Three-dimensional Discontinuity Research Articles

Discontinuity third harmonic generation microscopy for label-free imaging and quantification of intraepidermal nerve fibers

Label-free imaging methodologies for nerve fibers rely on spatial signal continuity to identify fibers and fail to image free intraepidermal nerve endings (FINEs). Here, we present an imaging methodology-called discontinuity third harmonic generation (THG) microscopy (dTHGM)-that detects three-dimensional discontinuities in THG signals as the contrast. We describe the mechanism and design of dTHGM and apply it to reveal the bead-string characteristics of unmyelinated FINEs. We confirmed the label-free capability of dTHGM through a comparison study with the PGP9.5 immunohistochemical staining slides and a longitudinal spared nerve injury study. An intraepidermal nerve fiber (IENF) index based on a discontinuous-dot-connecting algorithm was developed to facilitate clinical applications of dTHGM. A preliminary clinical study confirmed that the IENF index was highly correlated with skin-biopsy-based IENF density (Pearson's correlation coefficient R= 0.98) and could achieve differential identification of small-fiber neuropathy (p= 0.0102) in patients with diabetic peripheral neuropathy.

Sound radiated by three-dimensional discontinuities immersed in a turbulent boundary layer

Measurements of the far-field sound from two types of three-dimensional discontinuities – swept forward-facing steps and a circular embossment – immersed in a low Mach number (M<0.18) turbulent boundary layer were made. The discontinuity height in each case was smaller than the height of the approaching, undisturbed boundary layer and the Reynolds number based on the discontinuity height ranged between 8300–92,000. The swept forward-facing steps were immersed in a wall-jet boundary layer, whereas the embossment was placed in a conventional boundary layer. For the embossment, measurements of surface pressure fluctuations were also performed beneath the separated flow on the vertical and the downstream surfaces. The wall-jet measurements show that the far-field sound from swept forward-facing step scales on the step-normal component of the boundary-layer edge velocity. The far-field sound from the circular embossment is qualitatively similar to that from a forward-facing step of the same height, a fact which is also reflected in the surface pressure fluctuation measurements. The step-normal velocity scaling obtained for the swept forward-facing steps was combined with an existing prediction formula for the sound radiated by a forward-facing step to yield a straight-forward prediction formula for the embossment noise. The prediction formula shows good agreement with the measured sound spectra if it is assumed that the dominant source for the embossment is the separation/reattachment region concentrated in the forward arc of the embossment. This assumption is supported by oil flow visualization which shows that the separated flow region is concentrated between approximately ±45°on either side of the spanwise center.

Characterization of discontinuity surface morphology based on 3D fractal dimension by integrating laser scanning with ArcGIS

The fractal geometry method has been employed to quantitatively characterize the roughness of a rock discontinuity, which is one of the key factors affecting its shear strength and the seepage characteristics of a rock mass. However, the current fractal methods involving the three-dimensional discontinuity morphology suffer from one or more problems, such as a complicated calculation procedure, an inaccurate calculation result and an inability to characterize the undulation and anisotropy of a discontinuity. To cope with these problems, the discontinuities in artificial granite samples with irregular and undulating surfaces were taken as examples, and a quantitative study on the discontinuity morphology was conducted based on the method of three-dimensional laser scanning in combination with ArcGIS data processing, geographical research, theoretical calculations and regression analysis. After performing systematic research, we proposed an extensive 3D fractal dimension including three discontinuity morphological parameters, i.e. the fractal dimension of discontinuity morphology, the ratio between the maximal undulating amplitude and the discontinuity length, and the average value of all the apparent dip angles of the discontinuity surfaces dipping opposite the shear direction. The extensive 3D fractal dimension can comprehensively characterize the roughness, undulation and anisotropy of the discontinuity morphology. A set of theoretical calculation methods were then developed to determine the three discontinuity morphological parameters of the extensive 3D fractal dimension based on ArcGIS. We finally established a mathematical expression of the extensive 3D fractal dimension. Compared with the current fractal methods, the extensive 3D fractal dimension can effectively compensate for the inability to characterize the undulation and anisotropy of the discontinuity morphology. Its calculation methods have the advantages of simplification, low-time consumption and high precision.

Assessing error in the 3D discontinuity-orientation distribution estimated by the Fouché method

Realistic discontinuity network modelling of rocks requires the accurate input of the three-dimensional discontinuity orientation distribution (3DDOD), but in practice, access to such three-dimensional information is limited. Fouché recently developed a method for estimating the 3DDOD from one-dimensional scanline-mapping observations. However, little is known about the error in such 3DDOD estimates. This study focuses on (1) the error and its possible contributors, (2) error prediction, and (3) error mitigation. An error investigation based on synthetic discontinuity geometry data reveals that the 3DDOD estimates are not always accurate. The error is significantly impacted by three factors: the Fisher constant (к), the angle between the major discontinuity plane and the mapping scanline (θ), and the discontinuity sample size (n), with larger к, θ, or n values contributing to lower errors. This к-, θ-, and n-dependent error appears to be inherently related to the sample density, which is defined as the pole number per unit cell (1° × 1°) in the Fouché method.Additionally, a predictor of the error based on an error-and-factors empirical relationship is developed to provide precise error prediction and optimize the scanline sampling for error mitigation. The weakness of the developed predictor is also disclosed.

Meticulous three-dimensional demarcation and separate presentation of statistically homogeneous structural domains within a large-scale jointed rock mass

Traditional methods used to demarcate homogeneous structural domains are mainly concerned with small-scale discontinuities; this approach can be regarded as a process for evaluating the degree of homogeneity among stochastic discontinuities from one site to another. This process forms the basis for determining the distribution of stochastic discontinuities within a homogeneous structural domain; however, it neglects large-scale discontinuities. In this study, the statistically homogeneous structural domains within a large-scale jointed rock mass were meticulously demarcated in three dimensions, which was performed in four steps. 1) A three-dimensional discontinuity network model was constructed containing both deterministic and stochastic discontinuities. 2) The model was subsequently discretized into a large number of cubic grids, and the MBi values of all the grids were calculated. 3) Using the K-means algorithm, the structural domain boundaries within the tested rock mass were meticulously demarcated, and a composited model consisting of homogeneous structural domains was constructed. 4) Each homogeneous structural domain was separately visualized. This demarcation method can consider large-scale discontinuities and improve the identification sensitivity in a large-scale jointed rock mass with a high accuracy. Moreover, the separate presentation of structural domains directly reflects the morphological properties unique to each structural domain.

Geometric quality assessment of in situ blocks in dimension stone quarries

The profitability of stone quarries depends on the appearance (color and texture) and, more importantly, on the size of the blocks, which is determined by the three-dimensional discontinuity pattern of the rock mass. However, to date there is no comprehensive criterion by which to assess the size of blocks in quarries. The aim of the study reported here is to identify factors that facilitate decision-making on the desirability of an individual block and ultimately on all blocks for extraction as a dimension stone. To this end, we first developed a novel algorithm to calculate the largest rectangular parallelepiped that can be placed inside a block and then defined a regularity factor (α) and a quarry block shape factor (q) that enable the geometric quality of the block to be assessed more accurately than currently possible. The sensitivity analysis of these factors showed that the change in angle relative to the change in spacing has the greater effect on the geometric quality. The proposed factors displayed a good performance in a comparison analysis of four sites of the Dareh Bokhari travertine quarry complex located in Mahallat, Markazi province, Iran. Based on the results, blocks at sites 2 and 4 have the lowest and the highest geometric quality, respectively, which is in accordance with the operational conditions at the respective sites. Moreover, comparison of the values of factor q obtained from applying various cutting planes indicated that to maximally increase block production capacity, the current cutting direction at site 4 should be adjusted 33° eastward.

A new perspective for the directivity of Rock Quality Designation (RQD) and an anisotropy index of jointing degree for rock masses

The Rock Quality Designation (RQD) index is defined as the percentage of the borehole core or scanline that consists of intact lengths ≥0.1 m, and has been widely used as an engineering measure of the jointing degree of a rock mass. The directivity of RQD, which means the value of the RQD obtained from drilling cores or scanlines with different orientations can be different for a given site, is usually regarded as a drawback by researchers and engineers. This study presents a new perspective for the directivity of RQD: the directivity of RQD should be regarded as an advantage of the index because it essentially reflects the anisotropy of rock masses. Because a smaller value of the RQD index implies a better reproduction of the different jointing degrees of different rock masses, its minimum value is selected as the representative RQD (RQDr), and its corresponding direction is called as the representative direction of RQD. In this study, to obtain the RQDr and its representative direction, theoretical formulas are derived to calculate the RQD in any direction based on some known RQDs in different directions. The proposed formulas consider the variance of the discontinuity orientations and sizes for a discontinuity set, use the traditional concept of RQD, and possess the associated simplicity; they only require sampling a few additional discontinuity orientations. Therefore, the proposed approach is easy to understand and apply in engineering practices. An anisotropy index of jointing degree (AIjd) is defined to quantitatively describe the anisotropy of jointing degree for rock masses. The range of AIjd is from 0 to 100, and the larger value of AIjd reflects stronger anisotropy of jointing degree of a rock mass. In this study, a hypothetical three-dimensional discontinuity network is constructed by performing Monte Carlo simulations, based on which the generated data are analyzed to validate the proposed method. The comparison between the calculated and actual values of the RQD shows that (a) the difference between the calculated and actual values of the RQD is <4 for >86.0% of the 1368 scanlines examined with respect to 12 scanline groups, except SG4–1 group, for which the proportion is 66.2%; (b) the difference between the calculated and actual values of the RQD is <6 for >96.2% of the 1368 scanlines examined with respect to 12 scanline groups, except SG4–1, for which the proportion is 86.7%; and (c) the proposed method of calculating the RQD in any direction based on some known RQDs in different directions is satisfactory. In addition, the comparisons show that the calculated RQDr and representative direction exhibit a good agreement with the actual results, and hence, the proposed method for this purpose is satisfactory. Moreover, an actual rock mass of a slope around the Qingshan Lake National Forest Park in China is investigated, and the results show that the proposed method is valid for that rock mass.

Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface

Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface

Multi-slicing strategy for the three-dimensional discontinuity layout optimization (3D DLO).

SummaryDiscontinuity layout optimization (DLO) is a recently presented topology optimization method for determining the critical layout of discontinuities and the associated upper bound limit load for plane two‐dimensional and three‐dimensional (3D) problems. The modelling process (pre‐processing) for DLO includes defining the discontinuities inside a specified domain and building the target function and the global constraint matrix for the optimization solver, which has great influence on the the efficiency of the computation processes and the reliability of the final results. This paper focuses on efficient and reliable pre‐processing of the discontinuities within the 3D DLO and presents a multi‐slicing strategy, which naturally avoids the overlapping and crossing of different discontinuities. Furthermore, the formulation of the 3D discontinuity considering a shape of an arbitrary convex polygon is introduced, permitting the efficient assembly of the global constraint matrix. The proposed method eliminates unnecessary discontinuities in 3D DLO, making it possible to apply 3D DLO for solving large‐scale engineering problems such as those involving landslides. Numerical examples including a footing test, a 3D landslide and a punch indentation are considered, illustrating the effectiveness of the presented method. © 2016 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.

Numerical modeling of discontinuous rock slopes utilizing the 3DDGM (three-dimensional discontinuity geometrical modeling) method

The geometry of discontinuities in a rock mass is one of the most important influences on the behavior and characteristics of that rock mass. The geometry of discontinuities largely determines the stability of the rock mass, as well as appropriate methods for reinforcing and stabilizing that mass. This study introduces the 3DDGM (three-dimensional discontinuity geometrical modeling) method, which is based on the 3DGM (three-dimensional geometrical modeling) algorithm that was developed using the Mathematica software package. The 3DDGM method provides essential input data for the stability analysis of a discontinuous rock mass using block stability assessment techniques or block modeling codes. The 3DDGM method developed in the present work was designed to model discontinuities in rock masses and to provide accurate values for discontinuity parameters (i.e., location, spacing, separation, system, orientation, etc.). This algorithm was developed to increase the accuracy of the discontinuity model based on the Heliot algorithm. The 3DDGM algorithm was tested by applying it to a real case, the sloping discontinuous rock mass at the phase 7 gas flare site in the South Pars Gas Complex in Assalouyeh, Iran, and the algorithm was successful in providing a three-dimensional model of the discontinuities in the rock mass at the site.

Assessing the ability of rock masses to support block breakage at the TBM cutter face

In order for tunnel boring machines to efficiently cut or break rock, it is necessary that the block of rock in contact with the cutter be adequately supported by the surrounding rock mass. This support is provided by the interlocking of blocks and the friction of the surfaces. If blocks are inadequately supported or become free without breakage the result can be jamming at the TBM face. Such blocky ground conditions are typically assessed according to the spacing and orientation of discontinuities (including joints) within the rock mass, typically using a rock mass classification system. In laboratory tests on cuttability or abrasivity of rocks, test samples are typically supported securely in a frame or jig. Numerical models of rock breakage also assume boundary conditions in which the sample is completely supported. Therefore the applicability of the results from laboratory and numerical studies depends on the same degree of support of blocks in the ground. The conditions required to adequately support a block for breakage are investigated and related to rock mass parameters, in particular, the three-dimensional patterns of discontinuities. A rock mass can be capable of providing adequate support to a block of rock such that the cuttability is adequately described by conventional methods. However, there are some rock mass conditions where support of blocks is not well developed, potentially resulting in otherwise unexpected poor TBM progress or jamming of TBM with loose blocks. Three-dimensional discontinuity patterns can be assessed using stereographic methods or borehole (α–β) methods. It is proposed that problematic conditions may occur where: two or more oblique (α between 20° and 70°) discontinuity sets are present (and over-represented relative to a uniform distribution); one or more of these discontinuity sets are dipping into the opening (β=180°±90°) and additional discontinuities (in sets or randomly oriented) are present to form complete tetrahedral wedge blocks.

Approximation schemes of stresses on elements for the three-dimensional displacement discontinuity method

The displacement discontinuity method is a boundary element method. It uses the analytical expressions for displacements and stresses in an infinite isotropic homogeneous linear elastic body caused by difference (discontinuity) of displacements across small planar crack surfaces. The basic solution of the method is the displacement discontinuities (DDs) across the crack elements. After DDs are obtained, the displacement and stresses at other points in the body can be calculated. It discretises the crack without considering the individual surface of the crack, thus for crack propagation issues, it uses fewer (half) number of elements than normal BEM and therefore less computation time and computer memory requirement. However, it is found that the stresses calculated from the DDs for points on and close to the crack have large errors. Here we present two numerical schemes for approximation of stresses on the crack elements in three-dimensional problems, which are implemented in a code for fracture propagation. The schemes give a reasonably accurate approximation for elements where the crack surface is relatively smooth. It is found that for elements next to sharp kinking or at the corner of a crack, the results from the schemes are not satisfactory. A modification is proposed for these cases.

The lithosphere–asthenosphere boundary and the tectonic and magmatic history of the northwestern United States

This study explores the properties of the lithosphere–asthenosphere boundary (LAB) and other shallow mantle discontinuities across the diverse geologic provinces of the northwest United States. Sp phases were used to image three-dimensional discontinuity structure by common conversion point stacking with data from 804 temporary and permanent broadband stations, including 247 from EarthScope's USArray Transportable Array. Substantial variations in mantle discontinuity structure are apparent over a variety of spatial and temporal scales. To the west of the Sevier Thrust Belt, a coherent Sp negative phase coincides with the LAB depth range inferred from tomographic models. To the east, within the stable craton, multiple negative phases typically occur in the high velocity lithospheric layer, although in places a single mid-lithospheric discontinuity is imaged. Sub-cratonic LAB phases are often absent, indicating an LAB velocity gradient that is distributed over >50 km and that is consistent with effects of temperature alone. Where weak and intermittent LAB phases appear, they suggest more vertically localized velocity gradients produced by other factors such as bulk composition, volatile content, or contrasts in grain size or melt. In the tectonically active west, a positive Sp phase at depths consistent with the base of the asthenospheric low velocity zone in tomography models is intermittently observed. Beneath magmatic provinces in the west, the LAB Sp discontinuity deepens by ∼10 km from the High Lava Plains, where magmatism has occurred from 0–10.5 Ma, to the northern region of the Columbia River Basalts, which has been magmatically quiet since 15 Ma. Here we suggest a model in which the negative LAB velocity gradient is created by a layer of partial melt ponding beneath a solidus-defined boundary. This model predicts that higher temperatures associated with more recent magmatism would result in a shallowing of the intersection of the geotherm with the solidus. Beneath the Yellowstone Caldera, the absence of an LAB Sp phase suggests that the contrast in seismic velocity between the lithosphere and asthenosphere has been erased by intrusion of partial melt and heat into the lithosphere.

Fabrication of imitative stress corrosion cracking specimen using lithography and solid state bonding

This study proposes a method to fabricate artificial flaws whose response to non-destructive testing, especially electromagnetic non-destructive testing, is the same as that of stress corrosion cracking. Stacked metallic films containing through-grooves are sandwiched between metallic blocks and then bonded using solid state bonding. The profiles of the through-grooves are accurately controlled with the aid of the lithography technique, which allows a three-dimensional discontinuity to be embedded inside a material. Since the spatial distribution of the discontinuity is small compared to the spatial resolution of non-destructive testing methods, the method allows the response of stress corrosion cracking to be simulated. Specimens are fabricated using type 316 L austenitic stainless steel, and validated using eddy current testing.

Hybrid technique for analysing metallic waveguides containing isotropic chiral materials

A novel full-wave hybrid technique was formulated to treat three-dimensional discontinuities in isotropic chiral materials present in rectangular waveguides with perfect electrically conducting walls. Our formulation comprises (i) the coupled-mode method for obtaining the electromagnetic field in each section that is homogeneous in the propagation direction and (ii) the mode-matching method for obtaining the generalised scattering matrix of each discontinuity (formed by two adjacent homogeneous sections). In particular, we reformulated the coupled-mode method based on appropriate representations of E and B. This formulation ensures the satisfaction of the boundary conditions of the magnetic field on perfect electrically conducting walls, which is not ensured by the conventional formulation of the coupled-mode method as it is based on representations of E and H. A new expansion for H emerges indirectly from our formulation. Knowledge of this expansion is required by the mode-matching method for incorporating the effects of discontinuities between adjacent homogeneous sections. Calculations confirmed that our new formulation delivers correct results, whereas the conventional formulation may not.

On local tracking algorithms for the simulation of three-dimensional discontinuities

The present manuscript focuses on the algorithmic treatment of three-dimensional discontinuities within a purely displacement based finite element setting. In contrast to two-dimensional cracks, the local element based geometric representation of three-dimensional crack surfaces is non-unique and thus not straightforward. Accordingly, we compare different crack tracking strategies, one being algorithmically extremely efficient but yet somehow restrictive, the other one being more complex but rather general in nature. While the first method is able to represent entirely smooth discontinuity surfaces, the second approach introduces inter-element discontinuities in the overall crack surface representation. Both methods are compared systematically and additional comments about the algorithmic realization are provided. From the numerical results we conclude that neither of the two algorithms is able to solve all defined quality criteria satisfactorily, although both are mesh independent, computationally cheap and rather efficient. The ultimate solution might be an overall global crack surface representation that a priori circumvents a number of algorithmic deficiencies and at the same time provides a unique and smooth three-dimensional crack surface representation.

On two different formulations of the Coupled Mode Method: Application to 3D rectangular chirowaveguides

This paper develops two different formulations of the Coupled Mode Method (CMM), called EH-indirect and EH-direct formulations, for rectangular waveguides nonhomogeneously loaded with bi-isotropic media. The results of the EH-indirect formulation are compared with the more usual EH-direct formulation, and its range of applicability is established. When the EH-indirect or EH-direct formulations of the CMM are combined with the classical Mode Matching Method (MMM) a hybrid technique is obtained that is capable of solving three-dimensional (3D) discontinuities without restriction in number. The results of the generalized scattering matrix for the two formulations are compared and it is concluded that the indirect formulation gives more accurate results than the direct formulation.

P-Type Multiplicative Schwarz (pMUS) Method With Vector Finite Elements for Modeling Three-Dimensional Waveguide Discontinuities

This paper presents the application of a p-type multiplicative Schwarz (pMUS) method for solving three-dimensional waveguide discontinuity problems. The two major contributions of the proposed pMUS method are: 1) the use of hierarchical curl-conforming basis functions that incorporate a discrete Hodge decomposition explicitly and 2) the treatment of each polynomial space (or basis functions group) as an abstract grid/domain in the Schwarz method. These two features greatly improve the applicability of the curl-conforming vector finite-element methods (FEMs) for solving Maxwell equations. Various numerical examples are solved using the proposed approach. The performance of the pMUS method has been compared to commercial FEM software as well as the incomplete Choleski conjugate gradient method. It is found that the pMUS method exhibits superior efficiency and consumes far less memory and CPU times.

Application of the short‐open calibration technique to vector finite element method for analysis of microwave circuits

AbstractA novel numerical de‐embedding scheme called the short‐open calibration (SOC) technique, in conjunction with the vector finite element method (FEM), has been developed to characterize two‐port network of arbitrarily shaped, three‐dimensional discontinuities in microwave circuits. This SOC technique is effectively implemented into the FEM for boundary truncation of the unbounded circuit structures. In such a manner, fast convergence of iterative solver for large‐sparse linear matrix equations from FEM is achieved. The SOC technique is used to remove parasitic effects brought by the approximation of the impressed voltage source and also the problem of resulting consistency between the two‐ and three‐dimensional simulations. Scattering parameters of discontinuous sections are constructed from the definition of port voltages and currents. Numerical solutions are well compared with those published in the available literatures. It is demonstrated that the features of the SOC technique are advantageous when combined with FEM for electromagnetic problems. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Combined Scattering Matrix and Boundary Integral Equation Method for the Analysis and Field Computation in Three-Dimensional Multiport Waveguide Discontinuities

This paper discusses the analysis and numerical computation of time-harmonic electromagnetic fields in arbitrarily shaped three-dimensional waveguide discontinuities and junctions. The waveguides meeting at the junction may have different cross-sections. The developed method combines the boundary integral equation method, used for the discontinuities, with the analyticalwaveguide solutions. The RWG-functions are applied as basis and test functions in the method of moments solution of the electric field integral equation. In addition, all field solutions on the waveguide discontinuities are efficiently characterized by the scattering matrix of a multiport waveguide system. In a two-port case, for example, only two field solutions are required to determine all possible wave forms. The accuracy of the developed method is verified by considering numerical examples.