Curvilinear Triangle Research Articles

Analytical and HYDRUS solutions for exfiltration through inclined seepage faces

Steady, phreatic flows in wedge-shaped aquifers subtended by a tilted bedrock and discharged through a hillslope seepage face are studied by 2-D and 1-D analytical models for fully saturated and a FEM numerical model for saturated-unsaturated flows. The curvilinear triangle in the hodograph domain is conformally mapped onto a reference half-plane where the Hilbert problem is solved for two holomorphic functions. Free boundaries and sizes of seepage face segments are found for given flow rates of the incident flows seeping downhills. The approximate (Dupuit-Forchheimer) water table is conjugated with a saturated triangle where the exact 1-D solution gives a constant Darcian velocity as a function of the angles of the soil wedge. HYDRUS-2D model gives the distributions of the pressure head, volumetric moisture content, isotachs, vector-fields of velocity and other characteristics of seepage described by the Richards-Richardson equation. Applications to stability of exfiltrating zones near tailwaters of earth dams are discussed.

Siberian Animal Style: Stylistic Features as Generic Indication

This paper is devoted to the problems of differentiation of stylistic variants in the common phenomenon of the so-called Scythian and Siberian animal styles, which is one of the main distinctive features of Eurasian nomadic art. The animal style is a concept of more scale than an artistic style proper which distinguishes with some formal characteristics and depends directly on generic traditions and ethnic and cultural roots of art. Together with the technical-technological methods these formal features could be evidential indications of the origin of works of art. The Siberian collection of Peter the Great includes some different groups of golden ornaments decorated in animal styles of different origins. The paper focuses on a compact group of items originating from various mostly unknown sites from different territories in Asia including the Oxus treasure, several items from the Siberian collection of Peter the Great from Southern Siberia, a few jewelry pieces from other collections of the world museums as well as items made of leather and felt coming from the First and the Second Pazyryk kurgans. A distinctive feature of this group of zoomorphic images are colored inlays that accentuate a hind-leg or a shoulder of the animal; such inlays have the form of an intricate figure made up of a circle and a curvilinear triangle abutting to it or elongated round brackets. Genetically, such an ornamental motif, which is not generally typical for Persian art, may be linked to a periphery area of the Iranian world and nomadic culture, while the group of sites can be dated back to the 4th–3rd centuries BC. The paper considers a bracelet from the Siberian collection of Peter the Great which is the only item in this category of jewelry type of bracelets. It represents a rare type of ornament with a multi-component structure. It consists of three open-work strips with zoomorphic compositions in an animal style similar to the above-mentioned stylistic group. All three parts of the bracelet are created in a unified style, but obviously in different individual manners. There is no doubt, that the zoomorphic images show three different authors’ hands, and were made by different artisans. So, there is evidence of collective work on the object when each artisan makes his own operation to create a unique jewel at a workshop. Some parts of the composition on the bracelet are similar in style to zoomorphic images from kurgan Issyk in Kazakhstan which perhaps were made in the same workshop. This fact confirms the assumption of the origin of some of Siberian jewelry.

Large-Degree Asymptotics of Rational Painlevé-IV Solutions by the Isomonodromy Method

The Painlevé-IV equation has two families of rational solutions generated, respectively, by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann–Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painlevé-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painlevé-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.

АНАЛИЗ КОНСТРУКЦИЙ ИЗМЕЛЬЧИТЕЛЕЙ ЖМЫХА ПОДСОЛНЕЧНОГО

Grinding of the components of the feed mixture and granulation are the most important operations in the technology of feed production according to zootechnical requirements. The subject of the study are different designs for grinding protein feed. A comparison was made by patent search of the design features of grinders of sunflower cake and grain. It has been established that hammer-type crushers are most widely used for crushing sunflower cake. The analysis revealed the following shortcomings of hammer-type crushers: high metal and energy consumption, uneven particle size distribution of the crushed product, large release of dusty fraction, significant rotating masses, rapid wear of working bodies, heating of the product. The advantages of crushers include the quality of the crushing process. The following conclusions are made. Improvement of sunflower cake grinders should be aimed at expanding its functionality, reducing the energy intensity of the process and improving the quality of its grinding due to the presence of a set of removable knife blocks with crankshafts with different necks for processing material with different physical and mechanical properties or knives in the form of a curvilinear triangle Arbelos Archimedes with sawtooth teeth along the perimeter of the entire surface of the cutting edge, while the entire surface of the cutting edge of the knife is involved in the technological process and sliding cutting is carried out.

Increasing performance of feed grinder in obtaining pelleted sunflower cake

One of the options for improving the protein feed (the feed in the form of sunflower cake) grinders, for which Russian Federation patent No. 2648392 was obtained, is the modernization of the working surface of knife packages. They include knives in the form of an Archimedes Arbelo curvilinear triangle with saw-edged teeth along the perimeter of its entire surface of cutting edges. This will reduce the energy intensity of the process and improve the grinding quality. In literary sources, the values of the performance quantities and the energy intensity of the grinder in obtaining pelleted sunflower cake are absent. As a result of mathematical modeling of the grinding process of sunflower cake in the pelleted form, the optimal parameters of the studied factors were determined. In the indicated ranges of factors changing, the obtaining of pelleted sunflower cake is provided, it meets the zootechnical requirements for cattle with a minimum energy intensity of the grinding process E = 1.03 kWh/t and its performance W = 2.3 t/h.

Mathematical modeling of the sunflower cake grinding process

The design of an improved sunflower cake grinder is proposed, which consists in the modernization of knives in knife blocks in the form of an Archimedes Arbelo curvilinear triangle with saw-edged teeth along the perimeter of the entire surface of the cutting edge. This will allow obtaining cake in a crumbled form, depending on the type of the farm animal by constructing a mathematical model that considers physical and mechanical properties of the material, which describes the dependence of its design parameters on the factors under study that affect the technological process of its production and thereby improve the grinding quality. The article presents the mathematical model that allows determining the optimal values of the design parameters of the improved grinder. The improved sunflower cake grinder allows reducing its specific energy consumption compared to series-produced ones without deteriorating the quality of the material and to increase its performance. Based on the experimental studies conducted, the improved sunflower cake grinder allows it to be used in technological lines for the production of mixed feed with its adding in a crumbled form. The obtained design parameters of the improved grinder will be used in the further modernization of series-produced grinders for concentrated feed produced by Russian enterprises for various types of farm animals, which will allow replacing imported concentrated feed with Russian ones and thereby not depend on them due to the current situation in the world.

АРБЕЛОС І ПОВ'ЯЗАНІ З НИМ КОЛА

Геометрія як наука зародилася в стародавній Греції, її аксіоматичні побудови описані в "Началах" Евкліда. Евклідова геометрія займалася вивченням найпростіших фігур на площині та в просторі. Грецькомовні математики, які жили в період між VI століттям до н.е. і V століттям н.е., поставили та розв’язали багато цікавих геометричних задач. Більшість цих задач розв’язувалася графічним шляхом, що вимагало виконання великої кількості різноманітних складних побудов. На той час вважалося, що "істинно геометричними" є ті задачі, які розв’язувалися тільки за допомогою таких "наукових інструментів" як циркуль та лінійка. Особливу увагу древньогрецькі математики приділяли одному із найважливіших геометричних образів – колу, яке навіть у ті часи знаходило широке практичне застосування. Суттєвий внесок у дослідження кола зробив Архімед Сіракузький, який вперше ввів поняття арбелос. Під арбелосом він розумів плоску геометричну фігуру, утворену деяким півколом, з якого вирізані два менших півкола з діаметрами, що лежать на діаметрі вихідного кола і розбивають його на дві частини. Таким чином, утворювався криволінійний трикутник, обмежений трьома півколами. У пропонованій роботі розглядається питання розв’язання відомих старовинних геометричних задач із застосуванням сучасних методів інженерної графіки, аналітичної геометрії та числових методів, без проведення додаткових побудов, які використовуються при графічному розв'язанні розглянутих задач. При числовій реалізації поставлена задача зводилася до розв’язання нелінійного рівняння з однією змінною. Нелінійні рівняння пов’язані із знаходженням радіусів вписаних або описаних кіл та координат їх центрів. У роботі, зокрема, побудовані коло, вписане в арбелос, спарені кола Архімеда, відомі як кола-близнюки, ланцюг Паппи Олександрійського. Спираючись на дослідження сучасних математиків, присвячених арбелосу, розв’язані задачі побудови кіл Банкова (Bankoff), Шоха (Schoch), Ву (Woo).

An Explicit Time Marching Scheme for Efficient Solution of the Magnetic Field Integral Equation at Low Frequencies

An explicit marching-on-in-time (MOT) scheme to efficiently solve the time-domain magnetic field integral equation (TD-MFIE) with a large time step size (under a low-frequency excitation) is developed. The proposed scheme spatially expands the current using high-order nodal functions defined on curvilinear triangles discretizing the scatterer surface. Applying Nyström discretization, which uses this expansion, to the TD-MFIE, which is written as an ordinary differential equation (ODE) by separating self-term contribution, yields a system of ODEs in unknown time-dependent expansion coefficients. A predictor-corrector method is used to integrate this system for the samples of these coefficients. Since the Gram matrix arising from the Nyström discretization is a block-diagonal, the resulting MOT scheme replaces the matrix “inversion” required at each time step by a product of the inverse block-diagonal Gram matrix and the right-hand side vector. It is shown that, on the convergence of the corrector updates, this explicit MOT scheme produces the same solution as its implicit counterpart and is faster for large time step sizes.

CONSTRUCTING AN APPROACHED SWEEP OF THE SURFACE GELICODE OF THE WIND GENERATOR

It is well-known that experts estimate that traditional energy sources, such as oil, gas and coal, may be depleted in the near future. This raises the question of finding alternative sources. In addition to the use of atomic energy, recent attention has been given to developments in the use of solar and wind energy. In this paper, attention is paid to the design features of the blades of the wind turbine with the vertical axis of rotation. Such wind turbines have a significant advantage over structures with a horizontal axis of rotation, since they do not need to be oriented in the wind direction. The main feature of the design of the considered wind turbines is their blades, which, most often, are compartments of non-expandable helicoidal surfaces, which complicates their manufacture, especially for individual use. In view of all the above, it seems relevant to construct an approximate sweep of the blade, which can be performed using the triangulation method. This method makes it possible, from an engineering point of view, to erroneously construct the contours of the sweep and to determine the approximate area of the blade, and thus to determine the cost of the required material. The paper deals with the method of triangulation for construction of the residential vertical turbine blade sweep, which can be used in a private house. To do this, split the blade height into several equal parts. The second step is to break a certain strip with concentric cylinders. Joining the points of intersection of the edges of the strip with segments, we obtain curvilinear triangles, which after the necessary calculations are transformed into planes. Thus, we create an approximate sweep of one strip across the entire surface. By copying this sweep as many times as we need, we get an approximate sweep for the entire blade. For an approximate calculation of the area of the helical surface, it suffices to find the area of planar triangles with which we have replaced the corresponding curvilinear ones.

Compact Packings of the Plane with Three Sizes of Discs

A compact packing is a set of non-overlapping discs where all the holes between discs are curvilinear triangles. There is only one compact packing by discs of size $1$. There are exactly $9$ values of $r$ which allow a compact packing by discs of sizes $1$ and $r$. We prove here that there are exactly $164$ pairs $(r,s)$ allowing a compact packing by discs of sizes $1$, $r$ and $s$.

Analytic solutions for fresh groundwater lenses floating on saline water under desert dunes: The Kunin-Van Der Veer legacy revisited

A fully-saturated lens of steady fresh groundwater floating in a homogeneous and isotropic desert sandy aquifer is analytically studied based on a hydrological model by Kunin and interface solution by Van Der Veer. A static saline groundwater is beneath the lens. A phreatic surface of moving fresh water inside the lens is partially recharged (either naturally or by managed aquifer recharge) from the vadose zone and partially exfiltrates to it. A spatially focused recharge and intensive evapotranspiration preserve a steady downward-upward topology of fresh water motion. In terms of the 1-D Dupuit-Forchheimer approximation in a horizontal in-lens saturated flow a boundary value problem (BVP) for an ODE for the Strack potential is solved. The shape of the water table and, based on the Ghijben-Herzberg assumption, the interface are found. The total volume of the positive-pore pressure water flowing within the lens is evaluated. Constant infiltration and evaporation rates as well as evaporation linearly decreasing with depth of the water table (counted from the ground surface) are considered. The case of 2-D flow is tackled by the Toth model. A triangular analytic element approximates a half of the flow domain and consists of an isobaric side and two no-flow sides. Conformal mapping of this triangle onto a reference plane and solution of the Dirichlet BVP in a half-plane deliver the distribution of infiltration-exfiltration intensity along the water table, total flow rate and locus of the hinge point. A mathematically more cumbersome approximation of the flow domain assumes the water table to be a tilted straight line but the interface to be found as a free boundary. Solution of the corresponding BVP uses a curvilinear triangle in the hodograph plane.

Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

The Laplace-Beltrami problem ΔΓψ = f has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green’s function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calderon projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles.

Analytical Solution for Interface Flow to a Sink With an Upconed Saline Water Lens: Strack's Regimes Revisited

AbstractA study is made of a steady, two‐dimensional groundwater flow with a horizontal well (drain), which pumps out freshwater from an aquifer sandwiched between a horizontal bedrock and ponded soil surface, and containing a lens‐shaped static volume of a heavier saline water (DNAPL‐dense nonaqueous phase liquid) as a free surface. For flow toward a line sink, an explicit analytical solution is obtained by a conformal mapping of the hexagon in the complex potential plane onto a reference plane and the Keldysh‐Sedov integral representation of a mixed boundary‐value problem for a complex physical coordinate. The interface is found as a function of the pumping rate, the well locus, the ratio of liquid densities, and the hydraulic heads at the soil surface and in the well. The shape with two inflexion points and fronts varies from a small‐thickness bedrock‐spread pancake to a critical curvilinear triangle, which cusps toward the sink. The problem is mathematically solvable in a relatively narrow band of geometric and hydraulic parameters. A similar analytic solution for a static heavy bubble confined by a closed‐curve interface (no contact with the bedrock) is outlined as an illustration of the method to solve a mixed boundary‐value problem.

A stack of cards rebuilt with calculus

Previous work covers building a tower from a stack of homogeneous rectangular plates, each with a maximum shift in displacement. We suggest using plates shaped as curvilinear triangles bounded by segments of power-law functions. The masses of the plates and the position of their center of mass are calculated and measured experimentally after cutting them out from cardboard and aluminum sheets. A computer simulation of the displacement towers is combined with their live building. Individual maximum shifts of the plates in the stack prove to be much bigger the higher the power coefficient of the boundary curves. The resulting total overhang of such a displacement tower may exceed that of a stack of traditionally used homogeneous rectangular cards.

Free Surface Flow in a Microfluidic Corner and in an Unconfined Aquifer with Accretion: The Signorini and Saint-Venant Analytical Techniques Revisited

Steady, laminar, fully developed flows of a Newtonian fluid driven by a constant pressure gradient in (1) a curvilinear constant cross section triangle bounded by two straight no-slip segments and a circular meniscus and (2) a wedge bounded by two rays and an adjacent film bulging near the corner are studied analytically by the theory of holomorphic functions and numerically by finite elements. The analytical solution of the first problem is obtained by reducing the Poisson equation for the longitudinal flow velocity to the Laplace equation, conformal mapping of the corresponding transformed physical domain onto an auxiliary half-plane and solving there the Signorini mixed boundary value problem (BVP). The numerical solution is obtained by meshing the circular sector and solving a system of linear equations ensuing from the Poisson equation. Comparisons are made with known solutions for flows in a rectangular conduit, circular annulus and Philip’s circular duct with a no-shear sector. Problem (2) is treated by the Saint-Venant semi-inverse method: the free surface (quasi-meniscus) is reconstructed by a one-parametric family, which specifies a holomorphic function of the first derivative of the physical coordinate with respect to an auxiliary variable. The latter maps the flow domain onto a quarter of a unit disc where a mixed BVP for a characteristic function is solved by the Zhukovsky–Chaplygin method. Velocity distributions in a cross section perpendicular to the flow direction are obtained. It is shown that the change of the type of the boundary condition from no slip to perfect slip (along the meniscus) causes a dramatic increase of the total flow rate (conductance). For example, the classical Saint-Venant formulae for a sector, with all three boundaries being no-slip segments, predict up to four times smaller rate as compared to a free surface meniscus. Mathematically equivalent problems of unconfined flows in aquifers recharged by a constant-intensity infiltration are also addressed.

Determination of the modular elliptic function in problems of free-flow filtration

The calculated dependences in elementary functions for determining the modular elliptic function λ(τ) =λ1 + iλ2 obtained on the basis of consecutive (six) conformal mappings of a curvilinear triangle to a complex half-plane are presented. Comparison of the values of λ(τ) from the proposed dependences with the results of the Hamel–Gunter exact analytical solution for the boundary contour of the curvilinear triangle, i.e., the real axis of the complex half-plane, gives a very close coincidence (with the largest error of ≤1%). The use of the complex values of the function λ(τ) for the entire internal region of the curvilinear triangle makes it possible to solve one of the most difficult problems of the theory of filtration (filtration through a rectangular dam) in the direct formulation and, for the first time, to construct the pattern of an equal filtration-rate field (the family of isotaches) over the entire internal region of the dam. In this case, the boundary values of filtration rates for special cases (along the sides and along the base of the dam) completely coincide with the results of the Masket exact analytical calculations.

Numerical Integration Scheme for the Near-Singular Green Function Gradient on General Triangles

A new near-singularity cancellation transformation quadrature scheme for triangle domains is presented. It is tailored to the Green function gradient kernel (static and dynamic) multiplied with higher-order weighting functions on curvilinear triangles. The transformation is analytically invertible. Attention is paid to ensuring accuracy in both tangential and normal components. Such integrals must be routinely evaluated in modern method of moments (MoM) implementations. The scheme follows the standard three subtriangle splitting approach of which the benefits are single-parameter accuracy control, a geometry-independent total number of quadrature points and implementation simplicity. Extensive numerical results show superior efficiency and reliability to existing three-subdomain schemes. Consistent, dependable error convergence is demonstrated, with very low variation in accuracy for fixed quadrature order and singularity height. The scheme is suitable for practical use by MoM developers.

Large-degree asymptotics of rational Painlevé-II functions: critical behaviour

This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), of the rational solutions of the inhomogeneous Painlevé-II equation and associated rational solutions of the homogeneous coupled Painlevé-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behaviour. Our results display both a trigonometric degeneration of the rational Painlevé-II functions and also a degeneration to the tritronquée solution of the Painlevé-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann–Hilbert representation of the rational Painlevé-II functions, and supplies leading-order formulae as well as error estimates.

Outer billiard around a curvilinear triangle with a fixed diameter

We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes.

Analysis on spherical conformal microstrip antenna array by characteristic basis function method

Spherical conformal microstrip antenna array which is used widely in the field of aeronautics is analyzed by the full wave analysis in this paper. Using the surface integral equation with spherical dyadic Green's function can decrease the number of unknowns remarkably, and also reduce the demand of memory as compared with the method using volume-surface integral equation. The curvilinear triangle is proposed to mesh the surface of the microstrip antenna array, which can simulate the characteristic of spherical surface accurately. Firstly, the problem about probe feed model is solved successfully by introducing the half Rao-Wilton-Glisson function and boundary charge, and the image method is used to treat the line integral singularity problem. Then, the characteristic basis function method is employed to further save memory and computation time further by reducing the rank of impedance matrix. Finally, the input impedance and far-field spherical conformal microstrip antenna array of different sizes are analyzed. Results are in good agreement with those in the literature and simulation software, and thus the validity and effectiveness of the analysis method are demonstrated.