Concave Boundary Research Articles

R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green’s function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

Robust boundary flow in chiral active fluid.

We perform experiments on an active chiral fluid system of self-spinning rotors in a confining boundary. Along the boundary, actively rotating rotors collectively drive a unidirectional material flow. We systematically vary rotor density and boundary shape; boundary flow robustly emerges under all conditions. Flow strength initially increases then decreases with rotor density (quantified by area fraction ϕ); peak strength appears around a density ϕ=0.65. Boundary curvature plays an important role: flow near a concave boundary is stronger than that near a flat or convex boundary in the same confinements. Our experimental results in all cases can be reproduced by a continuum theory with single free fitting parameter, which describes the frictional property of the boundary. Our results support the idea that boundary flow in active chiral fluid is topologically protected; such robust flow can be used to develop materials with novel functions.

Study on the Characteristic Point Location of Depth Average Velocity in Smooth Open Channels: Applied to Channels with Flat or Concave Boundaries

Based on the flow partition theory, we derive a mathematical expression by using the log-law for the characteristic point location (CPL) of depth average velocity in channels with flat or concave boundaries. It can manifest the position of the characteristic points in the vertical direction relative to the channel side wall or bed. Taking rectangular and semi-circular channels as research objects, we put forward a method to calculate the discharge of channels with CPL. Additionally, we carried out some experiments on rectangular and semi-circular channel sections. CPL’s analytic expression is validated against experimental results through comparison of velocity and discharge. The proposed formulation of characteristic point location could be extensively employed in flow measurements of flat and concave boundary channels, which has practical application value in simplifying the flow measurement steps of open channels.

Positive solutions for ( p , 2 ) -equations with superlinear reaction and a concave boundary term

Positive solutions for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mi>,</mml:mi> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math>-equations with superlinear reaction and a concave boundary term

Nonlinear resonant problems with an indefinite potential and concave boundary condition

Nonlinear resonant problems with an indefinite potential and concave boundary condition

Darcy–Brinkman–Forchheimer flow over irregular domain using finite elements method

The finite element method of solution with curved triangles to solve the three-dimensional, fully-developed Darcy–Brinkman–Forchheimer flow equation in channel with curved side is solved using quasi-linearization and Gauss-Seidel iteration method. Exhaustive numerical computation and numerical experimentation reveals the parameters’ influence on the velocity distributions.. A salient feature of the method adopted in the present paper is that it ensures that the errors are almost equally distributed among all the nodes. It is found that the irregular cross-section channel with upward concave boundary decelerates the flow. Numerical experimentation involved different order curved triangular elements and extensive computation revealed that the quintic order curved triangular element yields the desired solution to an accuracy of 10−5. The finite element method is found to be very effective in capturing boundary and inertia effects in the three-dimensional, fully-developed flow through porous media. Further, it succeeds in giving the required solution for large values of Forchheimer number when shooting method fails to do so. The method can be easily employed in any other irregular cross-section channel.

Dynamics of an ultrasound contrast agent microbubble near spherical boundary in ultrasound field**Project supported by the National Natural Science Foundation of China (Grant Nos. 11774088 and 11474090), the Hunan-Provincial Natural Science Foundation of China (Grant No. 13JJ3076), and the Science Research Program of Education Department of Hunan Province of China (Grant No. 14A127).

The goal of this article is to establish the conditions of excitation where one has to deal with ultrasound contrast agent (UCA) microbubbles pulsating near biological tissues with spherical boundary in ultrasound field for targeted drug delivery and cavitation-enhanced thrombolysis, etc., and contributes to understanding of mechanisms at play in such an interaction. A modified model is presented for describing microbubble dynamics near a spherical boundary (including convex boundary and concave boundary) with an arbitrary-sized aperture angle. The novelty of the model is such that an oscillating microbubble is influenced by an additional pressure produced by the sound reflection from the boundary wall. It is found that the amplitude of microbubble oscillation is positively correlated to the curve radius of the wall and negatively correlated to the aperture angle of the wall and the sound reflection coefficient. Moreover, the natural frequency of the microbubble oscillation for such a compliable wall increases with the wall compliance, but decreases with the reduction of the wall size, indicating distinct increase of the natural frequency compared to a common rigid wall. The proposed model may allow obtaining accurate information on the radiation force and signals that may be used to advantage in related as drug delivery and contrast agent imaging.

Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills

In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ϕ from the original mesh M ∈ R 3 to the planar domain ϕ ( M ) ∈ R 2 . The mapping may preserve angles, areas, or distances. Distance-preserving parameterizations (i.e., isometries) are obviously attractive. However, geodesic-based isometries present limitations when the mesh has concave or disconnected boundary (i.e., holes). Recent advances in computing geodesic maps using the heat equation in 2-manifolds motivate us to revisit mesh parameterization with geodesic maps. We devise a Poisson surface underlying, extending, and filling the holes of the mesh M. We compute a near-isometric mapping for quasi-developable meshes by using geodesic maps based on heat propagation. Our method: (1) Precomputes a set of temperature maps (heat kernels) on the mesh; (2) estimates the geodesic distances along the piecewise linear surface by using the temperature maps; and (3) uses multidimensional scaling (MDS) to acquire the 2D coordinates that minimize the difference between geodesic distances on M and Euclidean distances on R 2 . This novel heat-geodesic parameterization is successfully tested with several concave and/or punctured surfaces, obtaining bijective low-distortion parameterizations. Failures are registered in nonsegmented, highly nondevelopable meshes (such as seam meshes). These cases are the goal of future endeavors.

Modeling and Analysis of hHoneycomb—A Polyhex Inspired Reconfigurable Tiling Robot

hHoneycomb, a self-reconfigurable cleaning robot, is designed based on tiling theory, to overcome the significant challenges experienced by the fixed morphology cleaning robot. It consists of four regular hexagonal units and the units are connected by a planar revolute joint which helps in reconfiguration. This platform attains six distinct configurations (bar, bee, arch, wave, worm, and pistol) and these configurations have circular arcs and irregular concave and convex boundary that would help in accessing various obstacles in the cleaning space. This work addresses the mechanical design, system-level modeling, reconfiguration of the platform via hinged joint mechanism, mobility of the platform, polyhex based tiling set, and power consumption during reconfiguration. The strength of the mechanical structure is studied based on the structural analysis of the system using finite element method. Based on the natural frequency and deformation pattern, the proposed design is validated and proven to overcome structural failure and system resonance. The kinematics formulation of the platform during locomotion and dynamics of each block during reconfiguration are derived. The robotic system is modeled in Simscape multibody toolbox of Matlab and the mobility of the platform is studied using the numerical simulation. Based on the real-time current consumption of each joint during reconfiguration, the energy efficient configuration and tiling set are addressed.

Geometric control of bacterial surface accumulation.

Controlling and suppressing bacterial accumulation at solid surfaces is essential for preventing biofilm formation and biofouling. Whereas various chemical surface treatments are known to reduce cell accumulation and attachment, the role of complex surface geometries remains less well understood. Here, we report experiments and simulations that explore the effects of locally varying boundary curvature on the scattering and accumulation dynamics of swimming Escherichia coli bacteria in quasi-two-dimensional microfluidic channels. Our experimental and numerical results show that a concave periodic boundary geometry can decrease the average cell concentration at the boundary by more than 50% relative to a flat surface.

Laplace\u2019s equation with concave and convex boundary nonlinearities on an exterior region

This paper studies Laplace’s equation -Delta u=0 in an exterior region Uvarsubsetneq {mathbb{R}}^{N}, when Ngeq 3, subject to the nonlinear boundary condition frac{partial u}{partial nu }=lambda vert u vert ^{q-2}u+mu vert u vert ^{p-2}u on ∂U with 1< q<2<p<2_{*}. In the function space mathscr{H} (U ), one observes that, when lambda >0 and mu in mathbb{R} arbitrary, then there exists a sequence {u_{k} } of solutions with negative energy converging to 0 as kto infty ; on the other hand, when lambda in mathbb{R} and mu >0 arbitrary, then there exists a sequence {tilde{u}_{k} } of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation -Delta _{p} u=0, the exterior p-harmonic Steklov eigenvalue problems are described.

Solution of Darcy–Brinkman–Forchheimer Equation for Irregular Flow Channel by Finite Elements Approach

The finite element method of solution with curved triangles to solve the three-dimensional, fully-developed Darcy–Brinkman–Forchheimer flow equation in channel with curved side is solved using quasi-linearization and Gauss-Seidel iteration method. Exhaustive numerical computation and numerical experimentation reveals the parameters’ influence on the velocity distributions. A salient feature of the method adopted in the present paper is that it ensures that the errors are almost equally distributed among all the nodes. It is found that the irregular cross-section channel with upward concave boundary decelerates the flow. Numerical experimentation involved different order curved triangular elements and extensive computation revealed that the quintic order curved triangular element yields the desired solution to an accuracy of 10−5. The finite element method is found to be very effective in capturing boundary and inertia effects in the three-dimensional, fully-developed flow through porous media. Further, it prevails with regards to giving the required answer for vast estimations of Forchheimer number when shooting technique fails to do as such. The technique can be effortlessly utilized in some other sporadic cross-area channel.

Turbulence structures and statistics of a supersonic turbulent boundary layer subjected to concave surface curvature

Supersonic turbulent flows at Mach 2.7 over concave surfaces for two different radii of curvature were investigated and compared with a flat plate turbulent boundary layer using direct numerical simulations. The streamwise velocity reduces in the outer part of the boundary layer due to compression, while it increases near the wall due to curvature, with a higher shape factor for the concave cases. The near-wall spanwise streak spacing reduces compared to the flat plate, with large-scale streaks and turbulence amplification also observed. Streamwise velocity iso-surfaces and streamlines show the generation of Görtler-like vortices, consistent with significant centrifugal effects. Abundant small vortices are shown to be associated with large baroclinic production of vorticity that is caused by the density and pressure gradients that are associated with concave compression. Profiles of turbulent kinetic energy and turbulent Mach number exhibit a characteristic two-layer structure in the concave boundary layer cases. In the outer layer, turbulence is greatly amplified, whereas a local balance exists in the inner layer. Turbulent energy budget analysis shows that both production and dissipation increase near the concave wall, whereas in the outer part of the boundary layer, the production is increased and ultimately balanced by convection and turbulent transport.

The amplification of large-scale motion in a supersonic concave turbulent boundary layer and its impact on the mean and statistical properties

Direct numerical simulation is conducted to uncover the response of a supersonic turbulent boundary layer to streamwise concave curvature and the related physical mechanisms at a Mach number of 2.95. Streamwise variations of mean flow properties, turbulence statistics and turbulent structures are analysed. A method to define the boundary layer thickness based on the principal strain rate is proposed, which is applicable for boundary layers subjected to wall-normal pressure and velocity gradients. While the wall friction grows with the wall turning, the friction velocity decreases. A logarithmic region with constant slope exists in the concave boundary layer. However, with smaller slope, it is located lower than that of the flat boundary layer. Streamwise varying trends of the velocity and the principal strain rate within different wall-normal regions are different. The turbulence level is promoted by the concave curvature. Due to the increased turbulence generation in the outer layer, secondary bumps are noted in the profiles of streamwise and spanwise turbulence intensity. Peak positions in profiles of wall-normal turbulence intensity and Reynolds shear stress are pushed outward because of the same reason. Attributed to the Görtler instability, the streamwise extended vortices within the hairpin packets are intensified and more vortices are generated. Through accumulations of these vortices with a similar sense of rotation, large-scale streamwise roll cells are formed. Originated from the very large-scale motions and by promoting the ejection, sweep and spanwise events, the formation of large-scale streamwise roll cells is the physical cause of the alterations of the mean properties and turbulence statistics. The roll cells further give rise to the vortex generation. The large number of hairpin vortices formed in the near-wall region lead to the improved wall-normal correlation of turbulence in the concave boundary layer.

Symplectic divisorial capping in dimension $4$

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $\omega$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.

A novel automated classification technique for diagnosing liver disorders using wavelet and texture features on liver ultrasound images

A novel automated classification technique for diagnosing liver disorders is contributed in this paper by utilizing the merits of wavelet and texture features of ultrasound images. In this automated classification technique, initially the diseased part of the ultrasound image is isolated based on the application of improved active contour-based segmentation scheme. Improved active contour-based segmentation is mainly for preventing the issue of worse convergence, which is prevalent in the concave boundary regions of ultrasonic images. After segmentation, shift variant bi-orthogonal wavelet transform is applied for decomposing the region of focus into diagonal, vertical and horizontal component images. This shift variant bi-orthogonal wavelet transform is used in this approach for reducing the degree of prediction errors that are most possible in the classical discrete wavelet transform schemas. Finally, an improved random forest classifier (IRFC) is used for classifying the features that are extracted from the wavelet filtered images using gray level run length matrix (GLRLM). The performance of this scheme is evaluated based on sensitivity, specificity and accuracy metrics and shows the comparison of each classifier performance. The results of the proposed scheme infer an overall classification accuracy rate of 97.8% and confirm better results using GLRLM.

Interaction of planar shock wave with three-dimensional heavy cylindrical bubble

Interactions of a planar shock wave with two-dimensional (2D), three-dimensional (3D) concave, and 3D convex SF6 cylindrical bubbles surrounded by air are studied both experimentally and numerically. The effects of initial interface curvature on the bubble deformation and wave propagation are highlighted. The cylindrical bubbles are generated by a wire-restriction method based on the soap-film technique, and their shapes are well controlled by adjusting the pressure difference across the interface. The high-speed schlieren results demonstrate that the evolving interfaces develop more symmetrically than previous studies as they are free of holder and fewer disturbance waves are generated. Typical evolution processes of the 2D bubble such as the jet formation and vortex pair formation are clearly captured. Compared with the 2D case, the oppositely (identically) signed principal curvatures of the concave (convex) boundary produce more complicated high pressure fields and 3D additional baroclinic vorticity. For 3D cases, the numerical results show that the wave patterns in the symmetry or boundary slice are distinct from the 2D case owing to the 3D movement of the generated waves, and the jet structure presents an evident three dimensionality. In particular, for the concave bubble, a certain slice between the boundary and symmetry slices presents the fastest-developing jet, while for the convex case the fastest jet emerges at the boundary slice. The upstream interface along the symmetry slice of the concave (convex) bubble moves faster (slower) than that of the 2D case, which is reasonably predicted by a 3D theoretical model.

Propagation characteristics of rotating detonation wave in plane–radial structure with different pressure conditions

This paper simulates the propagation characteristics of rotating detonation wave in the plane–radial structure for mixtures of 2H2 + O2 + 3.76N2. Two-dimensional numerical simulation was modeled, and two kinds of typical flow field and corresponding operating range were obtained under various pressure conditions. Due to the influence of curvature, the detonation wave is strengthened near the outer concave boundary and weakened near the inner convex one. The pressure ratio was varied from 1.6 to 10 by varying both stagnation and back pressure for detonation parameters and flow parameters. It is found that these parameters are dependent only on stagnation pressure for higher pressure ratio. While the pressure ratio is low, the back pressure also has an effect on them. The detonation wave height initially increases and then decreases as stagnation pressure increases, and the pressure ratio has a significant effect on it for lower pressure ratio. The inlet block ratio varies slightly from 14% to 21%. The exit average Mach number has small fluctuations between 0.89 and 1.05. The exit supersonic flow ratio varies from 14% to 74%, and the peak value is gained when pressure ratio is 6. The exit pressure amplifying ratio varies from 1.45 to 1.95, and the maximum value is obtained when pressure ratio is 2.5.

Active fluids at circular boundaries: swim pressure and anomalous droplet ripening.

We investigate the swim pressure exerted by non-chiral and chiral active particles on convex or concave circular boundaries. Active particles are modeled as non-interacting and non-aligning self-propelled Brownian particles. The convex and concave circular boundaries are used to model a fixed inclusion immersed in an active bath and a cavity (or container) enclosing the active particles, respectively. We first present a detailed analysis of the role of convex versus concave boundary curvature and of the chirality of active particles in their spatial distribution, chirality-induced currents, and the swim pressure they exert on the bounding surfaces. The results will then be used to predict the mechanical equilibria of suspended fluid enclosures (generically referred to as 'droplets') in a bulk with active particles being present either inside the bulk fluid or within the suspended droplets. We show that, while droplets containing active particles behave in accordance with standard capillary paradigms when suspended in a normal bulk, those containing a normal fluid exhibit anomalous behaviors when suspended in an active bulk. In the latter case, the excess swim pressure results in non-monotonic dependence of the inside droplet pressure on the droplet radius; hence, revealing an anomalous regime of behavior beyond a threshold radius, in which the inside droplet pressure increases upon increasing the droplet size. Furthermore, for two interconnected droplets, mechanical equilibrium can occur also when the droplets have different sizes. We thus identify a regime of anomalous droplet ripening, where two unequal-sized droplets can reach a final state of equal size upon interconnection, in stark contrast with the standard Ostwald ripening phenomenon, implying shrinkage of the smaller droplet in favor of the larger one.

Gradient Divergence of Fluid-Dynamic Quantities in Rarefied Gases on Smooth Boundaries

The behavior of fluid-dynamic (or macroscopic) quantities of rarefied gases is studied, with a special interest in its non-analytic feature near boundaries. It is shown that their gradients normal to the boundary diverge even if the boundary is smooth, irrespective of the value of the (nonzero) Knudsen number. The boundary geometry determines the diverging rate. On a planar or concave boundary, the logarithmic divergence $$\ln s$$ should be observed, where s is the normal distance from the boundary. In other cases, the diverging rate is enhanced to be the inverse-power $$s^{-1/n}$$ , where $$n({\ge }2)$$ is the degree of the dominant terms of the polynomial which locally represents the boundary. Some numerical demonstrations are given as well.