Isometric Grid Research Articles

Laminar premixed hydrogen/air flame dynamics with/without sidewall interactions

This paper investigates laminar premixed hydrogen/air flame-wall interactions (FWI) subject to flame stretch (curvature and strain rate). Direct numerical simulations of finite-rate chemical reactions and in-depth molecular diffusion are used to numerically study two-dimensional (2D) V-shaped flames, with one side functioning as an isothermal cold wall that interacts with the flame and the other as a symmetrical center for free-propagating flame. An isometric grid with a grid resolution of 10 μm is used to resolve the FWI zone. It is discovered that the wall at the flame tip experiences increased negative stretching and enhanced local preferential diffusion which reduces the intensity of local combustion. This effect is especially noticeable at low equivalent ratios. Additionally, there is a close relationship between quenching behavior and flame dynamics. The present results show that the Markstein numbers for both the sidewall quenching flame and the freely propagating flame (MaSWQ and MaFF) are negative in lean hydrogen flames and positive in rich hydrogen flames. Cold wall has less impact on Ma. Furthermore, an intersection in the equivalence ratio is observed in the lean-fuel hydrogen/air flame, located at Φ = 0.8–1.0, with a change in the sign of Ma.

Proposing Cubic-Grid System as a New Structural System for Tall Buildings

Background: Changes in the skyline of cities all over the world show that tall buildings are an interesting solution to accommodating growth more sustainably in today’s urban areas. Stability against lateral loading is the main issue for designing high-rise buildings. Diagrid systems or grid structures are one of these stable systems. For a long time, the only structural pattern used in this system was the triangular modulus (diagrid). Methods: In the present study, a new pattern for grid structures was introduced, and its performance was compared with the two previous common patterns (triangular and hexagonal). The new pattern is called the Isometric Cube grid, thanks to its particular shape. The design process was performed based on a simple stiffness criterion presented in the previous research. In order to calculate the stiffness of the structure for the new pattern, two types of existing structural nodes were distinguished, and the results were combined to determine the stiffness of the unit forming the entire grid. Results and Conclusion: From the results of analysis, it was found that the optimal angles of diagonal elements were in the range of 50-65 degrees. Although the behavior of the new pattern in terms of shear stiffness was between the two previous patterns (and closer to the hexagonal pattern), the shear stiffness ratio increased from 30° to 65° for cube pattern, while for hexagrid, it decreased over the same range of angles.

Elongation Patterns of the Superficial Medial Collateral Ligament and the Posterior Oblique Ligament: A 3-Dimensional, Weightbearing Computed Tomography Simulation.

Background:Although length change patterns of the medial knee structures have been reported, either the weightbearing state was not considered or quantitative radiographic landmarks that allow the identification of the insertion sites were not reported.Purpose:To (1) analyze the length changes of the superficial medial collateral ligament (sMCL) and posterior oblique ligament (POL) under weightbearing conditions and (2) to identify the femoral sMCL insertion site that demonstrates the smallest length changes during knee flexion and report quantitative radiographic landmarks.Study Design:Descriptive laboratory study.Methods:The authors performed a 3-dimensional (3D) analysis of 10 healthy knees from 0° to 120° of knee flexion using weightbearing computed tomography (CT) scans. Ligament length changes of the sMCL and POL during knee flexion were analyzed using an automatic string generation algorithm. The most isometric femoral insertion of the sMCL that demonstrated the smallest length changes throughout the full range of motion (ROM) was identified. Radiographic landmarks were reported on an isometric grid defined by a true lateral view of the 3D CT model and transferred to a digitally reconstructed radiograph.Results:The sMCL demonstrated small ligament length changes, and the POL demonstrated substantial shortening during knee flexion (P = .005). Shortening of the POL started from 30° of flexion. The most isometric femoral sMCL insertion was located 0.6 ± 1.7 mm posterior and 0.8 ± 1.2 mm inferior to the center of the sMCL insertion and prevented ligament length changes >5% during knee flexion in all participants. The insertion was located 47.8% ± 2.7% from the anterior femoral cortex and 46.3% ± 1.9% from the joint line on a true lateral 3D CT view.Conclusion:The POL demonstrated substantial shortening starting from 30° of knee flexion and requires tightening near full extension to avoid overconstraint. Femoral sMCL graft placement directly posteroinferior to the center of the anatomical insertion of the sMCL demonstrated the most isometric behavior during knee flexion.Clinical Relevance:The described elongation patterns of the sMCL and POL aid in guiding surgical medial knee reconstruction and preventing graft lengthening and overconstraint of the medial compartment. Repetitive graft lengthening is associated with graft failure, and overconstraint leads to increased compartment pressure, cartilage degeneration, and restricted ROM.

Construction of a family of flat curves according to the equations of isometric grids

The article reveals an analytical description of the formation of families of orthogonal flat curved lines in the implicit form based on the analysis of the parametric equation of a flat isometric grid constructed by separating the real and imaginary parts of the function of a complex variable. This problem is due to the fact that flat isometric grids, as two families of orthogonal coordinate lines with square cells, are used in conformal mappings, for example, when drawing images on curved surfaces with the least distortion. At the same time, families of flat parallel lines are widely used in geometric modeling of heat transfer, electric fields, fluid flow, etc. There is a connection between these geometric images, which is explained by specific examples. Analytical calculations of deriving the parametric equation of an isometric grid are quite time-consuming, so they are performed in the environment of symbolic algebra Maple. For this purpose, the corresponding software of the interactive model of derivation of parametric equations of isometric grids for any initial function of a complex variable with the subsequent separation of its real and imaginary parts was created. It was found that the values of the abscissa and ordinates of the parametric equation of a flat isometric grid can be represented as explicit surface equations. For integer values of the power of the exponential function of the complex variable, the values of the abscissa and the ordinate will be represented by algebraic surfaces in the explicit form. The projections of the cross sections of the abscissa and ordinate surfaces by horizontal cutting planes on the horizontal plane form two families of curved lines, the equations of which can be obtained only implicitly. By the example of the quadratic function of a complex variable, it is proved that these families of lines are mutually perpendicular. The practical application of building a family of lines for geometric modeling of fluid flow lines that flow around the barrier in the form of a semicircle is shown. Key words: isometric grids, functions of a complex variable, families of orthogonal lines, geometric flow modeling

The mapping of raster images on plane isometric grid

Drawing images on curvilinear shapes with the least distortion takes place in many design tasks. In most ways, build a grid, each elementary cell of which is painted a given color. In this problem it is necessary to solve two main problems: the first - to carry out the formation of a given curvilinear grid with elementary cells in the form of squares, which are called isometric (or isothermal); the second is to paint each cell of the curved area with the corresponding pixel color of the original raster The aim of the study is to reveal the way of displaying raster images on flat curvilinear areas represented by isometric grids, and with the help of a computer model in the Maple symbolic algebra to analyze the influence of isometric grid parameters on the position and size of displayed raster images. The mapping of images onto curvilinear forms with minimal distortion takes place in many design tasks. A method of conformal mapping of arbitrary raster images onto plane curvilinear region is proposed, which are represented by isometric (also called isothermal) grids. The essence of the proposed method is as follows. Any raster image, for example, digital photography in jpg format, is characterized by the dimensions N×M - the number of pixels in width and height. In addition, each pixel has a color and brightness, which are arranged in rows and columns. To apply a raster image to a curvilinear region, it is also necessary to divide the curvilinear domain into N×M, the number of elementary squares, each of which is assigned the corresponding color from the raster. The influence and arguments of the various isometric grids constructed on the sizes and positions of an arbitrary raster image are investigated in the article. It is shown how the isometric grid, depending on and localizes the raster image - it can be located both within the limits of the isometric grid coordinate lines and beyond it, can also be oriented in different directions with respect to the and coordinate lines. It is shown the possibility of scaling a raster image that can be performed relative to the relative dimensions of an isometric grid. Since there is a correspondence between the pixel matrix of the original raster image and the - cells of the isometric grid, the rotation of the image will affect its position in the isometric grid. For example, rotating the original bitmap image at an angle 90 degrees will change its location on a plane isometric grids – from along the coordinate lines to along the coordinate lines. Note that, the curvilinear cells of the constructed isometric grids differ somewhat from the shape of the squares because the values and of the corresponding arguments and of their coordinate lines were taken somewhat too large. Otherwise, cells would degenerate into points and the corresponding grid image would not be so clear.

Узагальнений спосіб формування плоских ізотропних кривих

The process of modeling the temperature distribution on surfaces, applying an image to curved areas with minimal distortion requires the formation of isometric grids on the plane and on the surface. One of the common ways to form planar isometric networks is to use the functions of a complex variable and planar isotropic curves, followed by separation of the real and imaginary parts. The development of computer models for the interactive search and analysis of isometric networks according to various initial geometric conditions provides a generalized method for their formation with the possibility of varying their shape and position. It is proposed to use an isotropic vector for the formation of flat isotropic curves, which ensured a single sequence of analytical calculations according to the following initial conditions: 1) selection of an arbitrary function of a real argument; 2) a given parametric equation of a plane curve; 3) a given polar equation of a plane curve. Since the analytical calculations of the derivation of the parametric equation of a plane isotropic curve and the corresponding isometric grid are rather laborious, their execution is carried out in the environment of the Maple symbolic algebra. To this end, the corresponding software has been created, which interactively allows you to select the function of a real argument, a parametric or polar equation of a plane guide curve. All subsequent stages of analytical transformations to form an isotropic curve and the corresponding isometric grid are carried out automatically. An interactive model for the formation and analysis of plane isotropic curves with various initial conditions has been created, which has shown its effectiveness, which is confirmed by the given examples of plane isometric grids for specific functions of the real parameter, plane curves in the parametric and polar form of their job.

Binary tomography on the isometric tessellation involving pixel shape orientation

In this study, a tomography reconstruction problem of binary images is considered on the isometric grid. On this grid, the triangle pixels have two types of orientations, accordingly, the authors call them delta or nabla shape pixels. The proposed reconstruction method uses data of projections of three natural directions. They are the lane directions of the triangular tessellation (these directions are somewhat analogous to row/column directions on the rectangular grids). The projection ray, penetrating through a grid lane, now not passing through the middle of pixels (i.e. through the middle line of triangle shape pixels), as usually taken, but little bit shifted from the middle parallel to the lane. This method provides the exact information about the number of nabla and delta shape triangle pixels in each lane of the image. This additional information is included in the reconstruction process to improve the quality of reconstruction. They formulate the suggested model into an energy-minimisation problem and apply a gradient-based approach for its minimisation. They show and analyse various experimental results on test images. The presented approach shows both better quality reconstructions and shorter running time than the earlier approaches.

Chamfer distances on the isometric grid: a structural description of minimal distances based on linear programming approach

Chamfer distances on the isometric grid are considered. A new method to compute the chamfer distances based on linear optimization is presented. In the LP model the starting pixel is the Origin, that is a triangle of the grid having co-ordinates (0, 0, 0). The co-ordinates of the end pixel of the path give the right-hand side of the model. The variables are the used numbers of the elementary steps. Each type of an elementary step has a uniquely defined weight. Our operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. We give directed graphs with feasible bases as nodes and arcs with conditions on the used weights such that the simplex method of linear programming may step from one feasible basis to another feasible basis. Thus, the possible course of the simplex method can be followed and the optimal bases can easily be captured. Thus, the final result of the analysis is an O(1) checking of the feasibility and optimality conditions. The optimal bases are summarized in a theorem which is the consequence of the general theory of linear programming. The method can be applied for other grids, but it needs to be adjusted for the particular grid.

Detail-preserving switching algorithm for the removal of random-valued impulse noise

This paper presents a new algorithm for the denoising of images corrupted with random-valued impulse noise (RVIN). It employs a switching approach that identifies the noisy pixels in the first stage and then estimates their intensity values to restore them. Local statistics of the textons in distinct orientations of the sliding window are exploited to identify the corrupted pixels in an iterative manner; using an adaptive threshold range. Textons are formed by using an isometric grid of minimum local distance that preserves the texture and edge pixels of an image, effectively. At the noise filtering stage, fuzzy rules are used to obtain the noise-free pixels from the proposed tri-directional pixels to estimate the intensity values of identified corrupted pixels. The performance of the proposed denoising algorithm is evaluated on a variety of standard gray-scale images under various intensities of RVIN by comparing it with state-of-the-art denoising methods. The proposed denoising algorithm also has robust denoising and restoration power on biomedical images such as, MRI, X-Ray and CT-Scan. The extensive simulation results based on both quantitative measures and visual representations depict the superior performance of the proposed denoising algorithm for various noise intensities.

On the Polygon Containment Problem on an Isometric Grid

This paper addresses the issue of placing a simple polygon (upon translation and rotation) on an isometric triangular grid such that the polygon contains the maximum number of triangles in its closure. This solves the problem left open in two recent papers titled “On the problem of the automated design of large-scale robot skin” and “An improved algorithm for the automated design of large scale robot skin” published in the IEEE Transactions on Automation Science and Engineering . Based on the properties of the grid, an improved algorithm is also presented. We also present some experimental results describing the use of this algorithm.

Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

Structural and biophysical simulation of angiogenesis and vascular remodeling.

The purpose of this report is to introduce a new computer model for the simulation of microvascular growth and remodeling into arteries and veins that imitates angiogenesis and blood flow in real vascular plexuses. A C++ computer program was developed based on geometric and biophysical initial and boundary conditions. Geometry was defined on a two-dimensional isometric grid by using defined sources and drains and elementary bifurcations that were able to proliferate or to regress under the influence of random and deterministic processes. Biophysics was defined by pressure, flow, and velocity distributions in the network by using the nodal-admittance-matrix-method, and accounting for hemodynamic peculiarities like Fahraeus-Lindqvist effect and exchange with extravascular tissue. The proposed model is the first to simulate interdigitation between the terminal branches of arterial and venous trees. This was achieved by inclusion of vessel regression and anastomosis in the capillary plexus and by remodeling in dependence from hemodynamics. The choice of regulatory properties influences the resulting vascular patterns. The model predicts interdigitating arteriovenous patterning if shear stress-dependent but not pressure-dependent remodeling was applied. By approximating the variability of natural vascular patterns, we hope to better understand homogeneity of transport, spatial distribution of hemodynamic properties and biomass allocation to the vascular wall or blood during development, or during evolution of circulatory systems.

An Extension: Gather Your Students around the Gathering Circles

You and your students can investigate arrangements of tightly packed circles by using drinking straws and string or rubber bands. Wrap the string or band loosely around a handful of straws, and pull it taut. Observe the way that the straws align on the end when the string is pulled as tightly as possible. See figure 1. Looking at the cross section of the bundle of straws, notice that the centers of the circles and the points of tangency form the lattice points of an isometric grid, as shown in figure 2. You can use isometric paper to record the “packed” arrangements of various numbers of circles (fig. 3).

Equilateral Triangles on an Isometric Grid

The problem in figure 1 appeared on the cover of the Virginia Mathematics Teacher (Fall 1985). An interesting related problem is “How many equilateral triangles of different sizes can be constructed on the isometric grid shown” (using the points as vertices)?

Taxicab Geometry—A New Slant

As anyone who has played with a bit of screen wire knows, a square is not a rigid geometrical figure. A slight push to the right or left will deform a square into a nonsquare rhombus. If such a transformation is performed on the points of a plane which has been coordinatized by a square grid so that the positive y-axis makes an angle of 600 with the positive x-axis and the shorter diagonals of each rhombus are drawn, an isometric grid results. Points will still be named by ordered pairs of real numbers with respect to the x-axis and the transformed y-axis (FIGURE la). There are only three regular polygons which will tessellate the plane: the equilateral triangle, the square, and the regular hexagon. The first two of these can be subdivided into smaller similar polygons (yielding the isometric grid and the square grid). Work has already been done on taxicab geometry using the square grid; this note considers taxicab geometry using the isometric grid. Square-taxi geometry arises because, for the two points A = (xi, Y1) and B = (x2, Y2), a new distance function is chosen:

An isometric triaxial reconstruction method for quantitative study of serial histological sections.

A method is described for the reconstruction of specimens from serial histological sections that permits measurements to be made on the two-dimensional representations of the reconstruction. With this technique, sections are plotted on an isometric triaxial grid and measurements may be obtained from the reconstruction and from transections of the reconstruction made in any plane required.

Extremal Problems on A Geoboard

There are many fascinating geoboard problems and patterns that are of interest to high school students. One of these is the nine-point circle, presented on an isometric grid geoboard by Robert L. Jones in the February 1976 issue of the Mathematics Teacher.

The Nine-Point Circle on a Geoboard

The nine-point circle theorem is beatifully illustrated on the isometric grid geoboard.