Surfaces Of Arbitrary Shape Research Articles

Repulsive van der Waals interaction between a quantum particle and a conducting toroid

We calculate the non-retarded dispersion force exerted on an electrically polarizable quantum particle by a perfectly conducting toroid, which is one of the most common objects exhibiting a non-trivial topology. We employ a convenient method developed by Eberlein and Zietal that essentially relates the quantum problem of calculating dispersion forces between a quantum particle and a perfectly conducting surface of arbitrary shape to a corresponding classical problem of electrostatics. Considering the quantum particle in the symmetry axis of the toroid, we use this method to find an exact analytical result for the van der Waals interaction between the quantum particle and the conducting toroid. Surprisingly, we show that for appropriate values of the two radii of the toroid the dispersive force on the quantum particle is repulsive. This is a remarkable result since repulsion in dispersive interactions involving only electric objects (and particles) in vacuum is rarely reported in the literature. Final comments are made about particular limiting cases as, for instance, the quantum particle-nanoring system.

A New Computational Modeling Framework for the 3D Flow and Shape Dynamics of Cellular Membranes

Cellular membranes are characterized by complex mechanical behavior that is very challenging to describe accurately at continuum length scales. They undergo large shape changes, they exhibit non-trivial surface flows, they undergo phase transitions, and they interact with surrounding objects. Modeling those requires solving coupled partial differential equations on evolving surfaces. This work presents a new, three-dimensional modeling framework to describe and simulate the mechanical behavior of cellular membranes. It is based on advanced, recently developed modeling techniques in mathematics, physics and computational analysis. Its mathematical description is entirely formulated in the general theory of differential geometry, since it is capable of describing arbitrary surface flows and shape changes. Its physical description is based on a rigorous thermodynamical treatment, since it provides coupling relations for the various physical fields. Its computational description is based on so-called isogeometric surface finite elements, since they are capable of representing surface flow, shape and constitution to high accuracy. The capabilities of the new framework are illustrated by several challenging numerical examples. Those include endocytosis, adhesive contact and deformation-induced surface flows.

Optical design of transmitter lens for asymmetric distributed free space optical networks

We present a method of transmitter lens design dedicated for light distribution shaping on a curved and asymmetric target. In this context, target is understood as a surface determined by hypothetical optical detectors locations. In the proposed method, ribbon-like surfaces of arbitrary shape are considered. The designed lens has the task to transform collimated and generally non-uniform input beam into desired irradiance distribution on such irregular targets. Desired irradiance is associated with space-dependant efficiency of power flow between the source and receivers distributed on the target surface. This unconventional nonimaging task is different from most illumination or beam shaping objectives, where constant or prescribed irradiance has to be produced on a flat target screen. The discussed optical challenge comes from the applications where single transmitter cooperates with multitude of receivers located in various positions in space and oriented in various directions. The proposed approach is not limited to optical networks, but can be applied in a variety of other applications where nonconventional irradiance distribution has to be engineered. The described method of lens design is based on geometrical optics, radiometry and ray mapping philosophy. Rays are processed as a vector field, each of them carrying a certain amount of power. Having the target surface shape and orientation of receivers distribution, the rays-surface crossings map is calculated. It corresponds to the output rays vector field, which is referred to the calculated input rays spatial distribution on the designed optical surface. The application of Snell’s law in a vector form allows one to obtain surface local normal vector and calculate lens profile. In the paper, we also present the case study dealing with exemplary optical network. The designed freeform lens is implemented in commercially available optical design software and irradiance three-dimensional spatial distribution is examined, showing perfect agreement with expectations.

Third-order polynomial model for analyzing stickup state laminated structure in flexible electronics

Laminated hard-soft integrated structures play a significant role in the fabrication and development of flexible electronics devices. Flexible electronics have advantageous characteristics such as soft and light-weight, can be folded, twisted, flipped inside-out, or be pasted onto other surfaces of arbitrary shapes. In this paper, an analytical model is presented to study the mechanics of laminated hard-soft structures in flexible electronics under a stickup state. Third-order polynomials are used to describe the displacement field, and the principle of virtual work is adopted to derive the governing equations and boundary conditions. The normal strain and the shear stress along the thickness direction in the bi-material region are obtained analytically, which agree well with the results from finite element analysis. The analytical model can be used to analyze stickup state laminated structures, and can serve as a valuable reference for the failure prediction and optimal design of flexible electronics in the future.

Approach to solution of coupled heat transfer problem on the surface of hypersonic vehicle of arbitrary shape

In this paper, an approach to solve conjugate heat- and mass-transfer problems is considered to be applied to hypersonic vehicle surface of arbitrary shape. The approach under developing should satisfy the following demands. (i) The surface of the body of interest may have arbitrary geometrical shape. (ii) The shape of the body can change during calculation. (iii) The flight characteristics may vary in a wide range, specifically flight altitude, free-stream Mach number, angle-of-attack, etc. (iv) The approach should be realized with using the high-performance-computing (HPC) technologies. The approach is based on coupled solution of 3D unsteady hypersonic flow equations and 3D unsteady heat conductance problem for the thick wall. Iterative process is applied to account for ablation of wall material and, consequently, mass injection from the surface and changes in the surface shape. While iterations, unstructured computational grids both in the flow region and within the wall interior are adapted to the current geometry and flow conditions. The flow computations are done on HPC platform and are most time-consuming part of the whole problem, while heat conductance problem can be solved on many kinds of computers.

Shim Coil Set for an Open Biplanar MRI System Using an Inverse Boundary Element Method

This paper proposes a numerical method with an inverse boundary element method (IBEM) to design a set of shim coils for an open bipanar superconducting magnetic resonance imaging (MRI) system. For the traditional cylindrical MRI system, claustrophobics will suffer the palpitations and anxiety attacks when they are enclosed in a small area during the scanning process. However, such bipanar structure provides an open room temperature bore, thus improving patients’ comfort. Based on the IBEM, the shim coil skeleton needs to be divided into some small flat triangular elements. By formulating the magnetization distribution by discrete current density, an objective function is constructed to evaluate the minimum error between axial magnetic field and desired magnetic field. In particular, the selfinduction is added into the objective function which is used to adjust the deviation from the ideal target field. Finally, the well-known stream function method is commonly used to map the current density into wires. The algorithm details have been exemplified through biplanar shim coil design examples in this paper. This method can be used for coils on arbitrary surface shapes.

Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence

This paper presents two key contributions; the first concerns the development of analytical expressions for the axial and transverse acoustic radiation forces exerted on a 2D rigid elliptical cylinder placed in the field of plane progressive, quasi-standing, or standing waves with arbitrary incidence. The second emphasis is on the acoustic radiation torque per length. The rigid elliptical cylinder case is important to be considered as a first-order approximation of the behavior of a cylindrical fluid column trapped in air because of the significant acoustic impedance mismatch at the particle boundary. Based on the rigorous partial-wave series expansion method in cylindrical coordinates, non-dimensional acoustic radiation force and torque functions are derived and defined in terms of the scattering coefficients of the elliptic cylinder. A coupled system of linear equations is obtained after applying the Neumann boundary condition for an immovable surface in a non-viscous fluid and solved numerically by matrix inversion after performing a single numerical integration procedure. Computational results for the non-dimensional force components and torque, showing the transition from the progressive to the (equi-amplitude) standing wave behavior, are performed with particular emphasis on the aspect ratio a/b, where a and b are the semi-axes of the ellipse, the dimensionless size parameter, as well as the angle of incidence ranging from end-on to broadside incidence. The results show that the elliptical geometry has a direct influence on the radiation force and torque, so that the standard theory for circular cylinders (at normal incidence) leads to significant miscalculations when the cylinder cross section becomes non-circular. Moreover, the elliptical cylinder experiences, in addition to the acoustic radiation force, a radiation torque that vanishes for the circular cylinder case. The application of the formalism presented here may be extended to other 2D surfaces of arbitrary shape, such as Chebyshev cylindrical particles with a small deformation, stadiums (with oval shape), or other non-circular geometries.

Confined systems within arbitrary enclosed surfaces

A new model of electronic confinement in atoms and molecules is presented. This is based on the electronic flux J which is assumed to vanish on some notional bounding surface of arbitrary shape. J is necessarily calculated using an approximate wave-function, whose parameters are chosen to satisfy the required surface conditions. This model embraces the results of all previous calculations for which the wave-functions or their derivatives vanish on conveniently shaped surfaces, but now extends the theory to more general surfaces. Examples include one-centre hydrogen-like atoms, the valence state of Li and the two centre molecular systems and .

A high-fidelity approach towards simulation of pool boiling

A novel numerical approach is developed to simulate the multiscale problem of pool-boiling phase change. The particular focus is to develop a simulation technique that is capable of predicting the heat transfer and hydrodynamic characteristics of nucleate boiling and the transition to critical heat flux on surfaces of arbitrary shape and roughness distribution addressing a critical need to design enhanced boiling heat transfer surfaces. The macro-scale of the phase change and bubble dynamics is addressed through employing off-the-shelf Computational Fluid Dynamics (CFD) methods for interface tracking and interphase mass and energy transfer. The micro-scale of the microlayer, which forms at early stage of bubble nucleation near the wall, is resolved through asymptotic approximation of the thin-film theory which provides a closed-form solution for the distribution of the micro-layer and its influence on the evaporation process. In addition, the sub-grid surface roughness is represented stochastically through probabilistic density functions and its role in bubble nucleation and growth is then represented based on the thermodynamics of nucleation process. This combination of deterministic CFD, local approximation, and stochastic representation allows the simulation of pool boiling on any surface with known roughness and enhancement characteristics. The numerical model is validated for dynamics and hydrothermal characteristics of a single nucleated bubble on a flat surface against available literature data. In addition, the prediction of pool-boiling heat transfer coefficient is verified against experimental measurements as well as reputable correlations for various roughness distributions and different surface orientations. Finally, the model is employed to demonstrate pool-boiling phenomenon on enhanced structures with reentrance cavities and to explore the effect of enhancement feature design on thermal and hydrodynamic characteristics of these surfaces.

The works of William Rowan Hamilton in geometrical optics and the Malus–Dupin theorem

The works of William Rowan Hamilton in Geometrical Optics are presented, with emphasis on the Malus-Dupin theorem. According to that theorem, a family of light rays depending on two parameters can be focused to a single point by an optical instrument made of reflecting or refracting surfaces if and only if, before entering the optical instrument, the family of rays is rectangular (\emph{i.e.}, admits orthogonal surfaces). Moreover, that theorem states that a rectangular system of rays remains rectangular after an arbitrary number of reflections through, or refractions across, smooth surfaces of arbitrary shape. The original proof of that theorem due to Hamilton is presented, along with another proof founded in symplectic geometry. It was the proof of that theorem which led Hamilton to introduce his \emph{characteristic function} in Optics, then in Dynamics under the name \emph{action integral}

An Efficient Modeling Technique for Analysis of AC Field Measurement Probe Output Signals to Improve Crack Detection and Sizing in Cylindrical Metallic Structures

The paper proposes an efficient modeling technique for predicting the output signal of an AC field measurement (ACFM) probe when encountering a fatigue crack in a cylindrical metallic structure. The proposed technique is used to determine the electromagnetic fields around an arbitrary-shape surface crack in a cylindrical metal, induced by a high frequency alternating-current-carrying coil of arbitrary shape. The modeling technique involves two steps. First, the finite-difference method is used to solve the governing Laplace equation inside the crack region. Then, the normal component of magnetic field at the crack opening on the metal is obtained to derive a closed-form solution in the Fourier domain for the regions outside the crack. Experimental and theoretical results associated with prototype ACFM probes confirm the validity of the proposed technique while demonstrating its remarkable computation efficiency. The latter is an important feature of the proposed modeling technique as a fast “forward” solver that results in a remarkable improvement in solving the “inverse” problem required for the sizing of fatigue cracks in cylindrical metallic structures, particularly in real-time applications.

Simplifying numerical ray tracing for two-dimensional non circularly symmetric models of the human eye.

Ray tracing is a powerful technique to understand the light behavior through an intricate optical system such as that of a human eye. The prediction of visual acuity can be achieved through characteristics of an optical system such as the geometrical point spread function. In general, its precision depends on the number of discrete rays and the accurate surface representation of each eye's components. Recently, a method that simplifies calculation of the geometrical point spread function has been proposed for circularly symmetric systems [Appl. Opt.53, 4784 (2014)]. An extension of this method to 2D noncircularly symmetric systems is proposed. In this method, a two-dimensional ray tracing procedure for an arbitrary number of surfaces and arbitrary surface shapes has been developed where surfaces, rays, and refractive indices are all represented in functional forms being approximated by Chebyshev polynomials. The Liou and Brennan anatomically accurate eye model has been adapted and used for evaluating the method. Further, real measurements of the anterior corneal surface of normal, astigmatic, and keratoconic eyes were substituted for the first surface in the model. The results have shown that performing ray tracing, utilizing the two-dimensional Chebyshev function approximation, is possible for noncircularly symmetric models, and that such calculation can be performed with a newly created Chebfun toolbox.

Fast circuit model for interaction of open‐ended rectangular waveguide probes with arbitrary‐shape surface cracks in metals

The study proposes a fast modelling technique to predict output signal of an open-ended rectangular waveguide when scanning an arbitrary-shape surface crack in a metal. In this technique, the crack is first segmented into an appropriate number of short rectangular waveguides each of which is modelled by a two-port network with an equivalent RLC circuit. The generalised scattering matrix technique is then used to predict the probe output signal. The efficiency of the proposed model is demonstrated by comparing the simulation results of several case studies with measurements and those obtained using the mode-matching method and a commercial finite integration technique code.

Influence of a surface in the nonretarded interaction between two atoms

In this work we obtain analytical expressions for the non-additivity effects in the dispersive interaction between two atoms and perfectly conducting surface of arbitrary shape in the non-retarded regime. We show that this three bodies quantum-mechanical problem can be solved by mapping it into a two-bodies electrostatic one. We apply the general formulas developed in this paper in several examples. Firstly we re-derive the London interaction as a particular case of our formalism. Then we treat two atoms in the presence of a plane, re-obtained the result displayed in the literature. After we add some new examples. A particularly interesting one is two atoms inside a plate capacitor, a situation where non-additivity is very manifest since the plates lead to the exponentially suppression of the interaction of the atoms, provided the atoms are separated by a distance of the order of the distance between the plates or greater. Our results holds even in the presence of other atoms inside the plate capacitor. As a last example we deal with two atoms in the presence of a sphere, both grounded and isolated. We show that for realistic experimental parameters the non-additivity may be relevant for the force in each atom.

Method to improve the accuracy of the geometrical parameters measurement in stereoscopic AOTF-based spectral imagers

The problem of non-contact geometrical measurements by the stereoscopic spectral AOTF-based imagers is considered. Major factors contributing to the measurement error are analyzed for the first time. Expressions are derived for area measurement inaccuracy on arbitrary shape surfaces, caused by measurement errors of individual points' 3D coordinates with and without a priori information about surface shape. Verification of the obtained expressions with the real experimental data showed that area measurement error for a complex figure, given by the set of points, is caused mainly by ignoring the fact that these points belong to the surface. It is shown that the use of a priori information about investigated surface shape, which is often available from the design documentation, in many cases would radically improve the accuracy of surface defects area measurement. The presented results can be effectively used for the development of new 3D spectral imagers and modernization of existing ones.

Fabrication of micro-scale wrinkles on a curved surface using weak-polymerization and surface shrinkage

We proposed a new process to fabricate micro-scale polymer wrinkles on a curved surface. A UV-curable resin layer on a substrate is weakly polymerized by a small dose of UV-light. By the exposure, there is a curing gradient through the thickness in the resin layer; fully-polymerized zone on the surface and sol-gel inside. Then, it is cured completely under temperature of 28°C. According to the different states between surface and inside of the layer, a compressive force is generated on the surface zone during a fully-curing process. Through this work, regular surface wrinkles are generated on the surface with a width of around 20 μm. To reduce the total processing time, an approach of thermal curing after weak-polymerization is proposed. We believed that the proposed method can be applied to make wrinkles on arbitrary-shaped surface for applications to diverse research fields such as heat transfer and catalyst enhancement.

Analytic Evaluation of the Matrix Entries for Linear Algebraic Systems in the Problem of Electromagnetic Scattering by Surfaces of Arbitrary Shape

The paper considers computation of the matrix entries for linear algebraic systems of equations occurring in the Rao–Wilton–Glisson solution by the method of moments for problems of electromagnetic scattering by surfaces of arbitrary shape. Using the Taylor series expansion and its truncation to four terms, all double integrals needed in computing the matrix entries are taken, and closed-form analytic expressions are obtained. It is demonstrated that these expressions provide for a faster and more accurate solution than numerical integration.

A compact shape descriptor for triangular surface meshes

Three-dimensional shape-based descriptors have been widely used in object recognition and database retrieval. In the current work, we present a novel method called compact Shape-DNA (cShape-DNA) to describe the shape of a triangular surface mesh. While the original Shape-DNA technique provides an effective and isometric-invariant descriptor for surface shapes, the number of eigenvalues used is typically large. To further reduce the space and time consumptions, especially for large-scale database applications, it is of great interest to find a more compact way to describe an arbitrary surface shape. In the present approach, the standard Shape-DNA is first computed from the given mesh and then processed by surface area-based normalization and line subtraction. The proposed cShape-DNA descriptor is composed of some low frequencies of the discrete Fourier transform of the processed Shape-DNA. Several experiments are shown to illustrate the effectiveness and efficiency of the cShape-DNA method on 3D shape analysis, particularly on shape comparison and classification.

基于有限元法的面形反演模型

In order to inversed compute the optical element surface profile using the finite element method and the surface shape interference test results, the composition of interference test results was decomposed, and the testing principles of the rotational averaging method were analysised. The feasibility of using simplified geometrical model to caculate the deformation of an optical element with arbitrary given surface shape was discussed. Surface profile inversion model was proposed based on finite element method. A finite element analysis model was then presented based on surface-surface contact modeling. A plane mirror supported by three rigid spheres as the simulation and experimental model, the rotationally asymmetrical surface deviation was quantitative analysis using the numerical method presented in this paper and the N-step rotational averaging method. The obtained results show that the root mean square of rotationally asymmetric surface is 2.933 nm obtained through numerical method and the root mean square of rotationally asymmetric surface is 2.75 nm obtained through N-step rotational averaging method. Subtracted result of the two rotationally saymmetric surface show that the root mean square of the subtraction are 6.31% of the numerical method result and 6.73% of the N-step rotational averaging method result. Finally, comparing the optical element surface profile computed by proposed inversion model with the surface profile tested by N-step rotational averaging method. Results show that the root mean square of optical element surface profile are 3.535 nm and 3.351 nm respectively. Subtracted result of the two surface profile show that the root mean square of the subtraction are 11.67% of the inversiong result and 11.06% of the result tested by N-step rotational averaging method. The proposed inversion model is accurate and reliable.

A generalized limit equilibrium method for the solution of active earth pressure on a retaining wall

In this paper, a generalized limit equilibrium method of solving the active earth pressure problem behind a retaining wall is proposed. Differing from other limit equilibrium methods, an arbitrary slip surface shape without any assumptions of pre-defined shapes is needed in the current framework, which is verified to find the most probable failure slip surface. Based on the current computational framework, numerical comparisons with experiment, discrete element method and other methods are carried out. In addition, the influences of the inclination of the wall, the soil cohesion, the angle of the internal friction of the soil, the slope inclination of the backfill soil on the critical pressure coefficient of the soil, the point of application of the resultant earth pressure and the shape of the slip surface are also carefully investigated. The results demonstrate that limit equilibrium solution from predefined slip plane assumption, including Coulomb solution, is a special case of current computational framework. It is well illustrated that the current method is feasible to evaluate the characteristics of earth pressure problem.