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The interface crack problems in the two-dimensional (2D) decagonal quasi-crystal (QC) coating are theoretically and numerically investigated with a displacement discontinuity method. The 2D general solution is obtained based on the potential theory. An analogy method is proposed based on the relationship between the general solutions for 2D decagonal and one-dimensional (1D) hexagonal QCs. According to the analogy method, the fundamental solutions of concentrated point phonon displacement discontinuities are obtained on the interface. By using the superposition principle, the hypersingular boundary integral-differential equations in terms of displacement discontinuities are determined for a line interface crack. Further, Green’s functions are found for uniform displacement discontinuities on a line element. The oscillatory singularity near a crack tip is eliminated by adopting the Gaussian distribution to approximate the delta function. The stress intensity factors (SIFs) with ordinary singularity and the energy release rate (ERR) are derived. Finally, a boundary element method is put forward to investigate the effects of different factors on the fracture.

Dynamics of nonlinear oscillators with discontinuous nonlinearities subjected to harmonic and random excitations is investigated. Impact, dry friction and Hertzian type compliant contact nonlinearities are considered. Stochastic bifurcations like the P-bifurcation and D-bifurcation are discussed. P and D bifurcations are characterized respectively by the joint probability density functions (jpdf) of the response and the largest Lyapunov exponent. The jpdf is obtained by the solution of the corresponding Fokker–Planck equation by the finite element and path integral methods. The results are verified by Monte Carlo simulation methods. Adaptive time step integration procedure (ATSP) is adopted which accurately determines the point of discontinuity. A bisection method and a Brownian tree approach are used in this process and direct the solution along the correct Brownian path. Numerical results are also obtained using non-smooth coordinate transformations like the Zhuvarlev and Ivanov transformations converting the discontinuous systems to equivalent smooth systems and compared with the results of the ATSP. The Filippov convex transformation is used in the case of the dry friction nonlinearity in the integration near the discontinuity. The results are discussed with respect to some examples like the Duffing and Van der Pol oscillators with impact and dry friction.

Underground engineering is becoming increasingly sophisticated in recent years, exemplified by shale gas recovery, nuclear waste disposal and enhanced geothermal systems. Displacement Discontinuity Method (DDM) and Fictitious Stress Method (FSM) are two branches of boundary element method which can efficiently and accurately simulate fractures and openings embedded in large spatial domains. Nevertheless, in literature, studies engaging DDM/FSM to perform the aforementioned hot-topic research are rare, and one of the key reasons lies in the inconvenient access to DDM-FSM programs. In this study, we provide DDFS3D, a set of open-source codes incorporating the commonly used constant triangular and quadrilateral elements to implement 3D DDM, FSM and the hybrid of the two. The constant triangular element is reproduced from early studies, while the constant quadrilateral element is an independent work presented in this study. Gaussian quadrature and Hadamard finite-part integral are utilized to evaluate the regular and strongly singular integrals, respectively. To the authors’ best knowledge, DDFS3D is the first and the only open-source codes for 3D DDM-FSM implementation. We hope it could promote the application of DDM-FSM and advance the role of DDFS in coping with the ever-increasing new challenges in underground engineering.

In layered media, the solution of the Maxwell equations has discontinuity of the derivative or the function itself at media interfaces. For the first time, finite-difference schemes providing convergence for discontinuous solutions across straight media interfaces are proposed for the one-dimensional formulation of the Maxwell equations. These are bicompact conservative schemes. They are two-point and layer boundaries are taken as mesh nodes. The scheme explicitly accounts for physically correct interface conditions at media interfaces. We propose an essentially new technique which accounts for medium dispersion. All these features provide the second order of accuracy even on discontinuous solutions. Calculation examples, which illustrate these results, are given. The proposed method is verified by comparison with a previously performed experiment on propagation of normally incident plane wave on one-dimensional photonic crystal. Calculated spectrum agrees well with the measured one within experimental error of 2-5%.

In this paper, the usability of the classical Harmonic Balancing method (cHB) to calculate the weakly nonlinear motion of phase- and amplitude controlled MEMS resonators is demonstrated. A polynomial mechanical stiffness description and linearized electrostatic effects are considered, which allow determining the Jacobian analytically. The effects of amplitude and phase control are taken into account by applying boundary conditions in the iteration process. With a customized cHB, it is possible to efficiently simulate the behavior of a MEMS gyroscope undergoing nonlinear resonance with higher parasitic modes. The presented results of a virtual model compare very well with solutions obtained with the Shooting method. Furthermore, we compare the measurement of a drive frequency discontinuity, taken from a different prototype, with the prediction from our linear extrapolated model and from a model with artificially fitted parameters. The cHB approach yields an order of magnitude speed-up in comparison to the transient simulation method. Besides some restrictions, customized cHB methods are interesting candidates for fast system-type simulation considering the MEMS-ASIC interface. [2021-0028]

Since proposed, the genuinely multidimensional Riemann solvers have been attracting more attentions because it considers the waves normal and parallel to the cell interfaces at the same time. However, all these methods are built upon the hyperbolic property of the compressible Euler equations. Therefore, they can not be applied to solve the Euler equations in steady subsonic cases which are with the elliptic property. In this study, a novel multidimensional Riemann solver called MULE-AUSMPWM (Multidimensional E-AUSMPW Modified) is proposed. It is built upon the Zha–Bilgen splitting procedure and adopts the Balsara's multidimensional wave model as the ME-AUSMPW scheme. In addition, it improves the ME-AUSMPW scheme's accuracy at subsonic speeds by avoiding simulating the convective fluxes in fully upwind manners. Extensive numerical tests, such as the two dimensional Euler problem, the one dimensional moving contact discontinuity case, the two-dimensional double Mach reflection case, the two-dimensional Riemann problem, and the turbulent flow past a 2d NACA0012 airfoil case, are conducted. Results illustrate that the MULE-AUSMPWM scheme proposed is capable of simulating compressible complex flows and improving the existing multidimensional Riemann solvers' accuracy at subsonic speeds remarkably.

Various Nash equilibrium results for a broad class of aggregative games are presented. The main ones concern equilibrium uniqueness. The setting presupposes that each player has $$\mathbb {R}_+$$ R + as strategy set, makes smoothness assumptions but allows for a discontinuity of stand-alone payoff functions at 0; this possibility is especially important for various contest and oligopolistic games. Conditions are completely in terms of marginal reductions which may be considered as primitives of the game. For many games in the literature they can easily be checked. They automatically imply that conditional payoff functions are strictly quasi-concave. The results are proved by means of the Szidarovszky variant of the Selten–Szidarovszky technique. Their power is illustrated by reproducing quickly and improving upon various results for economic games.

Within the incompressible three-dimensional Euler equations, we study the pancake-like high vorticity regions, which arise during the onset of developed hydrodynamic turbulence. We show that these regions have an internal fine structure consisting of three vortex layers. Such a layered structure, together with the power law of self-similar evolution of the pancake, prevents development of the Kelvin-Helmholtz instability., Comment: 4 pages, 2 figures

Since proposed, the self-similarity variables based genuinely multidimensional Riemann solver is attracting more attentions due to its high resolution in multidimensional complex flows. However, it needs numerous logical operations in supersonic cases, which limit the method’s applicability in engineering problems greatly. In order to overcome this defect, a hybrid multidimensional Riemann solver, called HMTHS (Hybrid of MulTv and multidimensional HLL scheme based on Self-similar structures), is proposed. It simulates the strongly interacting zone by adopting the MHLLES (Multidimensional Harten-Lax-van Leer-Eifeldt scheme based on Self-similar structures) scheme at subsonic speeds, which is with a high resolution by considering the second moment in the similarity variables. Also, it adopts the MULTV (Multidimensional Toro and Vasquez) scheme, which is with a high resolution in capturing discontinuities, to simulate the flux at supersonic speeds. Systematic numerical experiments, including both one-dimensional cases and two-dimensional cases, are conducted. One-dimensional moving contact discontinuity case and sod shock tube case suggest that HMTHS can accurately capture one-dimensional expansion waves, shock waves, and linear contact discontinuities. Two-dimensional cases, such as the double Mach reflection case, the supersonic shock / boundary layer interaction case, the hypersonic flow over the cylinder case, and the hypersonic viscous flow over the double-ellipsoid case, indicate that the HMTHS scheme is with a high resolution in simulating multidimensional complex flows. Therefore, it is promising to be widely applied in both scholar and engineering areas.

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.

Conventional methods for measuring local shear stress on the wetted perimeter of open channels are related to the measurement of the very low velocity close to the boundary. Measuring near-zero velocity values with high fluctuations has always been a difficult task for fluid flow near solid boundaries. To solve the observation problems, a new model was developed to estimate the distribution of boundary shear stress from the velocity distribution in open channels with different cross-sectional shapes. To estimate the shear stress at a point on the wetted perimeter by the model, the velocity must be measured at a point with a known normal distance to the boundary. The experimental work of some other researchers on channels with various cross-sectional shapes, including rectangular, trapezoidal, partially full circular, and compound shapes, was used to evaluate the performance of the proposed model. Optimized exponent coefficients for the model were found using the multivariate Newton method with the minimum of the mean absolute percentage error (MAPE) between the model and experimental data as the objective function. Subsequently, the calculated shear stress distributions along the wetted perimeter were compared with the experimental data. The most important advantage of the proposed model is its inherent simplicity. The mean MAPE value for the seven selected cross-sections was 6.9%. The best results were found in the cross-sections with less discontinuity of the wetted perimeter, including the compound, trapezoidal, and partially full circular pipes. In contrast, for the rectangular cross-section with an angle between the bed and walls of 90°, MAPE increased due to the large discontinuities.

We prove the stability of a planar contact discontinuity without shear, a family of special discontinuous solutions for the three-dimensional full Euler system, in the class of vanishing dissipation limits of the corresponding Navier–Stokes–Fourier system. We also show that solutions of the Navier–Stokes–Fourier system converge to the planar contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the planar contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the planar contact discontinuity is unique for the multi-D compressible Euler system. Our proof is based on the relative entropy method, together with the theory of a-contraction up to a shift and our new observations on the planar contact discontinuity.

This paper develops a low-diffusion robust flux splitting scheme termed TVAP to achieve the simulation of wide-ranging Mach number flows. Based on Toro-Vazquez splitting approach, the new scheme splits inviscid flux into convective system and pressure system. This method introduces Mach number splitting function and numerical sound speed to evaluate advection system. Meanwhile, pressure diffusion term, pressure momentum flux, interface pressure and interface velocity are constructed to measure pressure system. Then, typical test problems are utilized to systematically assess the robustness and accuracy of the resulting scheme. Matrix stability analysis and a series of numerical cases, such as double shear-layer problem and hypersonic viscous flow over blunt cone, demonstrate that TVAP scheme achieves excellent low diffusion, shock stability, contact discontinuity and low-speed resolution, and is potentially a good candidate for wide-ranging Mach number flows.

In this paper, we present a deep learning framework for solving two-dimensional elliptic equations with singular forces on arbitrary domains. This work follows the ideas of the physical-inform neural networks to approximate the solutions and the immersed boundary method to deal with the singularity on an interface. Numerical simulations of elliptic equations with regular solutions are initially analyzed in order to deeply investigate the performance of such methods on rectangular and irregular domains. We study the deep neural network solutions for different number of training and collocation points as well as different neural network architectures. The accuracy is also compared with standard schemes based on finite differences. In the case of singular forces, the analytical solution is continuous but the normal derivative on the interface has a discontinuity. This discontinuity is incorporated into the equations as a source term with a delta function which is approximated using a Peskin's approach. The performance of the proposed method is analyzed for different interface shapes and domains. Results demonstrate that the immersed boundary neural network can approximate accurately the analytical solution for elliptic problems with and without singularity.

A mesh refinement method is described for solving optimal control problems using Legendre-Gauss-Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targeted $h$-refinement approach whereby the discontinuity locations are bracketed with mesh points. The remaining smooth portions of the mesh are refined using previously developed techniques. The method is demonstrated on two examples, and results indicate that the method solves optimal control problems with discontinuous control solutions using fewer mesh refinement iterations and less computation time when compared with previously developed methods., Comment: 22 Pages, 8 Figures, 0 Tables

In this paper, we consider inverse problems for the Dirac operator with complex-valued weight and the jump conditions inside the interval. We prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

Various versions of equations of two-fluid plasma called the equations of electromagnetic hydrodynamics are considered. They are a generalization of the classical magnetic hydrodynamics equations obtained by adding dispersive terms. Application of finite-difference methods for solving these equations is analyzed. The Riemann problem is solved numerically, and various types of discontinuity structures that expand with time are considered. These are fast and slow magnetosonic structures and Alfven structures. In the case of moderate amplitudes, fast and slow magnetosonic structures are typical in the theory of nondissipative structures. It is found that due to the disappearance of dispersion, short waves can overturn under certain initial conditions, which requires the analysis of discontinuous solutions or the inclusion of additional dissipative or dispersive terms into the equations. In the case when the fluid dynamic viscosity is added, a shock-type structure is discovered. The evolutionary nature of the discontinuity and the conditions on the discontinuity are studied.

In this paper, we have considered a very general van der Waals equation of state to study the one-dimensional motion of magnetogasdynamic (MGD) equations. Transport equations for the discontinuity waves are obtained and studied in detail. The effect of van der Waals excluded volume and the intermolecular force of attraction on the evolution of these waves is studied. Explicit expressions for the jump in the mass density and the magnetic pressure are obtained and analyzed in detail. Finally, the interaction of the weak discontinuity with the characteristic shock is studied and the amplitudes of the reflected and the transmitted waves are obtained.

Beam-in-beam large sliding contact exists in simulations of drillstring-riser systems, offshore pipe-in-pipe systems, medical catheter-guidewire-stent systems, among others. Large sliding between the pipes may lead to discontinuity of contact force and non-sparsity of system Jacobian in numeric simulations, which then may result in non-physical disturbances and low efficiency. To overcome this phenomenon, a consistent-contact-force algorithm and a highly efficient implicit-integration scheme are proposed in this paper. Geometric treatments are applied in gap and overlap areas to obtain continuous contact force, in addition to contact-history-based fast detection. The contact force Jacobian is ignored as the flexible multibody system is solved with an inexact Newton iteration and an empirical formula is used to estimate the convergence conditions. Four numeric examples are computed to verify the presented method, including a buckling benchmark and a riser-recoil industrial application. The numeric results show that the proposed method is accurate and highly efficient.

This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. DUVAUT and J.-L. LIONS, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [ 6 ], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.

In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a jump in vorticity at the boundary of the cross-section of the vortex ring. Using the vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in several kinds of domains. The precise localization of the asymptotic singular vortex filament is proved to depend on the circulation and the velocity at far fields of the vortex ring, and the geometry of the domains. Some qualitative and asymptotic properties are also established.

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper. We would like to thank the anonymous referees for their valuable comments, which have helped to significantly improve this work. This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia), by the national research project MTM2015- 64382-P (MINECO/FEDER) and by NSF grant DMS-1719410.

This paper deals with hyperbolic conservation laws exhibiting a flux discontinuity at the origin and which does not admit a weak solution satisfying the Rankine–Hugoniot jump condition. We therefore seek unbounded solutions in the form of shadow waves supported by at the origin. The shadow waves are defined as nets of piecewise constant functions approximating a shock wave to which we add a delta function and possibly another unbounded part.

The Stefan two-phase problem with an internal source, as well as the heat conduction problem for rectangular and curvilinear regions, has been solved by the method of rapid sine-expansion, which involves the Fourier series. The problem of heat conduction in a body of rectangular form was solved in a general form with the Dirichlet conditions and an internal heat source by the method of rapid expansion, and thereafter the problem for a body of curvilinear shape. The solutions obtained were used in considering the Stefan problem with unknown curvilinear interface. Expressions are presented for matching the input data of the problem, on satisfaction of which the solution will be universally continuous everywhere in the region of the rectangle together with derivatives up to the sixth order inclusive. The nonfulfillment of these conditions leads to the discontinuity of the solution at the corner points. The algorithms of the solution of problems in rectangular and curvilinear regions, as well as of the Stefan twophase problem, are written. The solution for the rectangular region is represented in an explicit form, whereas the solution for the curvilinear region and the solution of the Stefan problem are reduced to a closed algebraic system of equations of small volume.

A common defect found when reproducing old vinyl and gramophone recordings with mechanical devices are the long pulses with significant low-frequency content caused by the interaction of the arm-needle system with deep scratches or even breakages on the media surface. Previous approaches to their suppression on digital counterparts of the recordings depend on a prior estimation of the pulse location, usually performed via heuristic methods. This paper proposes a novel Bayesian approach capable of jointly estimating the pulse location; interpolating the almost annihilated signal underlying the strong discontinuity that initiates the pulse; and also estimating the long pulse tail by a simple Gaussian Process, allowing its suppression from the corrupted signal. The posterior distribution for the model parameters as well for the pulse is explored via Markov-Chain Monte Carlo (MCMC) algorithms. Controlled experiments indicate that the proposed method, while requiring significantly less user intervention, achieves perceptual results similar to those of previous approaches and performs well when dealing with naturally degraded signals.

This paper presents a new stability analysis approach for time-varying delay continuous systems which formulates the stability problem by presenting an overlapped switching Lyapunov Krasovskii functional (OSLKF). An exponential stability assessment method is proposed which considers two sets of conditions; local and discontinuity conditions. The local conditions establish the stability of the system based on the assumption that the delay parameters belong to a subdomain of the original domain. The discontinuity conditions are used to deal with the discontinuity issue of the switching Lyapunov functions at their switching points. These conditions have two major properties; it is not required that the time derivative of the delay to be less than unity and that the global exponential stability analysis of the model is possible. To illustrate the performance of the proposed method, two numerical examples are presented which demonstrate the effectiveness of the proposed approach as compared to the existing methodologies.

The present work introduces a modified Incompressible Smoothed Particle Hydrodynamics (ISPH) model for simulation of free-surface flows and convection heat transfer. First, two new gradient and Laplacian models are proposed based on the Taylor series expansion and then used for discretization of the diffusion terms, Pressure Poisson's equation (PPE), and divergence of velocity. To maintain overall high-order accuracy, an explicit third-order TVD Runge-Kutta scheme is employed for discretization of the transient terms in Navier-Stokes and energy equations. Moreover, a new Hybrid Particle Shifting Technique (HPST) is developed by combining the classical PST and a collision model. A new kernel function is developed by combination of the Gaussian and polynomial functions and is then applied to the simulation of classical 1D Sod shock tube. Furthermore, a novel Hybrid Free-surface Detection (HFD) technique is proposed for accurate imposition of Dirichlet pressure boundary condition at the free surface area. The validity and applicability of proposed numerical schemes are verified against the several challenging benchmark cases including: dam-break flows with/without an obstacles, stretching water drop, rotating square patch of fluid, Rayleigh-Taylor instability, energy and exergy analysis of natural convection heat transfer in differentially heated cavity. The results show that, the newly constructed kernel function can successfully guarantee the stability and convergence of the numerical solution. Furthermore, it is found that, the proposed Hybrid Particle Shifting Technique (HPST) can efficiently resolve the unphysical discontinuity and suppress spurious pressure fluctuations.

We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2-energy methods.

Although classical WENOCU schemes can achieve high-order accuracy by introducing a moderate constant parameter C to increase the contribution of optimal weights, they exhibit distinct numerical dissipation in smooth regions. This study presents an extension of our previous research which confirmed that adaptively adjusting parameter C can indeed overcome the inadequacy of the usage of a constant small value. Cmin is applied near a discontinuity while Cmax is used elsewhere and they are switched according to the variation of the local flow-field property. This study provides the reference values of the adaptive parameter C of WENOCU4 and systematically evaluates the comprehensive performance of three different switches (labeled as the binary, continuous, and hyperbolic tangent switches, respectively) based on an optimized efficient WENOCU4 scheme (labeled as EWENOCU4). Varieties of 1D scalar equations, empirical dispersion relation analysis, and multi-dimensional benchmark cases of Euler equations are analyzed. Generally, the dissipation and dispersion properties of these three switches are similar. Especially, employing the binary switch, EWENOCU4 achieves the best comprehensive properties. Specifically, the binary switch can efficiently filter more misidentifications in smooth regions than others do, particularly for the cases of 1D scalar equations and Euler equations. Also, the computational efficiency of the binary switch is superior to that of the hyperbolic tangent switch. Moreover, the optimized scheme exhibits high-resolution spectral properties in the wavenumber space. Therefore, employing the binary switch is a more cost-effective improvement for schemes and is particularly suitable for the simulation of complex shock/turbulence interaction. This study provides useful guidance for the reference values of parameter C and the evaluation of adaptive switches.

The exact Riemann solutions for a macroscopic production model under the equation of state given by the Chaplygin gas are solved explicitly for all possible Riemann initial data. It is discovered interestingly that a composite hyperbolic wave is involved in Riemann solution under some specially designated initial conditions, which is made up of a rarefaction wave and a delta contact discontinuity attached on the wave front of the rarefaction wave. Furthermore, the constructions of global solutions to the perturbed Riemann problem for this system are also displayed in completely explicit forms when the initial data are taken to be three piecewise constant states under some suitable restrictive conditions by using the method of characteristics. During the process of constructing global solutions, the interactions of elementary waves are studied in detail. Moreover, it is proved rigorously that Riemann solutions are stable with respect to the specific small perturbations of Riemann initial data.

In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching., 11 pages, 1 figure. Final version, to appear on Geometric And Functional Analysis (GAFA)

In this paper, the fixed-time synchronization of complex-valued memristor-based neural networks with impulsive effects is investigated. We first separate these complex-valued networks into real and imaginary parts, and design the appropriate controllers. Then apply the set-valued map and the differential inclusion theorem to handle the discontinuity problems at the right-hand side of the drive-response systems. By constructing the comparison systems together with the Lyapunov function, we get the fixed-time synchronization conditions. Moreover, the estimate of the settling time is also explicitly obtained. Finally, two examples and their numerical simulations are presented to show the effectiveness of the obtained theoretical results.

Physical explosion causes large damages in the process industry and quite often escalates to chemical explosions. The shock waves generated by such events are challenging to model and they must be numerically captured without spurious oscillations in order to make an accurate estimative of the accidental effects. In this context, this paper investigates how a new high order numerical scheme models the physical explosion. We have considered a confined explosion in a spherical vessel and blast load throughout pipelines as the framework to investigate the performance of the numerical scheme. The developed numerical approach considers the effect of less smooth substencils when there is a discontinuity inside the stencil based on the local Mach number what avoids oscillations and instability. The numerical solution of the fundamental equations is coupled with the Modified Colebrook-White formulation in order to consider the blast load through the pipeline. Shock waves from experimental data and analytical model are used to validate the proposed model. The research provides an efficient method for prediction of blast loads from spherical vessels rupture to an open atmosphere and in pipelines.

In this paper, we study the exact solutions of Riemann problem for the compressible ideal fluid flow, where the external force is a continuous function of time. The exact expressions for contact discontinuity curve, shock wave curve and rarefaction wave curve are provided. Six types of exact solutions of the Riemann problem are also obtained. In particular, a vacuum arises at t > 0 although the initial data never involves the vacuum, and these solutions do not possess self-similarity in the ( t , x ) -plane.

The displacement discontinuity method is extended to study the fracture behavior of interface cracks in one-dimensional hexagonal quasicrystal coating subjected to anti-plane loading. The Fredholm integral equation of the first kind is established in terms of displacement discontinuities. The fundamental solution for anti-plane displacement discontinuity is derived by the Fourier transform method. The singularity of stress near the crack front is analyzed, and Chebyshev polynomials of the second kind are numerically adopted to solve the integral equations. The displacement discontinuities across crack faces, the stress intensity factors, and the energy release rate are calculated from the coefficients of Chebyshev polynomials. In combination with numerical simulations, a comprehensive study of influencing factors on the fracture behavior is conducted.

In layered media, the solution of Maxwell’s equations suffers a strong or weak discontinuity at the layer boundaries. Finite-difference schemes providing convergence on strong discontinuities have been proposed for the first time. These are conservative bicompact two-point schemes with mesh nodes lying on the layer boundaries. A fundamentally new technique for taking into account the medium dispersion is proposed. All these features ensure the second order of accuracy of the schemes on discontinuous solutions. Numerical examples illustrating these results are given.

Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm for solving the inverse problems. The key role in our method is played by the Riesz-basis property of a special vector-functional system in a Hilbert space. In addition, we obtain a new uniqueness theorem for recovering the potential on a part of the interval, by using a fractional part of the spectrum.

The stability of one-dimensional steady flows of an ideal (inviscid and non-heat-conducting) gas in channels of variable cross section with combustion in a structurally stable detonation wave is studied. The detonation wave represents a discontinuity surface propagating normally to the axis of the channel. It was previously established that, in this formulation, steady flows with combustion in a Chapman–Jouguet detonation wave are always unstable, in contrast to combustion flows in an overcompressed detonation wave. The stability analysis of such flows is reduced to the numerical solution of an initial-boundary value problem describing the evolution of finite flow perturbations between the moving detonation wave and the minimal nozzle cross section (in the case of a sudden contraction) or the exit nozzle cross section. The problem is solved by applying a modified Godunov scheme of higher order accuracy. More specifically, a Riemann solver with switching (between the overcompressed and Chapman–Jouguet detonation waves) with an explicitly represented detonation wave is created. Examples of stable flows and flows collapsing due to large initial perturbations are given, and their dynamics with transitions to a Chapman–Jouguet detonation wave and back are computed.

In this paper, we present a two-grid partially penalized immersed finite element (IFE) scheme for the approximation of semi-linear elliptic interface problems. Extra stabilization terms are introduced at interface edges for penalizing the discontinuity in IFE functions. Optimal error estimates in both H 1 and L p norms are obtained for IFE discretizations. To linearize the IFE equations, two-grid algorithm based on some Newton iteration approach is applied. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h 1 ∕ 4 ) . As a result, solving such a large class of non-linear equation will not be much more difficult than solving one single linearized equation.

In this paper, we prove that any $W^{2,1}$ strong solution to second-order non-divergence form elliptic equations is locally $W^{2,\infty}$ and piecewise $C^{2}$ when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be $C^{1,\text{Dini}}$. We also derive global weak-type $(1,1)$ estimates with respect to $A_{1}$ Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these surfaces of discontinuity of the coefficients., 41 pages; submitted

In this paper, two three-node triangular thin plate/shell elements are proposed based on the absolute nodal coordinate formulation. As the gradient deficient element, the thin plate/shell element does not possess a full Jacobian matrix for the mapping between different configurations. Thus, the formulation cannot be derived in the conventional method directly based on the continuum mechanics. The independent area coordinate gradients with obvious geometrical interpretation are introduced to simplify the derivation of the shape function. To account for the initially curved reference configuration, the curvilinear coordinate system is used as the global structure coordinate system to calculate the Green-Lagrange strain and formulate the elastic force. The tangent plane is built node-wise to transform the global curvilinear structural gradients to the local area gradients. In this way, the problem of the slope discontinuity associated with the area gradient is circumvented and the continuity of the structural gradient is guaranteed by the standard element assembly procedure. The generalized transformation between the vectors of the Bezier triangle control points and the nodal vectors of the triangular element is presented. Thus, the elements can be used for the integration of computer-aided design and analysis. Finally, the accuracy and convergence property of the new ANCF triangular plate/shell elements are verified by both static and dynamic numerical examples.

We propose a parameter estimation method for non-stationary Poisson time series with the abnormal fluctuation scaling, known as Taylor&rsquo, s law. By introducing the effect of Taylor&rsquo, s fluctuation scaling into the State Space Model with the Particle Filter, the underlying Poisson parameter&rsquo, s time evolution is estimated correctly from given non-stationary time series data with abnormally large fluctuations. We also developed a discontinuity detection method which enables tracking the Poisson parameter even for time series including sudden discontinuous jumps. As an example of application of this new general method, we analyzed Point-of-Sales data in convenience stores to estimate change of probability of purchase of commodities under fluctuating number of potential customers. The effectiveness of our method for Poisson time series with non-stationarity, large discontinuities and Taylor&rsquo, s fluctuation scaling is verified by artificial and actual time series.

We present a systematic method of removing endpoint-discontinuity artifacts in the Fourier analysis of spectral segments, enabling the more accurate extraction of information. This principal-component-removal approach differs from a previous version by using extrapolated (or extended) data outside rather than inside the spectral range. This not only allows coefficients to be accessed to the white-noise limit with no distortion of the segment, but also generates interpolated coefficients for improved analytic insight. Examples are provided.

We consider the random process at − v+(pt) + v−(−qt), t ∈ (−∞, −), where v− and v+ are independent standard Poisson processes if t ≥ 0 and v−(t) = v+(t) = 0 if t < 0. Under certain conditions on the parameters a, p, and q, we study the distribution function G = G(x) of the time of attaining the maximum for a trajectory of this process. In the present article, we find an exact asymptotics for the tails of G. We also find a connection between this problem and the statistical problem of estimation of an unknown discontinuity point of a density function.

In problems involving strong shock waves (the problem of a strong explosion, the motion of bodies with large supersonic speeds, and the problem of supersonic piston expansion), a significant increase in temperature occurs in the disturbed flow region. It is therefore necessary to take the real properties of the gas into account when determining the parameters of the gas flow behind the front of a strong shock wave. This significantly complicates the construction of approximate analytical solutions. However, studies show that the effect of the real properties of the gas on the gas-dynamic parameters of the flow can be taken into account by changing the adiabatic index, i.e., by introducing an effective adiabatic index. If the gas behind the shock wave is in a state of thermodynamic equilibrium, then the effective adiabatic index changes insignificantly in the entire flow zone. This allows us to simulate the flow behind the shock wave front using some perfect gas, whose adiabatic index is determined by the shock wave front depending on the Mach number and the thermodynamic state of the gas. To obtain more accurate solutions for problems with strong shock waves, the model must allow a discontinuity in the adiabatic index at the shock wave. An explicit expression for gas parameters behind the front of an intense shock wave is obtained in this study under the assumption that the adiabatic index undergoes discontinuity during the transition of gas particles through the shock wave surface. Plane and axisymmetric cases are considered.

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter " tend to zero, we analyze the stability of Riemann solutions.

For Ω ⊂ R N open bounded and with a Lipschitz boundary, and 1 ≤ p + ∞ , we consider the Poincare inequality with trace term C p ( Ω ) ‖ u ‖ L p ( Ω ) ≤ ‖ ∇ u ‖ L p ( Ω ; R N ) + ‖ u ‖ L p ( ∂ Ω ) on the Sobolev space W 1 , p ( Ω ) . We show that among all domains Ω with prescribed volume, the constant is minimal on balls. The proof is based on the analysis of a free discontinuity problem.

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.

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.