Roughness Metrics Research Articles

Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

The physical meaning of Grasselli's morphology parameters and its correlations with several other 2D fracture roughness parameters

Grasselli's 2D morphology parameter θ max ∗ / ( C + 1 ) 2 D has been widely used in rock engineering. However, the physical meaning of θ max ∗ / ( C + 1 ) 2 D is not clear, and the calculation is complex and time-consuming. By introducing a density function to characterize the distribution of the inclination, this study found that the physical meaning of θ max ∗ / ( C + 1 ) 2 D is the expected value of the inclination facing the shear direction. Based on this finding, this study found that θ max ∗ / ( C + 1 ) 2 D can be simplified as a statistical parameter. Besides, several new 2D roughness metrics expressed in discrete form were proposed. These new roughness metrics have clear physical meanings and can be obtained easily. This study found that two kinds of weight can be employed to calculate the average inclination facing the shear direction. Besides, this study found several generalized equivalent relationships among some 2D roughness metrics.

Using Ultrasound Color Doppler Twinkling to Identify Biopsy Markers in the Breast and Axilla

In breast radiology, ultrasound detection of biopsy markers or clips for localization purposes is often challenging, especially in the axilla. The purpose of this research was to test the hypothesis that the surface roughness of biopsy clips would elicit a twinkling signature on color Doppler, making them more readily identifiable by ultrasound. Ultrasound color Doppler imaging of 12 biopsy markers was performed and consensus scoring of the degree of twinkling (0 [no twinkling] to 4 [exuberant twinkling]) was obtained for each of the markers. The surface roughness characteristics of the markers were measured using 3-D coherence scanning interferometry. The 3 markers scoring at least 3 for twinkling in vitro were cork, Q and Vision. Of these 3 markers, only the cork marker scored a 4 ex vivo and in cadaveric tissue. Surface roughness metrics demonstrated a positive estimated correlation with the twinkling scores (rho = 0.33, 95% CI = [–0.48 to 0.84]). Of the 12 markers tested, the markers that twinkled corresponded to surface roughness measured with non-contact 3-D optical imaging. Qualitatively, lower color scales and color frequencies optimized twinkling, but the most specific qualitative predictor of confidence in twinkling was insensitivity to changes in color scale and color frequency values.

Going Beyond Traditional Roughness Metrics for Floor Tiles: Measuring Topography Down to the Nanoscale

Slipping is a major cause of injury and hospitalization in the USA and globally. Slipping occurs when the instantaneous friction between the shoe and floor is less than the required friction. While floor roughness is a key factor contributing to friction, prior investigations have primarily used stylus profilometry, which is incapable of measuring roughness at small scales, below approximately 1 micron in lateral size. In the present research, the small-scale roughness was quantified using cross-section scanning electron microscopy (SEM). Three different flooring materials were investigated, including tiles of ceramic and two different types of quarry stones, whose friction coefficients had been previously characterized. The surfaces were cross-sectioned, imaged at magnifications from 250 to 100,000 times, and then the surface profiles were extracted using image analysis. The SEM topography was combined with stylus profilometry measurements, using the power spectral density (PSD), to achieve multi-scale characterization of features ranging from a scan size of 4 mm down to a resolution of 10 nm. The results demonstrate meaningful differences in topography at different length scales, where surfaces with widely varying roughness at one scale were indistinguishable at another. The measurements further showed that floor-tile roughness has self-affine fractal-like character, with hierarchical roughness extending from the micron-scale down to the nanoscale, much of which is undetectable using conventional techniques. Overall, this research supports the investigation of small-scale roughness as a potential missing factor in the understanding of floor topography and its causal effect on slip-and-fall accidents.

CMP Pad Conditioning and Applications to Type 1 Impregnated Felt Polishing Pads

Pad conditioning is the process of “dressing” the surface of the polishing pad, generally with a highly engineered diamond abrasive disc. Conditioning not only refreshes the pad surface, the cutting characteristics of the conditioner determine the structure of the asperities that are in contact with the wafer. By tuning the cutting characteristics of the conditioner, polish performance can be manipulated over a wide range. With the most advanced pads, specific conditioning protocols are typically required to obtain the designed performance. Optimization of these protocols is enabled by an understanding of the pad surface statistics.Figure 1 illustrates the two types of polishing pads that will be compared in this talk. Suba 800 is an example of a Type 1 pad. Type 1 pads consist of a “non-woven” needled polyester felt impregnated with polyurethane. The pores in a Type 1 pad are formed from the spaces between the impregnated fibers and have a length scale of 100s of microns. The pores in a Type 1 pad tend to be interconnected. VP5000 is a Type 3 pad. Type 3 pads consist of discrete pores in a matrix of polyurethane. The matrix properties can be varied over a wide range and pore sizes range from single to 10s of microns. Type 3 pads should be isotropic in terms of their physical property and porosity distribution. Type 3 pads typically exhibit Shore D hardness in the range of 20-70. Type 1 pads are measured on softer hardness scales or in terms of compressibility.Pad conditioning of Type 3 pads in semiconductor wafer CMP is at a technically mature stage, and the knowledge base can be directly applied to substrate polishing with Type 1 pads. The decoupled Gaussian-Exponential model which has been successfully used in Type 3 pads can be directly applied to Type 1 pads. As Figure 2 illustrates, Type 1 pads exhibit an exponential porosity signature, but with a decay length about an order of magnitude larger than a typical Type 3 pad, consistent with the difference in length scales of the pores of the two pad types. The core of the distribution (accounting for most of the total surface) can be modeled as a Gaussian.Figure 3 provides examples of both Suba 400 and Suba 800 and differences in both the intrinsic porosity (exponential decay constant, Eτ) and Gaussian core width are evident. At least part of the difference in the porosity signature is due to the Suba 800 receiving a double impregnation of polyurethane, compared to a single impregnation for Suba 400, resulting in a comparatively shallower pore structure. The difference in the Gaussian core width is most likely the result of a different buffing process or media used to impart a surface finish at the factory.Figure 4 illustrates the range of texture that can be developed on the DuPont Suba 800 impregnated felt pad, one of the Type 1 pads most widely used in substrate polishing applications. As the figure illustrates, the hwhm of the Gaussian core can be driven from about 10 to 50 µm, while maintaining the deep background porosity that characterizes this pad. The ability to drive the texture into rougher and smoother regimes, compared to a new pad, implies a potential to drive the substrate process into previously unattainable performance spaces. Initial feedback from the field indicates that smoother pad surfaces can contribute to improved substrate surface finish. Rougher surfaces may contribute to improved throughput through polish rate increase.Figure 5 illustrates how key pad roughness metrics of Suba 800 are correlated with a basic conditioner diamond design parameter. Most roughness measurements are highly correlated, although one of the most widely used roughness parameters, the average roughness Ra, does not correlate well at all in this case. This is an excellent example where pad height distribution modeling provides critical insight into the drivers of the textural changes that effect polish performance, and how that performance might be manipulated through conditioning design.Signatures of surface textural degradation from the stress of polishing and accumulation of debris can be observed on Type 1 pad surfaces. It is well known that restoration of the pad surface structure through conditioning is a primary mechanism of stability in Type 3 pads. Data from the field indicates improvements in process performance and consumable life with Type 1 pads when conditioning has been implemented versus processes with no pad conditioning. The examples given here will be expanded upon and additional applications will be discussed. Figure 1

Effect of surface geometry on laser powder bed fusion defects

Laser Powder Bed Fusion (L-PBF) is an additive manufacturing (AM) technology which has enabled the fabrication of highly complex structures and components. A major challenge in producing these high complexity structures is predicting where defects are likely to occur, and how severe those defects will be. In this study build geometry is linked to manufacturability, using quantifiable AM defects and metrics. This was achieved by fabricating a variety of unsupported, thin-walled samples with different curvatures using Al-10Si-Mg alloy. A combination of X-Ray computed tomographic (XCT) and digital stereomicroscopy were used to capturing positional data about the total geometric error, surface quality and roughness metrics (Ra and Rz). This dataset was used to create empirical model which predicts geometric error based on the local curvature, orientation and thickness of the computer-aided-design (CAD) geometry. Local inclination angle was observed to have the strongest effect on manufacturability, heavily impacting the geometric error and surface quality metrics. Sample shape defined by curvature was observed to be a secondary factor, which became important in areas of low inclination angle. The empirical model was applied in a case study of a cellular structure, the double gyroid, which successfully identified problematic regions of the structure. The approach of linking fundamental geometry features to defects shown in this study provides additional guidance in design for AM, with potential applications in topology optimisation, optimising part orientation to reduce support, and for design of other intricate components such as the internal conduits of heat exchange devices, where supports are not feasible to use.

A Systematic Investigation Into the Control of Roughness on the Flow Properties of 3D‐Printed Fractures

AbstractHeterogeneous fracture aperture distribution, dictated by surface roughness, mechanical rock and fracture properties, and effective stress, limits the predictive capabilities of many reservoir‐scale models that commonly assume smooth fracture walls. Numerous experimental studies have probed key hydromechanical responses in single fractures; however, many are constrained by difficulties associated with sample preparation and quantitative roughness characterization. Here, we systematically examine the effect of roughness on fluid flow properties by 3D printing seven self‐affine fractures, each with controlled roughness distributions akin to those observed in nature. Photogrammetric microscopy was employed to validate the 3D topology of each printed fracture surface, enabling quantification using traditional roughness metrics, namely the Joint Roughness Coefficient (JRC). Core‐flooding experiments performed on each fracture across eight incremental confining pressure increases (11–25 bar), shows smoother fractures (JRC < 5.5) exhibit minor permeability variation, whilst rougher fractures (JRC > 7) show as much as a 219% permeability increase. Micro‐computed tomography imaging of the roughest fracture under varying effective stresses (5–13.8 bar), coupled with inspection into the degree of similarity between fracture closure behavior in 3D‐printed and natural rock fractures, highlight the capabilities of 3D‐printed materials to act as useful analogs to natural rocks. Comparison of experimental data to existing empirical aperture‐permeability models demonstrates that fracture contact area is a better permeability predictor than roughness when the mechanical aperture is below ∼20 μm. Such findings are relevant for models incorporating the effects of heterogeneous aperture structures and applied stress to predict fracture flow in the deep subsurface.

An investigation of additively manufactured coolant nozzles for cylindrical grinding applications and progression to high stock removal

This paper explores use of Additively Manufactured features in coolant nozzles for cylindrical grinding applications. In an external cylindrical grinding process, different stock removal rate regimes are trialed to assess printed nozzle features; 3D printed nozzles with special internal features are shown to generate an improvement in key surface roughness metrics (up to 0.3 µm Sa reduction in mean area surface roughness) and superior wheel condition in Q’ ≤ 2.61 mm2/s cylindrical plunge grinding applications. In addition, use of VIPER grinding principles achieved a stable cylindrical grind at Q’ = 20.47 mm2/s using a porous wheel and hook nozzle tooling.

Measuring the Electron Density Roughness of the D‐Region Ionosphere

AbstractWe present a method of characterizing the horizontal and vertical electron density roughness of the D‐region ionosphere using Nationwide Differential Global Position System (NDGPS) transmitters as low‐frequency (LF; 30–300 kHz) and medium‐frequency (MF; 300–3,000 kHz) signals of opportunity. The horizontal roughness is characterized using an amplitude cross‐correlation method, which yields the correlation length scale metric. The vertical roughness is characterized using a differential phase height, which is needed to mitigate the effects of transmitter phase instability. The ranges and typical values of roughness metrics are investigated using data from several field campaign measurements. Finally, the roughness metrics for an NDGPS transmitter and very low frequency (VLF) transmitter are compared. It is found that the roughness detected by the VLF transmitter is significantly smoother and demonstrates the utility of this method to complement traditional VLF measurements.

Heat kernels and regularity for rough metrics on smooth manifolds

AbstractWe consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

Turning of Thermoplastics Using Ceramic Cutting Tools

The question of evaluating the efficiency of turning polymer materials using cutting ceramics is examined. It is proved experimentally that turning caprolon and fluoroplastic using a cutting plate made from KNT16 cermet gives a higher-quality machined surface. This is attested by the lower values of the roughness metrics and type of chips as compared with turning using VK8 tungsten-cobalt alloy.

A model for roughness statistics of heterogeneous fibrous materials

We present a theory giving the standard roughness metrics R_{mathrm{a}} and R_{mathrm{q}} of stochastic fibrous materials in terms of their porosity and the thickness of the constituent fibres. Our treatment shows also that R_{mathrm{a}} and R_{mathrm{q}} are linearly dependent on each other with gradient depending on the skewness of the surface depth profile. Comparison of our theory with experiments on laboratory-formed paper samples and with data from the literature for industrially manufactured paper and carbon fibre nonwovens for use in fuel cells is excellent. The theory has applicability to generic families of stochastic fibrous materials and has relevance, for example, to printing of devices on paper.

Terrestrial structure-from-motion: Spatial error analysis of roughness and morphology

Structure-from-Motion (SfM) photogrammetry is rapidly becoming a key tool for morphological characterisation and change detection of the earth surface. This paper demonstrates the use of Terrestrial Structure-from-Motion (TSfM) photogrammetry to acquire morphology and roughness data at the reach-scale in an upland gravel-bed river. We quantify 1) spatially-distributed error in TSfM derived Digital Elevation Models (DEMs) and 2) identify differences in roughness populations acquired from TSfM photogrammetry versus TLS. We identify an association between local topographic variation and error in the TSfM DEM. On flatter surfaces (e.g. bar and terrace surfaces), the difference between the TSfM and TLS DEMs are generally less than ±0.1m. However, in areas of high topographic variability (>0.4m) such as berm or terrace edges, differences between the TSfM and TLS DEMs can be up to ±1m. Our results suggest that grain roughness estimates from the TSfM point cloud generate values twice those derived from the TLS point cloud on coarse berm areas, and up to four-fold those derived from the TLS point cloud over finer gravel bar surfaces. This finding has implications when using SfM data to derive roughness metrics for hydrodynamic modelling. Despite the use of standard filtering procedures, noise pertains in the SfM DEM and the time required for its reduction might partially outweigh the survey efficiency using SfM. Therefore, caution is needed when SfM surveys are employed for the assessment of surface roughness at a reach-scale.

2D and 3D Roughness Characterization

The quantification of surface roughness for the purpose of linking its effect to mechanical and hydrodynamic behavior has taken many different forms. In this paper, we present a thorough review of commonly used 2D and 3D surface roughness characterization methods, categorized as statistical, fractal, and directional. Statistical methods are further subdivided into parametric and functional methods that yield a single value and function to evaluate roughness, respectively. These statistical roughness metrics are useful as their resultant outputs can be used in estimating shear and flow behavior in fractures. Fractal characterization methods treat rough surfaces and profiles as fractal objects to provide parameters that characterize roughness at different scales. The directional characterization method encompasses an approach more closely linked to shear strength and is more suitable for estimating the influence of fracture roughness on mechanical responses. Overall, roughness characterization methods provide an effective objective measure of surface texture that describe its influence on the mechanics of surfaces without requiring qualitative description.

Observed Relationship Between BRF Spectral-Continuum Variance and Macroscopic Roughness of Clay Sediments

Spectral data offer a means of estimating the critical parameters of sediments, including sediment composition, moisture content, surface roughness, density, and grain-size distribution. Macroscopic surface roughness in particular has a substantial impact on the structure of the bidirectional reflectance factor (BRF) and the angular distribution of scattered light. In developing the models to invert the properties of the surface beyond just surface composition, roughness must also be accounted for in order to achieve reliable and repeatable results. This paper outlines laboratory studies in which the BRF and surface digital elevation measurements were performed on dry clay sediments. The results were used to explore the suitability of various roughness metrics to account for the radiometric effect of surface roughness. The metrics that are specifically addressed in this paper include random roughness and sill variance. Relative accuracy and tradeoffs between these metrics are described. We find that spectral variability, especially near spectral absorption features, correlates strongly with the quantified measures of surface roughness. We also find that spectral variability is sensitive to the sensor fore-optic size. The results suggest that roughness parameters might be directly determined from the spectrum itself. The relationship between spectral variability and macroscopic surface roughness was particularly strong in some broad spectral ranges of the visible, near infrared, and shortwave infrared, including the near-infrared region between 600 and 850 nm.

On the usability of different optical measuring techniques for joint roughness evaluation

The roughness of rock discontinuities is an important input parameter for mechanical models of rock masses. To reliably calculate roughness indices, adequate representations of the surfaces are required. Various optical measuring approaches have been applied in the past. However, many studies lack information on resolution and accuracy of the resulting surface meshes. These qualities are yet important, as they explicitly affect the deduced roughness metrics. Often, the sensors do not achieve the given precision and accuracy. Moreover, no technical standards presently exist for roughness evaluation from optical measuring approaches. Therefore, previous studies are difficult to compare. To overcome these issues, this study offers a comparison of four different techniques and sensors. Here, the focus lies on laboratory use and evaluation of micro-roughness, meaning sample sizes up to 20 cm in length. Stationary structured light scanning (SLS) serves as the reference method. As results, the surface models from dense image matching are very consistent with the reference. Their calculated roughness values accord to a high degree, both for 2D and 3D indices. In addition, roughness indices deduced from models acquired with manually operated SLS show deviations from the reference yet within an acceptable range. Instead, terrestrial laser scanning turned out to be not suitable for micro-roughness evaluation, at least at laboratory scale. Furthermore, in this contribution, an algorithm is applied, which can retrace all possible profile measurements directly from the triangulated surfaces. That way the ambiguity of the profile-based roughness measure joint roughness coefficient (JRC) is made visible.

Collapsing Complexity: Quantifying Multiscale Properties of Reef Topography

AbstractSeafloor topography affects a wide range of physical and biological processes; therefore, collapsing the three‐dimensional structure of the bottom to roughness metrics is a common challenge in studies of marine systems. Here we assessed the properties captured by metrics previously proposed for the seafloor, as well as metrics developed to characterize other types of rough surfaces. We considered three classes of metrics: properties of the bottom elevation distribution (e.g., standard deviation), length scale ratios (e.g., rugosity), and metrics that describe how topography varies with spatial scale (e.g., Hölder exponents). The metrics were assessed using idealized topography and natural seafloor topography data from airborne lidar measurements of a coral reef. We illustrate that common roughness metrics (e.g., rugosity) can have the same value for topographies that are geometrically very different, limiting their utility. Application of the wavelet leaders technique to the reef data set demonstrates that the topography has a power law scaling behavior, but it is multifractal so a distribution of Hölder exponents is needed to describe its scaling behavior. Using principal component analysis, we identify three dominant modes of topographic variability, or ways metrics covary, among and within reef zones. Collectively, the results presented here show that coral reef topography is both multiscale and multifractal. While individual metrics that capture specific topography properties relevant to a given process may be suitable for some studies, many applications will require a set of metrics that includes statistics that capture how topography varies with spatial scale.

Assessing Effects of Azimuthally Oriented Roughness on Directional Reflectance of Sand

We measured and compared hyperspectral bidirectional reflectance factors of a sand sample of varying roughness levels using the Goniometer of the Rochester Institute of Technology-Two. We developed a geotechnical routine to generate sand samples of approximately constant density and grain size distribution, but varying roughness levels. In addition, we developed sand samples of two different classes of surface roughness: wave-like and normally distributed. The samples exhibiting normally distributed roughness met several criterion outlined by Hapke in the roughness correction to his photometric model for a smooth sediment surface. We developed a method to empirically forward model the photometric model for a rough surface using roughness metrics derived from digital elevation models of the surface. Our results from empirically forward modeling the correction factor indicate that Hapke's shadowing function does not adequately describe the effects of macroscopic roughness at a subcentimeter scale. In addition, we also performed experiments in which we oriented the surface waves of the wave-like roughness profiles in different orientations to the principal plane of illumination. These results indicate that future photometric models of wave-like roughness should include a description of how multiple scattering increases reflectance within cavities, reducing the effects of shadowing within the cavities. Our results also suggest that since Hapke's model correction for macroscopic roughness assumes that the underlying distribution of surface slopes does not depend on azimuth, it cannot adequately characterize surface roughness when it is both structured and has a preferred orientation.

Hydrodynamic versus Surface Interaction Impacts of Roughness in Closing the Gap between Favorable and Unfavorable Colloid Transport Conditions.

Recent experiments revealed that roughness decreases the gap in colloid attachment between favorable (repulsion absent) and unfavorable (repulsion present) conditions through a combination of hydrodynamic slip and surface interactions with asperities. Hydrodynamic slip was calibrated to experimentally observed tangential colloid velocities, demonstrating that slip length was equal to maximum asperity relief, thereby providing a functional relationship between slip and roughness metrics. Incorporation of the slip length in mechanistic particle trajectory simulations yielded the observed modest decrease in attachment over rough surfaces under favorable conditions, with the observed decreased attachment being due to reduced colloid delivery rather than decreased attraction. Cumulative interactions with multiple asperities acting within the zone of colloid-surface interaction were unable to produce the observed dramatic increased attachment and decreased reversibility with increased roughness under unfavorable conditions, necessitating inclusion of nanoscale attractive heterogeneity that was inferred to have codeveloped with roughness. Simulated attachment matched experimental observations when the spatial frequency of larger heterodomains (nanoscale zones of attraction) increased disproportionately relative to smaller heterodomains as roughness increased, whereas attachment was insensitive to asperity properties, including the number of interactions per asperity and asperity height; colloid detachment simulations were highly sensitive to these parameters. These cumulative findings reveal that hydrodynamic slip moderately decreases colloid bulk delivery, nanoscale heterogeneity dramatically enhances colloid attachment, and multiple interactions among asperities decrease detachment from rough surfaces.

Effect of machining parameters and cutting tool coating on hole quality in dry drilling of fibre metal laminates

Fibre metal laminates (FMLs) are a special type of hybrid materials, which consist of sheets of metallic alloys and prepregs of composite layers stacked together in an alternating sequence and bonded together either mechanically using micro hooks or thermally using adhesive epoxies. The present paper contributes to the current literature by studying the effects of three types of cutting tool coatings namely TiAlN, AlTiN/TiAlN and TiN on the surface roughness and burr formation of holes drilled in an FML commercially known as GLARE®. While the cutting tool geometry is fixed, the study is also conducted for a range of drilling conditions by varying the spindle speed and the feed rate. The obtained results indicate that the spindle speed and the type of cutting tool coating had the most significant influence on the achieved surface roughness metrics, while tool coating had the most significant effect on burr height and burr root thickness. The most important outcome for practitioners is that the best results in terms of minimum roughness and burr formation were obtained for the TiN coated drills. However, such drills outperform the other two types of tools, i.e. with TiAlN and AlTiN/TiAlN coatings, only when used for short series of hole drilling due to rapid tool deterioration.