Equilateral Research Articles

Vibrational Normal Modes of an Equilateral Triangular Mechanical Molecule

Three point-like massive particles/atoms are connected with three springs forming an equilateral triangle replicating a prototype triatomic molecule. The triangle is inscribed within a stationary frame via three additional springs confining the vibrations of the molecule to a 2D space. It is the objective of this research flavored investigation to seek the normal vibrational modes for this three-body six-spring structure. The entire analysis including symbolic, numeric, and graphics is carried out by adapting a suitable Computer Algebra System (CAS), Mathematica. For a comprehensive understanding, the frequency of the normal mode is used for a visual animation; an actual mechanical replica of the “molecule” for the scenario on hand is fabricated.

Experimental determination of the optimum oil palm planting density in Western Africa

This article follows on from two articles published in 2014 and 2018 on the same trial conducted in an oil palm plantation in Nigeria which was aimed at assessing a range of different planting distances between oil palms (7.5 to 9.5 m) in an equilateral triangle design. The climate in the region is very stable, with two seasons and an average 2000 mm of rainfall per year. The soil is of the desaturated ferralitic type, sandy on the surface, deep, and without coarse elements. After continually monitoring the experimental palms for 16 years, there is now enough hindsight to propose an optimum planting density for oil palms in Western Africa. A plateau has been reached at between 143 and 160 palms per hectare for Pobè C1001F material in the aforementioned pedoclimatic context.

On the meniscus shape and marangoni flow in capillary tubes with noncircular cross-section shapes

The liquid–vapor interface remains a key point for improving the heat transfer efficiency of heat exchangers. In this paper, we report experimental results on the evaporation of ethanol in capillary tubes of different internal cross-sections as the bulk meniscus recedes inside the tube, since the chosen conditions allow for the development of a liquid film along tube's internal corners due to the presence of capillary forces. To achieve our aims, we used 3D video microscopy to monitor the behavior of the meniscus during evaporation and, also, calculated the capillary pressures for three channels with different cross-sectional shapes each, namely circular, square and equilateral triangle cross-sections, for three solid–liquid contact angles, i.e., 30°, 60°, and 90°. At the same time, we have combined infrared thermal imaging technology with particle imaging velocimetry (PIV) visualization technology to analyze the severe Marangoni convection at the included angle. To this end, the interfacial temperature distribution of Marangoni convection was measured by means of an infrared camera and the flow pattern by means of PIV, respectively, through the plane in various views. For capillary tubes with included angle, the temperature gradient at the meniscus reaches its maximum at the included angle. Moreover, both temperature and convective gradients exist in the liquid film with included angle. When the angle of the capillary tube is smaller, the capillary effect will increase, leading to the formation of a thicker film. Our investigation constitutes an “extreme” experiment of Marangoni flow at the corners of the capillary tube. Due to the capillary forces, liquid films can rise in the corners of polygonal tubes up to the entrance of the tubes. Furthermore, the wetting force of ethanol at the angle of the triangle tube is much larger than that in the case of the square tube. Finally, the Marangoni vortex flows from the relatively high-temperature region in the center of the liquid towards the included angle, where evaporation is faster and the recharge of material and energy is more concentrated. We anticipate that our study sheds light into the meniscus shape and Marangoni flow in capillary tubes of with cross-sections of different shapes, which is of fundamental importance for various applications and in particular microfluidic systems.

Research on Optimization of Tube Structure of Phase Change Heat Storage Device

In view of the low heat transfer efficiency of the phase-change heat storage device and the inability to quickly respond to temperature changes, the fluid tube structure in the heat storage device was optimized, and four types of structures: circular structure, square structure, regular hexagon structure and regular triangle structure were compared and studied. Numerical simulation and optimization analysis are carried out on the pipes with the same thermal fluid circulation area but different sizes. The simulation results show that the triangular structure has the highest heat transfer efficiency in the initial stage of heat transfer due to the longest perimeter, and the heat transfer efficiency is positively correlated with the perimeter under the heat transfer mode in which heat conduction is the main body. Under the heat transfer mode dominated by convective heat transfer in the later stage of melting, the time required for the complete melting of phase change materials outside the square tube is the shortest.

The Application of Leader Following Method and Cubic Polynomial Path Planning Algorithm with Formation Control on Multi-Robot Systems

In this study, the movement of mobile robots on a certain path by preserving their formation and with a leader-follower was examined. Knowing the initial and final positions and orientations of the leader robot, the cubic polynomial was used and the positions of the leader at all points on the path were found. Then, the positions, where the followers were found with the homogeneous transformations, were calculated in order to provide the equilateral triangle formation. The velocity profile of the leader was calculated with certain calculations by determining the maximum velocity for the leader. Controller has been designed for the leader's right and left motors. In order to find the velocity of the follower robots, it was calculated by using the position information that the robots should be at a certain moment. The resulting graphics showed how the robots would create a path at certain angles. The distance values that should be between the robots and the location information from the UWB (Ultra-wideband) were shown on the Matlab graphics. The calculated linear velocities and desired velocities of the leader and follower robots are shown graphically. In the control diagram created for the left and right motors of the leader robot, it has been shown how the linear and angular velocities are affected by PID.

Wave functions for high-symmetry, thin microstrip antennas, and two-dimensional quantum boxes

For a spinless quantum particle in a one-dimensional box or an electromagnetic wave in a one-dimensional cavity, the respective Dirichlet and Neumann boundary conditions both lead to non-degenerate wave functions. However, in two spatial dimensions, the symmetry of the box or microstrip antenna is an important feature that has often been overlooked in the literature. In the high-symmetry cases of a disk, square, or equilateral triangle, the wave functions for each of those two boundary conditions are grouped into two distinct classes, which are one- and two-dimensional representations of the respective point groups, $C_{\infty v}$, $C_{4v}$, and $C_{3v}$. Here we present visualizations of representative wave functions for both boundary conditions and both one- and two-dimensional representations of those point groups. For the one-dimensional representations, color contour plots of the wave functions are presented. For the two-dimensional representations, the infinite degeneracies are presented as common nodal points and/or lines, the patterns of which are invariant under all operations of the respective point group. The wave functions with the Neumann boundary conditions have important consequences for the coherent terahertz emission from the intrinsic Josephson junctions in the high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$: the enhancement of the output power from electromagnetic cavity resonances is only strong for wave functions that are not degenerate.

Selective transport of chiral particles by optical pulling forces

We propose a simple approach for chirality sorting based on the optical pulling forces. The scenario is to employ three simple plane waves with the same circular polarization. The three wave vectors have the same longitudinal positive z component while their corresponding transverse projections in the x-o-y plane form a regular triangle. In such an optical lattice, particles with certain chirality will be drawn by the transverse optical force to the periodic intensity maxima, where they are stably trapped and exerted by an optical pulling force along the negative z direction. On the contrary, particles with the opposite chirality are de-trapped transversely away from the intensity maxima and then dispersed in regions where they subject to a positive (pushing) force along the z direction. The phenomena therefore give rise to an alternative all-optical method for selective transport of chiral particles with a wide range of chirality parameters.

Unsupervised Machine Learning for Improved Delaunay Triangulation

Physical oceanography models rely heavily on grid discretization. It is known that unstructured grids perform well in dealing with boundary fitting problems in complex nearshore regions. However, it is time-consuming to find a set of unstructured grids in specific ocean areas, particularly in the case of land areas that are frequently changed by human construction. In this work, an attempt was made to use machine learning for the optimization of the unstructured triangular meshes formed with Delaunay triangulation in the global ocean field, so that the triangles in the triangular mesh were closer to equilateral triangles, the long, narrow triangles in the triangular mesh were reduced, and the mesh quality was improved. Specifically, we used Delaunay triangulation to generate the unstructured grid, and then developed a K-means clustering-based algorithm to optimize the unstructured grid. With the proposed method, unstructured meshes were generated and optimized for global oceans, small sea areas, and the South China Sea estuary to carry out data experiments. The results suggested that the proportion of triangles with a triangle shape factor greater than 0.7 amounted to 77.80%, 79.78%, and 79.78%, respectively, in the unstructured mesh. Meanwhile, the proportion of long, narrow triangles in the unstructured mesh was decreased to 8.99%, 3.46%, and 4.12%, respectively.

Three-dimensional composites with nearly isotropic negative Poisson's ratio by random inclusions: Experiments and finite element simulation

Composites with negative Poisson's Ratio (NPR), also known as auxetic composites, are of great interests due to excellent properties. Auxetic composites of various types have been developed, however most of them have anisotropic NPR behaviour. This paper reports a new type of three-dimensional (3D) composites with nearly isotropic NPR by randomly embedding rigid re-entrant equilateral triangle inclusions into soft matrix. Four types of inclusions with different structural parameters including re-entrant angle, prism length and rib thickness were first made from two types of carbon fibre prepregs, and then embedded into silicone rubber matrix with different inclusion numbers to fabricate 3D composites with various volume fractions of inclusions. Both experimental and finite element analyses were conducted to investigate the deformation behaviours under compression and the effects of different structural parameters and materials. The results show that the NPR of the composites can be realised with suitable structural parameters and properties. The lowest NPR of −0.19 was obtained via embedding 6 mm length inclusions made of one-layer 3K twill weave carbon fibre prepreg with an inclusion number of 800. The study provides a new approach for the development of 3D composites with isotropic NPR for the practical applications of auxetic materials.

Geometric approximation of the sphere by triangular polynomial spline patches

A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal approximation of equilateral spherical triangles by parametric polynomial patches if the measure of quality is the (simplified) radial error. As a consequence, optimal approximations of the unit sphere by parametric polynomial spline patches underlying on particular regular spherical triangulations arising from a tetrahedron, an octahedron and an icosahedron inscribed in the unit sphere are provided. Some low total degree spline patches with corresponding geometric smoothness are analyzed in detail and several numerical examples are shown confirming the quality of approximants.

A Novel Framework of Singularity Cancellation Transformations for Strongly Near-Singular Integrals

In the solution of frequency- and time-domain integral equations, the singularity cancellation transformations are well-known for the treatments of the singular integrals involving Green’s functions and their gradients. The cancellation transformations for strongly near-singular integrals become inaccurate and inefficient for an extremely deformed triangular patch, and this phenomenon is referred to as shape-dependent problem in this article. As the degree of the singularity increases, the shape-dependent problem of strong near-singularity is more severe than that of the weakly near-singular integrals. Furthermore, if the source triangle is deformed, the accuracy of the cancellation methods decreases when no near singularity exists. In this work, we first investigate the reason for these problems via the theoretical analysis with numerical verification. Second, an updated framework of singularity cancellation methods for strongly near-singular integrals is proposed, which has a fast and consistent convergence rate for both regular and irregular triangles. Third, some numerical experiments are presented to illustrate the effectiveness of the theoretical framework and the proposed transformations.

How to Lay Bricks: Local-Global Alignment Predicts Preference for Shifted Tile Patterns.

We examine the aesthetic characteristics of row tile patterns defined by repeating strips of polygons. In experiment 1 participants rated the perceived beauty of equilateral triangle, square and rectangular tilings presented at vertical and horizontal orientations. The tiles were shifted by one-fourth increments of a complete row cycle. Shifts that preserved global symmetry were liked the most. Local symmetry by itself did not predict ratings but tilings with a greater number of emergent features did. In a second experiment we presented row tiles using all types of three- and four-sided geometric figures: acute, obtuse, isosceles and right triangles, kites, parallelograms, a rhombus, trapezoid, and trapezium. Once again, local polygon symmetry did not predict responding but measures of correspondence between local and global levels did. In particular, number of aligned polygon symmetry axes and number of aligned polygon sides were significantly and positively correlated with beauty ratings. Preference was greater for more integrated tilings, possibly because they encourage the formation of gestalts and exploration within and across levels of spatial scale.

Two- and Three-Way Filtering Power Dividers With Harmonic Suppression Using Triangle Patch Resonator

This paper presents a new approach to design filtering power dividers (FPD) with regular triangle patch resonator (RTPR). According to the electric-field (E-field) distributions of the resonator modes, the TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10</sub> and TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> modes have been selected to realize two- and three-way power dividers respectively, showcasing the flexible characteristic of the RTPR. Afterwards, an in-phase multiway excitation scheme for the RTPR is developed to well excite the designated modes and suppress multiple spurious modes, simultaneously. Subsequently, to further reject the harmonics so as to extend the upper stopband greatly, shorting posts are deployed at the locations with null electric field distribution for the specified operation mode. Ultimately, to validate the design concept, a two-way and a three-way second-order FPD employed at TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10</sub> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{10}= 1.27$ </tex-math></inline-formula> GHz) with fractional bandwidth (FBW) of 7% and at TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{11}= 2.2$ </tex-math></inline-formula> GHz) with FBW of 4.5% are designed and measured. The measured results of the two fabricated circuits well agree with the simulated ones. Results indicate that both of the two FPDs exhibit properties of good port-to-port isolation, high selectivity with transmission zeros near both sides of the passband, as well as wide stopband extended to 3.35 and 2.9 times of fundamental operation frequencies, respectively.

3D Reconstruction Methods Purporting 3D Visualization and Volume Estimation of Brain Tumors

This work proposes the Crust algorithm for 3D reconstruction of brain tumor, an effective mechanism in the visualization of tumors for presurgical planning, radiation dose calculation. Despite the promising performance of Crust algorithm in reconstruction of Stanford models, it has not yet been considered in 3D reconstruction of brain tumor. Validation of the results is done using the comparison of the 3D models from two cutting edge techniques namely the Marching Cube and the Alpha shape algorithm. The obtained result shows that Crust algorithm provides the brain tumor model with an average quality of triangle meshes ranging from 0.85 to 0.95. Concerning the visual realism, the quality of Crust algorithm models is higher on comparison to the other models. Precision of tumor volume measurement by convex hull method is analysed by repeatability and reproducibility. The standard deviations of repeatability were between 2.03 % and 3.97 %. The experimental results show that Linear Crust algorithm produces high quality meshes with average quality of equilateral triangles close to 1.

A Surface Energy Approach to Developing an Analytical Model for the Underfill Flow Process in Flip-Chip Packaging

Abstract This paper presents a new approach to formulating an analytical model for the underfill process in flip-chip packaging to predict the flow front and the filling time. The new approach is based on the concept of surface energy along with the energy conservation principle. This approach avoids the need of modeling the flow path to predict the flow front and the filling time, and thus it is suitable to different configurations of solder bumps, including different shapes and arrangements of solder bumps in flip-chip packaging. An experiment along with the computational fluid dynamics simulation was performed based on a proprietarily developed testbed to verify the effectiveness of this approach. Both the experimental and simulation results show that the proposed approach along with its model is accurate for flip-chip packages with different configurations besides the configuration of a regular triangle arrangement of solder bumps and a spherical shape of the solder bump.

Modeling Local Scour around a Cylindrical Pier with Circular Collar with Tilt Angles (Counterclockwise around the Direction of the Channel Cross-Section) in Clear-Water

This research explores how a circular collar with a tilt angle (counterclockwise around the direction of the channel cross-section) could affect the local scour depth around a single cylindrical pier in clear-water based on Large Eddy Simulation (LES) in six cases. The results show that a horizontal circular collar is the best for reducing the local scour depth. With the increases of the tilt angle, the effect on reducing the local scour depth decreases gradually and is even counterproductive at the scour equilibrium. At the early stage of scouring, cases with circular collars show obvious scouring depth reductions. The smaller the tilt angle is, the better and longer-lasting the protection that the circular collar can provide. When the tilt angle is smaller than 5°, the location of the maximum local scouring is around 90–115° (the angle is measured clockwise from the flow direction) on both sides of the pier. When the tilt angle is greater than 5°, the depth of local scouring in the range around −115° to 115° is close to the maximum local scouring depth. Significantly larger areas reach the maximum scouring depth when the tilt angle increases. Compared to Case 1 (the pier without a circular collar), in the cases with a circular collar, the topographies downwards the pier in 1.0D (D is the diameter of the bridge pier) are changed to siltation from scouring. The topography downwards the pier changes from scouring to siltation with the increase of the tilt angle, and the shape of siltation changes from a long-narrow rectangle to an equilateral triangle. This study may provide valuable insights into the protection of the local scour of the pier.

Imbibition of Newtonian Fluids in Paper-like Materials with the Infinitesimal Control Volume Method

Paper-based microfluidic devices are widely used in point-of-care testing applications. Imbibition study of paper porous media is important for fluid controlling, and then significant to the applications of paper-based microfluidic devices. Here we propose an analytical approach based on the infinitesimal control volume method to study the imbibition of Newtonian fluids in commonly used paper-like materials. Three common paper shapes (rectangular paper strips, fan-shaped and circular paper sheets) are investigated with three modeling methods (corresponding to equivalent tiny pores with circle, square and regular triangle cross section respectively). A model is derived for liquid imbibition in rectangular paper strips, and the control equations for liquid imbibition in fan-shaped and circular paper sheets are also derived. The model is verified by imbibition experiments done using the mixed cellulose ester filter paper and pure water. The relation of imbibition distance and time is similar to that of the Lucas−Washburn (L−W) model. In addition, a new porosity measurement method based on the imbibition in circular paper sheets is proposed and verified. Finally, the flow rates are investigated. This study can provide guidance for the design of different shapes of paper, and for better applications of paper-based microfluidic devices.

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

Comparison of oblique triangular configuration and inverted equilateral triangular configuration of three cannulated screws in treating unstable femoral neck fracture: A finite element analysis

BackgroundThe cross-sectional area of three parallel screws might affect the stability of the internal fixation of femoral neck fractures. The screws fixed in the oblique-triangle configuration (OTC) were assumed to have a larger cross-sectional area, but the biomechanical stability has not yet been validated. In this study, finite element analyses were performed to compare the biomechanical properties of the internal fixation fixed by the OTC and the traditional Inverted Equilateral Triangle Configuration (IETC). MethodPauwels type III fracture was established on the three-dimensional femoral model and three cannulated screws with the OTC and traditional IETC methods were applied. The oblique-triangle configuration with the largest area inscribed the femoral neck isthmus by the three screws was determined, the area and circumference of the cross-section formed by the OTC and IETC model were compared. Stress, strain, and displacement peaks of the two configuration models under different loads were compared. Twelve pairs of nodes on the fracture ends were selected and the displacement of the fracture ends was evaluated through the displacement between these nodes. ResultsThe area and circumference of the cross-section formed by the OTC were larger than those in the IETC model. The degree of stress dispersion around the screw holes in the OTC model was better than that of the IETC, but the stress distribution order of the three screws in the two models was consistent. The maximum stress, strain, displacement, and displacement of the fracture end in the OTC model were smaller than those in the IETC model. The stress, strain, displacement, and fracture end displacement peaks of the two fixed models gradually increase with the increase of loads. ConclusionThe oblique-triangle configuration showed superior mechanical properties than the IETC in finite element analyses. This study suggests that when three screws are fixed in parallel method, the larger the cross-sectional area of the screw configuration, the better stability of the internal fixation might be obtained. Furthermore, the biomechanical properties of various spatial configurations and screw holes of the three parallel screws need to be considered before clinical practice.

On some peculiar fractures observed in tests of the Upper-Cretaceous fossil echinoid Micraster

Many skeletons (“tests”) of the fossil echinoid Micraster from the marl limestone strata of the Upper Cretaceous formation of western Navarra (Spain) show small triangular holes on their surface, caused by the loss of a pyramidal fragment. The base of the pyramidal borings is often close to an equilateral triangle. The test of echinoids is a mosaic structure constituted of interlocking monocrystalline plates of bio-calcite which possess a preferred crystalline orientation. In the case of the genus Micraster, the surface of the plates is near-parallel to the basal calcite plane (0001). The geometry of the observed holes, at first sight surprising, is simply explained by the intersection of cracks along the three easy 101¯4 cleavage planes of the calcite single crystalline plates constitutive of the skeletons. The fracture process leading to the pyramidal fractures is most likely due to stresses of thermo-elastic origin induced by the heterogeneous dilatational response of the anisotropic tests adhered to the isotropic marly limestone in which they are embedded and which is constitutive of their inner molds.