Equilateral Research Articles

Dispersion reduction in Feng and Wu’s IPDG method

We give an analytical justification of the determination of an important parameter involved in Feng and Wu’s solution of the Helmholtz equation by an Interior Penalty Discontinuous Galerkin (IPDG) method (Feng and Wu, 2009). This parameter was determined in this reference from a large number of numerical tests for a mesh composed of equal equilateral triangles at a fixed frequency. It plays an essential role in the reduction the dispersion error for a polynomial approximation of degree 1. We show how this justification makes it possible to slightly improve it and to handle other structured meshes. The analytical results are evaluated by several numerical experiments, showing in particular that the parameter determined in principle for equal equilateral triangles also eliminates dispersion for unstructured, well-smoothed meshes.

NUMERICAL SIMULATION OF PASSIVE MICROMIXER WITH NOVEL OBSTACLE STRUCTURE

The micromixer is an important prerequisite for realizing the function of the microfluidic chip system. Micromixers are mainly used in protein crystallization, biological reactions, drug delivery and other fields. Efficient micromixers enable fast bioreaction processes. In this paper, according to the principle of plane geometry, micromixers with three different structural obstacles, circular, equilateral triangle and square, are designed to study how to improve the mixing efficiency of the micromixer. In this paper, the influence of the obstacle structure on the mixing efficiency and pressure drop of the micromixer is analyzed. Through numerical simulation, the results show that when Re = 100, the mixing efficiency of the circular obstacle micromixer is up to 97.4%, and the pressure drop is 15,789.597 Pa. In addition, this paper analyzes that the use of cylindrical obstacles can make the fluid generate horizontal and vertical vortex flow, break the usual laminar flow, and significantly improve the mixing efficiency.

Equilateral triangles have minimal area and perimeter among all triangles containing a given circle in Hilbert planes

Equilateral triangles have minimal area and perimeter among all triangles containing a given circle in Hilbert planes

On the cubic equation with its Siebeck–Marden–Northshield triangle and the quartic equation with its tetrahedron

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck–Marden–Northshield equilateral triangle and regular tetrahedron. The Viète trigonometric formulæfor the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic x4+ax3+bx2+cx+d is proposed for which the conditions are imposed on the individual coefficients a, b, c, and d. The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of a and b. The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than (3/4)3a2−8b from −a/4. The real roots of the quartic are localized by finding intervals containing at most two roots. The end-points of these intervals depend on a and b and are roots of quadratic equations — which makes this localization helpful for quartic equations with complicated parametric coefficients.

Development of a 3D Computer Simulation Model Using C++ Methods

The article presents modelling of spherical elements based on the developed computer model. We recorded the main combinations of spherical particles during filling, which are formed in the hopper. It was found that the most likely combination that occurs when modelling spherical elements consists of three balls. It should be noted that in the cross-section of such a combination passing through the center of the balls, an equilateral triangle is formed. And in the cross-section of the structure, which consists of four spherical balls, a rhombus is formed, if you connect the centers of these spherical elements. It is worth noting that from this formed combination of spherical elements, it can be seen that the rhombus forms two smaller equilateral triangles that fix the process of pushing the spherical balls apart. In turn, the process of pushing spherical elements apart made it possible to fix the contact between spherical elements, as well as to state the stable position of each (individual) particle. This paper also presents the main fragments of encoding the source text of a 3D computer model for modelling spherical elements, which made it possible to optimize the model parameters. It was found that from the obtained data on the distribution of coordination numbers for different volume fillings of spherical elements, it follows that the largest filling was 72 %, which corresponds to the state when 112 lobules have an average coordination number of 3,92.

Investigation of various absorber surface shapes for performance improvement of solar chimney power plant

Solar chimney power plants are important for the development of renewable energy. The absorber surface shape can play a critical role in the solar chimney’s performance. In the relevant literature, there is no comprehensive study on the effect of various absorber surface shapes on the performance of solar chimney. The present study aims to numerically investigate the effect of absorber surface shapes on the power generated by a solar chimney. The investigated absorber shapes include a right triangle, an equilateral triangle, a rectangle, and a semi-circle. In addition, the effect of the absorber surface height is studied. The modelling of heat transfer and fluid flow in the chimney is based on conservation equations. The finite volume method is used to solve the set of equations using ANSYS Fluent software. The model is verified by comparison with experimental data from the literature, and good agreement is obtained. The outcomes of this study show that the maximum effect is found in the case of a rectangular-shaped absorber surface. Furthermore, increasing the height of the modified absorber surface increases the generated power. The output power improves by up to 67.4% in the case of a rectangular absorber with a height of 7.5 cm when compared to the baseline flat-chimney. These findings are useful for designing efficient solar chimney.

Reverse Isoperimetric Inequality for the Lowest Robin Eigenvalue of a Triangle

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.

Experimental study on energy harvesting maximization from the flow-induced vibration of a highly confined bluff body

Flow-induced vibrations of rigid prisms supported elastically were studied experimentally in a free-surface water channel with a high blockage (2/5). The study focused on finding the prism cross-sectional shape that maximizes the efficiency of energy harvesting. Seven cross-sectional shapes were tested: square, circular, 45° tilted square, equilateral triangle, isosceles 120° triangle, D-section, and C-section. All other dimensionless parameters of the problem, mass ratio, damping, blockage ratio, reduced velocity range, and the Reynolds (Re) number (characteristic velocity times characteristic length divided by kinematic viscosity) range (400–1070), were kept unchanged. By doing so, the effect of the cross-sectional shape was isolated. D-section proved to be the geometry with the highest values of energy transfer efficiency. A hysteresis loop was present in its oscillatory response (dimensionless oscillation amplitude vs reduced velocity). This loop was characterized by two branches, (+) and (−), meaning a bi-valued amplitude response for each reduced velocity. Regarding temporal patterns of wake topology and body motion, it was found that synchronization occurs in the (+) branch, but not in the (−). Regarding vortex shedding modes, particle image velocimetry was used for identification purposes, and it was found that the 2P mode is the dominant mode in the (+) branch, while the 2S mode pervades the (−). Finally, a new relative reduced velocity definition was introduced, and, when re-plotting the experimental results, it was found that the hysteresis loop disappears, thereby providing a more compact mathematical description of the observed phenomena.

Shear wave modes in an equilateral triangular prismatic bar.

Shear wave excitation of an equilateral triangular solid bar shows the existence of horizontally polarized (SH) and vertically polarized (SV) shear waves. The SH wave modes are Lamé solutions of the Neumann problem for displacement and Dirichlet problem for stress in the equilateral triangle. It is shown that Lamé's solutions are obtained from superposition of three equal plane waves and their reflections. The Neumann SH modes have cutoff wavenumber 4mπ/(3a), where m=1,2,3,…, a being the side length of triangular cross section. The first Dirichlet SH mode has a higher cutoff wavenumber 4π7/(3a). The SV modes are related to the 30°-60°-90° and 30°-120°-30° sub triangles. Their cutoff wavenumbers lie between those of the first (m = 1) and second (m = 2) Neumann SH modes. The standing wave condition in each sub triangle is related to three unequal plane waves which decompose via symmetrical component analysis into two different sets of equal plane waves. A nonlinear product solution based on the new plane wave set leads to modal SV wave solutions. Dispersion curves calculated using COMSOL confirm the correct cutoff wavenumbers of both SH and SV waves. Normal displacement mode shapes are calculated analytically and verified experimentally using laser vibrometer with good agreement for all modes.

Ethnomathematics: Utilization and Introduction of Geometry Buildings Using Janur (Coconut Leaf)

One area of mathematics that requires a good understanding and depiction of abstract concepts is geometry. Therefore it is necessary to apply realistic learning. One type of realistic learning is ethnomathematics. Ethnomathematics is a bridge between mathematics and culture by linking culture and students' everyday experiences. Traditional crafts that can be lifted are creations from coconut leaves (janur) which can be formed into various kinds of craft creations that contain geometric shapes. The type of research used in this research is descriptive qualitative research. Data collection was used through literature and interviews. The results of the research show that creations from coconut leaves (janur) can be geometric shapes that can be used as ethnomathematics objects. These forms include (1) coins that can be made into hexagons, (2) kerises that have elements of equilateral triangles, right triangles, and parallelograms, (3) diamond bricks that have elements of square geometry lengths and blocks, (4) balls (balans) which have square and cube geometry elements, (5) sinta diamonds which have rhombus geometry elements and rhombus upright prisms. Other geometric elements contained in the janur creations are congruence, angles, and symmetry.
 Keywords: ethnomathematics, geometry, coconut leaves.

A new window function for the spectral analysis of effect of celestial body on Taiji mission below 0.1 mHz

The Taiji project is a space gravitational wave detection mission consisting of three satellites that form a large equilateral triangle with a side length of approximately 3×109 m in heliocentric orbit. It may have the potential to detect gravitational waves below 0.1 mHz. Gravitational wave signals in the μHz range have a strain sensitivity as large as 2.9×10-14, or approximately 8.7×10-5 m variation in inter-satellite distance. However, inter-satellite distances are affected by the celestial bodies in the solar system, introducing changes that may reach magnitudes of 3×107 m; thus, the effect of celestial bodies on these distances may exceed the gravitational signals below 0.1 mHz. Therefore, it is necessary to accurately estimate the effect of the gravity of celestial bodies below 0.1 mHz. In this study, based on the inter-satellite distances in the Taiji mission, a window function design method is proposed and a six-term cosine window function is designed. It has a high decay rate of h-12 and a good peak sidelobe level of −88 dB. It improves the attenuation of spectral leakage while maintaining an acceptable minimum number of signal periods, which is beneficial for signals where it is difficult to obtain more periods, such as for the Taiji mission. Based on the analytical results of the states of satellites in cases when the satellites move only in the gravitational field of the Sun, the ability of the new window function is validated.

Interaction with the dental patient

Oral pathology must be successfully treated, but it depends on good planning from the beginning of the consultation and culminates with good functionality, aesthetics, and occlusal stability. In addition, it is essential to maintain this situation for as long as possible, an indeterminate time that will depend on the excellent performance of those who occupy the three vertices of an equilateral triangle: professional, team, and patient.

Cloaking of Equilateral Triangle Patch Antennas and Antenna Arrays with Planar Coated Metasurfaces

We have proposed an effective metasurface design to accomplish the cloaking of equilateral patch antennas and their array configuration. As such, we have exploited the concept of electromagnetic invisibility, employing the mantle cloaking technique with the intention to eliminate the destructive interference ensuing between two distinct triangular patches situated in a very congested arrangement (sub-wavelength separation is maintained between the patch elements). Based on the numerous simulation results, we demonstrate that the implementation of the planar coated metasurface cloaks onto the patch antenna surfaces compels them to become invisible to each other, at the intended frequencies. In effect, an individual antenna element does not sense the presence of the other, in spite of being in a rather close vicinity. We also exhibit that the cloaks successfully reinstate the radiation attributes of each antenna in such a way that it emulates its respective performance in an isolated environment. Moreover, we have extended the cloak design to an interleaved one-dimensional array of the two patch antennas, and it is shown that the coated metasurfaces assure the efficient performance of each array in terms of their matching as well as radiation characteristics, which in turn, enables them to radiate independently for various beam-scanning angles.

Physical simulation of aerodynamic flow characteristics in a vertical diffuser with air supply through different pipe configurations

RELEVANCE. Vertical cone diffusers are used in various technical applications: heat exchangers, gas cleaning units, boilers, industrial furnaces, dryers, ventilation devices, nozzle systems and others. For their efficient operation, it is necessary to ensure a uniform supply of the working medium to the device, which is determined by the characteristics of the flow in thediffuser. Thus, the study of the aerodynamics of technological devices with conical diffusers is an urgent task for gas-dynamic improvement and the search for ways to control flow characteristics. THE PURPOSE. To establish the evolution of the velocity field along the height of the cylindrical part of the diffuser for different configurations of the supply tubes, and also to determine the magnitude of the change in the intensity of turbulence along the height of the diffuser under different initial conditions based on experimental data on the instantaneous values of the air flow velocity. METHODS. Measurement of instantaneous values of air flow velocity is carried out using a constant temperature hot-wire anemometer. The article provides data on velocity fields and turbulence intensity along the height and along the diameter of the cylindrical part of the diffuser when air is supplied through tubes of different configurations. Feed tubes with cross sections in the form of a circle, a square and an equilateral triangle were used. RESULTS. The article provides a detailed description of the experimental stand (including key geometric dimensions), instrumentation and measurement system, and data processing techniques. The ranges of changes in the initial conditions for the experiments are presented. A comparison of the aeromechanical characteristics of flows in a vertical diffuser when air issupplied through different tube configurations is carried out. CONCLUSION. It is shown that in the diffuser there is a drop in the average velocity upstream, which is typical for all configurations of the supply tubes. It has been established that profiled tubes influence the shape of the velocity field. It was found that the values of turbulence intensity vary from 0.05 to 0.39 (the highest values were typical when air was supplied through profiled tubes). It is shown that the intensity of turbulence has its maximum values at a height of 300-350 mm, which is typical for all investigated tube configurations.

Optical Control of the Localized Surface Plasmon Resonance in a Heterotype and Hollow Gold Nanosheet.

The remote excitation and remote-controlling of the localized surface plasmon resonance (LSPR) in a heterotype and hollow gold nanosheet (HGNS) is studied using FDTD simulations. The heterotype HGNS contains an equilateral and hollow triangle in the center of a special hexagon, which forms a so-called hexagon-triangle (H-T) heterotype HGNS. If we focus the incident-exciting laser on one of the vertexes of the center triangle, the LSPR could be achieved among other remote vertexes of the outer hexagon. The LSPR wavelength and peak intensity depend sensitively on factors such as the polarization of the incident light, the size and symmetry of the H-T heterotype structure, etc. Several groups of the optimized parameters were screened out from numerous FDTD calculations, which help to further obtain some significant polar plots of the polarization-dependent LSPR peak intensity with two-petal, four-petal or six-petal patterns. Remarkably, based on these polar plots, the on-off switching of the LSPR coupled among four HGNS hotspots could be remote-controlled simply via only one polarized light, which shows promise for its potential application in remote-controllable surface-enhanced Raman scattering (SERS), optical interconnects and multi-channel waveguide switches.

An Algorithm Design about Psychological Counseling Platform Using the Derivative Works

The psychological counseling platform using the derivative works in this study includes four main modules and two sub modules. The first main one is a content database that stores a plurality of derivative works in which keywords are set. The second is a message input module including a request message providing unit that displays a request message input from the client terminal on the counselor's terminal, and a counseling message input unit that receives a counseling message written based on the request message from the counselor's terminal. The third main module is a keyword extraction module that extracts keywords by analyzing the counseling message. Final one is an answer providing module that provides derivative works corresponding to the extracted keyword and the consultation message to the client terminal. As the sub modules, this system consists of a region division module and a chart generation module. The region division module allows the screen of the client terminal to correspond to the central region where the counseling message is placed and the vertices of an equilateral triangle having the central point of the central region as the outer center. The chart generation module consists of a chart display unit, an expected value calculation unit, and a variable value setting unit. Based on the proposed algorithm, this system has the effect of increasing the effectiveness of psychological counseling and improving the concentration of counseling by providing recreated secondary copyrighted content using copyright-protected videos, images, music, and photos together with psychological counseling at the same time.

Hydrodynamics with triangular point group

When continuous rotational invariance of a two-dimensional fluid is broken to the discrete, dihedral subgroup D_6D6 - the point group of an equilateral triangle - the resulting anisotropic hydrodynamics breaks both spatial-inversion and time-reversal symmetries, while preserving their combination. In this work, we present the hydrodynamics of such D_6D6-symmetric fluids, identifying new symmetry-allowed dissipative terms in the hydrodynamic equations of motion. We propose two experiments - both involving high-purity solid-state materials with D_6D6-invariant Fermi surfaces - that are sensitive to these new coefficients in a D_6D6-invariant electron fluid. In particular, we propose a local current imaging experiment (which is present-day realizable with nitrogen vacancy center magnetometry) in a hexagonal device, whose D_6D6-exploiting boundary conditions enable the unambiguous detection of these novel transport coefficients.

Estimation of phase velocity using array observation of microtremors with arbitrary shape

To estimate the phase velocity using the array observations of microtremors, some algorithms for the estimation include constraints on the array shape, such as equilateral triangles or the placement of receivers on a circle, in order to reduce the estimation error of the phase velocity. In the present study, a direct estimation technique is introduced for the phase velocity using records obtained through an array with an arbitrary shape based on a complex coherency function (CCF), where CCF is defined as the normalized cross spectrum of the microtremor records observed simultaneously by two receivers. The particle swarm optimization (PSO) method, one of metaheuristic optimization methods, is applied and optimal values are provided for the phase velocity and other unknown parameters. Approximate representations of the stochastic properties for the unknown variables are analytically derived based on the discrete representation of the CCF, for a case where the arrival directions of microtremors are treated as random variables following a uniform distribution. Furthermore, the validity of the proposed method is confirmed using numerical simulations and actual observation records.Graphical

Clough’s Theorem

Summary We offer a visual proof of Clough’s theorem on the semiperimeter of an equilateral triangle.

Fully developed flow in a long triangular channel under an applied magnetic field

Full closed form solutions are provided for the fluid flow impacted by a perpendicularly applied uniform magnetic field inside a pipe of triangular cross section. The governing equation of pressure gradient induced flow under the external magnetic field is reduced to the Helmholtz partial differential equation with Dirichlet boundary conditions on the scaled side lines of the equilateral triangle. The velocity solution is then derived in terms of elementary exponential functions involving the magnetic strength parameter, or the Hartmann number, controlling the retarding effect of magnetic field on the velocity profiles. The Lorentz force effects are clarified on the two-variable velocity variations as well as on the centerline velocity, volumetric flow rate and wall shears. The control of flow field and recirculating zone is visualized graphically increasing the strength of magnetic field. At the Hartmann number 625, it is anticipated that the velocity is nearly flattened at the vertical positions. The provided exact solutions are beneficial to the research community studying the triangular shaped pipes under different physical scenarios. The presented mathematical foundation can be used in recently popular nanotechnology facilitated by miniature channels.