Elliptical Boundary Research Articles

Mechanics of pressurized planar hyperelastic membranes.

The paper examines the mechanics of inflation of incompressible planar hyperelastic membranes that are rigidly fixed at their boundary and subjected to a uniform pressure. Strain energy functions characterized by the neo-Hookean, Mooney-Rivlin and the Ogden forms are used. Fixity is provided along either circular or elliptical boundaries. The computational results indicate that the strain energy function has a significant influence on the pressure versus inflated volume response of the deformed membrane. When the strain energy function corresponds to a Mooney-Rivlin form, the circular membrane displays no tendency to develop any instability. The equivalent circular membranes composed of both the neo-Hookean and Ogden-type strain energy functions developed an initial 'Wrinkling Instability'. For planar membranes with an elliptical planform, the wrinkling instability is more pronounced; membranes composed of hyperelastic materials with a Mooney-Rivlin form of the strain energy function continue to deform without the development of an initial instability point, whereas membranes composed of both the neo-Hookean and Ogden materials exhibit wrinkling behaviour at critical locations at the interior of the fixed boundary region. The dependency of the strain energy function on the second invariant of the Cauchy-Green strain tensor has an influence in the suppression of hyperelastic effects. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Numerical computation of the effective thermal conductivity of two-phase composite materials by digital image analysis

We consider the numerical determination of the effective thermal conductivity of 2D, two-phase composite materials, whose microstructural geometries are described by digital images. More specifically, key microstructural data pertaining to the spatial distribution and volume fractions of the constituent phases in the material are extracted, along with the shapes of the interphase boundaries, using computer vision techniques. For inclusions of simple shapes (polygons, circles, ellipses), the extracted interphase boundary data is easily obtained and can be exploited immediately (e.g. polygonal vertices) or can be fitted with smooth curves, using least square error minimization techniques in order to reconstruct parametric representations in the case of circular or elliptical boundaries. For more intricate microstructure geometries characterized by clustered or agglomerated intersecting inclusions, piecewise cubic spline functions are adopted to obtain a least-square fitting of the interphase boundaries. The effective thermal conductivities are then calculated based on a boundary integral formulation defined over the reconstructed inclusion boundaries. They are expressed in terms of power series expansions of a conductivity contrast parameter of the constituent phases in the material. To accelerate the convergence of these expansions, nonlinear transformations (known as Padé Approximants) are used to yield rational approximations of the effective thermal conductivity tensor. Tests conducted on a variety of 2D, two-phase microstructures yielded accurate results over a wide range of microstructure image resolution, regardless of the contrast parameters of the constituent phases, thus, validating the efficacy and robustness of the proposed scheme.

Observation of a half-illuminated mode in an open Penrose cavity

The illumination problem in mathematics questions the existence of a bounded region in which light rays from a point light source do not illuminate the whole region. Since Penrose disproved the illumination problem with elliptical reflective boundaries, the interest has mostly remained in ray optics mainly because there can be no completely dark region for light waves due to diffraction. Here, in a two-dimensional Penrose cavity with elliptical boundaries, we report experimental observation of a symmetry-broken mode in the long-wavelength regime with the half of the cavity region with reflection symmetry almost unilluminated in the steady state. The half-illuminated mode (HIM) was observed in an acoustic cavity by using the schlieren method. The HIM originated from the coherent superposition of near-degenerate modes, among which two scarred modes with opposite parities played a major role. The illuminated part of the HIM could be even flipped by choosing different coefficients in the coherent superposition of the participating modes. The HIM of the Penrose cavity provides new perspective to the illumination problem in an open system.

Magnetic mixed convection of a Casson hybrid nanofluid due to split lid driven heat generated porous triangular containers with elliptic obstacles

The current study concerns with the mixed convection process of Casson hybrid nanofluids within split lid-driven porous inclined triangular enclosures with elliptic obstacles. The top wall of triangular containers is split to two-parts; those move to left and right directions and four cases are considered based on the direction of the movements. The inner elliptic shape has various thermal conditions, namely, hot, cold and adiabatic. An inclined magnetic field and a heat generating porous medium are considered. The considered non-Newtonian hybrid nanofluids are methano-Ti6Al4V + AA7075, ethylene glycol-Ti6Al4V + AA7075 and engine oil- Ti6Al4V + AA7075. Comprehensive analyses for the heat transfer at the top moving and the inner elliptic boundaries as well as the entropy generation are performed. FEM-CBS algorithm is applied to solve the governing system. The major findings revealed that the cases of UL=-1,UR=1 and cold ellipse give the highest rate of the heat transfer. Also, engine oil- Ti6Al4V + AA7075 augments the Nusselt number more than the other types of the non-Newtonian hybrid nanofluids. Furthermore, using methanol as a based hybrid nanofluids is a better for obtaining the higher irreversibility rate.

Interacting stress of multiple random elliptical holes under remote and arbitrary surface stresses

Study of multiple elliptical hole problem is of great significance for optimization design and safety assessment in both geotechnical engineering and perforated materials concerned with elliptical holes. Currently, the existing kind of literature mainly focus on calculating the interacting stress of multiple elliptical holes under the remote stresses, less on that under the surface stresses. In this study, based on the derived fundamental solutions, the integral equations and superposition technique, a new analytical–numerical method is established to calculate the interacting stresses of an infinite elastic plane containing multiple elliptical holes with any size and orientation, under both remote stresses and arbitrarily distributed surface stresses applied on the elliptical holes. The new method is proved to be valid by existing methods (including the conformal mapping method, Laurent series method, complex function method, boundary element alternating method, and finite element method). Some computational examples of two elliptical holes (i.e., horizontal and oriented elliptical holes) under biaxial stresses at infinity (=) are given to analyze the influencing factors on the interacting tangential stresses at the elliptical boundaries. Research results show that the oriented elliptical hole usually has an amplifying effect on the horizontal one, while the horizontal elliptical hole has an amplifying or shielding effect on the oriented elliptical one, mainly depending on the orientation angle (α 2) of the oriented one. When the major axes of two elliptical holes are vertical to their center connecting-line, the plate with two elliptical holes has smaller stress concentration than that only with one elliptical hole, which is helpful for improving the strength of underground engineering and perforated materials. The proposed method has not only simple formulation (without singularity), high accuracy (due to the exact fundamental solution), and wider applicability (under both remote stresses and arbitrarily distributed surface stresses) than the common conformal mapping method, Laurent series method, complex function method and boundary element alternating method, but also the potential to be further developed for the isotropic/anisotropic problem of multiple arbitrarily shaped holes under the same complex stresses.

Uniform in‐plane elastic field inside a monoclinic elliptical inhomogeneity with non‐confocal elliptical interphase layer

We employ the Stroh quartic formalism to study the in-plane deformations of a three-phase composite consisting of an elliptical inhomogeneity embedded in an infinite matrix via an intermediate interphase layer itself bounded by two elliptical boundaries. The matrix is subjected to uniform remote in-plane loading. The two corresponding elliptical interfaces (one between the inhomogeneity and the interphase layer and the other between the interphase layer and the surrounding matrix) are non-confocal. The inhomogeneity and the matrix are assumed to be monoclinic materials with the symmetry plane at x3 = 0 whilst the intermediate interphase layer is composed of a mathematically degenerate orthotropic material. The non-confocal character of the two elliptical interfaces is carefully designed according to the orthotropic property of the interphase layer. Two simple conditions are found that ensure that the internal elastic field of stresses and strains inside the elliptical inhomogeneity is uniform. For given material and geometric parameters of the composite, these two conditions give two simple relationships among the three remote in-plane stresses. In addition, the internal uniform elastic field inside the elliptical inhomogeneity and the non-uniform elastic field in the interphase layer are determined in real-form in terms of the two 4 × 4 fundamental elasticity matrices for the inhomogeneity and the matrix and the three 2 × 2 Barnett-Lothe tensors for (only) the matrix.

The effect of exposure to elevated temperatures on the microstructure and hardness of Mg–Ca–Zn alloy

Abstract The die cast Mg-5 wt.% Ca-6 wt.% Zn ternary alloy was exposed to 160 °C for different times of up to 40 days. Microstructural analysis and microhardness and hardness measurements of the samples after different exposure times enabled the thermal stability of the cast alloy to be monitored. The as-cast structure is composed mainly of α-Mg solid solution, elliptical grain boundary particles of CaMg2, and an intergranular eutectics of Mg and Ca2Mg6Zn3. The grain size of α-Mg and the structure of the alloy did not change during the treatment, but the amount of the intergranular phases increased slightly. Exposure to 160 °C resulted in a decrease in micro-hardness of the α-Mg grains, but no change in the overall hardness of the samples was observed. Microstructural analyses indicated diffusion of the solute elements from the α-Mg grains to the grain boundaries, resulting in a decrease in hardness of the α-Mg which is, however, compensated by an increase in hardness related to intergranular precipitates. Using a simple diffusion model, the diffusion coefficient of Ca atoms in Mg at 160 °C was estimated to be 2.1 · 10 –17 m2 s– 1, which is greater than that of Zn by two orders of magnitude.

Preparation of (B,N)-codoped TiO2 micro-nano composite powder using TiBN powder as a precursor

(B,N)-codoped TiO2 was prepared by the oxidation method using TiBN powder as a precursor. The oxidation of TiBN powder particles started with the surface, producing a high-Ti low-O compound Ti2O3. Meanwhile, accompanied by the migration and oxidation of B, the oxidative process gradually extended towards the interior, giving rise to the formation of (B,N)-codoped TiO2. XRD analysis revealed that, in oxidative products, the residual TiBN phase was structurally similar to Ti4N3B2, suggesting that Ti had diffused and migrated outwards from TiBN. On this basis, an oxidative model was built for TiBN particles. The oxidation of TiBN consisted of two processes: chemical reaction (TiBN→Ti2O3→TiO2) and organizational structure evolution (TiBN→Ti4N3B2). The prepared TiO2 was dominated by rutile form and also contained small amounts of anatase and residual TiBN. (B,N) Codoping increased the lattice parameter of rutile TiO2. Organizational structure was found to be related to the parameters of the oxidative process. Pure rutile TiO2 could be obtained from TiBN powder under 800°C + 2h oxidative treatment, and the interplanar spacings of its (110), (101), (211), and (111) planes were 0.324877 nm, 0.24875 nm, 0.16874 nm, and 0.21873 nm, respectively. These interplanar spacings were greater than the interplanar spacings of TiO2 produced through the oxidation of TiN under equal conditions, which was attributable to (B,N) codoping in TiO2. According to direct observations by high-resolution transmission electron microscopy (HRTEM), the organizational structure of TiO2 particles demonstrated the fine size and elliptical boundary and contained amorphous layers. This study offers a new approach to (B,N) codoping in TiO2. The (B,N)-codoped TiO2 with high photocatalytic efficiency obtained in this study can be applied in environmental and energy fields.

Solution of Exterior Quasilinear Problems Using Elliptical Arc Artificial Boundary

In this paper, the method of artificial boundary conditions for exterior quasilinear problems in concave angle domains is investigated. Based on the Kirchhoff transformation, the exact quasiliner elliptical arc artificial boundary condition is derived. Using the approximate elliptical arc artificial boundary condition, the finite element method is formulated in a bounded region. The error estimates are obtained. The effectiveness of our method is showed by some numerical experiments.

Solution of Two-Dimensional Stokes Flow With Elliptical Coordinates and Its Application to Permeability of Porous Media

AbstractIn this paper, analytical developments of the biharmonic equation representing two-dimensional Stokes flow are performed with elliptical coordinates. It is found that the streamfunction is expressed with series expansions based on Gegenbauer polynomials of first and second kinds with order one Cn1 and Dn1 for n natural integer number. Application to an elliptical fiber enclosed in an elliptical boundary with uniform flow is made. It is found that a particular solution responsible of the drag must be added to the general solution. Following this, the flow through rectangular arrays of elliptical fibers is studied, and the permeability of this medium is determined as a function of porosity.

Statistical Analysis of Microarray Data Clustering using NMF, Spectral Clustering, Kmeans, and GMM.

In unsupervised learning literature, the study of clustering using microarray gene expression datasets has been extensively conducted with nonnegative matrix factorization (NMF), spectral clustering, kmeans, and gaussian mixture model (GMM)are some of the most used methods. However, there is still a limited number of works that utilize statistical analysis to measure the significances of performance differences between these methods. In this paper, statistical analysis of performance differences between ten NMF, six spectral clustering, four GMM, and the standard kmeans algorithms in clustering eleven publicly available microarray gene expression datasets with the number of clusters ranges from two to ten is presented. The experimental results show that statistically NMFs and kmeans have similar performances and outperform spectral clustering. As spectral clustering can be used to uncover hidden manifold structures, the underperformance of spectral methods leads us to question whether the datasets have manifold structures. Visual inspection using multidimensional scaling plots indicates that such structures do not exist. Moreover, as the plots indicate that clusters in some datasets have elliptical boundaries, GMM methods are also utilized. The experimental results show that GMM methods outperform the other methods to some degree, and thus imply that the datasets follow gaussian distributions.

Numerical determination of the contours of the area watering from the fluid source

The paper presents the results of numerical determination of water-flooding contours (or as they are called, depression curves) of a flat porous region from a source with a smooth convex boundary. Based on the Green integral formula, formulas are obtained for determining the values of the harmonic function of the fluid pressure at each point in the region. Areas with constant permeability and variable are considered. Based on the obtained integral identity, a numerical algorithm for determining filtration rates for these and other areas are constructed using the boundary element method is constructed, and depression curves are constructed for sources with a round and elliptical boundary. The eccentricities of the region’s water-flooding conditions are determined. To compare the numerical and analytical results, calculations were made for the test example. You can control the accuracy of calculations by changing the number of nodes on the boundary of the simulated area.

Torsion of Transversely Isotropic Plate with a Non-Circular Cylindrical Hole

The problem of the stress state of an unbounded transversely isotropic plate with a noncircular cylindrical hole is solved by expanding functions into Fourier–Legendre series in the thickness coordinate and using the boundary shape perturbation method. The surface of the hole is free from external forces, and the plate is subject to constant torques at infinity. The stress distribution in the vicinity of the hole with elliptical boundary or triangular boundary with rounded corners in the midsurface is analyzed. The dependence of stresses on the relative thickness of the plate and elastic constants is established.

Fuzzy-set analysis of models of temperature deformation of thin-walled elements with elliptic boundaries in industrial and aerospace structures

A theoretical numerical-analytical fuzzy-set methodology to take into account the influence of scatter errors in the values of the initial physical, mechanical and geometric parameters on the estimates of the endogenous characteristics of temperature stresses in models of bending and flat temperature deformation of plate-type thin-walled structural elements with elliptical boundaries in industrial and aerospace equipment was developed. The methodology is based on the use of the alpha-level form of the heuristic principle of generalization in the transition to fuzzy-multiple arguments in the analytical representations of deterministic models for the estimated indefinite indicators of temperature bending moments and circumferential thermomechanical stresses at the boundaries of elliptical holes in thin plates. The results of calculations using the developed technique are presented. The proposed method is intended for use in theoretical and applied research on assessing the reliability and strength of thin-walled plate structural elements of aerospace engineering, industrial equipment and building structures under thermomechanical influences for conditions of uncertainty.

IMU-Based Locomotion Mode Identification for Transtibial Prostheses, Orthoses, and Exoskeletons.

Active transtibial prostheses, orthoses, and exoskeletons hold the promise of improving the mobility of lower-limb impaired or amputated individuals. Locomotion mode identification (LMI) is essential for these devices precisely reproducing the required function in different terrains. In this study, a terrain geometry-based LMI algorithm is proposed. The environment should be built according to the inclination grade of the ground. For example, when the inclination angle is between 7 degrees and 15 degrees, the environment should be a ramp. If the inclination angle is around 30 degrees, the environment is preferred to be equipped with stairs. Given that, the locomotion mode/terrain can be classified by the inclination grade. Besides, human feet always move along the surface of terrain to minimize the energy expenditure for transporting lower limbs and get the required foot clearance. Hence, the foot trajectory estimated by an IMU was used to derive the inclination grade of the terrain that we traverse to identify the locomotion mode. In addition, a novel trigger condition (an elliptical boundary) is proposed to activate the decision-making of the LMI algorithm before the next foot strike thus leaving enough time for performing preparatory work in the swing phase. When the estimated foot trajectory goes across the elliptical boundary, the decision-making will be executed. Experimental results show that the average accuracy for three healthy subjects and three below-knee amputees is 98.5% in five locomotion modes: level-ground walking, up slope, down slope, stair descent, and stair ascent. Besides, all the locomotion modes can be identified before the next foot strike.

Computation of effective thermo-piezoelectric properties of porous ceramics via asymptotic homogenization and finite element methods for energy-harvesting applications

We consider the linear thermo-piezoelectric properties of a ceramic matrix with cylindrical empty pores distributed periodically. The asymptotic homogenization method is applied to an elliptical tensor-weighted boundary value problem in the Stress-Charge-Entropy formulation of the constitutive relations with rapidly oscillating coefficients and free boundary conditions on the surfaces of the pores. For different shapes of the pore cross section, we solve the local problems via finite element method to compute the effective coefficients as functions of the physical properties of the matrix, the shape of the pore cross section and their volume fraction. The numerical results show excellent agreement with analytical formulae. When the effective coefficients are transformed to the Strain-Charge-Entropy formulation of the constitutive relations, they become independent of the shape of the cross section, which further validates the importance of the analytical formulae. We compute the piezoelectric and pyroelectric figures of merit for energy-harvesting applications, which depend on the effective coefficients and are compared with recent experimental results. This contribution could be useful for fine-tuning the properties of this class of materials for energy-harvesting applications.

Representation of the Reynolds stress tensor through quadrant analysis for a near-neutral atmospheric surface layer flow

To disseminate the role of the eddy motions on the anisotropic states of the Reynolds stress tensor ($$\varvec{B}$$), we devise a novel methodology based on the quadrant analysis, where the distributions of the invariants of $$\varvec{B}$$ are studied on a $$u^{\prime }$$–$$w^{\prime }$$ quadrant plane, with $$u^{\prime }$$ and $$w^{\prime }$$ being the turbulent fluctuations in the streamwise and vertical velocities. We apply this methodology to a near-neutral atmospheric surface layer (ASL) flow, derived from a field experiment dataset having multi-level turbulence measurements. The results show that in a near-neutral ASL flow, the anisotropic states of $$\varvec{B}$$ are determined by the distribution of the streamwise and cross-stream velocity variances ($$\sigma _{u}^2$$ and $$\sigma _{v}^2$$) on the $$u^{\prime }$$–$$w^{\prime }$$ quadrant plane. By studying the contour maps of the invariants of $$\varvec{B}$$ on the $$u^{\prime }$$–$$w^{\prime }$$ quadrant plane, we discover three distinct zones with elliptical boundaries in the $$u^{\prime }$$–$$w^{\prime }$$ plane, across which the anisotropic states of the eddy motions evolve. We find that the eddy motions which occur We also notice that the distinction between these three zones in the $$u^{\prime }$$–$$w^{\prime }$$ plane is prominent at the lowest measurement level, but becomes progressively indistinguishable as the measurement height increases.

Investigation about elliptical region observed on CdZnTe wafers after etching treatment

An elliptical region was observed on the wafer cut from the ingot along (1 1 1) face after etching and the ingot was grown by the Vertical Bridgman method. The wafers with the elliptical region were characterized with zinc concentration distribution scanner and infrared transmission microscope. Infrared transmission images revealed the morphology evolution of Te inclusions from irregular droplets to regular tetrahedrons and different stages from the Te-rich threads to the pearl-string Te inclusions inside the elliptical region. Te atoms preferentially diffused and aggregated in the dislocation direction, which resulted in the formation of Te-rich droplets and Te-rich threads. The Zn component map indicated that Zn concentration varied significantly at the elliptical boundary, the Zn concentration inside the ellipse was lower than other normal region. Based on the above experiments, it was considered that such an elliptical region created because of the occurrence of constitutional supercooling, in other words, it is caused by a drastic change of the Zn concentration gradient in the liquid phase at the front of the solid-liquid interface. The segregation of the Zn concentration in the liquid phase at the front of the interface was related to the growth rate. During crystal growth, when approaching the region where the Zn concentration changed drastically, the growth rate should be appropriately lowered to meet the normal Zn concentration gradient to avoid the generation of elliptical region.

A semi-analytical solution for electric double layers near an elliptical cylinder

A theoretical analysis on the electric double layer formed near the surface of an infinite cylinder with an elliptical cross section and a prescribed electric potential in an ionic conductor was performed using the linearized Gouy–Chapman theory. A semi-analytical solution in terms of the Mathieu functions was obtained. The distributions of the electric potential, cations, anions, and electric field were calculated. The effects of various physical and geometric parameters were examined. The fields vary rapidly near the elliptical boundary and are nearly uniform at far field. Electric field concentrations were found at the ends of the semi-major and semi-minor axes of the ellipse. These concentrations are sensitive to the physical and geometric parameters.

Reducibility of the ternary forms of unitary bordering matrices

In this paper, necessary and sufficient conditions are presented for reducibility of the ternary forms associated to 3×3 unitary bordering matrices. A one-parameter family of n×n unitary bordering matrices is given for which its associated ternary form is the product of linear and quadratic forms for n≤6. As a consequence, the higher rank numerical ranges of this family of unitary bordering matrices have elliptical boundaries.