Concave Boundary Research Articles

Numerical study on the effect of the concave rigid boundaries on the cavitation intensity

In this paper, we study the motion of a cavitation bubble near a concave boundary from a theoretical perspective. To illustrate the e ect of the surface concavity of the boundary, boundary integral and nite di erence methods are utilized to investigate the more detailed process of jet formation. Bubble shapes and its lifetime, movement of the bubble centroid, pressure contours, and velocity vectors are used to demonstrate thenumerical results. The velocity of a liquid jet impacting on the far side of the bubble surface tends to increase with decreasing boundary concavity. This result suggests that higher pressures can occur when cavitation bubbles collapse near a concave boundary. With the increase of boundary concavity, the time evolutions of the bubble growth and collapse tend to increase, and the liquid jet is formed later.

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

ABSTRACTWe prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.

Scintillation kinks, bumps and wiggles in the radio spectrum of the quasar PMN J1106−3647

We report radio observations of the quasar PMN J1106-3647. Our data, taken with the Australia Telescope Compact Array, show large variations in the amplitude and shape of its spectrum, on a short time-scale. A great variety of spectral features is evident, including: sharp kinks; broad spectral peaks; and wiggles. No two spectra are alike. We interpret the variations as interstellar scintillation of a radio source that is compact, but not point-like. Under this interpretation, complex spectral structure can arise purely refractively, under high magnification conditions, or from interference between waves that have been scattered by small-scale density fluctuations (diffractive scintillation). Both effects may be playing a role in J1106-3647, and we tentatively identify kinks with the former, and wiggles with the latter. Diffractive scintillation of AGN is uncommon, as the fringe visibility is low for all but the most compact radio sources. Refractive interpretation of the kink implies that the source has a sharp, concave boundary. Our data are consistent with a mildly boosted synchrotron source, provided the scattering material is at a distance ~50 pc from us.

Local and global analysis of nodal sets

This is a survey for the JDG 50th Anniversary conference of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are `local', i.e. only assuming eigenfunctions are defined on small balls, and in part the techniques are `global', i.e. exploiting dynamics of the geodesic flow. The local part begins with a review of doubling indices and freqeuency functions as local measures of fast or slow growth of eigenfunctions. The pattern of boxes with maximal doubling indices plays a central role in the results of Logunov-Malinnokova, giving upper and lower bounds for hypersurface measures of nodal sets in the setting of general $C^{\infty}$ metrics. The proofs of both their polynomial upper bound and sharp lower bound are sketched. The survey continues with a global proof of the sharp upper bound for real analytic metrics (originally proved by Donnelly-Fefferman with local arguments), using analytic continuation to Grauert tubes. Then it reviews results of Toth-Zelditch giving sharp upper bounds on Hausdorff measures of intersections of nodal sets with real analytic submanifolds in the real analytic setting. Last, it goes over lower bounds of Jung-Zelditch on numbers of nodal domains in the case of $C^{\infty}$ surfaces of non-positive curvature and concave boundary or on negatively curved `real' Riemann surfaces, which are based on ergodic properties of the geodesic flow and eigenfunctions, and on estimates of restrictions of eigenfunctions to hypersurfaces. The last section details recent results on restrictions.

Strichartz and localized energy estimates for the wave equation in strictly concave domains

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet conditions, the present approach provides an avenue for treating these estimates when Neumann conditions are imposed. The method also treats Schr{\"o}dinger equations with time independent coefficients.

Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter [Formula: see text] varies. We also show the existence of a smallest positive solution [Formula: see text] and investigate the properties of the map [Formula: see text]. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

On the impact of adverse pressure gradient on the supersonic turbulent boundary layer

By employing the particle image velocimetry, the mean and turbulent characteristics of a Mach 2.95 turbulent boundary layer are experimentally investigated without the impact of curvature. The physical mechanism with which the streamwise adverse pressure gradient affects the supersonic boundary layer is revealed. The data are compared to that of the concave boundary layer with similar streamwise distributions of wall static pressure to clarify the separate impacts of the adverse pressure gradient and the concave curvature. The logarithmic law is observed to be well preserved for both of the cases. The dip below the logarithmic law is not observed in present investigation. Theoretical analysis indicates that it could be the result of compromise between the opposite impacts of the compression wave and the increased turbulent intensity. Compared to the zero pressure gradient boundary layer, the principal strain rate and the turbulent intensities are increased by the adverse pressure gradient. The shear layer formed due the hairpin packets could be sharpened by the compression wave, which leads to higher principal strain rate and the associated turbulent level. Due to the additional impact of the centrifugal instability brought by the concave wall, even higher turbulent intensities than that of the adverse pressure gradient case are introduced. The existence of velocity modes within the zero pressure gradient boundary layer suggests that the large scale motions are statistically well organized. The generation of new velocity modes due to the adverse pressure gradient indicates that the turbulent structure is changed by the adverse pressure gradient, through which more turbulence production that cannot be effectively predicted by the Reynolds-stress transport equations could be brought.

Polyfolds, cobordisms, and the strong Weinstein conjecture

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.

An experimental investigation of the supersonic turbulent boundary layer subjected to concave curvature

By employing particle image velocimetry, the response of a Mach 2.95 turbulent boundary layer to the concave curvature is experimentally investigated. The radius of the concave wall is 350 mm, and the turning angle is 20∘. Logarithmic law is well preserved in the profile of streamwise velocity at all streamwise positions despite the impact of curvature. The varying trend of principal strain rate is found to be different at different heights within the boundary layer, which cannot be explained by the suggestion given by former researchers. Based on the three-layer model proposed in this paper, distribution of the principal strain rate is carefully analyzed. The streamwise increase of wall friction is suggested to be brought by the increase of velocity gradient in the thin subsonic layer. Increases of the static temperature and the related sound speed are responsible for that. Larger correlated turbulent motions could be introduced by the concave curvature. The probability density histograms of streamwise velocity reveal that the large scale hairpin packets are statistically well organized. The concave curvature is found to have the potential of reinforcing the organization, which explains the increase of turbulent level in the supersonic concave boundary layer.

On Lagrangian embeddings into the complex projective spaces

We prove that for any closed orientable connected [Formula: see text]-manifold [Formula: see text] and any Lagrangian immersion of the connected sum [Formula: see text] either into the complex projective [Formula: see text]-space [Formula: see text] or into the product [Formula: see text] of the complex projective line and the complex projective plane, there exists a Lagrangian embedding which is homotopic to the initial Lagrangian immersion. To prove this, we show that Eliashberg–Murphy’s [Formula: see text]-principle for Lagrangian embeddings with a concave Legendrian boundary and Ekholm–Eliashberg–Murphy–Smith’s [Formula: see text]-principle for self-transverse Lagrangian immersions with the minimal or near-minimal number of double points hold for six-dimensional simply connected compact symplectic manifolds.

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.

The Basic Results on Reinitiation Processes in Diffracting Multifront Detonations. Part I.

The basic results of experimental investigations on reinitiation processes in diffracting multifront detonation wave (DW) are discussed in connection with problem of practical application of new ecological technology for worn-out tire destruction, where the cooled worn-out tire destroys with the help of gaseous detonation. The experimental results for various fuel-oxygen and fuel-air mixtures are presented at wide range of mixture compositions, initial pressure and temperature, geometrical sizes of experimental equipment, symmetry types, dilution of inert gases, etc. Classical and nontraditional schema of DW-diffraction are investigated, such as multipointed initiation, DW-excitation by circular charge, initiation space-oriented longitudinal charges, initiation by circular charge, diffraction on concave boundary, diffraction on contact surface of different mixtures, flame diffraction, etc. The main characteristic parameters are identified for each diffraction schema. The physical processes taking place directly in the DW-front plays the governing role in reinitiation. The most important among these processes are collisions of transverse waves, which stick out as microscopic initiators. The optimization problem of DW-initiation from spatial and temporary distribution of energy is discussed carefully. This data can be used at hazard estimation also.

Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

It is an open problem in general to prove that there exists a sequence of $$\Delta _g$$ -eigenfunctions $$\varphi _{j_k}$$ on a Riemannian manifold (M, g) for which the number $$N(\varphi _{j_k}) $$ of nodal domains tends to infinity with the eigenvalue. Our main result is that $$N(\varphi _{j_k}) \rightarrow +\infty $$ along a subsequence of eigenvalues of density 1 if (M, g) is a non-positively curved surface with concave boundary, i.e., a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.

High speed observation of HIFU-induced cavitation cloud near curved rigid boundaries

This paper focuses on the experimental study of the influence of surface curvature to the behaviour of HIFU-induced cavitation cloud. A Q-switched ruby pulse laser is used to induce cavitation nuclei in deionized water. A piezoelectric ultrasonic transducer (1.7 MHz) provides a focused ultrasound field to inspire the nucleus to cavitation cloud. A PZT probe type hydrophone is applied for measuring the HIFU sound field. It was observed that the motion of cavitation cloud located near the boundary is significantly influenced by the distance between cloud and boundary, as well as the curvature of the boundary. The curvature was defined by parameters λ and ξ. Convex boundary, concave boundary, and flat boundary correspond to ξ <1, ξ >1 and ξ = 1, respectively. Different behaviours of the cloud, including the migration of the cloud, the characteristics of oscillation, etc., were observed under different boundary curvatures by high-speed photography. Sonoluminescence of the acoustic cavitation bubble clouds were also studied to illustrate the characteristics of acoustic streaming.

A mesh regularization scheme to update internal control points for isogeometric shape design optimization

This paper presents a variational method to update internal control points in isogeometric shape optimization. The important properties of domain parameterization such as bijective mapping between parametric and physical domains, uniform mesh, and orthogonal mesh are enforced simultaneously. The bijective mapping is achieved by minimizing a Dirichlet energy functional. To prevent the divergent behavior of the minimizing process due to the severely distorted initial mesh, a constraint is introduced to enforce the positive Jacobian of mapping from parametric to physical domains. In spite of adding the constraint that might increase computational costs, the proposed method is more efficient due to the convexity of Dirichlet energy functional, compared with the other unconstrained methods. Also, it turns out that the proposed method is more effective to achieve the bijective mapping, especially near a concave boundary. The uniform parameterization of the domain is achieved by minimizing the Dirichlet energy functional and the orthogonality of mesh is performed by minimizing a dimensionless functional. The required design sensitivity of the employed functional and constraint is derived with respect to the position of internal control points. The developed scheme of mesh regularization is effective to maintain the high quality of domain parameterization during the shape optimization process.

Manipulation of cell mechanotaxis by designing curvature of the elasticity boundary on hydrogel matrix

Directional cell migration induced by the stiffness gradient of cell culture substrates is known as a subset of the mechanical-cue-induced taxis, so-called mechanotaxis, typically durotaxis toward hard region. To establish the general conditions of biomaterials to manipulate the mechanotaxis, the effect of the shape of the elasticity transition boundary between hard and soft regions of a substrate on mechanotaxis should be systematically determined as well as the conditions of elasticity gradient strength. Here, as a simplified factor of expressing variations in the shape of the elasticity boundary in living tissues, we focus on the curvature of the elasticity boundary. Mask-free photolithographic microelasticity patterning of photocurable gelatin gel was employed to systematically prepare elasticity boundaries with various curvatures, and the efficiency of mechanotaxis of fibroblast cells around each curved boundary was examined. Highly efficient usual durotaxis was induced on a convex boundary with 100 μm in radius and on a concave boundary with 750 μm in radius of curvature. Interestingly, biased migration toward soft regions of the gel, i.e., inverse durotaxis, was first observed for concave boundaries with 50 μm or 100 μm in radius of curvature, which was named as “negative mechanotaxis”. The curvature of the elasticity boundary was found to markedly affect the efficiency of induction and the direction of mechanotaxis. The mechanism responsible for this phenomenon and the implication for the curvature effect in in vivo systems are discussed.

An Art Gallery Approach to Submap Meshing

The Art Gallery Theorem and its lemmas have been used since the 1990s in the area of computer graphics to decompose orthogonal polygons into convex sub-regions. This paper extends and improvises Art Gallery concepts to non-orthogonal, generic polygons to perform a multiblock decomposition of faces into a set of maximal, single-loop, convex sub-faces. Concepts such as staircase, dent and notch are used to categorize face concavities and virtual vertices are inserted on smoother concave boundary bends. Multiblocking is performed without the need for Delaunay triangulation with the aid of a notch diagram. A light-weight, mesher-native topology builder that uses virtual topological elements is proposed to construct a virtual topology network from a notch diagram. Subsequently, virtual faces are processed for transfinite meshing. Results indicate the advantage of the proposed technique over existing procedures in terms of orthogonality and anisotropy.

Spanwise velocity component in nonlinear region of Görtler vortices

The spanwise velocity component of Görtler vortices in a concave surface boundary layer flow was experimentally studied by means of hot-wire anemometer measurements. For this purpose, the wavelength of the Görtler vortices was pre-set to the most amplified wavelength of 12 mm at free-stream velocity of 2.8 m/s, by a series of thin perturbation wires placed near to the concave surface leading edge. The streamwise and spanwise velocity components, respectively, u and w, in the nonlinear region of Görtler vortices were measured using an X-wire probe. The secondary motion induced by the free-stream disturbances in the vortex's head region is observed to be amplified by the appearance of secondary instability. The ∂w/∂y iso-shear contours form alternate positive and negative shear layers across the boundary layer in the upwash region. On the other hand, two pairs of anti-symmetric structures are observed in the ∂w/∂z iso-shear contours near the wall and vortex's head. The $\overline {w^{\prime 2 }}$w′2¯ contours have two intense regions at the stem and head of the vortex structures. The most unstable characteristic frequency of secondary instability is found exactly at the maxima of the contours. The Reynolds shear stress $-\overline {u^{\prime} w ^{\prime}}$−u′w′¯ contour is anti-symmetric in z-direction about the upwash with a pair of two intense regions of positive and negative shears at both the stem and head of the vortices. The $ - \overline {u^{\prime} w^{\prime} } \partial u/\partial z$−u′w′¯∂u/∂z contours reveal that the energy conversion mechanism occurs intensely on the sides of the vortex's stem.

Boundary Segmentation Based on Improved GVF Snake Model

The gradient vector flow (GVF) Snake models are active contour models: they lock onto nearby edges, localizing them accurately. GVF refers to the definition of a bidirectional external force that can capture the object boundaries from either sides and can deal with concave regions. However, we find that there is a main disadvantage of GVF Snakes---- there are some critical points in the model, the critical points must be included in the initial contour or must not be contained internal, and otherwise, it could not converge to the correct boundary. To solve this problem, we propose an improved GVF Snake model: we construct a new driving force to improve the initialization of contour. The experimental result shows that the new model we proposed not only avoid the critical points of the initial contour, but also converge to the concave boundary better.

Lagrangian caps

We establish an $h$-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball $B$ in the standard symplectic $\R^{2n}, 2n\geq 6$, there exists an embedded Lagrangian $n$-disc transversely attached to $B$ along its Legendrian boundary.