Single Discontinuity Research Articles

Tributary mitigates river discontinuity by dam depending on the distance from dam to the tributary confluence

AbstractRiver ecosystems host a continuum of various discontinuities, such as dams and tributary inflows, which shape the habitats of aquatic organisms; this makes it important to evaluate the interactions among discontinuities and effects of a single discontinuity point. Herein, we focused on tributaries in dammed rivers to elucidate a manner to mediate discontinuities and hypothesized that the effects of the tributary inflow on the downstream river were mediated by the (1) distance between the dam and tributary inflow, (2) size ratio between the dam and tributary, and (3) ability of the tributary to transport fine sediments, such as sand and gravel. We conducted field investigations to measure 24 environmental factors, including the water column, riverbed, and river topography, along eight dammed rivers to test the three hypotheses. Only five factors differed significantly between the upstream and downstream sites of the tributary confluence, whereas 12 factors differed significantly between the tributary and mainstem. We also classified the variation pattern using principal component analysis (PCA). The correlation between each PCA axis and environmental factors suggests that the modifications of the water quality factors by tributary inflow were significantly correlated with the distance between the dam and the tributary inflow, and that the flux of various riverbed factors, that is, was significantly correlated with the ability to resupply of fine riverbed sediments by the tributary. Overall, tributary‐inflow downstream of a dam has the potential to be applicable in environmental restoration, especially in terms of resupplying fine sediment.

Tunneling-induced longitudinal responses of existing tunnel with discontinuities in structure stiffness

Owing to a change in construction method, the existence of a reinforcement section, or a settlement joint, an existing tunnel is likely to have discontinuities in its longitudinal stiffness. An analytical solution for the new tunneling-induced responses of the existing tunnel with stiffness discontinuities is derived in this paper. The existing tunnel is modelled as a series of Timoshenko beams connected by elastic springs for considering various connection types and possible discontinuous deformations at the locations of the stiffness discontinuities. The accuracy and applicability of the proposed solution are successfully verified by numerical models. Two hypothetical examples are used to further study the normalized response of the existing tunnel with single stiffness discontinuity or with a reinforcement section. The results show that when the location of the new tunnel is around the position of the stiffness discontinuity, the normalized internal force of the small-stiffness section of the existing tunnel becomes larger. The reinforcement length and the longitudinal stiffness of the reinforcement section should match each other to control abrupt increases in the normalized internal force and ensure the effectiveness of the reinforcement measures.

Conformal dispersion relations for defects and boundaries

We derive a dispersion relation for two-point correlation functions in defect conformal field theories. The correlator is expressed as an integral over a (single) discontinuity that is controlled by the bulk channel operator product expansion (OPE). This very simple relation is particularly useful in perturbative settings where the discontinuity is determined by a subset of bulk operators. In particular, we apply it to holographic correlators of two chiral primary operators in \mathcal{N}=4𝒩=4 Super Yang-Mills theory in the presence of a supersymmetric Wilson line. With a very simple computation, we are able to reproduce and extend existing results. We also propose a second relation, which reconstructs the correlator from a double discontinuity, and is controlled by the defect channel OPE. Finally, for the case of codimension-one defects (boundaries and interfaces) we derive a dispersion relation which receives contributions from both OPE channels and we apply it to the boundary correlator in the O(N) critical model. We reproduce the order \epsilon^2ϵ2 result in the \epsilonϵ-expansion using as input a finite number of boundary CFT data.

Analytical solution and free vibration analysis for beams with step changes in stiffness

PurposeThis paper aims to investigate analytical solutions of natural frequencies and mode shapes of Euler-Bernoulli beams with step changes in the stiffness.Design/methodology/approachIn this work, analytical solutions for a beam with a single discontinuity was performed. Subsequently, based on an effective matrix formulation, the closed-form expressions of the single discontinuity beam could be conveniently extended to stepped beams with multiple stiffness discontinuities.FindingsThe results of the study show that the natural frequency of the beam can be adjusted by the local stiffness variation, and step location plays a significant role in free vibration responses.Originality/valueThe effects of the stiffness of the segment and step location on the natural frequencies of the stepped beams under different boundary conditions were examined using the proposed analytical scheme. This study provides insights into the design of variable-stiffness beam structures with the capability to adjust natural frequencies.

Sliding modes of high codimension in piecewise-smooth dynamical systems

We consider piecewise-smooth dynamical systems, i.e., systems of ordinary differential equations switching between different sets of equations on distinct domains, separated by hyper-surfaces. As is well-known, when the solution approaches a discontinuity manifold, a classical solution may cease to exist. For this reason, starting with the pioneering work of Filippov, a concept of weak solution (also known as sliding mode) has been introduced and studied. Nowadays, the solution of piecewise-smooth dynamical systems in and close to discontinuity manifolds is well understood, if the manifold consists locally of a single discontinuity hyper-surface or of the intersection of two discontinuity hyper-surfaces. The present work presents partial results on the solution in and close to discontinuity manifolds of codimension 3 and higher.

A dispersion relation for defect CFT

We present a dispersion relation for defect CFT that reconstructs two-point functions in the presence of a defect as an integral of a single discontinuity. The main virtue of this formula is that it streamlines explicit bootstrap calculations, bypassing the resummation of conformal blocks. As applications we reproduce known results for monodromy defects in the epsilon-expansion, and present new results for the supersymmetric Wilson line at strong coupling in mathcal{N} = 4 SYM. In particular, we derive a new analytic formula for the highest R-symmetry channel of single-trace operators of arbitrary length.

Experimental Study on the Effect of Single Structural Plane on the Stability of Shallow Tunnel Surrounding Rock

There are a large number of structural planes distributed in the surrounding rock of a tunnel, and this is one of the key factors causing a tunnel’s instability. Due to different geological and historical conditions, the distribution characteristics and the occurrence of structural planes in the rock mass also have significant differences. In engineering, it is common to encounter structural planes that cut across the tunnel section and have a significant impact on the stability. The occurrence of structural planes is a key factor controlling the mechanical behavior of the surrounding rock. Based on this, laboratory uniaxial compression tests were carried out by constructing a small tunnel physical model with single structural planes of different inclination angles. A related numerical simulation analysis was also carried out. This research indicates that: (1) Under the influence of a single structural plane, the dip direction with 30°–60° is the most dangerous situation, and when the dip angle of the structural plane is between 38 and 88°, it will slip along the structural plane. (2) According to the mechanical mechanism, there are three types of cracks: tensile cracks, shear cracks, and tensile shear cracks. According to the deformation characteristics, there are four types: tension, friction, bending, and shear. (3) There is a certain correlation between the strength of a single discontinuity rock mass and that of a multi-jointed rock mass. When the dip angle of the joints’ combination is 45°+60°, the rock mass is in its weakest state.

Formation of delta shock and vacuum state for the pressureless hydrodynamic model under the small disturbance of traffic pressure

The formation of delta shock and vacuum state for the pressureless hydrodynamic model is investigated in detail under the small disturbance of traffic pressure. Exact Riemann solutions for the perturbed system can be constructed explicitly for four different possible structures. Asymptotically, the perturbed Riemann solution involving two shocks will collapse to a single delta shock, and the perturbed Riemann solution involving two rarefaction waves will degenerate into a solution containing two contact discontinuities along with the vacuum state between them when the perturbed parameter goes to zero. It should be stressed here that the internal state in each of the two rarefaction wave-fans turns out to be the vacuum state gradually in such a limiting case, which differs obviously from the previous result that each of the two rarefaction wave-fans is compressed globally to be a single contact discontinuity. Additionally, some typical numerical results exhibiting the formation process of delta shock and vacuum state are presented to verify our theoretic results.

Failure strength and fracture characteristics of rock with discontinuity under indirect tension

Large-scale discontinuities can significantly affect the mechanical properties of rock masses. However, the tensile behavior of rock discontinuities is often less investigated. To study the statistical characteristics of failure strength and fracture characteristics of rock discontinuities, Brazilian disc tests were conducted on limestone specimens with a single natural discontinuity at different load-discontinuity angles ( β ). In this study, β = 0 ° and β = 90 ° correspond to the discontinuity parallel and perpendicular to loading direction, respectively. The results show that Brazilian failure strength (BFS) can reasonably represent the tensile strength of rock with discontinuities, by comparing the BFS and tensile stress in the disc center at peak force. The two-parameter Weibull distribution can capture the statistical BFS characteristics of rock discontinuities parallel to loading direction ( β = 0 ° ) and at different load-discontinuity angles ( β ≠ 0 ° ). All specimens with discontinuity at different load-discontinuity angles show more plastic deformational behaviour than intact rock specimen. With increasing β , the mean BFS of limestone with discontinuity increases before reaching a plateau at β = 45 ° . The single plane of weakness theory best explains the BFS of fractured limestone with β . Only a specific segment of pre-existing rock discontinuity could affect the fracture process. When β = 0 ° , interfacial cracks and alternative cracks formed. When β ≠ 0 ° , mixed failure mode with shear and tensile failure occurred, particularly when β = 30 ° and β = 60 ° . The findings can contribute to better understanding the failure and fracture characteristics of rock with discontinuities, particularly the interaction of pre-existing discontinuities with stress-induced fracturing.

A δ

Over the last decade, automatic differentiation (AD) has profoundly impacted graphics and vision applications --- both broadly via deep learning and specifically for inverse rendering. Traditional AD methods ignore gradients at discontinuities, instead treating functions as continuous. Rendering algorithms intrinsically rely on discontinuities, crucial at object silhouettes and in general for any branching operation. Researchers have proposed fully- automatic differentiation approaches for handling discontinuities by restricting to affine functions, or semi- automatic processes restricted either to invertible functions or to specialized applications like vector graphics. This paper describes a compiler-based approach to extend reverse mode AD so as to accept arbitrary programs involving discontinuities. Our novel gradient rules generalize differentiation to work correctly, assuming there is a single discontinuity in a local neighborhood, by approximating the prefiltered gradient over a box kernel oriented along a 1D sampling axis. We describe when such approximation rules are first-order correct, and show that this correctness criterion applies to a relatively broad class of functions. Moreover, we show that the method is effective in practice for arbitrary programs, including features for which we cannot prove correctness. We evaluate this approach on procedural shader programs, where the task is to optimize unknown parameters in order to match a target image, and our method outperforms baselines in terms of both convergence and efficiency. Our compiler outputs gradient programs in TensorFlow, PyTorch (for quick prototypes) and Halide with an optional auto-scheduler (for efficiency). The compiler also outputs GLSL that renders the target image, allowing users to interactively modify and animate the shader, which would otherwise be cumbersome in other representations such as triangle meshes or vector art.

Retaining an Upper Bound for the Capillary Retention of Two-Phase Flow in Porous Media

At the sub-meter scale, capillary retention occurs when two immiscible phases flow across two heterogeneous media. It happens in the upstream medium, and the phase retained depends on the heterogeneity order and the system wettability. This phenomenon has a major impact on the oil recovery and hence requires comprehension. The analysis of the spatial variation in the phase saturation along the flowing domain is well established. However, the literature lacks the general characteristics of capillary retention caused by such variation at various flowing conditions. The continuum model for the flow of two immiscible phases in porous media at steady-state conditions is examined. The analysis is limited to the medium with a single discontinuity in rock properties. In this examination, the characteristics of the capillary retention upstream of the discontinuity are the primary consideration. Three retention regions related to the flow regime are identified: two plateaus and a transition zone occurring, in general, at intermediate capillary number. The width and location of the transition depend on the fractional flow. The crucial finding is related to the identification of an upper bound for the capillary retention and its dependency on the capillary variable ratio and the form of the Leverett J-curve. Two J-models are investigated, and it is shown that, for a given capillary variable ratio, the upper bound depends on the level of flatness of the J-curve at intermediate saturations. Potential implications of the current analysis to reservoir characterization are also discussed.Article HighlightsThree retention regions related to the ow regime are identified: two plateaus and a transition zone occurring at intermediate capillary numbers. The width and position of the transition depend on the fractional flow.An upper bound for the capillary retention is derived and its dependency on the heterogeneity ratio and the form of the Leverett-J curve is discussed..Potential implication to reservoir characterization: In the case of steeper Leverett-J curves with low to moderate random heterogeneity ratios, the upper bound is minimal regardless of the various lithologies and their congurations. In such conditions, the upper bound can be used to assert the insignicance of the eld-scale capillary retention.

OPERATOR METHOD IN DIFFRACTION PROBLEM OF WAVES OF CIRCULAR WAVEGUIDE BY THE ANNULAR DISCONTINUITIES

Purpose. Diff raction problem of the H11 and E11 waves of circular waveguide by the fi nite and semi-infi nite system of similar discontinuities is considered. Four types of discontinuities are considered: iris, disk, ring, annular slot. Th e distance between irregularities is the same. Th e waveguide is fi lled by the dielectric with losses. Design/methodology/approach. To solve the problem we chose the operator method. While so-called key problem, the diff raction problem by a single discontinuity, is solved by the method of moments. The field in the domain of the obstacle is represented as a series in terms of eigenwaves of infi nite waveguide which cross section coincides with the cross section of discontinuity, with unknown amplitudes. To fi nd the amplitudes we obtain the infi nite system of equations, which is solved by the reduction. As a result, the transmission and refl ection operators of a key-problem are obtained. The properties of fi nite sequence are determined from the operator equations relatively amplitudes of the scattered field. To write these equations the iterative procedure is used. The properties of the structure, which consists of N discontinuities are obtained under assumption that the properties of the structure, which consists of N – 1 discontinuity are known. The reflection operator of the semi-infi nite system is obtained from known non-linear operator equation of the second kind. Findings. The systems of equations for every single discontinuity are obtained. The operator equations relatively amplitudes of the scattered fields for fi nite system are presented. With the help of the created model the dependences of the transmission, refl ection coeffi cients of the H11 wave as well as transformation coeffi cients of the H11 to E11 wave of the circular waveguide on the wavenumber, geometrical and material parameters for fi nite and semi-infi nite structures are studied. The comparison of the results obtained by presented method with the results obtained in HFSS is made. Conclusions. With the use of the operator method the scattering model of the H1n and E1n eigenwaves by the system of annular discontinuities of zero thickness in a circular waveguide is built. Th e study of the scattering characteristics on the parameters is made. The coincidence of the behavior of the curves obtained by the presented method with those obtained in HFSS allows us to draw a conclusion about the correctness of the results. Th e results can be used during creation of a series of the microwaves and optic devices.

Hybrid high-order steepness-adjustable harmonic scheme based on combined discontinuity sensors

Extending discontinuity-capturing schemes to high resolution using a hybrid approach has become popular in recent times. The hybrid schemes reported in literature usually combine high-order compact/explicit-schemes in smooth regions and the weighted essentially non-oscillatory (WENO) scheme in discontinuities by a discontinuity sensor. However, there are two major concerns: (1) the significant dissipation of the WENO scheme will smear the contact discontinuities, and (2) the accuracy of the discontinuity sensor will significantly influence the performance of the hybrid scheme. However, these two aspects are typically coupled. Therefore, in this study, we attempt to replace the discontinuity-capturing schemes (WENO) with discontinuity-preserving schemes (steepness-adjustable harmonic (SAH) scheme). Then, we extend the SAH scheme to high-order accuracy by hybridizing it with a linear fourth-degree polynomial (P4) scheme. The new P4-SAH scheme was employed to evaluate the nine different discontinuity sensors. The numerical results revealed that a single discontinuity sensor can hardly adapt to different numerical cases. Therefore, we propose a new class of discontinuity sensors that combines MP-WGVC discontinuity sensors. Several classical one- and two-dimensional Euler equation benchmark tests were simulated. The numerical results demonstrate that the new hybrid P4-SAH scheme with a combined MP-WGVC sensor not only sharply captures shock waves and contact discontinuities, but also capture small vortex structures.

Bootstrapping holographic defect correlators in mathcal{N} = 4 super Yang-Mills

We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in mathcal{N} = 4 SYM. We use modern analytic bootstrap in the form of an inversion formula, that reconstructs the CFT data starting from a single discontinuity of the correlator. In the planar strong coupling limit only a finite number of conformal blocks contributes to the discontinuity, which allows us to obtain elegant closed-form expressions for two-point functions of single-trace operators mathcal{O} J of weight J = 2, 3, 4. Our bootstrap approach guarantees that the correlator satisfies all the expected properties: it has the correct discontinuity, it satisfies the superconformal Ward identities, and it has a sensible expansion in both defect and bulk OPEs. The result is also consistent with available results coming from localization. The method is completely algorithmic and can be implemented to calculate correlators of arbitrary weight.

Variational Stress Analysis of Laminates Containing Regions with Different Material Properties in Loading Direction

In this study, a variational stress analysis was performed on a laminate that contained regions with varying material properties in the loading direction. Investigation of material properties and damage behavior of laminates with fiber discontinuity (resin-rich region) has been the main motivation for this study. The analysis considered an arbitrary number of regions with different material properties. For simplicity, we assumed a plane stress condition or stress distribution, we assumed that the normal stress in the loading direction was constant along the direction of thickness in each ply. The admissible stress state that satisfied both equilibrium and boundary/continuity conditions was constructed with unknown functions that were determined using the principle of minimum complementary energy. The analysis was applied to laminates with single fiber discontinuity and with distributed fiber discontinuities. These results were compared using finite element analyses, which obtained good agreement. The results show the validity of this analysis, which can be used to investigate the effects of fiber discontinuity on the laminate mechanical properties and damage behavior in laminates with fiber discontinuities.

Boosting the Model Discovery of Hybrid Dynamical Systems in an Informed Sparse Regression Approach

Abstract We present an efficient data-driven sparse identification of dynamical systems. The work aims at reconstructing the different sets of governing equations and identifying discontinuity surfaces in hybrid systems when the number of discontinuities is known a priori. In a two-stage approach, we first locate the switches between separate vector fields. Then, the dynamics among the manifolds are regressed, in this case by making use of the existing algorithm of Brunton et al. (2016, “Discovering Governing Equations From Data by Sparse Identification of Nonlinear Dynamical Systems,” Proc. Natl. Acad. Sci., 113(15), pp. 3932–3937). The reconstruction of the discontinuity surfaces comes as the outcome of a statistical analysis implemented via symbolic regression with small clusters (microclusters) and a rigid library of models. These allow to classify all the feasible discontinuities that are clustered and to reduce them into the actual discontinuity surfaces. The performances of the sparse regression hybrid model discovery are tested on two numerical examples, namely, a canonical spring-mass hopper and a free/impact electromagnetic energy harvester (FIEH), engineering archetypes characterized by the presence of a single and double discontinuity, respectively. Results show that a supervised approach, i.e., where the number of discontinuities is pre-assigned, is computationally efficient and it determines accurately both discontinuities and set of governing equations. A large improvement in the time of computation is found with the maximum achievable reliability. Informed regression-based identification offers the prospect to outperform existing data-driven identification approaches for hybrid systems at the expense of instructing the algorithm for expected discontinuities.

A generalized machine learning workflow to visualize mechanical discontinuity

Accurate detection and mapping of mechanical discontinuity in materials has widespread industrial and research applications. We developed a generalized machine-learning framework for visualizing single mechanical discontinuity embedded in material of any composition, velocity, density, porosity, and size with limited data. The proposed visualization of discontinuity requires accurate estimations of the length, location, and orientation of the embedded discontinuity by processing multipoint wave-transmission measurements. k-Wave simulator is used to create a large dataset of elastic waveforms recorded during multi-point wave-transmission measurements through materials containing single mechanical discontinuity. k-Wave simulator considers the wave attenuation, dispersion, and mode conversion in wave motion. Discrete wavelet transform (DWT) and statistical feature extraction are essential for data preprocessing prior to the data-driven model development. DWT also minimizes the effect of noise. Using hyper-parameter tuning and cross validation, gradient boosting regression can visualize the mechanical discontinuity with an accuracy of 0.85, in terms of coefficient of determination. A double-layered neural network-based regression has better performance with an accuracy of 0.95. Use of convolutional neural network converts the predictive task from a waveform processing to an image processing problem. Convolutional neural network achieved a generalization performance of 0.91. The proposed generalized workflow requires robust simulation of wave propagation, signal processing, feature engineering, and model evaluation. Sensors closest to the source and those located opposite the source are the most significant for the desired visualization. Notably, the sensors closest to the source capture the non-linear associations, whereas the sensor on the border opposite to the source capture the linear associations between the measured waveforms and the properties of the mechanical discontinuity.

Phase‐field modeling of rock fractures with roughness

AbstractPhase‐field modeling—a continuous approach to discontinuities—is gaining popularity for simulating rock fractures due to its ability to handle complex, discontinuous geometry without an explicit surface tracking algorithm. None of the existing phase‐field models, however, incorporates the impact of surface roughness on the mechanical response of fractures—such as elastic deformability and shear‐induced dilation—despite the importance of this behavior for subsurface systems. To fill this gap, here we introduce the first framework for phase‐field modeling of rough rock fractures. The framework transforms a displacement‐jump‐based discrete constitutive model for discontinuities into a strain‐based continuous model, without any additional parameter, and then casts it into a phase‐field formulation for frictional interfaces. We illustrate the framework by constructing a particular phase‐field form employing a rock joint model originally formulated for discrete modeling. The results obtained by the new formulation show excellent agreement with those of a well‐established discrete method for a variety of problems ranging from shearing of a single discontinuity to compression of fractured rocks. It is further demonstrated that the phase‐field formulation can well simulate complex crack growth from rough discontinuities. Consequently, our phase‐field framework provides an unprecedented bridge between a discrete constitutive model for rough discontinuities—common in rock mechanics—and the continuous finite element method—standard in computational mechanics—without any algorithm to explicitly represent discontinuity geometry.

Asymptotical quasinormal mode spectrum for piecewise approximate effective potential

It was pointed out that the black hole quasinormal modes resulting from a piecewise approximate potential are drastically distinct from those pertaining to the original black hole metric. In particular, instead of lining up parallel to the imaginary axis, the spectrum is found to stretch out along the real axis. In this work, we prove that if there is a single discontinuity in the effective potential, no matter how insignificant it is, the asymptotical behavior of the quasinormal modes will be appreciably modified. Besides showing numerical evidence, we give analytical derivations to support the above assertion even when the discontinuity is located significantly further away from the maximum of the potential and/or the size of the step is arbitrarily small. Moreover, we discuss the astrophysical significance of the potential implications in terms of the present findings.

An accessible calculation method of the bearing capacity of shallow foundations on anisotropic rock masses

Numerical modeling of the bearing capacity of foundations on rock masses having a high degree of anisotropy needs to represent, in most cases, a large number of discontinuities, thus making the numerical analysis difficult and, in most cases, unapproachable.In this research, a simplified numerical procedure by using the Discontinuity Layout Optimization method (DLO) which proves to be appropriate when studying the limit bearing capacity of foundations involving slip line formation and failure wedges is presented.A single discontinuity simplified procedure is proposed maintaining the accuracy of the full model. The isolated discontinuity is located in the proper position and has the correct slope to represent the whole family of discontinuities.DLO models using the proposed procedure can represent the different possible failure mechanisms that may occur in the rock mass. Besides, the ultimate bearing capacity of a particular case is checked against the available analytical solution.