Triangular Grid Research Articles

A Conservative Semi-Lagrangian Finite Volume Method for Convection–Diffusion Problems on Unstructured Grids

A conservative semi-Lagrangian finite volume method is presented for the numerical solution of convection–diffusion problems on unstructured grids. The new method consists of combining the modified method of characteristics with a cell-centered finite volume discretization in a fractional-step manner where the convection part and the diffusion part are treated separately. The implementation of the proposed semi-Lagrangian finite volume method differs from its Eulerian counterpart in the fact that the present method is applied at each time step along the characteristic curves rather than in the time direction. To ensure conservation of mass at each time step, we adopt the adjusted advection techniques for unstructured triangular grids. The focus is on constructing efficient solvers with large stability regions and fully conservative to solve convection-dominated flow problems. We verify the performance of our semi-Lagrangian finite volume method for a class of advection–diffusion equations with known analytical solutions. We also present numerical results for a transport problem in the Mediterranean sea.

A stable hybrid Roe scheme on triangular grids

AbstractNumerical shock instability is a common problem for shock‐capturing methods that try to resolve contact and shear waves with minimal diffusion. Most flux‐difference splitting and the AUSM family of schemes produce the carbuncle phenomenon on both structured and unstructured grids. The original Roe scheme is well known to generate shock anomalies and can lead to nonentropic weak solutions to the Euler equations. A simple and robust approach for healing these numerical instabilities is to apply the hybrid technique incorporated with an efficient weighting switch function to control the amount of dissipation in the vicinity of shock waves. This article proposes a simple, robust, and accurate hybrid Roe scheme (Roe+scheme) by hybridizing the Roe scheme and the modified AUSMV+scheme. A new normalized pressure/density‐based weighting switch function is proposed and applied to the scheme to minimize the numerical dissipation and maintain the robustness of the hybridization. The linearized discrete analysis is performed to evaluate the proposed scheme according to the perturbation damping mechanism of an odd–even decoupling problem. The resulting recursive equations indicate that the hybridized mechanism damps all perturbations effectively. Finally, several numerical examples demonstrated that the Roe+scheme provides an accurate, robust, and carbuncle‐free solution on both structured and unstructured triangular grids.

A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids

In this paper, a finite volume element (FVE) method is proposed for the time fractional Sobolev equations with the Caputo time fractional derivative. Based on the L1-formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson FVE scheme is established by using an interpolation operator Ih*. The unconditional stability result and the optimal a priori error estimate in the L2(Ω)-norm for the Crank–Nicolson FVE scheme are obtained by using the direct recursive method. Finally, some numerical results are given to verify the time and space convergence accuracy, and to examine the feasibility and effectiveness for the proposed scheme.

A Splitting Mixed Covolume Method for Viscoelastic Wave Equations on Triangular Grids

A new splitting mixed covolume (SMCV) method is proposed for the viscoelastic wave equations by combining the splitting positive definite mixed finite element (SPDMFE) method with the mixed covolume (MCV) method. Based on the idea of the SPDMFE method, the difference between this method and the MCV method is that the proposed method does not need to solve the coupled system, thus reducing the scale of linear equations. By introducing a transfer operator, the semi-discrete and fully discrete SMCV schemes are proposed. The existence and uniqueness analysis of the semi-discrete scheme are given, and optimal a priori error estimates in different norm for these two schemes are derived. Finally, the feasibility and effectiveness of the proposed scheme are verified by some numerical results.

Smooth reconstruction method for surfaces defined by a continuous set of triangles

The paper considers the problem of triangular grids distortion in three-dimensional space, approximating the boundaries and interfaces of interacting solid, liquid and gaseous media. Unacceptable deformations of the mesh cells size and shape can occur during the numerical solution of continuous media dynamics problems. A method and algorithm for rebuilding grids using smooth piecewise polynomial interpolation by algebraic polynomials of the third degree is proposed.

On Triangular Secure Domination Number

Let T_m=(V(T_m), E(T_m)) be a triangular grid graph of m ϵ N level. The order of graph T_m is called a triangular number. A subset T of V(T_m) is a dominating set of T_m if for all u_V(T_m)\T, there exists vϵT such that uv ϵ E(T_m), that is, N[T]=V(T_m). A dominating set T of V(T_m) is a secure dominating set of T_m if for each u ϵ V(T_m)\T, there exists v ϵ T such that uv ϵ E(T_m) and the set (T\{u})ꓴ{v} is a dominating set of T_m. The minimum cardinality of a secure dominating set of T_m, denoted by γ_s(T_m) is called a secure domination number of graph T_m. A secure dominating number γ_s(T_m) of graph T_m is a triangular secure domination number if γ_s(T_m) is a triangular number. In this paper, a combinatorial formula for triangular secure domination number of graph T_m was constructed. Furthermore, the said number was evaluated in relation to perfect numbers.

Dynamics of interacting particle systems: Modeling implications of the repulsive interactions and experiments on magnetic prototypes

In this work, we investigate the dynamics of interacting particle systems subjected to repulsive forces, such as lattices of magnetized particles. To this end, we first develop a general model capable of capturing the complete dynamical behavior of interacting particle systems governed by arbitrary potentials. The model elucidates the important role played by the static repulsive forces exchanged between particles in the initial equilibrium configuration, which is distilled and mathematically captured by a dedicated component of the stiffness matrix. The implications of the model are then examined through the simple illustrative example of a magnetic particle oscillator, by which we show that the effect associated with the initial static forces is germane to two- or higher-dimensional particle systems and vanishes for 1D chains. In the context of wave propagation, we show that this type of effect manifests as modal-selective corrections of the dispersion relation of 2D repulsive lattices. To corroborate these findings, we perform laser vibrometry experiments on a lattice prototype consisting of a triangular grid of magnets supported by an elastic foundation of thin pillars. The tests unequivocally confirm the emergence of distinctive dispersive regimes in quantitative accordance to the model.

A curved boundary treatment for discontinuous Galerkin method applied to Euler equations on triangular and tetrahedral grids

For Euler equations defined in a domain Ω with boundary conditions specified on a curved boundary Γ, we consider a discontinuous Galerkin (DG) method with high order polynomial basis functions on a geometry fitting triangular and tetrahedral grids. We derive a modification to the DG scheme defined on a boundary triangle and tetrahedron based on the general curvilinear element DG method. In the modified DG scheme, even though integration along the curved boundary is still necessary, integrals over any curvilinear element are avoided. To further simplify the calculation of integration along the curved boundary, the modified DG scheme is constructed by the relationship between normal vector of geometric boundary and surface Jacobian. In the new modified DG scheme, not only integrals over any curvilinear element are avoided, but also the line and face integrations along the curved boundary are not required. Numerical tests suggest that such a new modified DG scheme is as accurate as the basic modified DG scheme and full curvilinear DG scheme and more efficient than latter, and is stable on reasonably coarse, as well as more refined, grids. It is shown that this new modified DG scheme can be extremely useful to solve the problems defined in a domain with boundary conditions specified on a curved boundary.

Use of symmetry groups for generation of complex space grids and group-theoretic vibration analysis of triple-layer grids

Space grids often exhibit a highly regular structure which is formed by repeated structural units. In the past, various algorithms have been developed to generate more complex structural assemblies using simple commands. In this paper, we explore the innovative use of symmetry groups to create various configurations of space grids, ranging from rectangular grids and their variants, to triangular and hexagonal grids, as well as grid forms of more complex shape. The advantage of this approach is that the generated space grid has pre-determined symmetry properties; this represents a shift in current design philosophy. A second objective is to show how group theory may be used to simplify the vibration analysis of layered space grids. The procedure is applied to the transverse vibrations of triple-layer space grids of D2h and D4h symmetries, modelled as discrete-parameter systems. A study of the symmetry of subspace basis vectors allows some important predictions to be made on the pattern of the motions and the location of stationary nodes. It is found that the lowest and highest frequencies of space grids of this type occur in the first subspace of the system, a finding that has significant computational importance.

Discrete finite volume approach for multidimensional agglomeration population balance equation on unstructured grid

In this work, a discrete finite volume scheme (FVS) is developed for approximating a multidimensional agglomeration population balance equation on an unstructured grid. Previous applications of the FVS were restricted only to structured rectangular and triangular grids. The main hindrance of using the unstructured grid is less knowledge of the relation between the quality of solution and the nature of elements. The testing of the proposed FVS is conducted against the FVS on a structured grid using analytically tractable kernels. The numerical results reveal that the FVS with unstructured grid estimates the numerical results with higher accuracy and efficiency than the existing method. The use of unstructured grid offers improved numerical results due to the flexibility of placing the pivot in the space. Moreover, the mixing of components for a bicomponent population is examined using the variance of excess solute for the unstructured grid.

Dual-Polarized Planar-Printed Ultrawideband Antenna Array on a Triangular Grid

This article presents a dual-polarized planar-printed ultrawideband antenna array on an optimally sampled triangular (hexagonal) lattice. The design is based on the dual-polarized planar ultrawideband modular antenna (PUMA). In this article, the dual-polarized square lattice PUMA array is adapted to an equilateral triangular lattice for grating-lobe-free scanning at a 15.5% reduced element count. A prototype triangular lattice PUMA array is built and compared with a square lattice PUMA aperture of similar size. As configured, the triangular lattice array demonstrates an 8% larger aperture area with 7% fewer elements than the square lattice array. Modeling predictions are validated with measured results for embedded element performance. The triangular lattice PUMA is shown to compare favorably to the square lattice PUMA with modest performance compromises.

Formex algebra adaptation into parametric design tools and rotational grids

Abstract:This paper describes the adaptation of the formex configuration processing to the computer program Grasshopper 3D and focuses on the applied mathematical solutions. Formex algebra is a mathematical system, primarily used for planning structural systems like truss-grid domes and vaults, together with the programming language Formian. The goal of the research is to allow architects to plan truss-grid structures easily with parametric design tools based on the versatile formex algebra mathematical system. To produce regular structures, coordinate system transformations are used. Owing to the abilities of the parametric design software, it is possible to apply further modifications on the structures and gain special forms. The paper covers the basic dome types, and it introduces additional dome-based structures using special coordinate-system solutions based on a spherical coordinate system, vault structures and their modifications based on a cylindrical coordinate system and circular structures and their modifications based on polar coordinates. Moreover two rotational grid tools are introduced, which uses coordinate system transformations on a unique way to create surfaces of revolutions based on the given generating curve and create grid structures on these surfaces. It also describes the solution technique to implement the triangular grid version of every one of these tools based on diamatic domes. The adaptation of formex algebra and the parametric workflow of Grasshopper together give the possibility of quick and easy design and optimization of special truss-grid domes.

On Multigrid Methods for Solving Two-Dimensional Boundary-Value Problems

Various methods for constructing algebraic multigrid type methods for solving multidimensional boundary-value problems are considered. Two-level iterative algorithms in Krylov subspaces based on approximating the Schur complement obtained by eliminating the edge nodes of the coarse grid are described on the example of two-dimensional rectangular grids. Some aspects of extending the methods proposed to the multilevel case, to nested triangular grids, and also to three-dimensional grids are discussed. A comparison with the classical multigrid methods based on using smoothing, restriction (aggregation), coarse-grid correction, and prolongation is provided. The efficiency of the algorithms suggested is demonstrated by numerical results for some model problems.

Digitized rotations of 12 neighbors on the triangular grid

There are various geometric transformations, e.g., translations, rotations, which are always bijections in the Euclidean space. Their digital counterpart, i.e., their digitized variants are defined on discrete grids, since most of our pictures are digital nowadays. Usually, these digital versions of the transformations have different properties than the original continuous variants have. Rotations are bijective on the Euclidean plane, but in many cases they are not injective and not surjective on digital grids. Since these transformations play an important role in image processing and in image manipulation, it is important to discover their properties. Neighborhood motion maps are tools to analyze digital transformations, e.g., rotations by local bijectivity point of view. In this paper we show digitized rotations of a pixel and its 12-neighbors on the triangular grid. In particular, different rotation centers are considered with respect to the corresponding main pixel, e.g. edge midpoints and corner points. Angles of all locally bijective and non-bijective rotations are described in details. It is also shown that the triangular grid shows better performance in some cases than the square grid regarding the number of lost pixels in the neighborhood motion map.

Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations

In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann–Liouville fractional derivative and an interpolation operator I_{h}^{*}, a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L^{infty }(L^{2}(varOmega)) and L^{2}(H^{1}(varOmega)) norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness.

An improved global-direction stencil based on the face-area-weighted centroid for the gradient reconstruction of unstructured finite volume methods**Project supported by the National Key Project, China (Grant No. GJXM92579).

The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.

Polygon Based Topology Formation and Information Gathering in Satellite Based Wireless Sensor Network

The satellite network is one of the major source of information and these days small satellites are gaining lot of focus. The group of small satellites form a distributed network that work collaboratively to accomplish the mission task. These networks are very similar to the terrestrial wireless sensor network in terms of restricted resources and constrained capabilities. Sometimes, network of small satellites is also called as space based wireless sensor network (SBWSN). In any distributed network, topology formation and its control plays a significant role. This is true for SBWSN also. The topology in SBWSN decides the area of coverage (field of view), time of coverage, data gathered and data transmitted to ground station. The proposed topology is a distributed network of small satellites formed by trivalent, toroidal and spherical polyhedron graph forming a fullerene, which is called as polygon based network topology (PBNT). It comprises of both pentagonal (Fp) and hexagonal (Fh) faces with K regular graphs such that K ≥ 3 with genus equal to 1. It also satisfies Eulers formula with n vertices. The fullerene comprises of simple rings and each of this ring forms the cluster. Each cluster is further represented as a triangular grid, that is linearly convex or non-linear, with K-connected graph along with Hamiltonian extendible cycle. The nodes/satellites on the triangular grid represent sensing nodes (low capability nodes/satellites), while the vertices of the ring are sink nodes (higher capability nodes/satellites). In this work, the topology is formed by small satellites (pico or nano satellites). In the proposed topology formation, the network is considered as virtual network with logical neighbours forming the cluster. Each node in the cluster covers a particular swath for a particular time interval based on the mission payload and on the p3 tiling. In the simulation, we consider n small satellites being placed in low earth orbit (LEO), (where n ranges from 3 to 150). The performance enhancements are seen during simulation in the following parameters, (1) Coverage Area: The coverage area increases as multiple satellites have different field of view at different times. (2) Reduces Gaps:The proposed distributed network also minimises uncovered areas as multiple satellites cover the target location at different time stamp which is not possible by a single large satellite. (3) Increase in Data Throughput: Each satellite in the network transmits data, when it is at perigee. The data throughput of the network increases, as data is transmitted by multiple satellites. Therefore, the throughput is increased by n-fold. (4) Continuous Connectivity: The data captured by one satellite in the network is made available to other using multi-hop communication. Thus the proposed topology also increases the continuous connectivity between satellites and also with the ground station. (5) Increases lifetime and Network Reliability: The SBWSN accomplishes its mission task even when one/more satellites encounters functional failure. The satellites in the network can reconfigure themselves and continue the mission task. Thus SBWSN also reduces the risk of mission failure and ensures mission reliability. Due to reconfiguration the lifetime of the network is also increased. The proposed topology is used for small satellites (specially nano and pico satellites) network, which permit the single board satellite weighing less than 10 kg (Pico satellite less than 1 kg and nano satellites less than 10 kg). The advantage of these small satellites network over single large satellite is low cost and reduced development time, as it uses commercially of the shelf (COTS) components. In this paper, we propose network architecture formed by the spherically embedded clusters formed by polyhedron. The vertices of polyhedron have both pentagonal (Fp) and hexagonal (Fh) faces with K regular graphs such that K ≥ 3, with genus equal to 1. The vertices of these polyhedron form the sink nodes and the other nodes are sensing nodes. Here satellites and nodes are interchangeably used. Sensing nodes are used for data gathering (pico/nano), while sink nodes are higher capability nodes which perform computational extensive operations in the network (nano or macro satellites). The sensing and sink nodes transmit data to the ground station when they are at perigee. The main objective of the proposed work is, technology demonstration of low cost, distributed small satellites network for earth observations replacing single huge satellite.

Effects of cost optimised grid configuration on earthing system performance: a comparative assessment

This study presents the comparative analysis of earthing grid configurations to determine the optimum design in terms of safety parameters and cost for 66/11 kV substation. For different grid configurations, a fixed earthing area of 9000 m2 was considered. Rectangular, triangular, L-shaped and T-shaped grids buried in two-layer soil model were considered for this study. Three methods, i.e. ANSI/IEEE standard 80-2000, finite element method and genetic algorithm, were used to design the grids. All safety constraints as mentioned in the IEEE standard were followed for designing the earthing grid. A new mathematical cost function (CF) was proposed for designing an effective and economic L-shaped, T-shaped and triangular earthing systems. The CFs used in this study include grid depth, dimensions of earthing rod and horizontal conductor, number of rods and conductors, revetment and excavation area. The economic earthing grid obtained through the proposed method reduces the total cost related to grid materials, installation and excavation. Analysis also showed that the rectangular grid showed preferred results in terms of safety parameters while the triangular grid showed optimum cost design for the earthing system under study. Good agreements were found with the rectangular earthing grid in comparison with triangular, T-shape and L-shape configurations.

An unstructured finite volume method based on the projection method combined momentum interpolation with a central scheme for three-dimensional nonhydrostatic turbulent flows

This paper presents a three-dimensional nonhydrostatic model to solve the Navier–Stokes equations using an unstructured finite volume method. The physical domain could be geometrically arbitrary. To avoid the checkerboard problem caused by non-staggered grids, a momentum interpolation method is used by introducing face-normal velocities at the mid-points of the cell faces. As the Large Eddy Simulation (LES) requires at least second-order accuracy in time and in space for all the terms, a central scheme combined with an explicit Adams–Bashforth scheme is proposed in this model. The projection method is applied to decouple the velocity field and pressure. Several benchmark test cases are used to validate the second-order accuracy, the numerical stability and the performance of the model. Analysis on divergence noise using an unstructured collocated triangular grid, as well as on the ratio between vertical and horizontal spacing steps have been done to show the reliability of the model. The proposed model has been used to simulate backward-facing step flows, lid-cavity flows, turbulent open channel flows and the turbulent flows around a vertical cylinder. The convergence of the linear solver is analyzed in terms of the iterations and CPU time. The results are fairly in agreement with the references in the literature. The proposed model is able to correctly reproduce the characteristic flow features in all the test cases.

On triangular paperfolding patterns

We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute eigenvalues and eigenvectors of the corresponding matrices. We also prove that densities of all basic triangles are equal in these patterns.