Sinusoidal Surface Research Articles

A Metasurface-Based LTC Polarization Converter with S-Shaped Split Ring Resonator Structure for Flexible Applications.

This paper presents a metasurface-based linear-to-circular polarization converter with a flexible structure for conformal and wearable applications. The converter consists of nested S- and C-shaped split ring resonators in the unit cell and can convert linearly polarized incident waves into left-handed circularly polarized ones at 12.4 GHz. Simulation results show that the proposed design has a high polarization conversion rate and efficiency at the operating frequency. Conformal tests are also conducted to evaluate the performance under curvature circumstances. A minor shift in the operating frequency is observed when the converter is applied on a sinusoidal wavy surface.

Extended-resolution of a single-camera synthetic Schlieren method for measurement of free liquid surfaces

In a synthetic Schlieren method of measuring the height of a dynamic free liquid surface, the smallest wavelength of the measured surface topography is determined by the spatial resolution of the displacement field of markers. Currently, a displacement vector is obtained for each interrogation window, including a few markers, with a cross-correlation algorithm. In this study, the measurement resolution is extended by obtaining the displacement of individual markers. This simple and rational change, however, brings technical difficulties in numerically solving the governing equation over spatially randomly distributed markers. For this, the governing equation, which associates the surface height to be measured with the virtual displacements of the markers, is solved by the finite-volume scheme for the unstructured meshes, where a marker locates at the center of each triangular mesh face. The present method is examined with the synthetic generated data, in laboratory experiments of a transparent solid carved with a sinusoidal surface, and in experiments of water ripples. The measurement uncertainty is discussed.

Simulation of the Pressure Bonding Process Using the Phase-field Crystal Method

The present study simulated the pressure bonding process by the phase-field crystal method for a two-dimensional system. Two crystals with different orientations and sinusoidal surfaces were placed relative to each other, and their uppermost and lowermost layers were gradually approached as chuck regions. The dependencies of the atomic structure and energy of the system on the relative position of the sinusoidal surface, relative azimuth angle of the two crystals and the moving velocity of the chuck (V), were investigated. The voids at the interface remained when V was slow, suggesting that there was an upper limit to the chuck moving velocity to obtain a good bonding state. This would be the optimum value of the chuck moving velocity from an engineering viewpoint. The relative position of the upper and lower crystals had no significant effects on the atomic structure and energy of the system, while the relative azimuth angle between the crystals had a crucial effect on the change in the atomic structure and energy of the system, i.e., a smaller orientational difference was necessary for good bonding.

Numerical investigation of effect of surface pattern and rotation on power-law fluid flow and heat transfer around a cylinder in laminar flow regime

We have studied the effect of surface topography, fluid behavior, and rotation on fluid flow and heat transfer phenomena over a cylinder. In this study, we have incorporated a sinusoidal surface topography to account for the impact of surface patterns and a power-law model to include fluid behavior. The governing equations are solved numerically for a range of pattern frequency (ω=5 and 11), pattern amplitude (δ=0.01 and 0.1), power-law index (0.4≤n≤1.6), Prandtl number (1≤Pr≤100), rotational speed (0.5≤α≤2), and Reynolds number (5≤Re≤40). In particular, the study aims to determine the degree to which various macroscopic parameters, such as the drag and lift coefficients, the average Nusselt number in relation to the Reynolds number, the Prandtl number, the rotating speed, and the power-law index, vary. In this range of Reynolds number, the flow around a circular cylinder is steady, with two symmetric vortices in the rear side. The sliding mesh method is used to deal with dynamic interface between solid and fluid. The streamlines are drawn to visualize the flow field around the patterned cylinder. For a non-rotating patterned cylinder, small recirculation zones are observed over the trough, which are absent in circular cylinders. The size of these recirculation regions increases on increase in Reynolds number and power-law index. On adding rotation to the cylinder, these recirculation zones move away from the cylinder and appear over the crest. On increasing the rotating speed of the cylinder, the frontal vortices disappear. The enveloping vortex gets larger with increase in rotating speed and power-law index. The size of the rear detached vortex increases with power-law index and Reynolds number and decreases with rotating speed. It is observed that the results are contrasted with the previous studies on smooth circular cylinder. The drag force acting on the patterned cylinder is seen to be reduced. Compared to a circular cylinder, a significant reduction in drag can be achieved by choosing suitable value of pattern frequency (ω) and amplitude (δ). Overall, there is a decrease in the amount of drag reduction as the Reynolds number increases. The behavior of the fluid has a considerable influence on the reduction of drag. It has been observed that shear-thickening fluid significantly contributes to the enhancement of drag reduction. For a higher value of pattern frequency and amplitude (ω=11,δ=0.1), the drag force reduces significantly for Newtonian and shear-thickening fluids at higher rotating speed (α=2). Also, the pattern frequency and amplitude substantially impact the average Nusselt number. On increase in pattern frequency and amplitude, a progressive decrease in the average Nusselt number is observed. Compared to shear-thinning and Newtonian fluid, shear-thickening fluid exhibits a greater reduction in average Nusselt number. One correlation is provided at the end to show the relationship between the average Nusselt number, the Prandtl number, the Reynolds number, the rotating speed, and the power-law index.

Effects of dutasteride and tamsulosin on penile morphology in a rodent model.

To evaluate the penile morphology after the isolated and combined administration of dutasteride and tamsulosin in a rodent model. Forty male rats were assigned into the following groups: Control group (C, receiving distilled water, n=10); Dutasteride group (D, receiving 0.5 mg/Kg/day of dutasteride, n=10); Tamsulosin group (T, receiving 0.4 mg/Kg/day of tamsulosin, n=10); and Dutasteride associated with Tamsulosin group (DT, receiving both drugs n = 10). All drugs were administered via oral gavage. After 40 days, the animals were submitted to euthanasia and their penises were collected for histomorphometric analyses. Data were compared using one-way ANOVA followed by Bonferroni's post-test, considering p<0.05 as significant. The sinusoidal space and smooth muscle fiber surface densities (Sv), and the cross-sectional penile areas of rats in groups D, T and DT were reduced in comparison to controls with the most notable reductions in the combined therapy group. The connective tissue and elastic system fibers Sv were augmented in groups D, T and DT in comparison with the control group, again with the most pronounced changes observed in animals receiving the combined therapy. Both treatments with dutasteride or tamsulosin promoted penile morphometric modifications in a rodent model. The combination therapy resulted in more notable modifications. The results of this study may help to explain the erectile dysfunction observed in some men using these drugs.

Beam-Like Topologically Interlocked Structures With Hierarchical Interlocking

Abstract Topologically interlocked materials and structures, which are assemblies of unbonded interlocking building blocks, are promising concepts for versatile structural applications. They have been shown to exhibit exceptional mechanical properties, including outstanding combinations of stiffness, strength, and toughness, beyond those achievable with common engineering materials. Recent work has established a theoretical upper limit for the strength and toughness of beam-like topologically interlocked structures. However, this theoretical limit is only attainable for structures with unrealistically high friction coefficients; therefore, it remains unknown whether it is achievable in actual structures. Here, we demonstrate that a hierarchical approach for topological interlocking, inspired by biological systems, overcomes these limitations and provides a path toward optimized mechanical performance. We consider beam-like topologically interlocked structures that present a sinusoidal surface morphology with controllable amplitude and wavelength and examine the properties of the structures using numerical simulations. The results show that the presence of surface morphologies increases the effective frictional strength of the interfaces and, if well-designed, enables us to reach the theoretical limit of the structural carrying capacity with realistic friction coefficients. Furthermore, we observe that the contribution of the surface morphology to the effective friction coefficient of the interface is well described by a criterion combining the surface curvature and surface gradient. Our study demonstrates the ability to architecture the surface morphology in beam-like topological interlocked structures to significantly enhance its structural performance.

Experimental and Numerical Study of the Effect of Double Row Slot Injection Locations on Film Cooling Performance of a Corrugated Surface

ABSTRACT Based on experimental and numerical studies the present work proposes an effective injection configuration for film cooling applications on sinusoidal corrugated surfaces. The experimental study is performed for different double-row injection locations (per wavelength) viz. (L0-L25), (L0-L50), (L0-L75), (L25-L50), (L25-L75) and (L50-L75) on the sinusoidal corrugated surface at blowing ratio (1), fixed injection angle (45°), and density ratio (1.095). The numerical study is conducted for a wide range of operating parameters such as blowing ratios on (0.5–2), and density ratios (1.095, 1.5, and 2.5). To access the effect of corrugation geometry three different amplitude to wavelength viz. 0.05, 0.075, and 1 are considered in the present numerical study. The outcome of the present double-row slot injection investigations on the sinusoidal corrugated surface indicates that the flow pattern and film cooling strongly depend on secondary injection locations. The (L50-L75) injection case shows the best performance among all selected cases. It shows a 15–19% improvement in film cooling effectiveness (average over one wavelength). The increase of amplitude to wavelength ratio is reported to be a detrimental effect and decreases the film cooling effectiveness. The increase of corrugation amplitude to wavelength ratio increases the depth of the corrugation valley, further restricts the coolant spread, and promotes coolant mixing with hot mainstream. The coolant injection in mainstream flow was also found to be a distinct impact on flow and the nondimensional velocity profile. Apart from that a significant impact of the blowing ratio and density ratio is reported.

Investigation of nanofluid physical behavior inside porous container utilizing mathematical numerical method

The numerical method for modeling the behavior of nanofluid within the closed container in the presence of Lorentz force was dissected in this paper. The working fluid contains water and two shapes of nanoparticles. The porous geometry has one sinusoidal inner surface which faces constant heat flux. Influences of gravity and vertical Lorentz was complicated in modeling and momentum equations have been edited with adding related source terms. The precision of code was tested and respectable agreement has been obtained. The influences of loading nanomaterial and strength of two important forces (gravity and buoyancy) have been analyzed in the result section. Loading nanoparticles leads to a growth of Nu around 26.99% and 41.04% when [Formula: see text] and 15. Increasing buoyancy leads to augment of Nu about 44.34% and 1.46% in the absence and existence of Ha. Applying Lorentz force makes Nu to decrease about 40.39% when [Formula: see text].

Influence of skewed three-dimensional sinusoidal surface roughness on turbulent boundary layers

The impact of roughness skewness (ksk) on turbulent boundary layer (TBL) flow with a zero pressure gradient over three-dimensional (3D) sinusoidal rough surfaces was experimentally investigated using a single hotwire anemometer. Nine 3D sinusoidal profiles were manufactured with positive, negative, and zero roughness skewness values. Measurements were taken at three different freestream velocities for each surface and compared with smooth wall TBL results. This study covered a range of friction Reynolds numbers (Reτ) from approximately 1000 to 4000, with δ/k≈20 ± 2, where δ represents the local boundary layer thickness, and k is the maximum height of the roughness, measured from the valley to peak. The results indicate that the wall-unit normalized streamwise mean velocity profiles for all rough surfaces exhibit a downward shift compared to the smooth wall profiles. Surfaces with positive roughness skewness produced the highest drag, leading to the largest downward shift. The friction coefficient (Cf) decreased as ksk decreased. The percentage increase in Cf and ΔU+ (the roughness function) was much larger when moving from negative to zero roughness skewness than when moving from zero to positive roughness skewness. The small differences in turbulence intensity profiles and higher-order turbulence statistics in the outer region of the TBL support the outer layer similarity hypothesis for the roughness considered in this study. The autocorrelation study revealed that surfaces with positive roughness skewness tend to shorten the average length of turbulence structures in the near-wall region.

Near-Complete Suppression of Harmonic Currents in SPMSMs Caused by Back EMF and Dead Time

This paper introduces a novel algorithm forsuppression of phase current harmonics in three-phase sinusoidal surface mounted permanent magnet synchronous machines (SPMSMs) caused by non-ideal back-emf waveform and dead-time effects. Proposed feedback acquisition chain obtains an exact information on all the relevant harmonics within each period of the fundamental. Design of the harmonic current controller based on the internal model control principle is given. The paper outlines the relevant details of implementation and the results of verification performed by both computer simulations and experimentally, using a laboratory prototype machine. Experimental results, obtained in presence of non-sinusoidal back-emf and with erroneous dead-time compensation, prove the ability of the proposed solution to remove the stator current harmonics quickly, in just two fundamental periods, and thus eliminate torque ripple.

Viscoelastic hybrid nanofluid flow over a vertical plate with sinusoidal surface temperature variations

Natural convection of a viscoelastic hybrid nanofluid along a vertically heated plate with sinusoidal surface temperature variations is investigated. The current investigation explores the non-similar boundary layer flow patterns and heat transfer of second-grade viscoelastic flow of hybrid nanofluid. Effects of magnetic field and thermal radiation are considered. The governing dimensional equations are converted into a non-dimensional form taking suitable transformations. Resulting equations are solved with the aid of finite difference method. It is discovered that the momentum boundary layer lessens while the thermal boundary layer grows for higher radiation parameters, surface temperature parameters, Eckert numbers, magnetic field parameters and amount of nanoparticles. For larger Deborah numbers (De1), shear stress (τ) and heat transfer rate (q) accelerate, but momentum and thermal boundary decline near the leading edge of the vertical plate. However, the effects of Deborah number (De2) show opposite results. Increase in magnetic field parameters causes a reduction in shear stress. The higher volume fraction of nanoparticles (φ1, φ2) enhances q as it was expected. Moreover, τ and q were increased with larger surface temperature parameters and decrease with higher Eckert numbers. This is because higher surface temperature boost up the fluid temperature, but higher Eckert numbers admit the fluid to spread over the surface. An increase in the amplitude of surface temperature oscillation enhances the shear stress and heat transfer rate.

Double diffusion and Hall effects on MHD sinusoidal natural convection flow of silver water-based nanofluid from a porous vertical plate

This paper scrutinizes Hall current, Soret and Dufour impacts on natural convective flow of nanofluid attributable to variation in sinusoidal surface temperature over a vertical plate through a porous medium with strong transverse magnetic field applied normally to the flow. In this regard, the silver metal is considered as nanoparticles with water as base fluid. The overlapping multi-domain bivariate spectral local linearization method (OMD-BSLLM) has been utilized to solve the dimensionless governing equations which are attained by means of appropriate transformations. The obtained results are portrayed via graphical and tabular formations to inspect flow fields, shear stresses, heat and mass transmission characteristics for varying thermo-physical parameters. We found that there is an enhancement in the flow fields, shear stresses and heat transportation with the use of silver nanoparticles. Thermal and concentration boundary layer thickness enhance by using porous material, whereas diminish with improvement in Hall effect and ratio of buoyancy forces. The flow characteristics decline with the inclusion of porous medium while elevates with increment in Hall parameter. Moreover, an upgrade in thermal-diffusion correspond to a substantial growth in concentration field along with mass transfer rate. Current analysis can be useful in MHD energy generators, and industrial applications such as heating and cooling procedures owing to the involvement of nanoparticles having superior thermal conductivity features.

Squirmer hydrodynamics near a periodic surface topography.

The behaviour of microscopic swimmers has previously been explored near large-scale confining geometries and in the presence of very small-scale surface roughness. Here, we consider an intermediate case of how a simple microswimmer, the tangential spherical squirmer, behaves adjacent to singly and doubly periodic sinusoidal surface topographies that spatially oscillate with an amplitude that is an order of magnitude less than the swimmer size and wavelengths that are also within an order of magnitude of this scale. The nearest neighbour regularised Stokeslet method is used for numerical explorations after validating its accuracy for a spherical tangential squirmer that swims stably near a flat surface. The same squirmer is then introduced to different surface topographies. The key governing factor in the resulting swimming behaviour is the size of the squirmer relative to the surface topography wavelength. For instance, directional guidance is not observed when the squirmer is much larger, or much smaller, than the surface topography wavelength. In contrast, once the squirmer size is on the scale of the topography wavelength, limited guidance is possible, often with local capture in the topography troughs. However, complex dynamics can also emerge, especially when the initial configuration is not close to alignment along topography troughs or above topography crests. In contrast to sensitivity in alignment and topography wavelength, reductions in the amplitude of the surface topography or variations in the shape of the periodic surface topography do not have extensive impacts on the squirmer behaviour. Our findings more generally highlight that the numerical framework provides an essential basis to elucidate how swimmers may be guided by surface topography.

Newton-GMRES-Method for the Scritinization of electric double layer and surface roughness on EHL line contact problem

The electric double layer phenomenon exists on the solid interface under the water-liquid condition. The water molecules are ionized and adhered in the interface forming the sturn layer is a diffused layer in which molecules can move with the movement of bulk of molecules. Because of these two characteristics, the boundary layer of water molecules is called the electric double layer. The aim of present study is to explore the impact of two fraction surfaces of electric double layer (EDL) on a thin water lubricating film on an elastohydrodynamic lubrication (EHL) line contact problem with a sinusoidal surface roughness. The governing modified Reynolds and film thickness equations are based on mathematical model of electro-viscosity of asymmetrical electrical double layer is analyzed numerically. The viscosity-pressure relation of water and theoretical evaluation pertaining to the effect of electric double layer on film-thickness and pressure distribution of EHL with water film of line contact problem is discussed in detail. The effect of zeta potential on film thickness and pressure is determined using Newton’s-GMRES method with Daubechies D6 wavelet as a pre-conditioner. The results predict that, EDL has less impact on pressure distribution and significant impact on film thickness. The obtained results are compared with results of Dowson and Higginson which are comparable.

Heat-transfer scaling at moderate Prandtl numbers in the fully rough regime

In the fully rough regime, proposed models predict a scaling for a roughness heat-transfer coefficient, e.g. the roughness Stanton number ${St}_k \sim (k^+)^{-p} {Pr}^{-m}$ where the exponent values $p$ and $m$ are model dependent, giving diverse predictions. Here, $k^+$ is the roughness Reynolds number and ${Pr}$ is the Prandtl number. To clarify this ambiguity, we conduct direct numerical simulations of forced convection over a three-dimensional sinusoidal surface spanning $k^+ = 5.5$ – $111$ for Prandtl numbers ${Pr} = 0.5$ , 1.0 and 2.0. These unprecedented parameter ranges are reached by employing minimal channels, which resolve the roughness sublayer at an affordable cost. We focus on the fully rough phenomenologies, which fall into two groups: $p=1/2$ (Owen &amp; Thomson, J. Fluid Mech., vol. 15, issue 3, 1963, pp. 321–334; Yaglom &amp; Kader, J. Fluid Mech., vol. 62, issue 3, 1974, pp. 601–623) and $p=1/4$ (Brutsaert, Water Resour. Res., vol. 11, issue 4, 1975b, pp. 543–550). Although we find the mean heat transfer favours the $p=1/4$ scaling, the Prandtl–Blasius boundary-layer ideas associated with the Reynolds–Chilton–Colburn analogy that underpin the $p=1/2$ can remain an apt description of the flow locally in regions exposed to high shear. Sheltered regions, meanwhile, violate this behaviour and are instead dominated by reversed flow, where no clear correlation between heat and momentum transfer is evident. The overall picture of fully rough heat transfer is then not encapsulated by one singular mechanism or phenomenology, but rather an ensemble of different behaviours locally. The implications of the approach to a Reynolds-analogy-like behaviour locally on bulk measures of the Nusselt and Stanton numbers are also examined, with evidence pointing to the onset of a regime transition at even-higher Reynolds numbers.

On the oxidation resistance of sinusoidal microstructure fabricated by elliptical vibration cutting

The sinusoidal microstructure fabricated by the conventional slow tool servo (STS) cutting has low oxidation resistance and large fluctuation of surface roughness, which affects fluid instability in Rayleigh–Taylor experiments. This study is devoted to reducing roughness fluctuation and improving the oxidation resistance of sinusoidal microstructure by adopting the elliptical vibration (EV) of tool in the STS cutting. First, the kinematic principle of EVSTS cutting is demonstrated by highlighting the time-varying orientation of the elliptical tool trajectory relative to instantaneous nominal cutting velocity. Next, the geometric roughness model of sinusoidal surface is developed to analyze the roughness fluctuation along the wavelength direction. Then, flat surface cutting and accelerated oxidation experiments are performed to explore the deterministic effects factors on machined surfaces' surface roughness and oxidation resistance using elliptical vibration cutting (EVC). The experimental results show that the EVC can reduce the oxide thickness of the machined surface after accelerated oxidation by 35 % on average, owing to the reduced roughness and surface strengthening effects. The effects of the vibration trajectory, edge radius, and wavelength-direction nominal cutting velocity on the surface roughness and surface oxidation resistance are established. Finally, the processing parameters for the EVSTS cutting of sinusoidal microstructure are optimized by further considering the time-varying orientation of the elliptical tool trajectory. Surface texturing tests with results demonstrates that the EVSTS can significantly improve the oxidation resistance of the microstructured surface, suppressing the roughness fluctuation along the wavelength direction by 25–60 %.

Non-planar multinozzle additive manufacturing of thermoset composite microscaffold networks

Material extrusion-based non-planar additive manufacturing (AM) using 6-axis industrial robot arms shows great promise for three-dimensional (3D) printing of non-planar parts, despite its typical use of a single large coarse nozzle and high printing time. Here, a non-planar multinozzle AM method capable of achieving high resolution large-scale 3D printing with short printing time using 6-axis robots is reported. A custom Multinozzle Toolpath Generator (MTG), based on non-planar parameterized surface projection, is used to compute the orientation of a multinozzle extrusion array along the curvature of a 3D scanned substrate. The method is demonstrated by the fast 3D printing of large periodic structures (up to an area of ∼5.2 × 104 mm2 within 20 min) consisting of interconnected epoxy composite microscaffolds on a single curvature sinusoidal surface and a double curvature surface of an actual aircraft casing, with an error of 15 μm. The method could be used for AM in various fields such as tissue engineering, microfluidic channels, and porous acoustic coatings of aircraft components.

Nano-jet impingement cooling of a sinusoidal wavy hot surface by using an oriented L-shaped object under magnetic field with two phase mixture formulation

In this study, slot nano-jet impingement cooling of a wavy sinusoidal surface is analyzed by combined utilization of an oriented L-shaped object and magnetic field. A two phase mixture model of nanofluid is adopted. It is observed that the presence of the L-shaped object and its orientation are very influential on the local and average heat transfer performance. Magnetic field suppresses the vortices near the object and on the bottom all. The average Nusselt number (Nu) rises by 17.5% and 8.6% when its strength is varied for object orientations of te=−90 and te=90. When magnetic field inclination is concerned, the cooling performance improvements are in the range of 25.6%–53.9% depending upon the orientation of the object. The orientation of te=0 and te=45 are the most favorable cases in terms of thermal performance while significant variations in the average Nu are seen in the absence of magnetic field when varying the L-shaped object orientation. When compared to reference case of slot jet impingement (without magnetic field using pure fluid and without object), using nanofluid improves the cooling performance by about 8.4%. When object is installed at Ha=0, it rises by 27% at orientation of te=45. When magnetic field is used, additional rise of thermal performance with nanofluid is obtained up to 10% when compared to case by using only pure fluid.

Leakage Model of Tubing and Casing Premium Connection Based on Sinusoidal Contact Simulation between Rough Surfaces

This paper proposed a semi-theoretical model to quantitatively predict leakage rate of tubing and casing premium connections. The geometric parameters of the sealing surface profile approximated by a sinusoidal micro-convex surface were first obtained based on the random normal distribution sampling method. With the actual area prediction formula for elastic–plastic contact of an axisymmetric sinusoidal micro-convex body based on the equivalent simulation principle, the circumferential leakage width and radial average leakage height of the micro-leakage channel between sealing surfaces were then acquired with the surface roughness and geometric mean contact pressure. At last, the actual micro-leakage rate of the premium connection was derived by considering the non-uniform contact pressure distribution between sealing surfaces. An example was investigated to validate the model and reveal the sealing and leakage characteristics, and anti-leakage measures were proposed. The results show that average contact pressure, circumferential leakage width, and radial average leakage height between sealing surfaces were non-uniformly distributed. The leakage rate of a premium connection decreases exponentially with an increase in radial interference between sealing surfaces. In order to reduce leakage rate, it is beneficial to increase radial interference and lower sealing surface roughness.

Sputter yields of surfaces with nanoscale textures: Analytical results and Monte Carlo simulations

We find the spatially averaged sputter yield Y¯ analytically for non-planar surfaces that have slowly varying surface heights h=h(x,y). To begin, nonlocal effects like redeposition of sputtered material and secondary sputtering are neglected. We show that the leading order corrections to Y¯ are proportional to the spatial averages of (∂h/∂x)2 and (∂h/∂y)2. The constants of proportionality can be written in terms of the first and second derivatives of the sputter yield of a flat surface with respect to the ion incidence angle θ. For a range of θ values, Y¯ is a decreasing function of the amplitude of the surface texture. We also determine how the contribution of redeposition to Y¯ depends on the amplitude and characteristic lateral length scale of the surface morphology. As a test of our theory and to quantify the roles of redeposition and secondary sputtering, we performed Monte Carlo simulations of sputtering from Si targets with sinusoidal surfaces by 1 keV Ar+ ions. The theory agrees remarkably well with our Monte Carlo simulations. Our simulations also lead to the notable result that atoms that are sputtered and then strike the surface can themselves cause significant sputtering.