Cone Boundary Research Articles

Local transmitting boundaries for the generalized interpolation material point method

SummaryThe boundary‐value problems of mechanics can be solved using the material point method with explicit solver formulations. In explicit formulations, even quasi‐static problems are solved as if dynamic, which means that waves are reflected at computational boundaries, generating spurious oscillations in the solution to the boundary‐value problem. Such oscillations can be reduced to a level such that they are barely noticeable with the use of transmitting boundaries. Current implementations of transmitting boundaries in the material point method are limited to the standard viscous boundary. The absence of any stiffness component in the standard viscous boundary may lead to an undesirable finite rigid‐body motion over time. This motion can be minimized through the adoption of the transmitting cone boundary that approximates the stiffness of the unbounded domain. This paper lays out the implementation of the transmitting cone boundary for the generalized interpolation material point method. The cone boundary reflection‐canceling tractions can be applied to either the edges or the centroids of material points; this paper discusses the implications of both approaches.

Azimuthal Variation of Instabilities Generated on a Flared Cone by Laser Perturbations

To study the azimuthal development of boundary-layer instabilities, a controlled, laser-generated perturbation was created in the freestream of the Boeing/U.S. Air Force Office of Scientific Research Mach 6 Quiet Tunnel. The freestream perturbation convected downstream in the wind tunnel to interact with a flared-cone model. The flared cone is a body of revolution bounded by a circular arc with a 3 m radius. Pressure transducers were used to measure a wave packet generated in the cone boundary layer by the freestream perturbation. Nine of these sensors formed three stations of azimuthal arrays and were used to determine the azimuthal variation of the wave packets in the boundary layer. The freestream laser-generated perturbation was positioned upstream of the model in three different configurations: along the centerline axis, offset from the centerline axis by 1.5 mm, and offset from the centerline axis by 3.0 mm. When the freestream perturbation was offset from the centerline of a flared cone with a 1.0 mm nose radius, a larger wave packet was generated on the side toward which the perturbation was offset. As a result, transition occurred earlier on that side. The offset perturbation did not have as large of an effect on the boundary layer of a nominally sharp flared cone.

Dynamic analysis in three layered conical shells utilising numerical methods

In this work, dynamic composite analysis truncated sandwich conical shells (STCS) with different boundary conditions under low velocity impact was investigated analytically, according higher order sandwich panel theory. Impact occurred at the top face sheet and the contact force history is predicted using two solution models calculated applying principal of Hamilton by considering the displacement continuity conditions between the layers. For obtaining the displacement and contact force histories, the differential quadrature method (DQM) is utilised. In this investigation, the effects of layer lamina number, cone semi vertex angle, boundary conditions and impact velocity on the impact response are studied.

Dynamic analysis in three layered conical shells utilising numerical methods

In this work, dynamic composite analysis truncated sandwich conical shells (STCS) with different boundary conditions under low velocity impact was investigated analytically, according higher order sandwich panel theory. Impact occurred at the top face sheet and the contact force history is predicted using two solution models calculated applying principal of Hamilton by considering the displacement continuity conditions between the layers. For obtaining the displacement and contact force histories, the differential quadrature method (DQM) is utilised. In this investigation, the effects of layer lamina number, cone semi vertex angle, boundary conditions and impact velocity on the impact response are studied.

Electrically induced structure transition in nematic liquid crystal droplets with conical boundary conditions.

Polymer-dispersed liquid crystal composites have been a focus of study for a long time for their unique electro-optical properties and manufacturing by "bottom-up" techniques at large scales. In this paper, nematic liquid crystal oblate droplets with conical boundary conditions (CBCs) under the action of electric field were studied by computer simulations and polarized optical microscopy. Droplets with CBCs were shown to prefer an axial-bipolar structure, which combines a pair of boojums and circular disclinations on a surface. In contrast to droplets with degenerate planar boundary conditions (PBCs), hybridization of the two structure types in droplets with CBCs leads to a two-minima energy profile, resulting in an abrupt structure transition and bistable behavior of the system. The nature of the low-energy barrier in droplets with CBCs makes it highly sensitive to external stimuli, such as electric or magnetic fields, temperature, and light. In particular, the value of the electric field of the structure reorientation in droplets with CBCs was found to be a few times smaller than the one for droplets with PBCs, and the droplet state remained stable after switching off the voltage.

Orientational structures in nematic droplets with conical boundary conditions

Oblate nematic droplets encapsulated in a polymer specifying conical boundary conditions have been considered. Calculations by the extended Frank elastic continuum approach show that a number of various structures can be formed in such droplets under the variation of their size. Polarizing optical microscopy studies of composite film samples confirm the results of calculation and demonstrate the formation of the following orientational structures in the considered system: (i) a radial-bipolar structure with a twisted hedgehog defect and two hyperbolic boojums, (ii) an axial-bipolar structure with a circular disclination and two radial boojums, and (iii) a structure with a hedgehog defect, a hyperbolic boojum, and a radial boojum. Such a diversity of possible topologies of droplets is due to a complex balance between the energies of elasticity of the director field, disclinations, and anchoring with the surface, which is ensured by conical boundary conditions.

Flow and self-noise around bypass transition strips

To analyze the downstream effects of bypass transition strips on a laminar incoming flow, a direct numerical simulation of the fully transient, explicit and compressible Lattice Boltzmann equations is performed. Near wake analysis of a staggered grid of cubic blocks as transition device is compared with a more conventional zigzag strip, to ensure transition to a fully developed conical boundary layer. The staggered grid of blocks is more efficient in stopping the flow and creating large, coherent, flow structures of the size of the blocks, which results in a stronger transition. However, the downstream merging of spanwise created structures is relatively long resulting in higher correlated boundary layers. If the spanwise variation of the zigzag strip is small, the streamwise vortices created merge quicker, resulting in an earlier uncorrelated boundary layer. The self-noise created by zigzag strip is also significantly less than the noise from the staggered grid of blocks and becomes only dominant at high frequency compared to the predicted trailing edge noise.

The Weyl–Van Der Pol Phenomenon in Acoustic Diffraction by a Wedge or a Cone with Impedance Boundary Conditions

The paper deals with the asymptotic description of a diffraction pattern similar to the classical Weyl–Van der Pol phenomenon (the Weyl–Van der Pol formula). The latter arises in the problem of diffraction of waves generated by a source located near an impedance plane. An incident wave illuminates an impedance wedge or cone. The singular points of the wedge’s (the edge points) or cone’s (the vertex of the cone) boundary play the role of an imaginary source, giving rise to a specific boundary layer in some neighborhood of the corresponding impedance surface, provided that the surface impedance is relatively small. From the mathematical point of view, the description of the phenomenon is given by means of the far field asymptotics for the Sommerfeld integral representations of the scattered field. For small impedance of the scattering surface, the singularities describing the surface wave, which propagates from the edge (or from the vertex) along the impedance surface, may be located in a neighborhood of saddle points. The latter are responsible for a cylindrical wave from the edge of the wedge (or for a spherical wave from the vertex of the cone). As a result, the asymptotics of the Sommerfeld integral are uniformly represented by a Fresnel type integral for the wedge problem or by a parabolic cylinder type function for the cone problem.

On finite marked length spectral rigidity of hyperbolic cone surfaces and the Thurston metric

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichmuller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichmuller space with the Teichmuller space of complete hyperbolic surfaces with punctures, by showing that the two spaces (endowed with the Thurston metric) are almost isometric.

Development of Instabilities Generated by Freestream Laser Perturbations in a Hypersonic Boundary Layer

A controlled, laser-generated perturbation was created in the freestream of the Boeing/U.S. Air Force Office of Scientific Research Mach 6 Quiet Tunnel. The freestream perturbation convected downstream in the wind tunnel to interact with a flared cone model. The model is a body of revolution bounded by a circular arc with a 3.0 m radius. Two different nose tips were used in this experiment: a 1.0 mm nose tip and a 0.16 mm nose tip. Fourteen fast pressure transducers were used to measure a wave packet generated in the cone boundary layer by the freestream perturbation. This wave packet grew large and became nonlinear before experiencing natural transition in quiet flow. Breakdown of this wave packet occurred when the amplitude of the pressure fluctuations at the most unstable second-mode frequency was approximately 10% of the surface pressure for a nominally sharp nose tip. The amplitude of the second-mode instability at the neutral point on the flared cone is estimated to be on the order of times the surface pressure for the 1.0 mm nose tip and times the surface pressure for the 0.16 mm nose tip.

The hydrogen atom confined by one and two hard cones

The bound states of the H atom in a semi-infinite space limited by one or two conical boundaries are studied. The exact solution when the nucleus is located at the apex of the conical boundaries is obtained. A rapid increase of the energy when the cone angle opens and tends to π/2 is found. A second situation with the atom separated from the summit of the cone is considered. The changes on the energy and the electronic structure are analyzed. The quantum force is evaluated by calculating the energy derivative versus the distance to the cone vertex. One of the forces exerted on the tip of an Atomic Force Microscope can be modelized by a hard cone probing the electron cloud in the contact mode. Our numerical results show that the quantum force present an important dependence with the cone angle and it vanishes rapidly as the distance increases.

Field effects on inversion walls in nematic films: A computer simulation study

Monte Carlo simulations of lattice spin models are employed to investigate the effects of an external field on the creation and evolution of inversion walls in nematic films with conical boundary conditions. The simulations, based on the well known Lebwohl-Lasher potential, allow to produce simulated polarized microscopy images which are followed during their evolution when the field is applied.

Convergence of shock waves between conical and parabolic boundaries

Convergence of shock waves, generated by underwater electrical explosions of cylindrical wire arrays, between either parabolic or conical bounding walls is investigated. A high-current pulse with a peak of ∼550 kA and rise time of ∼300 ns was applied for the wire array explosion. Strong self-emission from an optical fiber placed at the origin of the implosion was used for estimating the time of flight of the shock wave. 2D hydrodynamic simulations coupled with the equations of state of water and copper showed that the pressure obtained in the vicinity of the implosion is ∼7 times higher in the case of parabolic walls. However, comparison with a spherical wire array explosion showed that the pressure in the implosion vicinity in that case is higher than the pressure in the current experiment with parabolic bounding walls because of strong shock wave reflections from the walls. It is shown that this drawback of the bounding walls can be significantly minimized by optimization of the wire array geometry.

Study of effect of a smooth hump on hypersonic boundary layer instability

Effect of a two-dimensional smooth hump on linear instability of hypersonic boundary layer is studied by using parabolized stability equations. Linear evolution of mode S over a hump is analyzed for Mach 4.5 and 5.92 flat plate and Mach 7.1 sharp cone boundary layers. Mean flow for stability analysis is obtained by solving the parabolized Navier–Stokes equations. Hump with height smaller than local boundary layer thickness is considered. The case of flat plate and sharp cone without the hump are also studied to provide comparable data. For flat plate boundary layers, destabilization and stabilization effect is confirmed for hump located at upstream and downstream of synchronization point, respectively. Results of parametric studies to examine the effect of hump height, location, etc., are also given. For sharp cone boundary layer, stabilization influence of hump is also identified for a specific range of frequency. Stabilization influence of hump on convective instability of mode S is found to be a possible cause of previous experimental observations of delaying transition in hypersonic boundary layers.

Particle tracking velocimetry investigations on density dependent velocity vector profiles of a swirling fluidized bed

ABSTRACTSwirling fluidized bed (SFB) is a newer version of the well-known bubbling bed and very little know. An insight study is therefore required for complete understanding of the hydrodynamics of a SFB operation. The current study was conducted on stable regime of a SFB operated at different blade fin angles, blade inclination angles, particle densities and superficial air velocities. Roughly one quarter of the fluidized bed was photographed and its velocity vector field plots were generated using a MATLAB supported particle tracking velocimetry (PTV) technique. At lower superficial velocities, Gaussian distribution of the velocity vectors was predicted along the radius of the bed. Particles in the vicinity of the bed walls moved relatively slower than those marching in the middle of the bed. However, at higher superficial velocities, the particles closer to the cone boundary were moving with velocities comparable to the particles in the middle of the bed. Unlikely, the particles closer to the outer bed wall kept on moving with lower velocities regardless of increasing superficial air velocity. A further look into individual velocity vector profiles revealed negligible influence of smaller blade angles (9° and 12°) on particles’ motion. The overall velocity magnitude decreased by 6% with 3° increase in blade fin angle and by 9% with 5° increase in inclination angle. Contrarily, the particle velocity underwent a monotonic decrease with particle density.

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.

THE MODERN HIGH FREQUENCY METHODS FOR SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS (Invited Paper)

The high frequency scattering problems of electromagnetic fields scattered from electrically large scatterers are important and challenging. On the calculation of the reflected and diffracted wave fields, the high frequency methods could be classified into the current based method and the ray based method. In this paper, first, we give a review on the progress of the modern high frequency methods for solving the electromagnetic scattering problems. Next, due to the highly oscillatory property of the high frequency electromagnetic scattered fields, we propose the numerical steepest descent path method. Finally, we comprehensively address the high frequency wave physics, including the high frequency critical point contributions, the Keller’s cone, the shadow and reflection boundaries and the creeping wave fields.

Guided mode extraction in monolayer colloidal crystals based on the phase variation of reflection and transmission coefficients

An accurate and fast method for guided modes extraction in monolayer colloidal crystals and their inverse replicas is presented. These three-dimensional structures are composed of a monolayer of spherical particles that can easily and simply be prepared by self-assembly method in close packed hexagonal lattices. In this work, we describe how the guided modes, even or odd modes and light cone boundary can be easily determined using phase variations of reflection and transmission coefficients. These coefficients are quickly calculated by Fourier modal method. The band structures are obtained for a monolayer of polystyrene particles and two-dimensional TiO2 inverse opal by this proposed method.

The influence of substrate bias voltages on structure, mechanical properties and anti-corrosion performance of Cr doped diamond-like carbon films deposited by steered cathodic arc evaporation

Cr-doped diamond-like carbon films (Cr-DLC) were deposited on 304 stainless steel (304SS) substrates by a steered cathodic arc plasma evaporation with various negative bias voltages (−50 to −550V). The Cr layer (~90nm), CrN layer (~400nm), transition layer and Cr-DLC layer (~3.2μm.) were deposited sequentially onto the substrate. The Raman spectra showed that the sp3/sp2 ratio increased when the magnitude of bias voltage increased from 50 to 450V and then it slightly decreased with further increase in the voltage. A dome-shaped morphology was found on the film deposited at low bias voltage due to incorporation of Cr droplets into the films and low mobility of adatoms. In addition, the crack in the conical structure and boundaries between the regular film and conical structure were also observed. The size of dome-shaped morphology and surface roughness decreased with increasing the bias voltage, which is attributed to enhanced surface mobility, increase in re-sputtering events above the droplets, and repulsion of droplets at higher bias voltages. The hardness significantly increased from 8.3 to 25.3GPa when the bias voltage was increased from 50 to 450V and then slightly decrease to 23.7GPa as further increased to 550V. The best corrosion current density of the film deposited at 450V was 2.69×10−8A/cm2, which is about 2-order of magnitude less than that of the bare 304SS plate (1.19×10−6A/cm2). The anti-corrosion properties and hardness were improved at higher substrate bias voltage due to the decrease in surface defects. The breakdown potentials of 304SS plates can be significantly improved from 0.75V for bare samples to 1.3V for those coated with Cr-DLC films. The passive current density of the Cr-DLC coated 304SS plate significantly decreased to around 1×10−7A/cm2, because the pits did not penetrate through underlying Cr-DLC film. The optimum anti-corrosion performance and hardness was achieved for the samples deposited at a bias of −450V.

The Behavior of Solutions to the Dirichlet Problem for Second Order Elliptic Equations with Variable Nonlinearity Exponent in a Neighborhood of a Conical Boundary Point

We study the Dirichlet problem for the p-Laplacian in a conical domain with the homogeneous boundary condition on the lateral surface of a cone with vertex at the origin. We assume that the variable exponent p = p(x) is separated from 1 and ∞ and denote by Ω the intersection of the cone with the unit (n − 1)-dimensional sphere. We prove that (i) if p satisfies the Lipschitz condition and ∂Ω is of class C 2+β, then the solution to the Dirichlet problem is O(|x| λ ) in a neighborhood of the origin, where λ is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the p(0)-Laplacian and (ii) if p satisfies the Holder condition, p(0) = 2, and ∂Ω is of class C 1+β, then the solution to the Dirichlet problem is O(|x| λ0) in a neighborhood of the origin, where λ 0 is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the Laplace operator. Bibliography: 18 titles.