Triangular Grid Research Articles

An efficient numerical method for reaction–diffusion equation on the general curved surfaces

In this paper, we propose an efficient numerical algorithm for reaction–diffusion equation on the general curved surface. The surface is discretized by a mesh consisting of triangular grids. The partial differential operators are defined based on the surface mesh and its dual surface polygonal tessellation. The proposed method has three advantages including intrinsic geometry, conservation law, and convergence property. The proposed method only needs the information of 1-ring of neighboring vertices for the divergence of a vector field and the Laplace–Beltrami operators, while the numerical conservation laws still hold. The proposed method avoids the global surface triangulation and its implementation is simple since we can explicitly define the Laplace–Beltrami operator by using the information of the neighborhood of each triangular grid. In order to obtain second-order temporal accuracy, we utilize the Crank–Nicolson formula to the reaction–diffusion system. The discrete system is solved by the biconjugate gradient stabilized method. The proposed algorithm is simple to implement and is second-order accurate both in space and time. Various numerical experiments are presented to demonstrate the efficiency of our algorithm.

High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H({\rm div})-P_k (k\geq 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, which can be viewed as the Hu-Zhang elements enriched with higher order interior bubble functions. The displacement field is approximated by $C^{-1}-P_{k-1}$ polynomial vectors enriched with higher order terms to ensure the stability condition. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress with $H(\rm div)$ norm and the displacement with $L^2$ norm. Numerical results confirm the theoretical analysis.

A novel monotone finite volume element scheme for diffusion equations

We present a novel monotone finite volume element scheme for the diffusion problem on triangular grids. Firstly, one fictitious triangular element is introduced for each edge of any element, and another vertex of this fictitious element is on the outward extended straight line passing through two points: The vertex opposite to this edge and the barycenter of the element. The diffusive tensor and the solution in this element are continuously extended to its corresponding fictitious element. The fictitious element and the continuous extension not only avoid the phenomenon that the control volume segment passes through other elements, but also are helpful to the construction and theoretical analysis for our monotone scheme. Secondly, one new nonlinear two-point flux formulation is obtained for some part control volume edge whose another endpoint is the barycenter of the fictitious element, and this straight segment crosses through the common edge of the triangular element and its fictitious one. The approximation of κ∇u on this straight control volume edge is constructed by some weighted average with the gradients of two finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we proved that the new scheme is monotone under some conditions: the diffusion tensor and exact solution functions are properly smooth, and the partition step size h is properly small, and the solutions on three nodes of any element are not nonnegative, and furthermore not simultaneously zero, and the nodes of the fictitious element are properly chosen. We also proved that the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and confirm its accuracy and stability.

An Automatic Isotropic Triangular Grid Generation Technique Based on an Artificial Neural Network and an Advancing Front Method

To improve the efficiency and automation of the traditional advancing front method (AFM) of unstructured grid generation, a novel isotropic triangular generation technique is developed based on an artificial neural network (ANN). First, some existing high-quality triangular grids are used as data sources, and then an automatic extraction method of training dataset is proposed. Second, the dataset is input into the ANN to train the network by the back-propagation (BP) algorithm, and then some typical patterns are identified through iterative learning. Third, after inputting the initial discretized fronts, the grid generator starts from the shortest front, and the adjacent front information is collected as the input of the neural network to choose the most proper pattern and predict the coordinates of the new point until the grid covers the whole computational domain. Finally, the initial grid is smoothed to further improve the grid quality. Some typical two-dimensional (2D) geometries are tested to validate the capability of the ANN-based advancing front triangle generator. The experimental results demonstrate that the efficiency of the proposed ANN-based triangular grid generator is about 30 percent higher than that of the traditional AFM, and grid quality has also been improved significantly.

Investigating the Impact of Triangle and Quadrangle Mesh Representations on AGV Path Planning for Various Indoor Environments: With or Without Inflation

In a factory with different kinds of spatial atmosphere (warehouses, corridors, small or large workshops with varying sizes of obstacles and distribution patterns), the robot’s generated paths for navigation tasks mainly depend on the representation of that environment. Hence, finding the best representation for each particular environment is necessary to forge a compromise between length, safety, and complexity of path planning. This paper aims to scrutinize the impact of environment model representation on the performance of an automated guided vehicle (AGV). To do so, a multi-objective cost function, considering the length of the path, its complexity, and minimum distance to obstacles, is defined for a perfect circular robot. Unlike other similar studies, three types of representation, namely quadrangle, irregular triangle, and varying-size irregular triangle, are then utilized to model the environment while applying an inflation layer to the discretized view. Finally, a navigation scenario is tested for different cell decomposition methods and an inflation layer size. The obtained results indicate that a nearly constant coarse size triangular mesh is a good candidate for a fixed-size robot in a non-changing environment. Moreover, the varying size of the triangular mesh and grid cell representations are better choices for factories with changing plans and multi-robot sizes due to the effect of the inflation layer. Based on the definition of a metric, which is a criterion for quantifying the performance of path planning on a representation type, constant or variable size triangle shapes are the only and best candidate for discretization in about 59% of industrial environments. In other cases, both cell types, the square and the triangle, can together be the best representation.

Assessment of Environmental Flow Requirements for Four Major Chinese Carps in the Lower Reaches of the Jinsha River, Southwest China

An appropriate environmental flow (AEF) is one of the necessary conditions for fish spawning. The operation of cascade reservoirs in the lower reaches of the Jinsha River has changed the discharge process downstream, which in turn has directly affected the spawning habitat of fishes. Determining how to quantitatively evaluate the impact of different outflows from Xiangjiaba Reservoir on the spawning habitat of four major Chinese carp species in the downstream area is a key problem that must be solved immediately. The Yibin River section was selected as the study area, and the velocity and water depth were measured. A physical habitat simulation model (PHABSIM) was used to analyze the main hydrodynamic conditions that affect the spawning of the four major Chinese carp species and to quantify the weighted usable area (WUA) for the fishes based on different outflows from Xiangjiaba Reservoir. Hydrodynamic suitability curves for the four major Chinese carp species were established based on observed data from 2012 to 2020, and a two-dimensional hydrodynamic model was developed using a triangular grid to obtain the hydrodynamic distribution characteristics of the fish habitat. Then, the AEF range was obtained based on the outflow-WUA relationship. The results showed that the velocity during the spawning period was 0.6–1.3 m/s, the most appropriate velocity was 0.9–1.0 m/s, the water depth range was 3.0–18.0 m, and the most appropriate water depth was 9.0–12.0 m. Additionally, the AEF range was 2,000–4,500 m3/s, and the optimal AEF was, m3/s. The research results provide a scientific basis for the ecological operation of cascade reservoirs in the lower reaches of the Jinsha River.

An ALE residual distribution scheme for the unsteady Euler equations over triangular grids with local mesh adaptation

This work presents a novel interpolation-free mesh adaptation technique for the Euler equations within the arbitrary Lagrangian–Eulerian framework. For the spatial discretization, we consider a residual distribution scheme, which provides a pretty simple way to achieve high order accuracy on unstructured grids. Thanks to a special interpretation of the mesh connectivity changes as a series of fictitious continuous deformations, we can enforce by construction the so-called geometric conservation law, which helps to avoid spurious oscillations while solving the governing equations over dynamic domains. This strategy preserves the numerical properties of the underlying, fixed-connectivity scheme, such as conservativeness and stability, as it avoids an explicit interpolation of the solution between different grids. The proposed approach is validated through the two-dimensional simulations of steady and unsteady flow problems over unstructured grids.

A new three-triangle based method to linearly concave hydropower output in long-term reservoir operation

The nonlinearity and non-convexity of the hydropower output function (HOF) make it very challenging to search for the optimal solution to the hydropower scheduling problem, which however can be more easily solved with consistency by mathematical programming if the HOF can be properly linearized with high accuracy. This paper presents a new three-triangle based method to linearly concave the HOF without introducing any integer variables, and mathematically proves its equivalence in fitting accuracy to the all-triangle based method in which any active plane needs to be compared with all the other active planes to ensure it is active within its triangular grid, making the number of constraints be increased dramatically. The two methods are applied in approximating the HOFs of 4 hydropower reservoirs located in the Lancang River. The case studies show that the present linearization method can achieve a root-mean-square error (RMSE) at 2.05% on average of the installed power capacity, the same as the all-grid concaving method, but takes only less than 0.097s in solving the problem, 80 times faster than the all-grid concaving method. The present method should be very promising in solving the real-world hydropower scheduling problems, especially when the linearization needs to be done frequently during the solution procedure.

Experiment and Modelling of Triangular Holes Array by Non-traditional Machining Process

Triangular holes array is a common structure of the specially shaped holes that can be used for various products in aerospace vehicles. These are commonly produced in the electrochemical machining (ECM) process. The high-quality and accuracy of a triangular grid are closely related to the processing parameters of cathode structure and sidewall insulation sleeve, power supply, electrolyte flow rate, feed speed, and machining gap. Simulations have been carried out to consider the influence of parameters including structure and dimension of the cathode, initial machining gap, feed speed, and current on the homogeneity of the flow field in the machining gap. Consequently, the modeling can support the design of the cathode. The paper is divided into two parts. Firstly, the tool cathode is simulated and fabricated. Secondly, the triangular holes array is produced by using the designed cathode tool. It was found that higher feed speed and lower machining gap increase the stability, accuracy, and efficiency of the machining process and the quality of the triangular holes array.

On estimating the interface normal and curvature in piecewise linear interface calculation‐volume of fluid approach for three‐dimensional arbitrary meshes

AbstractVolume of fluid (VOF) method with its Piecewise Linear Interface Calculation (PLIC) reconstruction algorithm is one of the most popular approaches in numerical simulation of interfacial flows with a wide range of applications in different areas. In an effort to evaluate the similarity of the PLIC‐generated planes in comparison with the exact interface, a point‐cloud, based on the polygon centers of PLIC planes is extracted, which later is used to form a triangular grid that represents the estimated interface. The main objective of this article is to evaluate the interface geometrical properties based on the extracted triangular grid of the interface. The methods presented in this article, characterized by a higher spatially convergence ratio, are compared with the commonly used methods. The proposed methods are tested for two 3‐dimensional general test cases, where an evident improvement is seen in calculation accuracy and spatial convergence of the errors of interface normal vector and curvature.

Volume calculation of goaf based on 3D scanning

The goaf formed by underground mining poses a great safety hazard to the production of the mining area. The accurate calculation of goaf volume is the data basis for the filling and blasting of the goaf, and it also has certain reference significance for the monitoring of collapse accidents. In this paper, the data detected by the 3D ranging scanner is used to build a network model on the Self - developed 3D modeling platform, and the network model is divided into the smallest triangular grid. The algorithm uses the scanner’s laser probe as the common point, accumulates all the minimum tetrahedral volumes constructed by the common point and the smallest triangular grid, and completes the accurate solution of the total volume of the scanned goaf. Experiments and practical applications show that the volumetric algorithm has high accuracy, and has universal applicability to the volume calculation of similar types of space scanning equipment.

Decentralised optimal deployment of mobile underwater sensors for covering layers of the ocean

This paper presents the problem of sensing coverage of layers of the ocean in three dimensional underwater environments. We propose distributed control laws to drive mobile underwater sensors to optimally cover a given confined layer of the ocean. By applying this algorithm at first the mobile underwater sensors adjust their depth to the specified depth. Then, they make a triangular grid across a given area. Afterwards, they randomly move to spread across the given grid. These control laws only rely on local information also they are easily implemented and computationally effective as they use some easy consensus rules. The feature of exchanging information just among neighbouring mobile sensors keeps the information exchange minimum in the whole networks and makes this algorithm practicable option for undersea. The efficiency of the presented control laws is confirmed via mathematical proof and numerical simulations.

Geometry of anonymous binary social choices that are strategy-proof

We consider a set of voters who express preferences about two alternatives, indifferences included. We characterize the class of all strategy-proof and anonymous social choice functions by means of a geometric property related to an integer triangular grid representing the number of voters supporting each of the two alternatives.

Accurate Time-Domain Modeling of Arbitrarily Shaped Graphene Layers Utilizing Unstructured Triangular Grids

The accurate modeling of curved graphene layers for time-domain electromagnetic simulations is discussed in the present work. Initially, the advanced properties of graphene are presented, focusing on the propagation of strongly confined surface plasmon polariton waves at the far-infrared regime. Then, the implementation of an unstructured triangular grid was examined, based on the Delaunay triangulation method. The electric-field components were placed at the edges of the triangles, while two different techniques were proposed for the sampling of the magnetic ones. Specifically, the first one suggests that the magnetic component is placed at the triangle’s circumcenter providing more accurate results, although instability may occur for nonacute triangles. On the other hand, the magnetic field was sampled at the triangle’s centroid, considering the second technique, ensuring the algorithm’s stability, but further approximations were required, leading to a slight accuracy reduction. Moreover, the updating equations in the time-domain were extracted via an appropriate approximation of Maxwell equations in their integral form. Finally, graphene was introduced in the computational domain as an equivalent surface current density, whose location matches the corresponding electric components. The validity of our methodology was successfully performed via the comparison of graphene surface wave propagation properties to their theoretical values, whereas the global error determination indicates the minimal triangle dimensions. Additionally, an instructive setup comprising a circular graphene scatterer was analyzed thoroughly, to reveal our technique’s advantages compared to the conventional staircase discretization.

Finite volume element methods for two-dimensional time fractional reaction–diffusion equations on triangular grids

In this paper, time fractional reaction–diffusion equations with the Caputo fractional derivative are solved by using the classical -formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability results and optimal a priori error estimate in -norm are derived, but it is difficult to obtain the corresponding results in -norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness.

Emergence of a geometric pattern of cell fates from tissue-scale mechanics in the Drosophila eye.

Pattern formation of biological structures involves the arrangement of different types of cells in an ordered spatial configuration. In this study, we investigate the mechanism of patterning the Drosophila eye epithelium into a precise triangular grid of photoreceptor clusters called ommatidia. Previous studies had led to a long-standing biochemical model whereby a reaction-diffusion process is templated by recently formed ommatidia to propagate a molecular prepattern across the eye. Here, we find that the templating mechanism is instead, mechanochemical in origin; newly born columns of differentiating ommatidia serve as a template to spatially pattern flows that move epithelial cells into position to form each new column of ommatidia. Cell flow is generated by a source and sink, corresponding to narrow zones of cell dilation and contraction respectively, that straddle the growing wavefront of ommatidia. The newly formed lattice grid of ommatidia cells are immobile, deflecting, and focusing the flow of other cells. Thus, the self-organization of a regular pattern of cell fates in an epithelium is mechanically driven.

Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids

Non-staggered triangular grids have many advantages in performing river or ocean modeling with the finite-volume method. However, horizontal divergence errors may occur, especially in large-scale hydrostatic calculations with centrifugal acceleration. This paper proposes an unstructured finite-volume method with a filtered scheme to mitigate the divergence noise and avoid further influencing the velocities and water elevation. In hydrostatic pressure calculations, we apply the proposed method to three-dimensional curved channel flows. Approximations reduce the numerical errors after filtering the horizontal divergence operator, and the approximation is second-order accurate. Numerical results for the channel flow accurately calculate the velocity profile and surface elevation at different Froude numbers. Moreover, secondary flow features such as the vortex pattern and its movement along the channel sections are also well captured.

Study on Structural Characteristics of Composite Smart Grille Based on Principal Component Analysis.

In recent years, many scholars have conducted in-depth and extensive research on the mechanical properties, preparation methods, and structural optimization of grid structural materials. In this paper, the structural characteristics of composite intelligent grid are studied by combining theoretical analysis with experiments. According to the existing conditions in the laboratory, the equilateral triangular grid structure experimental pieces were prepared. In this paper, principal component analysis combined with nearest neighbor method was used to detect the damage of composite plates. On this basis, the multiobjective robustness optimization of the structure is carried out based on artificial intelligence algorithm, which makes the structure quality and its sensitivity to uncertain parameters lower. Particle swarm optimization (PSO) is used in neural network training. The damage characteristics of different grid structures, different impact positions, and different impact energies were studied. The results show that the structural damage types, areas, and propagation characteristics are very different when the structure is impacted at different positions, which verifies that the grid structure has a good ability to limit the damage diffusion and shows that the grid structure has a good ability to resist damage.

An Accurate and Robust Hybrid Hllc Riemann Solver on Triangular Grids

An Accurate and Robust Hybrid Hllc Riemann Solver on Triangular Grids

Diagrams based on the Hexagonal and Triangular Grids

Diagrams based on the Hexagonal and Triangular Grids