Regular Triangular Grid Research Articles

Resistance values under transformations in regular triangular grids

In Evans and Francis (2022) and Hendel (2021) the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced s times provided the original grid has at least 4s rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously.

Weighted distances and distance transforms on the triangular tiling

AbstractDigital geometry is a field in the intersection of discrete mathematics and geometry having various applications including geographical information systems (GIS). In digital spaces, in grids, distances can be defined based on steps in paths in somewhat similarly as in graph theory. However, the grids have more definite structures, thus one may obtain more concrete results, for example, close formulae, than on arbitrary graphs. In this article, the weighted (also called chamfer) distances, and based on them, the distance transform are investigated on the regular triangular grid. Three types of neighborhood relations are used on the grid, and therefore, three weights are used to define a distance function. Natural conditions are used on the weights such as they are positive and a larger step (in the usual and also in the Euclidean sense) cannot have a smaller weight than a smaller one. Some properties of the weighted distances are discussed; for example, they are proven to be metrics. We also give algorithms and formulae that compute the weighted distance of any point pair on a triangular grid. Algorithm for weighted distance transform is provided based on wave‐front propagation. Therefore, these new distance functions are ready for further applications in GIS, in image processing tasks, in computer vision, in graphics, in networking, and also in other applied fields.

Non-oscillatory butterfly-type interpolation on triangular meshes

This paper proposes and analyses a non-oscillatory interpolatory subdivision scheme for data on regular triangular grids, a non-linear analogue of the well-known butterfly subdivision scheme. The scheme is obtained, first, by re-interpreting the butterfly refinement rule as a linear combination of ‘smaller’linear rules and, then, by introducing a particular type of non-linear average in place of the linear ones. We end up with a non-linear interpolatory subdivision scheme that, applied to discrete data with large gradients, shows no Gibbs-like oscillatory phenomenon, while behaves similarly to the butterfly scheme when applied to smooth data. Convergence, reproduction and approximation properties of the proposed scheme are investigated and several numerical examples are discussed.

A face-area-weighted ‘centroid’ formula for finite-volume method that improves skewness and convergence on triangular grids

This paper proposes a face-area-weighted ‘centroid’ as a superior alternative to the geometric centroid for defining a local origin in a cell-centered finite-volume method on triangular grids. It is demonstrated theoretically and numerically that the face-area-weighted ‘centroid’ can reduce grid skewness and improve iterative convergence for triangular grids. It is also shown that source terms do not have to be integrated over a cell and can be evaluated simply at the local origin without losing the design order of accuracy. Numerical results demonstrate that the face-area-weighted ‘centroid’ improves iterative convergence of an implicit defect-correction second-order finite-volume solver for inviscid and viscous flow problems on regular and irregular triangular grids.

Finite volume approximation of nonlinear agglomeration population balance equation on triangular grid

In this present work, a finite volume scheme for approximating a multidimensional nonlinear agglomeration population balance equation on a regular triangular grid is developed. The finite volume schemes developed in literature are restricted to a rectangular grid [43]. However, the accuracy and efficiency of finite volume scheme can be enhanced by considering triangular grids. The triangular grid is generated using the concept of ‘Voronoi Partitioning’ and ‘Delaunay Triangulation’. To test the accuracy and efficiency of the scheme on a triangular grid, the numerical results are compared with the sectional method, namely Cell Average Technique [38] for various analytically tractable kernels. The results reveal that the finite volume scheme on a triangular grid is computationally less expensive and predicts the number density function along with the different order moments more accurately than the cell average technique. Furthermore, the numerical comparison is extended by comparing the finite volume scheme on a rectangular grid. It also demonstrates that the finite volume scheme with a regular triangular grid computes the numerical results more accurately and efficiently than the finite volume scheme with a rectangular grid.

Investigation of sparse near‐field focusing array antenna with different topologies

The method to efficiently design a sparse near-field focusing sparse array antenna is introduced here. First, the calculation model and analysis method of three array topologies including the square grid, the circular grid, and the regular triangular grid are established. The evaluation criterion is the lowest sparsity under the condition that the focusing performance of the sparse array approaches to that of a referenced focusing array with the element spacing of a half of one wavelength. After comparing the focusing performance of these array antennas with different grids, the circular-grid array is the optimal one and selected to further discuss the influence of different topology parameters. The optimal sparse near-field focusing array should have the constant distance between two adjacent rings and increase three elements compared with the last ring. Finally, the method based on the proposed selection criteria is employed to design a sparse near-field focusing sparse array. The correctness of the method has been validated by the full-wave simulated results.

Effects of high-frequency damping on iterative convergence of implicit viscous solver

This paper discusses effects of high-frequency damping on iterative convergence of an implicit defect-correction solver for viscous problems. The study targets a finite-volume discretization with a one parameter family of damped viscous schemes. The parameter α controls high-frequency damping: zero damping with α=0, and larger damping for larger α(>0). Convergence rates are predicted for a model diffusion equation by a Fourier analysis over a practical range of α. It is shown that the convergence rate attains its minimum at α=1 on regular quadrilateral grids, and deteriorates for larger values of α. A similar behavior is observed for regular triangular grids. In both quadrilateral and triangular grids, the solver is predicted to diverge for α smaller than approximately 0.5. Numerical results are shown for the diffusion equation and the Navier–Stokes equations on regular and irregular grids. The study suggests that α=1 and 4/3 are suitable values for robust and efficient computations, and α=4/3 is recommended for the diffusion equation, which achieves higher-order accuracy on regular quadrilateral grids. Finally, a Jacobian-Free Newton–Krylov solver with the implicit solver (a low-order Jacobian approximately inverted by a multi-color Gauss–Seidel relaxation scheme) used as a variable preconditioner is recommended for practical computations, which provides robust and efficient convergence for a wide range of α.

Reprint of Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.

Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.

First-, second-, and third-order finite-volume schemes for diffusion

In this paper, we present constructions of first-, second-, and third-order schemes for diffusion by the method introduced in Nishikawa (2007) [10]. In this method, numerical schemes for diffusion are constructed by advection schemes via an equivalent hyperbolic system. This paper demonstrates that the method enables straightforward constructions of diffusion schemes for finite-volume methods on unstructured grids. In particular, it is demonstrated that a robust first-order upwind scheme leads to a robust first-order diffusion scheme, and a high-order advection scheme leads to a high-order diffusion scheme. It is shown that first-, second-, and third-order schemes are capable of producing first-, second-, and third-order accurate solution gradients, respectively, on irregular grids. Accuracy, Fourier stability, and the energy stability of the developed schemes are discussed. A new hyperbolic diffusion system having virtually no source terms is also introduced to simplify the construction of the third-order scheme. Numerical results are presented for regular and irregular triangular grids to demonstrate not only the superior accuracy but also the accelerated steady convergence over a traditional method.

Divergence formulation of source term

In this paper, we propose to write a source term in the divergence form. A conservation law with a source term can then be written as a single divergence form. We demonstrate that it enables to discretize both the conservation law and the source term in the same framework, and thus greatly simplifies the construction of numerical schemes. To illustrate the advantage of the divergence formulation, we apply the new formulation to construct a uniformly third-order accurate edge-based finite-volume scheme for conservation laws with a source term. Third-order accuracy is demonstrated for regular and irregular triangular grids for the linear advection and Burgers’ equations with a source term.

Impact of wave interactions effects on energy absorption in large arrays of wave energy converters

This paper presents a parametric study on arrays of wave energy converters (WECs). Its goal is to assess the influence of interactions between bodies on the overall yearly energy production of the array. Generic WECs (heaving cylinder and surging barge) are considered. Nine to twenty-five WECs are installed along regular square and triangular grids; the influence of the separating distance between the WECs is investigated. Results show that constructive and destructive interactions compensate each other over the considered range of wave periods. The influence of the separating distance can be limited, especially if the damping of the power take-off is tuned properly, and if the WECs have a large bandwidth. It is found that grouping the devices into arrays have generally a constructive effect. Diffracted and radiated waves in the array lead to a sufficient increase in the energy absorption which overcomes the reduction due to masking effects.

Application of Semitotalistic 2D Cellular Automata on a Triangulated 3D Surface

This paper presents the application of semi-totalistic, also called outer totalistic cellular automata on any three-dimensional (3D) surface. Cellular automata (CA) can be applied in controlling the state of a freeform building envelope. Such an intelligent ‘skin of a building’ can have any shape and a certain ‘organic’ appearance which can be dynamically controlled in response to the changes of the external conditions or users’ requirements. Any 3D surface can be triangulated. This means that it can become a grid of topologically identical elements. With the exception of boundary conditions, applicable to the elements positioned at the edge of the surface or around holes, every triangle in the grid has exactly three neighboring triangles. As with a CA, every element of a triangulated surface can be individually assigned with characteristics such as color or transparency level, which is analogous to a CA ‘state’. Therefore, it is possible to control to some degree the state of the whole surface taking advantage of the emergent behavior of the CA. The concept of CA on a triangular tessellation is discussed followed by discussion about the entities of irregularity of a grid, called ‘holes’ and ‘edges’. The concept of an ‘organic’ pattern in the context of CA is also briefly discussed. Two-dimensional (2D) CA in the triangular tessellation is discussed and implemented. A brief study covers the entities of neighborhood, type of rules (semi-totalistic and general), rule encoding and search for rules that meet given criteria. An implementation is performed on a regular triangular grid, irregular triangular grid and an imported triangulated 3D mesh which is irregular and has holes. A selected 2D triangular CA is applied on an imported triangulated 3D model.

Numerical modelling of regular structures with randomly perturbed structural elements

The problem of the effective elastic properties of regular composites with randomly perturbed geometric and mechanical parameters is formulated and solved numerically. Mean sample values and standard deviations of compliances are used to characterize the elastic properties. Compliance data are obtained by solving reduction problems for each of a set of realizations of random perturbations, and here the number of realizations is increased until the values of the statistical means become stable (within a given tolerance). Calculations for each realization are carried out by numerical solution of the complex hypersingular boundary integral equations obtained for a doubly periodic structure. The principal cell of this structure, containing a fairly large number of perturbed elements, is identified with a representative volume when a further increase in the number of perturbed elements do not alter the statistical means (again, within a given tolerance). Calculations are carried out for square and triangular grids with different densities of circular inclusions or holes, the centre coordinates of which are given random perturbations (weak, medium and strong). The results of the calculations are summed up in tables showing the effective compliances with an accuracy to at least three significant digits. An analysis of the values obtained for the holes shows’ that, with a tolerance of 5%, the principal cell of a square grid with four holes determines the representative volume for all the geometric parameter combinations investigated. For rigid inclusions this cell is the representative volume at a considerably greater tolerance than for compliant inclusions (4% as against 0.9%). Data on the effective properties of perturbed structures indicate that the difference between their compliance and that of the initial regular structures depends substantially on the relative stiffness of the inclusions. It is most marked for holes and rigid inclusions (9.5 and 12.6% respectively). It is established that, for a square grid, random perturbations have a stronger effect on the normal components of the compliance than on the shear component, and the opposite for a regular triangular grid – perturbations have a greater effect on the shear compliance. Calculations also show that symmetrical perturbations of holes (rigid inclusions) along one of the coordinates lead to a marked increase (reduction) in compliance in the orthogonal direction. The established dependence of the additional effective compliance on the amplitude of the perturbation enables the inverse problem to be solved: to find the parameters of the perturbed structure using data on its effective statistical properties.

Multigrid finite element methods on semi‐structured triangular grids for planar elasticity

AbstractMultigrid methods for a stencil‐based implementation of a finite element method for planar elasticity, using semi‐structured triangular grids, are presented. Local Fourier analysis (LFA) is applied to identify the correct multigrid components. To this end, LFA for multigrid methods on regular triangular grids is extended to the case of the problem of planar elasticity, although its application to other systems is straightforward. For the discrete elasticity operator obtained with linear finite elements, different collective smoothers, such as three‐color smoother and some zebra‐type smoothers, are analyzed, and LFA results for these smoothers are shown. The multigrid method is constructed in a block‐wise form. In particular, different smoothers and different numbers of pre‐ and post‐smoothing steps are considered in each triangle of the coarsest triangulation of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

On the behavior of upwind schemes in the low Mach number limit. IV: P0 approximation on triangular and tetrahedral cells

Finite Volume upwind schemes for the Euler equations in the low Mach number regime face a problem of lack of convergence toward the solutions of the incompressible system. However, if applied to cell centered triangular grid, this problem disappears and convergence toward the incompressible solution is recovered. Extending the work of [3] that prove this result for regular triangular grid, we give here a general proof of this fact for arbitrary unstructured meshes. In addition, we also show that this result is equally valid for unstructured three dimensional tetrahedral meshes.

Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal's position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat's velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.

PREDICTION OF THE POSITION OF AN ANIMAL BASED ON POPULATIONS OF GRID AND PLACE CELLS: A COMPARATIVE SIMULATION STUDY

The grid cells of the rodent medial entorhinal cortex (MEC) show activity patterns correlated with the animal's position. Unlike hippocampal place cells that are activated at only one specific location in the environment, MEC grid cells increase firing frequency at multiple regions in space, or subfields, that are arranged in regular triangular grids. It has been recently shown that a conjunction of MEC grid cells can lead to unique spatial representations. However, it remains unclear what the key properties of the grids are that allow for an accurate reconstruction of the position of the animal and what the comparison with hippocampal place cells is. Here we use a theoretical approach based on data from electrophysiological recordings of the MEC to simulate the neural activity of grid cells. Our simulations account for the accurate reproduction of grid cell mean firing rates, based on only three grid parameters, that is grid phase, spacing and orientation. The analysis of the key properties of the grids first reveals that for an accurate position reconstruction, it is necessary to combine cells with different grid spacings (which are found at different dorsoventral locations of the MEC) or orientations. Second, the relationship between grid spacing and subfield size observed in physiological data is optimal to predict the animal's position. Third, the regular triangular tessellating patterns of grid cells lead to the best position reconstruction results when compared with all other regular tessellations of two-dimensional space. Finally, the comparison of grid cells with place cells shows that populations of MEC grid cells can better predict the animal's position than equally-sized populations of hippocampal place cells with similar but unique spatial fields. Taken together, our results suggest that the MEC provides highly compact representations of the animal's position, which may be subsequently integrated by the place cells of the hippocampus.

A MODEL OF GRID CELLS BASED ON A TWISTED TORUS TOPOLOGY

The grid cells of the rat medial entorhinal cortex (MEC) show an increased firing frequency when the position of the animal correlates with multiple regions of the environment that are arranged in regular triangular grids. Here, we describe an artificial neural network based on a twisted torus topology, which allows for the generation of regular triangular grids. The association of the activity of pre-defined hippocampal place cells with entorhinal grid cells allows for a highly robust-to-noise calibration mechanism, suggesting a role for the hippocampal back-projections to the entorhinal cortex.

Quantified symmetry for entorhinal spatial maps

General navigation requires a spatial map that is not anchored to one environment. The firing fields of the “grid cells” found in the rat dorsolateral medial entorhinal cortex (dMEC) could be such a map. dMEC firing fields are also thought to be modeled well by a regular triangular grid (a grid with equilateral triangles as units). We use computational means to analyze and validate the regularity of the firing fields both quantitatively (using summary statistics for geometric and photometric regularity) and qualitatively (using symmetry group analysis). Upon quantifying the regularity of real dMEC firing fields, we find that there are two types of grid cells. We show rigorously that both are nearest to triangular grids using symmetry analysis. However, type III grid cells are far from regular, both in firing rate (highly non-uniform) and grid geometry. Type III grid cells are also more numerous. We investigate the implications of this for the role of grid cells in path integration.