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A function f: ℝ → ℝ is internally cliquish (internally quasi-continuous) if it is cliquish (quasi-continuous) and the set of discontinuity points of f is nowhere dense. We prove that the set of int...

The normal impingement of the rotational stagnation-point flow on a surface executing bi-axial stretching is studied. This problem is governed by two parameters, α the dimensionless stretching along the x -axis, and β the dimensionless stretching along the y -axis. Consideration is given to both stretching and shrinking surfaces. At fixed β turning points are found at α = α c for which no solutions exist for α α c and dual solutions exist for all α > α c . Shear stress results are presented in graphical form as are sample similarity velocity profiles. The boundary for the existence of solutions is presented over a range of the governing parameters. Shrinking in both directions occurs only over the relatively small region − 1 . 212 ≤ α 0 and − 1 . 212 ≤ β 0 . An investigation of the zero stress values of F ′ ′ ( 0 ) reveals a bifurcation in solutions for β = − 0 . 075 at α = − 0 . 07675 that leads to triple solutions for α ≥ 0 . 21 . This bifurcation is responsible for a discontinuity that appears in the lower branch of the F ′ ′ ( 0 ) = 0 curve at this value of β .

The problem of plane discontinuities of relatively small amplitude propagating in a layer of nonuniformly charged gas in the variable electric and constant gravity fields is considered within the framework of ideal electrohydrodynamics model. The transport equations of variational nature are derived for their amplitudes and an exact solution of the problem of weak discontinuity is given. Then, the problem of weak shock wave is solved in the approximation to the solution obtained. Examples of propagation of a wave through an arbitrary equilibrium initial state and along a gas layer moving as a solid body in a variable electric field are considered.

We obtain a general result on the constancy of separately continuous mappings and their analogs, which yields the well-known Sierpinski theorem. By using this result, we study the set of continuity points of separately continuous mappings with at most countably many range of values including, in particular, the mappings of the square of Sorgenfrey line into the Bing plane.

Cost and other practically related reasons can mean that velocity sensors are not available on an underwater vehicle. For such cases, the results in this brief are developed on an observer-based fault-tolerant control for underwater vehicles in the presence of external disturbances and unknown thruster faults. An adaptive sliding mode observer is developed to achieve finite-time convergence where, in comparison to a high-gain-based design for the observer, a nonlinear feedback is constructed based on the position estimation error. Unlike alternatives, a discontinuity term in the developed fault tolerant controller is avoided, and the stability of the controlled dynamics is characterized using the Lyapunov theory. Finally, these new results are supported by both a simulation-based study and experimental verification.

Also after many years of research, stereo matching remains to be a challenging task in photogrammetry and computer vision. Recent work has achieved great progress by formulating dense stereo matching as a pixel-wise learning task to be resolved with a deep convolutional neural network (CNN). However, most estimation methods, including traditional and deep learning approaches, still have difficulty to handle real-world challenging scenarios, especially those including large depth discontinuity and low texture areas. To tackle these problems, we investigate a recently proposed end-to-end disparity learning network, DispNet (Mayer et al., 2015), and improve it to yield better results in these problematic areas. The improvements consist of three major contributions. First, we use dilated convolutions to develop a context pyramidal feature extraction module. A dilated convolution expands the receptive field of view when extracting features, and aggregates more contextual information, which allows our network to be more robust in weakly textured areas. Second, we construct the matching cost volume with patch-based correlation to handle larger disparities. We also modify the basic encoder-decoder module to regress detailed disparity images with full resolution. Third, instead of using post-processing steps to impose smoothness in the presence of depth discontinuities, we incorporate disparity gradient information as a gradient regularizer into the loss function to preserve local structure details in large depth discontinuity areas. We evaluate our model in terms of end-point-error on several challenging stereo datasets including Scene Flow, Sintel and KITTI. Experimental results demonstrate that our model decreases the estimation error compared with DispNet on most datasets (e.g. we obtain an improvement of 46% on Sintel) and estimates better structure-preserving disparity maps. Moreover, our proposal also achieves competitive performance compared to other methods.

The present paper deals with the localization of strains in those structures consisting of materials exhibiting plastic softening response. It is assumed that strain localization develops in a finite thickness band separated from the remaining part of the structure by weak discontinuity surfaces. In view of the small thickness of the band with respect to the dimensions of the structure, the interphase concept is used for the mechanical modeling of the localization phenomenon. We propose a formulation for the quasi-static modeling of strain localization based on a phase-field approach. In this sense, the localization band is smeared over the volume of the structure and a smooth transition between the fully broken and the sound material phases is introduced. The mechanical performance of the model is illustrated for the case of uniaxial tensile test, discussing the instability of the force-displacement response, and for the case of three-point bending test, comparing the analytical results with the experimental ones.

We study properties of functions with bounded variation in Carnot-Caratheodory spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R , we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative.

We consider six variants of (local) Whittle estimators of the fractional order of integration d. They follow a limiting normal distribution under stationarity as well as under (a certain degree of) nonstationarity. Experimentally, we observe a lack of continuity of the objective functions of the two fully extended versions at $$d=1/2$$ that has not been reported before. It results in a pileup of the estimates at $$d=1/2$$ when the true value is in a neighborhood to this half point. Consequently, studentized test statistics may be heavily oversized. The other four versions suffer from size distortions, too, although of a different pattern and to a different extent.

A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation. The idea gives a design of a special heat flux on the surface of the body such that from the corresponding temperature field on the surface one can extract the smallest radius of the sphere centered at an arbitrary given point in the whole space and enclosing unknown inclusions. Unlike before, the designed flux is free from a large parameter. An application of the idea to a coupled system of the elastic wave and heat equations are also given., 23 pages, the final version

Purpose This paper aims to proposes and analyzes a novel recovery-based posteriori error estimator for the stationary natural-convection problem based on penalized finite element method. Design/methodology/approach The optimal error estimates of the penalty FEM are established by using the lower-order finite element pair P1-P0-P1 which does not satisfy the discrete inf-sup condition. Besides, a new recovery type posteriori estimator in view of the gradient recovery and superconvergent theory to deal with the discontinuity of the gradient of numerical solution. Findings The stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations. Originality/value The provided reliability and efficiency analysis is shown that the true error can be effectively bounded by the recovery-based error estimator.

This paper developed a novel circular trajectory weaving welding control algorithm based on the space transformation principle. The three-dimensional weaving control problem was simplified into two-dimensional by establishing a basic plane. First, the transformation matrix between the basic plane and the weaving plane was solved by using the teaching parameters. The basic weaving point in the basic plane was obtained by using multiple weaving parameters. The basic weaving point was mapped to the weaving plane by the transformation matrix to obtain the final trajectory for the weaving welding. The above algorithm in combination with Ferguson Smooth algorithm was used as a novel control strategy which can effectively solve the problem of velocity discontinuity at sharp corners. The simulation and welding experiment results showed that this strategy can process more input weaving parameters and generate a more accurate circular trajectory. Therefore, this weaving welding control algorithm has a high practical value.

The cycling safety research literature has proposed methods to analyse safety and case studies to better understand the factors that lead to cyclist crashes. Surrogate measures of safety (SMoS) are being used as a proactive approach to identify severe interactions that do not result in an accident and interpreting them for a safety diagnosis. While most cyclist studies adopting SMoS have evaluated interactions by counting the total number of severe events per location, only a few have focused on the interactions between general directions of movement e.g. through cyclists and right turning vehicles. However, road users perform maneuvers that are more varied at a high spatiotemporal resolution such as a range of sharp to wide turning movements. These maneuvers (motion patterns) have not been considered in past studies as a basis for analysis to identify, among a range of possible motion patterns in each direction of travel, which ones are safer, and which are more likely to result in a crash. This paper presents a novel movement-based probabilistic SMoS approach to evaluate the safety of road users' trajectories based on clusters of trajectories representing the various movements. This approach is applied to cyclist-vehicle interactions at two locations of cycling network discontinuity and two control sites in Montréal. The Kruskal-Wallis and Kolmogorov-Smirnov tests are used to compare the time-to-collision (TTC) distribution between motion patterns in each site and between sites with and without a discontinuity. Results demonstrate the insight provided by the new approach and indicate that cyclist interactions are more severe and less safe at locations with a cycling network discontinuity and that cyclists following different movements have statistically different levels of safety.

In this work, we developed a code of finite volume method for 1D unsteady Euler equations to carry out the flow field problem due to moving contact discontinuity interface between two different species on both sides of the shock tube. The derived additional energy conservation equation would produce a non-conservative term. It is important to reduce the non-physical numerical oscillation problem and improve the capturing-shock ability and computing accuracy of high resolution problem. This study focuses on the following two parts. The first part, we expanded the energy equation of unsteady Euler equations into the form of i species. Further, we verified the feasibility and reliability of the new numerical energy model under the dual-species condition, and then we executed a series of systematic research to carry out the some experimental verification. Moreover, we adopted the most suitable numerical scheme for new Model B3 and validated with the old energy model. Second, we developed 9 new models of flux limiter and validated these limiters through the condition of dual species. For the validated results, the new energy Model B3 can effectively improve the numerical oscillations and capture phenomenon quite well, which just adopt the lower-order difference method. The validated results of new developed limiter models show that their capturing effects of some limiters are very well.

Many techniques were developed for creating true color images from satellite solar reflective bands, and the so-derived images have been widely used for environmental monitoring. For the newly launched Fengyun-3D (FY-3D) satellite, the same capability is required for its Medium Resolution Spectrum Imager-II (MERSI-II). In processing the MERSI-II true color image, a more comprehensive processing technique is developed, including the atmospheric correction, nonlinear enhancement, and image splicing. The effect of atmospheric molecular scattering on the total reflectance is corrected by using a parameterized radiative transfer model. A nonlinear stretching of the solar band reflectance is applied for increasing the image contrast. The discontinuity in composing images from multiple orbits and different granules is eliminated through the distance weighted pixel blending (DWPB) method. Through these processing steps, the MERSI-II true color imagery can vividly detect many natural events such as sand and dust storms, snow, algal bloom, fire, and typhoon. Through a comprehensive analysis of the true color imagery, the specific natural disaster events and their magnitudes can be quantified much easily, compared to using the individual channel data.

In this study we present a method for approximating the solution of a Singularly Perturbed Boundary Value Problem (SPBVP) containing two parameters ( e 1 , e 2 ), which multiply the diffusion coefficient and the convection term, respectively. Moreover, we consider that the convection coefficient and the source term present a discontinuity at an intermediate point. Theoretical bounds for the solution and its derivatives are derived for two complementary cases. A parameter uniform numerical scheme is constructed, which involves an upwind finite difference method with an appropriate piecewise uniform mesh. The error estimation and convergence analysis are presented, which show that the scheme provides a parameter uniform convergence of almost first order. Some numerical examples are discussed to illustrate the performance of the present method.

In this research, by means of a discontinuity indicator to detect troubled cells, we propose hybrid finite volume weighted essentially non-oscillatory schemes in combination with linear central schemes for hyperbolic conservation laws. In smooth regions, we apply the simple linear central schemes to save CPU time. While in discontinuous regions, we adopt WENO schemes to maintain the essentially non-oscillatory property near discontinuities. Extensive numerical examples strongly suggest that the proposed hybrid schemes can save computational cost considerably in comparison with the same order pure WENO schemes and keep steep discontinuity transition at the same time.

An iterative process for the grid problem of conjugation with iterations on the boundary of the discontinuity of the solution is considered. Similar grid problem arises in difference approximation of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions. The study of iterative processes for the states of such problems is of independent interest for theory and practice. The paper shows that the numerical solution of boundary problems of this type can be efficiently implemented using iterations on the inner boundary of the grid solution discontinuity in combination with other iterative methods for nonlinearities separately in each of the grid subregions. It can be noted that problems for states of controlled processes described by equations of mathematical physics with discontinuous coefficients and solutions arise in mathematical modeling and optimization of heat transfer, diffusion, filtration, elasticity theory, etc. The proposed iterative process reduces the solution of the initial grid boundary problem for a state with a discontinuous solution to a solution of two special boundary problems in two grid subdomains at every fixed iteration. The convergence of the iteration process in the Sobolev grid norms to the unique solution of the grid problem for each initial approximation is proved.

Purpose A finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl., 1(4)(1980)217–224 (in Chinese)], which is the so-called nine point scheme on structured quadrilateral meshes. The scheme has both cell-centered unknowns and vertex unknowns which are usually expressed as a linear weighted interpolation of the cell-centered unknowns. The critical factor to obtain the optimal accuracy for the scheme is the reconstruction of vertex unknowns. However, when the mesh deformation is severe or the diffusion tensor is discontinuous, the accuracy of the scheme is not satisfactory, and the author hope to improve this scheme. Design/methodology/approach The authors propose an explicit weighted vertex interpolation algorithm which allows arbitrary diffusion tensors and does not depend on the location of discontinuity. Both the derivation of the scheme and that of vertex reconstruction algorithm satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The vertex interpolation algorithm can be easily extended to 3 D case. Findings Numerical results show that it maintain optimal convergence rates for the solution and flux on 2 D and 3 D meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Originality/value This paper proposes a linearity preserving and explicit weighted vertex interpolation algorithm for cell-centered finite volume approximations of diffusion equations on general grids. The proposed finite volume scheme with the new interpolation algorithm allows arbitrary continuous or discontinuous diffusion tensors; the final scheme is applicable to arbitrary polygonal grids, which may have concave cells or degenerate ones with hanging nodes. The final scheme has second-order convergence rate for the approximate solution and higher than first-order accuracy for the flux on 2 D and 3 D meshes. The explicit weighted interpolation algorithm is easy to implement in three dimensions in case that the diffusion tensor is continuous or discontinuous.

The problem of real-time imaging in synthetic aperture radar (SAR) system is addressed here. Sub-aperture processing algorithm is usually used in the engineering applications. However, high azimuth sidelobes and edge aliasing problem usually occur in conventional sub-aperture algorithm, which degenerates the quality of the SAR image. To solve this problem, the reason for high azimuth sidelobes and edge aliasing problem is analysed, and a modified sub-aperture processing algorithm is proposed. The proposed algorithm minimises the effect of the phase discontinuity to improve the SAR image quality. Therefore, the azimuth sidelobes can be improved, and the edge aliasing phenomenon will be eliminated. Real measured airborne SAR datasets demonstrate the effectiveness of the proposed algorithm.

Driven by the ever increasing demand in function integration, more and more next generation high value-added products, such as head-up displays, solar concentrators and intra-ocular-lens, etc., are designed to possess freeform (i.e., non-rotational symmetric) surfaces. The toolpath, composed of high density of short linear and circular segments, is generally used in computer numerical control (CNC) systems to machine those products. However, the discontinuity between toolpath segments leads to high-frequency fluctuation of feedrate and acceleration, which will decrease the machining efficiency and product surface finish. Driven by the ever-increasing need for high-speed high-precision machining of those products, many novel toolpath interpolation and smoothing approaches have been proposed in both academia and industry, aiming to alleviate the issues caused by the conventional toolpath representation and interpolation methods. This paper provides a comprehensive review of the state-of-the-art toolpath interpolation and smoothing approaches with systematic classifications. The advantages and disadvantages of these approaches are discussed. Possible future research directions are also offered.

In this paper, we prove some fixed point theorems under a convex combination of generalized ( $$\epsilon -\delta $$ ) type rational contractions in which the fixed point may or may not be a point of discontinuity. As a by-product we explore some new answers to the open question posed by Rhoades (Contemp Math 72:233–245, 1988). Furthermore, we consider geometric properties of the fixed point set of a self-mapping on a metric space. We define a new kind of contractive mapping and prove that the fixed point set of this kind of contraction contains a circle (resp. a disc). Several non-trivial examples are given to illustrate our results. Apart from these, an application of discontinuous activation functions, frequently used in neural networks is also given.

International audience; In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately their result is only concerned with fixed sampling times, and thus it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally a linear-quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times.

This note gives a first sufficient condition that insures a non-negative, locally bounded, local solution to a logarithmically singular parabolic equation is continuous at a vanishing point and an estimate of the modulus of continuity is given. Moreover, an estimate of the Hausdorff measure of the set of discontinuity is established.

The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory. The process of studying the solution of cmKdV equation can be reduced to explore four quasi-linear equations, which predicts the evolution of dispersive shock wave. The results obtained here are quite different from the defocusing nonlinear Schrodinger equation: the bidirectionality of defocusing nonlinear Schrodinger equation determines that there are two basic rarefaction and shock structures while in the cmKdV case three basic rarefaction structures and four basic dispersive shock structures are constructed which lead to more complicated classification of step-like initial condition, and wave patterns even consisted of six different regions while each of wave patterns is consisted of five regions in the defocusing nonlinear Schrodinger equation. Direct numerical simulations of cmKdV equation are agreed well with the solutions corresponding to Whitham theory.

Generating matching meshes for finite element analysis is not always a convenient choice, for instance, in cases where the location of the boundary is not known a priori or when the boundary has a complex shape. In such cases, enriched finite element methods can be used to describe the geometric features independently from the mesh. The Discontinuity‐Enriched Finite Element Method (DE‐FEM) was recently proposed for solving problems with both weak and strong discontinuities within the computational domain. In this paper, we extend DE‐FEM to treat fictitious domain problems, where the mesh‐independent boundaries might either describe a discontinuity within the object, or the boundary of the object itself. These boundaries might be given by an explicit expression or an implicit level set. We demonstrate the main assets of DE‐FEM as an immersed method by means of a number of numerical examples; we show that the method is not only stable and optimally convergent but, most importantly, that essential boundary conditions can be prescribed strongly.

Friction phenomena exist in almost every mechanical device. Due to its complicated nature and influence on the system performance, extensive dynamic simulations are often required in the early system design stage. In this work, a novel approach for eliminating the numerical discontinuity in the classical Coulomb law and its extension is developed. Specifically, the method improves the computation process instead of modifying the Coulomb friction model. The estimated error of this procedure is derived under a simple and idealized model with an externally applied sinusoidal force. Two application examples of a single-body system are used to verify the proposed method, namely, 1-DOF mass-spring system with a moving belt and static ground. Results indicate that the proposed approach reveals the characteristics of the classical Coulomb friction law and its development and eliminates oscillations in the simulated friction force. Furthermore, a 3-DOF multi-body system is simulated to investigate the difference between the LuGre model and the proposed approach.

Meteorological data providers release updated forecasts several times per day – at the forecast epochs. The first time step (t=0) of each forecast, the so-called analysis step, is updated by a data-assimilation process so that the meteorological fields at this time in general do not match the fields from the previous forecast. Seen from the perspective of oceanographic modelling, the analysis step represents a possible discontinuity in the model forcing. Unless care is taken, this “meteorological discontinuity” may generate spurious waves in the ocean model. The problem is examined and quantified for a single meteorological model: the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System (IFS). A simple straightforward solution is suggested to overcome the forcing discontinuity and the effect on two particular ocean models is examined: the FCOO NA3 (North Atlantic 3 nm) storm-surge model and the NS1C (North Sea–Baltic Sea 1 nm) circulation model.

he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.

This paper aims at investigating fracture behavior of single and multiple cracks in two-dimensional solids by an adaptive extended isogeometric analysis (XIGA) based on locally refined (LR) B-splines. The adaptive XIGA is capable of modeling cracks without considering the location of crack faces due to the local enrichment technique based on partition-of-unity concept. The XIGA approximation is locally enriched by Heaviside function and crack tip enrichment functions to capture the discontinuity across crack faces and singularity in the vicinity of crack tips. The LR B-splines, which are generalized by B-splines and NURBS, not only inherent desirable properties of the B-splines and NURBS but also can be locally refined, ideally suitable for adaptive analysis. Structured mesh refinement strategy is applied to perform local refinement for LR B-splines based on a posteriori error estimator. According to the recovery technique proposed by Zienkiewicz and Zhu, the smoothed strain field is obtained to construct the posteriori error estimation based local refinement. The stress intensity factors (SIFs) are evaluated using the contour interaction integral technique. Several benchmark numerical examples are illustrated in comparison to analytical or reference solutions to verify the accuracy and efficiency of the developed approach. The proposed adaptive XIGA method is also applied to a curved crack, multiple cracks and complicated structure with a crack, which sufficiently presents the applicability of the proposed method in crack modeling. In addition, the convergence rate of the adaptive local refinement strategy is numerically studied and compared with that of the global refinement approach. The convergence rate of the adaptive local refinement is shown to be faster than that of the global refinement.

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.

Based on layer-by-layer analysis, we consider the construction of a model of two types of finite elements (FE) with natural curvature: a two-dimensional FE model of the moment of load-bearing layers and the three-dimensional FE of a filler for a more refined study of the stress-strain state (SSS) in the layers using the example of three-layer conical, generally irregular shells. The algorithm for constructing the model can avoid the errors caused by a discontinuity in the generalized displacements on the docking surfaces of the FEs of the bearing layers and filler. The model can more adequately to take into account the features of layered-heterogeneous structure (for example, a continuity violation of one or more layers), the moment state of bearing layers, the three-dimensional SSS in the filler, as well as different conditions for fixing and loading layers. In this case, it is possible to calculate the sandwich shells with variable geometric and physicomechanical properties and take into account the change in these properties and parameters of the SSS not only by meridional and circumferential coordinates, but also by the thickness of the shell and filler layer. As an example, the problem of determining the SSS parameters of a sandwich conical shell with blind cutouts in the inner bearing layer is solved, while the outer bearing layer and the filler layer remain continuous.

The problem of stationary Mach reflection of a Shockwave in a plane channel is considered within the framework of the Euler model. Emphasis is placed on an investigation of the flow parameters at the triple point. In an analytical investigation the local theory for curvilinear shock waves is employed. The conditions on the input data of the problem, at which singularity is realized at the triple point, are derived. Under the singularity conditions the flow parameter gradients and the curvatures of the shock fronts and tangential discontinuity at the triple point increase without bound. In the numerical investigation the second-order Godunov method is used, together with a new technology of contracting the grid toward the triple point in combination with the discontinuity fitting. The calculations confirm the theoretically predicted singularity boundary. Additional numerical experiments show that the singularity boundary is conserved, when artificial source terms are introduced into the energy equation. These results allow one to put forward the hypothesis that the singularity within the same boundaries is also realized for other two-dimensional flows with irregular shock-wave reflection.

The extended displacement discontinuity method is developed to analyze an interface line crack in a two-dimensional decagonal quasicrystal bimaterial. According to the general solutions and Fourier transform, fundamental solutions for phonon and phason displacement discontinuities and phonon and phason stresses are obtained. The delta function in the phonon and phason stresses is replaced by the Gaussian distribution function to eliminate the oscillatory singularity near the crack front. The expressions of phonon and phason stress intensity factors are given in terms of phonon and phason displacement discontinuities. The finite element analysis software COMSOL is used to validate the developed method.

A small-scale plastic zone at the point of intersection of microplastic deformation lines is identified. The problem on the plastic zone is reduced to a symmetric elasticity problem for a plane with four tangential displacement discontinuity lines emerging from this point. Two of them are semi-finite, while the other two are of finite length. The exact solution of the problem is found using the Wiener–Hopf method.

The paper presents the evolution of discontinuous control systems starting from a relay with only two output constant values. The relay systems were widely used at the early stage of the feedback control system history. The analysis and design methods for them were developed by Ya. Tsypkin and discussed in his monograph “Theory of relay control systems,” published in 1956. It is shown how a relay function is modified in the so-called variable structure systems, when the relay output cab be equal to one of two continuous state functions. The next step is made in the framework of variable structure systems with vector control. The design procedure for systems with vector relay control relies on selection of a discontinuity surface for each control component. High efficiency of such designed systems results from enforcing sliding modes on the surfaces. Finally, the vector relay unit control is offered. The method is free of the component-wise design and proved to be applicable for infinite-dimensional systems.

Mathematical defects of general cam curves such as high-order discontinuity and overlarge peak values often bring excessive transmission errors and vibrations to high-speed cam-follower systems. A comprehensive method is proposed in this study to develop high-speed cam curves with preferable mathematical characteristics. A high-order differential interpolation curve is first constructed by using nodal basis functions to accurately match with the multi-order derivatives on multiple nodes. A flexible cam curve is then reconstructed by adding variable shape factors on critical nodes for further shape adjustments. Numerical cases indicate that the proposed method can not only help construct a faithful cam curve satisfying all the given constraints for displacement, velocity, acceleration and jerk, but also effectively reconstruct a high-order continuous cam curve with lower peak values for high-speed precision cam mechanisms. The proposed method has been demonstrated to be effective and applicable in user-oriented reverse design, modification, and optimization of high-speed cam curves.

The reliable detection of defects is one of the most important tasks in nondestructive tests, mainly in the radiography test (RT) and ultrasonic test (UT). Non-Destructive Testing (NDT) is the most efficient as internal and external discontinuities are inspected while maintaining the target’s original structure (form and function). Therefore, in this paper, the Phased Array Ultrasonic Testing (PAUT) method was applied to propose inspection method that can address certain key PAUT Parameters and image processing to enhance the precision of discontinuities and the reliability of calculations of discontinuity size and characterization and also transfer 2D S-Scan images into 3D images (volumetric scan) which is more reliable and easily interpretation.

We study a four-parameter family of 2D piecewise linear maps with two discontinuity lines. This family is a generalization of the discrete-time version of the fashion cycle model by Matsuyama, which was originally formulated in continuous time. The parameter space of the considered map is characterised by quite a complicated bifurcation structure formed by the periodicity regions of various attracting cycles. Besides the standard period adding and period incrementing structures, there exist incrementing structures with some distinctive properties, as well as novel mixed structures, which we study in detail. The boundaries of many periodicity regions associated with border collision bifurcations of the related cycles are obtained analytically. Several mixed structures are qualitatively described.

We state and study a transmission boundary value problem for two-dimensional filtration flows in a piecewise anisotropic inhomogeneous layer of a porous medium. The layer is characterized by a generally nonsymmetric conductivity (permeability) tensor with components that undergo discontinuity on some smooth curve (the transmission line). The tensor components are modeled by a function of the coordinates that undergoes a discontinuity on the transmission line but is continuously differentiable outside of it. We consider a layer with separated anisotropy and inhomogeneity. Using a nonsingular affine transformation of the coordinates, we state the problem for a complex potential in canonical form, which considerably simplifies the analysis of the problem. The sources–sinks of the flow are set arbitrarily; they do not lie on the transmission line and are modeled by the singular points of the complex potential. The problem is reduced to a system of two singular integral equations if the discontinuity in the layer conductivity along the transmission line is variable and to one singular integral equation if the discontinuity is constant. The problem is of practical interest, for example, in extracting water (or oil) from natural piecewise anisotropic inhomogeneous layers (strata) of soil.

We propose a singular integral equation whose definition is extended to a singular point by additional conditions. In the space of smooth functions, this equation becomes equivalent, according to the indicated extended definition, to an ordinary differential equation. At the same time, in the space of piecewise discontinuous functions, it becomes equivalent to an impulsive differential equation. For smooth solutions of the singular equation, we substantiate the method of successive approximations. For the ordinary differential equation, this method turns into a new algorithm for the construction of successive approximations. For the investigated equation, we define a solution of new type equivalent, for the impulsive differential equation, to a solution with discontinuity of the second kind (“solution with needle”). We propose an algorithmic formula for the general solution of the initial-value problem for the impulsive differential equation.

This article outlines the objectives, activities, and achievements of using intuitive logics scenarios to improve strategic decision-making in situations of discontinuity. Pierre Wack's work at Royal Dutch/Shell illustrates the use of exploratory, reframing, and decision scenarios to change the mental models of decision-makers in advance of the 1973 oil crisis. Thomas S. Kuhn's The Structure of Scientific Revolutions offers a promising theoretical account of paradigm shifts in science, but its application is limited by the ambiguity of the concept of paradigm. Margaret Masterman's distinction of three senses of paradigm – introduced as construct paradigm, worldview paradigm, and action paradigm – clarifies and extends these accounts to support a preliminary framework and systematic approach. Overall, this investigation expands our understanding of strategic decision-making as guided not only by analytic achievements and developmental investigations but also by discontinuous breaks that can be explored and addressed using scenario methodology. Section 1 provides an introduction and literature review. Section 2 examines the scenarios developed at Royal Dutch/Shell. Section 3 summarizes Kuhn's new image of science, clarified by Masterman's three senses of paradigm in Section 4. Section 5 presents implications for scenario methodology, followed by conclusions and areas for research in Section 6.

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.

The Riemann problem for the shallow water equations with discontinuous topography is considered. In a general case the exact solution of this problem is not unique, which complicates the application of an exact Riemann solver in numerical methods, since it is not clear which solution should be chosen. In the present work it is shown that involving an additional physical assumption makes it possible to prove the existence and uniqueness of the solution. The assumption is that the discharge at the bottom discontinuity should continuously depend on the initial conditions. The proven uniqueness opens up a possibility to use an exact Riemann solver for a numerical solution of the shallow water equations with complex discontinuous topography.

Ray effect is one of the major shortcomings of discrete ordinate method (DOM). It is a consequence of angular discretization, resulting in unphysical oscillations in the solutions, and often occurs in the problems where the temperature of walls is discontinuous. In this paper, the modified DOM (MDOM) is developed for the radiative heat transfer in an axisymmetric cylindrical enclosure, which contains emitting, absorbing and scattering medium, to overcome this shortcoming. The radiative intensity is decomposed into two components, the wall-related and the medium-related ones. The former is determined analytically, whereas the latter is still solved by DOM. In MDOM, the most important step is the evaluation of wall-related incident radiation and radiative heat flux. Two methods to evaluate them are comparatively given: one is the semi-analytical method based on coordinate transformation which is more accurate, and the other is the numerical method with segments integration to treat the discontinuity which is mathematically simpler. Results show that both are effective to mitigate the ray effect, and cost computational time comparable to DOM. Exact semi-analytic solutions for the pure absorbing cases are also given in tabular form for convenience. Besides, alternative boundary conditions for the MDOM proposed by Sakami and Charette (JQSRT, 64(2000) 275–298) are investigated. This type of boundary conditions is proved to have simplified solution procedure, but results in slight unphysical oscillations in the MDOM solutions.

Dispersion curve, and displacement modeshapes of multilayer structures can be obtained using Semi-Analytical Finite Element (SAFE) method. Stress mode shapes calculated from SAFE were discontinuous at the interface, because SAFE method formulation does not apply stress continuity at the interface. A case study of 1 mm aluminum-1 mm steel double-layer plate is presented, showing the SAFE stress predictions at the interface. The results observed were discontinuity of out-of-plane stress modeshapes at the interface. Mesh refinement was performed at the interface to study the convergence of stress mode shapes at the interface, yet the stress discontinuity problem was not solved. A fine mesh at the interface was created by using variable mesh techniques. This also did not provide continuous stress mode shape at the interface. Therefore, in this paper, a novel hybrid SAFE-GMM (Global Matrix Method) approach was used to obtain stress mode shapes accurately and efficiently. GMM develops the displacement and stress equations for individual layers in a multilayered structure and assembles a global matrix by applying the boundary and interface continuity conditions. A case of practical importance, CFRP strengthened concrete structure was analyzed using the SAFE-GMM approach. The drawback of the SAFE technique to this case is also presented, and it shows that the hybrid SAFE-GMM approach gave the interface stresses accurately.

The aim of this paper is to deal with the problem of limit cycles for a general planar piecewise linear differential system of saddle–focus type. By using the Lienard-like canonical form with five parameters and dividing the total parameter space into several regions, the number of limit cycles is discussed in detail. In particular, we give parameter regions where there are at least two limit cycles. Moreover, we investigate the existence and stability of exactly two nested limit cycles in some parameter regions.

The paper is concerned with the minimization of the k-th eigenvalue of the Laplace operator with Robin boundary conditions, among all open sets of R N satisfying a volume constraint. We prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets. The main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation. One of the key difficulties (for k ≥ 3 ) comes from the fact that the eigenvalues are critical points.

Ultrafast heating has gained serious attention due to technological advancements in recent years. The thermoelastic process accounting for the finite speed of thermal signals is therefore becoming increasingly important. With the help of extended thermodynamics, Yu et al. (Meccanica 53(10):2543–2554, 2018) have established a thermoelastic model by including the strain rate and temperature rate in the constitutive equations. This model is also an attempt to remove the discontinuity in the displacement field under temperature rate-dependent thermoelasticity theory. The present work seeks to derive the representation of a Galerkin-type solution in the context of this recently proposed model in terms of elementary functions. The representation theorem of the Galerkin-type solution of the system of equations for steady oscillation is also proved. In accordance with this theorem, we finally establish a theorem which expresses the general solution of the system of homogeneous equations of steady oscillation in terms of metaharmonic functions.