A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. ChandlerWilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

A uniqueness result for scattering by infinite rough surfaces

Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Static and dynamic contact angle measurement on rough surfaces using sessile drop profile analysis with application to water management in low temperature fuel cells

Fuel Cells are a promising alternative energy technology. One of the biggest problems that exists in fuel cell is that of water management. A better understanding of wettability characteristics in the fuel cells is needed to alleviate the problem of water management. Contact angle data on gas diffusion layers (GDL) of the fuel cells can be used to characterize the wettability of GDL in fuel cells. A contact angle measurement program has been developed to measure the contact angle of sessile drops from drop images. Digitization of drop images induces pixel errors in the contact angle measurement process. The resulting uncertainty in contact angle measurement has been analyzed. An experimental apparatus has been developed for contact angle measurements at different temperature, with the feature to measure advancing and receding contact angles on gas diffusion layers of fuel cells.

A Physically Based Transmission Model of Rough Surfaces

Transparent and translucent objects involve both light reflection and transmission at surfaces. This paper presents a physically based transmission model of rough surface. The surface is assumed to be locally smooth, and statistical techniques is applied to calculate light transmission through a local illumination area. We have obtained an analytical expression for single scattering. The analytical model has been compared to our Monte Carlo simulations as well as to the previous simulations, and good agreements have been achieved. The presented model has potential applications for realistic rendering of transparent and translucent objects.

A literature survey of boundary-layer development on smooth and rough surfaces at zero pressure gradient.

"Project sponsored by the Office of Naval Research under Contract N8onr-500."

Fibrillar Elastomeric Micropatterns Create Tunable Adhesion Even to Rough Surfaces

Acknowledgements V.B., N.K.G., and E.A. contributed with conception and experimental design. V.B. performed the experiments. V.B., R.H., A.G., and R.M.M. carried out analysis and interpretation of data. V.B., R.H., A.G., and E.A. wrote the manuscript. V.B. and R.H. contributed equally to this work. V.B. acknowledges funding by SPP 1420 of the German Science Foundation DFG. E.A., N.K.G., and R.H. acknowledge funding from the European Research Council under the European Union/ERC Advanced Grant “Switch2Stick,” Agreement No. 340929.

Acoustic Propagation over Periodic and Quasi-Period Rough Surfaces Proceedings

Numerical techniques such as the Boundary Element Method, Finite Element Method and Finite Difference Time Domain have been used widely to investigate plane and curved wave-front scattering by rough surfaces. For certain shapes of roughness elements (cylinders, semi-cylinders and ellipsoids) there are semi-analytical alternatives. Here, we present a theory for multiple scattering by cylinders on a hard surface to investigate effects due to different roughness shape, the effects of vacancies and variation of roughness element size on the excess attenuation due to a periodically rough surfaces.

Joint statistical distribution of two-point sea surface elevations in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a joint statistical distribution of two-point sea surface elevations is derived by using the characteristic function expansion method. It is found that the joint distribution depends on five parameters. These five parameters can all be determined by the water depth, the relative position of two points and the wave-number spectrum of ocean waves. As an illustrative example, for fully developed wind-generated sea, the parameters that appeared in the joint distribution are calculated for various wind speeds, water depths and relative positions of two points by using the Donelan and Pierson spectrum and the nonlinear effects of sea waves on the joint distribution are studied. (C) 2003 Elsevier B.V. All rights reserved.

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length xi and the roughness exponent alpha, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with alpha = 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

Models and methods for the direct simulation of rough and micropatterned surfaces

Friction in hydrodynamic bearings are a major source of losses in car engines ([69]). The extreme loading conditions in those bearings lead to contact between the matching surfaces. In such conditions not only the overall geometry of the bearing is relevant, but also the small-scale topography of the surface determines the bearing performance. The possibility of shaping the surface of lubricated bearings down to the micrometer ([57]) opened the question of whether friction can be reduced by mean of micro-textures, with mixed results. This work focuses in the development of efficient numerical methods to solve thin film (lubrication) problems down to the roughness scale of measured surfaces. Due to the high velocities and the convergent-divergent geometries of hydrodynamic bearings, cavitation takes place. To treat cavitation in the lubrication problem the Elrod- Adams model is used, a mass-conserving model which has proven in careful numerical ([12]) and experimental ([119]) tests to be essential to obtain physically meaningful results. Another relevant aspect of the modeling is that the bearing inertial effects are considered, which is necessary to correctly simulate moving textures. As an application, the effects of micro-texturing the moving surface of the bearing were studied. Realistic values are assumed for the physical parameters defining the problems. Extensive fundamental studies were carried out in the hydrodynamic lubrication regime. Mesh-converged simulations considering the topography of real measured surfaces were also run, and the validity of the lubrication approximation was assessed for such rough surfaces.

Can one electrochemically measure the statistical morphology of a rough electrode

We have developed a theory for an electrochemical way of measuring the statistical properties of a nonfractally rough electrode. We obtained the expression for the current transient on a rough electrode which shows three times regions: short and long time limits and the transition region between them. The expressions for these time ranges are exploited to extract morphological information about the surface roughness. In the short and long time regimes, we extract information regarding various morphological features like the roughness factor, average roughness, curvature, correlation length, dimensionality of roughness, and polynomial approximation for the correlation function. The formulas for the surface structure factors (the measure of surface roughness) of rough surfaces in terms of measured reversible and diffusion-limited current transients are also obtained. Finally, we explore the feasibility of making such measurements.

An apparatus for evaluating adhesion between soils and solid surfaces

An apparatus in the direct shear mode has been developed to conduct soil-soil and soil-solid material interface tests in the undrained condition. Evaluation of the apparatus showed that all the requirements for simulating the undrained condition of shear are satisfied. The interface test results show that the adhesion factor a increases with the surface roughness of the solid material. In the case of the normally consolidated state, alpha is practically independent of the undrained shear strength of the clay for a given surface. For the overconsolidated state, alpha depends on the undrained shear strength and the overconsolidation ratio for smooth surfaces but for rough surfaces; alpha is independent of both undrained shear strength and overconsolidation ratio.