundary aquifer systems. The governing equation is discretized in space using the finite volume method to ensure flux conservation both locally and globally. The dispersion term is discretized by the multipoint flux approximation (MPFA) method which is a finite volume method that is suitable on arbitrary polygon grids. A total variation diminishing (TVD) scheme is used to control numerical oscillation in the advection term. The functionality and performance of the new model are demonstrated by comparing the simulated results between our model and MT3DMS model in five case studies. The comparisons show that the new model can well represent the solute transport processes. In a real-world watershed with complex-shaped boundaries, the new model outperforms the conventional groundwater solute transport model with smaller computed root-mean-square errors. These modeling results suggest that our model can provide robust modeling solutions for simulating groundwater solute transport processes, especially in aquifer systems with complex shaped boundaries. Furthermore, our model can provide a more flexible discretization solution for coupling surface water models and groundwater solute transport models.

Research on the Coverage of Ultraviolet Communication Network in the Arbitrary Polygon Area

Given that the current ultraviolet (UV) networking model is established in a regular circular area, this article studies the coverage of a UV non-line-of-sight (NLOS) communication network creatively in the arbitrary polygon area. In this paper, the UV communication model and the basic concepts of network coverage are introduced first. Then the influence parameters of the UV node communication radius are studied, and the changes of the communication radius under different work patterns are analyzed. Finally, the coverage of the square target area is simulated under different communication parameters (transmitted power, data rate and node density). The results illustrate that the smaller the transceiver elevation angles are, the better the network coverage performance is. Additionally, we numerically compare the UV network models of polygonal and circular regions, which can be used as a reference for actual networking.

WITHDRAWN: Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

WITHDRAWN: Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the [Formula: see text] and [Formula: see text] norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.

Numerically stable form factor of any polygon and polyhedron

Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes's theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near the singularities causes a loss of precision that can be avoided by using series expansions. An important application domain is small-angle scattering by nanocrystals.

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.

Predictive accuracy of wall-modelled large-eddy simulation on unstructured grids

The predictive accuracy of wall-modelled LES is influenced by a combination of the subgrid model, the wall model, the numerical dissipation induced primarily by the convective numerical scheme, and also by the density and topology of the computational grid. The latter factor is of particular importance for industrial flow problems, where unstructured grids are typically employed due to the necessity to handle complex geometries. Here, a systematic simulation-based study is presented, investigating the effect of grid-cell type on the predictive accuracy of wall-modelled LES in the framework of a general-purpose finite-volume solver. Following standard practice for meshing near-wall regions, it is proposed to use prismatic cells. Three candidate shapes for the base of the prisms are considered: a triangle, a quadrilateral, and an arbitrary polygon. The cell-centre distance is proposed as a metric to determine the spatial resolution of grids with different cell types. The simulation campaign covers two test cases with attached boundary layers: fully-developed turbulent channel flow, and a zero-pressure-gradient flat-plate turbulent boundary layer. A grid construction strategy is employed, which adapts the grid metric to the outer length scale of the boundary layer. The results are compared with DNS data concerning mean wall shear stress and profiles of flow statistics. The principle outcome is that unstructured simulations may provide the same accuracy as simulations on structured orthogonal hexahedral grids. The choice of base shape of the near-wall cells has a significant impact on the computational cost, but in terms of accuracy appears to be a factor of secondary importance.