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s The perforated stiffened panel is generally found as a sub-component of sophisticated structures. The fundamental purpose of this panel is to withstand against buckling under complicated loading and environmental conditions. Hence, an accurate knowledge of critical buckling behaviour of stiffened panels is very much essential for a reliable and lightweight structural design. In this paper, the focus is on quasi-laminated panels with different cutout shapes of various sizes and their responses to hygrothermal environments under nonlinearly varying edge loads and is compared with the locally stiffened panels. Towards this, the modelling of the panel and stiffener is done by adopting nine-noded heterosis plate elements and three noded beam elements respectively. The stiffener formulation is suitably modified in order to take the torsional effect also into consideration along with the effect of shear deformation. Initially, the plate and the stiffener elements are treated separately, and then the displacement compatibility is maintained between them by using the transformation matrix. For a given loading and geometric discontinuity, the stress distribution within the perforated panel is highly non-uniform in nature and hence a dynamic approach has been used to calculate buckling loads by adopting two sets of boundary conditions, one set for pre-buckling stress analysis and the second set for buckling analysis. Four different quasi-isotropic stacking sequences are deliberated in this work by varying different ply-orientation in each scheme. The study also addresses the effect of various parameters such as nonlinear loads, hygro-thermal loads, cutout size and shapes, position of cutout, stiffener parameters, stacking sequences, thickness of plate and boundary conditions.

The characterization of ( S , N ) -implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [ 0 , 1 ] 2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [ 0 , 1 ] 2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of ( S , N ) -implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [ 0 , 1 ] 2 or a strict t-conorm.

The fixed-time control problem for a class of interconnected systems is studied in this article via an asynchronous event-triggered control strategy, in which all subsystems are interconnected by discontinuous interactions. Especially, due to the discontinuity of interactions, the existence of solutions for the concerned systems is solved via the framework of differential inclusion. Unlike the common ones, the asynchronous event-triggered mechanism in this article generates samplings, triggering events, and control updates asynchronously among multiple subsystems. More significantly, due to the discontinuity of the interactions, a fixed-time discontinuous control law and a specific event-triggered scheme are designed to stabilize this class of interconnected systems in a fixed time. In addition, the interexecution time is lower bounded by a positive constant, and it is assured that the Zeno behavior will not happen. Finally, the effectiveness of the obtained results is illustrated by an example.

The characterizing set-valued functions were introduced for uninorms with continuous underlying functions and these set-valued functions were useful for the complete characterization of such uninorms. In this work we study the characterizing functions of n-uninorms with continuous underlying t-norms and t-conorms. We will show that an n-uninorm with continuous underlying functions possesses n characterizing set-valued functions, where the graphs of these characterizing set-valued functions cover the set of all points of discontinuity of the respective n-uninorm. Moreover, for i=1,…,n, the i-th characterizing set-valued function divides the unit square into the two sets, where below the graph of the i-th characterizing set-valued function the n-uninorm attains values smaller than the local neutral element ei and above the graph of the i-th characterizing set-valued function the n-uninorm attains values greater than the local neutral element ei.

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto like theorems characterizing semi-local structural stability., 46 pages, 16 figures; new title, abstract, introduction and conclusion; improved statements and argumentation

In this work, the structure of uninorms not locally internal on the boundary is discussed. We first study a conjunctive uninorm U with neutral element e, where the boundary function U ( 1 , ⋅ ) of U only has two points of discontinuity and one of them is 0 or e. We partially obtain the structure of the conjunctive uninorm U. Its complete structure, under some additional conditions, is achieved as well. For the structure of a disjunctive uninorm U, whose boundary function U ( 0 , ⋅ ) only has two points of discontinuity and one of them is 1 or e, we obtain similar results.

The recalibration of the sunspot number series, the primary long-term record of the solar cycle, requires the recovery of the entire collection of raw sunspot counts collected by the Zurich Observatory for the production of this index between 1849 and 1980. Here, we report about the major progresses accomplished recently in the construction of this global digital sunspot number database, and we derive global statistics of all the individual observers and professional observatories who provided sunspot data over more than 130 years. First, we can announce the full recovery of long-lost source-data tables covering the last 34 years between 1945 and 1979, and we describe the unique information available in those tables. We then also retrace the evolution of the core observing team in Zurich and of the auxiliary stations. In 1947, we find a major disruption in the composition of both the Zurich team and the international network of auxiliary stations. This sharp transition is unique in the history of the Zurich Observatory and coincides with the main scale-jump found in the original Zurich sunspot number series, the so-called "Waldmeier" jump. This adds key historical evidence explaining why methodological changes introduced progressively in the early $20^{th}$ century could play a role precisely at that time. We conclude on the remaining steps needed to fully complete this new sunspot data resource., Comment: 22 pages, 9 figures

Fast factorized back-projection (FFBP) is a classic fast time-domain algorithm (FTDA), which is not limited by the assumption of azimuth-invariant of echo signal and is suitable for the bistatic synthetic aperture radar (BiSAR) process of arbitrary geometric configuration. However, when the conventional FFBP processing is employed for continuous imaging of multiple full-apertures, the processing efficiency will be decreased significantly, and difficulty will be introduced in motion compensation (MOCO) development. The main contributions in this article include the following two aspects: 1) a new FTDA framework based on FFBP implementation is developed for continuous imaging where echo data are divided into several full-aperture data blocks and then processed separately by FFBP implementation to reduce redundant BP operations for achieving high efficiency and 2) an efficient and effective data-driven MOCO methodology is developed based on the new FTDA framework for high focusing quality. In MOCO, because the phase error functions of subimages are estimated in the phase history domain from different local polar coordinate systems, these phase error functions are actually discontinuous in the spatial domain, which will bring significant discontinuity and defocusing into the final image. To address this problem, the correspondence of error functions between the spatial domain and the wavenumber domain is revealed based on which the phase error functions are reconstructed to remove the discontinuity for high focusing quality. Promising results from both simulation and raw data experiments are provided and analyzed to validate the high performance of the proposed algorithm.

Phase unwrapping is a critical step of interferometric synthetic aperture radar (InSAR) processing, and its accuracy directly determines the reliability of subsequent applications. Many phase unwrapping methods have been proposed, most of which assume that the phase has spatial continuity, while decorrelation noise and aliasing fringes invalidate the assumptions, resulting in poor performance of these methods. To obtain more reliable unwrapping results, in this paper, a deep convolutional neural network, called DENet, is proposed for predicting the probabilities of phase discontinuities in interferograms. The main advantages of DENet are: (1) using branching structure to extract detailed and high-level features separately, to retain details while making full use of contextual information; (2) using multi-channel input, including interferogram, range/azimuthal phase gradients, and residues map, to provide effective guidance for discontinuity prediction; and (3) using a single network to estimate phase discontinuities in both range and azimuth directions simultaneously. To train the network, a data set simulation strategy is proposed to generate enough training samples. The strategy considers a variety of phase components such as terrain-related phase, random deformation, atmospheric turbulence, and noise. The phase discontinuity estimated by DENet is then converted to costs in the minimum cost flow solver of SNAPHU to obtain the final unwrapped phase. Based on validations of simulated and real interferograms, the proposed method exhibits excellent performance compared to traditional and deep learning unwrapping methods. The proposed method can effectively unwrap large-scale, low-quality interferograms, which is expected to significantly improve the accuracy of InSAR applications.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

In this paper, we consider the inverse scattering problem for Sturm-Liouville operator with discontinuity conditions at some point on the positive half line. The scattering data of this boundary value problem is examined. The resolvent operator is constructed and the expansion formula with respect to the eigenfunctions of this boundary value problem is obtained. The main equation or modified Marchenko equation of the inverse scattering problem is derived and an algorithm of the construction of the potential function according to scattering data of this boundary value problem is given.

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

In this paper, a new method for controlling battery emulators in the shape of bidirectional step-down converters driving constant power loads (CPL) is presented. The CPL introduces nonlinear behavior into the system and causes unstable system dynamics. This property and the discontinuity introduced by the switches makes it challenging to develop a suitable control strategy. Usually, switching frequencies are assumed to be fast enough for the controller to neglect the fact that the switches are either on or off. As the desired dynamics of the system approaches the switching frequency of the converter, the discontinuity of the switches can no longer be ignored and the controller has to generate a suitable piecewise constant control input. A novel approach for arriving at a piecewise constant control input is presented and compared to state of the art approaches. The advantages of the new control method are demonstrated with simulations as well as measurements from a hardware-in-the-loop testbed.

Speech production requires temporal coordination between the actions of different functional groupings of muscles in the human body. Crucially, such functionally organized units, or “modules”, may be susceptible to disruption by an external stimulus such as a startling auditory stimulus (SAS; >120dB), enabling a possible window into the internal structure of learned speech movements. Following on the observation that SAS is known to accelerate the release of pre-planned actions, the current study examines lip kinematics in SAS-induced responses during speech movements to test whether this accelerated release applies on the scale of entire syllables or on the scale of smaller functional units. Production measures show that SAS-elicited bilabial movements in [ba] syllables are prone to disruption as measured by discontinuity in velocity profiles. We use a 3D finite element method (FEM) biomechanical model to simulate the temporal interaction between muscle groupings in speech. Simulation results indicate that this discontinuity can be accounted for as an instance of temporally decoupled coordination across neuromuscular modules. In such instances, the muscle groupings controlling lip compression and jaw opening, which normally fire sequentially, appear more likely to be activated synchronously.

We study the ill-posedness issue for the compressible viscous heat-conductive flows in two dimensions. In the scaling invariant spaces, negative regularity of the temperature causes an essential problem for the well-posedness, and the ill-posedness is obtained for all integrability indices except for the $$L^1$$ case.

Soil settlement is an essential criterion in the design and performance assessment of geotechnical structures, often evaluated using results from one-dimensional compression test. However, assessment of the results from this test involves graphical methods, which produces inconsistent interpretations on the parameters to be used in design. The nonlinear compressibility curve is also subjected to bilinear idealization with divergence from the actual behavior at high stresses. In order to overcome these problems, a nonlinear mathematical formulation is developed to represent the soil compressibility with high precision over a wide range of stresses. The adequacy of the formulation is validated against a large and diverse dataset comprised of different geological materials, including undisturbed and reconstituted soils as well as stabilized soils. The proposed formulation is dependent on two fitting coefficients, which are defined in terms of commonly used compressibility parameters compression index, recompression index and preconsolidation pressure. A single expression is also derived to calculate soil settlement without numerical discontinuity. Finally, a method for evaluation of swelling pressure of soils is also developed using the parameters of the same mathematical formulation.

This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈ (1, 3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, consisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.

The methods and performance of a 3D pin-by-pin neutronics code based on the 2D/1D decoupling method are presented. The code was newly developed as an effort to achieve enhanced accuracy and high calculation performance that are sufficient for the use in practical nuclear design analyses. From the 3D diffusion-based finite difference method (FDM) formulation, decoupled planar formulations are established by treating pre-determined axial leakage as a source term. The decoupled axial problems are formulated with the radial leakage source term. To accelerate the pin-by-pin calculation, the two-level coarse mesh finite difference (CMFD) formulation, which consists of the multigroup node-wise CMFD and the two-group assembly-wise CMFD is implemented. To enhance the accuracy, both the discontinuity factor method and the super-homogenization (SPH) factor method are examined for pin-wise cross-section homogenization. The parallelization is achieved with the OpenMP package. The accuracy and performance of the pin-by-pin calculations are assessed with the VERA and APR1400 benchmark problems. It is demonstrated that pin-by-pin 2D/1D alternating calculations within the two-level 3D CMFD framework yield accurate solutions in about 30 s for the typical commercial core problems, on a parallel platform employing 32 threads.

Seismic coherence is of the essence for seismic interpretation as it highlights seismic discontinuity features caused by the deposition process, reservoir boundaries, tectonic movements, etc. Since its appearance in 1995, seismic coherence has become one of the most popular and highly recognized interpretation tools. In the last 25 years, there have been different kinds of approaches to calculate seismic coherence attributes, such as cross-correlation, semblance, and eigen-structure. We have also seen different coherence enhancement techniques and modified ways for coherence calculation to address different problems in diverse field applications. In this survey, we provide a general overview of seismic coherence, which is commonly used to delineate structural and stratigraphic discontinuities. We cover the development of the seismic coherence attributes via introducing existing approaches for coherence calculation, enhancement, spectral band-limited coherence methods, and offset-/azimuth-limited coherence methods. In addition, we also discuss the possible coherence artifacts and pitfalls. To better compare different techniques, field applications are provided, where different disciplinary coherence attributes with enhancement techniques are calculated for the channel, fault, carbonate karst, and volcano interpretation.

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.

A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.

This paper deals with the discontinuous dynamic behaviors of a class of three-degree-of-freedom friction and collision system with symmetric bilateral rigid constraints by using the flow switching theory of discontinuous dynamical systems. The model takes into account the inequality of static and dynamic friction forces, which is more common in practice. The external excitation of the object contains a negative feedback, which can change with the variation of the velocity of the object, so that the system can adapt to different external environment. And the connection between the objects adopts nonlinear springs and viscous dampers in order to achieve a better vibration reduction effect. The model can be applied to mechanical equipment such as shock absorbers etc. According to the discontinuity caused by friction and collision, the dynamic domain and boundary of the object’s motion are defined in phase space, which requires the introduction of absolute coordinates and relative coordinates respectively to discuss the motion between objects. The vector field for the respective domain in the system is given, which can control the motion of object in each domain. The corresponding normal vector on the discontinuous boundary is introduced to determine the positive direction of the flow. The sufficient and necessary conditions for the switching motion of such dynamical system are given based on $$\mathrm{G}$$ -function and its higher order. These switching conditions can be used to predict or even control the motion of the object on the separation boundary. In addition, several typical motions, such as sliding, stick, grazing, impact motions and periodic motion, are numerically simulated to better explain the complexity of the switching motion in the discontinuous dynamic system. Finally, the stick bifurcation scenarios for excitation frequency or amplitude are presented. The research results obtained can provide a theoretical basis for better use or control of friction and collision in practical applications.

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.

We investigate the emergence of nonphysical solutions to the steady-state Wigner transport equation on finite-sized simulation domains with inflow boundary conditions. We find that inflow boundary conditions are generally valid, but the wave number uncertainty of injected wave packets has a lower bound that can be significantly higher than usually assumed. Large values of the so-called quantum evolution term (which captures spatial nonlocality in the Wigner transport equation) near simulation-domain boundaries are the cause of spurious reflections, the often-reported discontinuity of the Wigner function at zero wave vector, and negative probabilities. We offer a simple relationship between the lower bound of the wave number uncertainty and simulation parameters that will ensure physical results with inflow boundary conditions.

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.

Depth-dose profiles of electron radiation in continuous and discontinuous materials of water, plastic and food-like dummy irradiated by 10 MeV electron beam have been evaluated based on Monte Carlo simulations and measurements. In the discontinuous mediums, material layers were separated by air gaps. The discrepancy of the absorbed doses in the continuous and discontinuous materials is within 12%. No significant effect of the discontinuity on the main characteristics such as practical range and double-sided optimal thickness was found. The results indicated that the double-sided optimal thicknesses of the materials can be increased to greater than the reference values of 8.5–8.8 g/cm $$^2$$ to improve the efficiency of the irradiation process.

Исследованы возможности численного метода коллокации и наименьших квадратов (КНК) на примерах кусочно-полиномиального решения задачи Дирихле для уравнений Пуассона и типа диффузии-конвекции с особенностями в виде больших градиентов и разрыва решения на границах раздела двух подобластей. Предложены и реализованы новые hp-варианты метода КНК, основанные на присоединении внутри области малых и/или вытянутых нерегулярных ячеек, отсекаемых криволинейной границей раздела от исходных прямоугольных ячеек сетки, к соседним самостоятельным ячейкам. Выписываются с учетом особенности условия согласования между собой кусков решения в ячейках, примыкающих с разных сторон к границе раздела. Проведено сравнение результатов, полученных методом КНК и другими высокоточными методами. Показаны преимущества и достоинства метода КНК. Для ускорения итерационного процесса применены современные алгоритмы и методы: предобуславливание; свойства локальной системы координат в методе КНК; ускорение, основанное на подпространствах Крылова; операция продолжения на многосеточном комплексе; распараллеливание. Исследовано влияние этих способов на количество итераций и время расчетов при аппроксимации полиномами различных степеней. The capabilities of the numerical least-squares collocation (LSC) method of the piecewise polynomial solution of the Dirichlet problem for the Poisson and diffusion-convection equations are investigated. Examples of problems with singularities such as large gradients and discontinuity of the solution at interfaces between two subdomains are considered. New hp-versions of the LSC method based on the merging of small and/or elongated irregular cells to neighboring independent cells inside the domain are proposed and implemented. They cut off by a curvilinear interface from the original rectangular grid cells. Taking into account the problem singularity the matching conditions between the pieces of the solution in cells adjacent from different sides to the interface are written out. The results obtained by the LSC method are compared with other high-accuracy methods. Advantages of the LSC method are shown. For acceleration of an iterative process modern algorithms and methods are applied: preconditioning, properties of the local coordinate system in the LSC method, Krylov subspaces; prolongation operation on a multigrid complex; parallelization. The influence of these methods on iteration numbers and computation time at approximation by polynomials of various degrees is investigated.

In this work, a robust corrected weakly compressible smoothed particle hydrodynamics scheme combined with several improved interface treatments (CWCSPH-IT) is proposed to investigate viscous or viscoelastic two-phase flows of high density ratios. The proposed viscoelastic two-phase CWCSPH-IT model is mainly derived from three aspects: (a) a combined elastic stress approximation in the momentum equation based on the weighted idea is proposed; (b) the transition region near the interface of two fluids is regarded as a weak polymer fluid by approximating the relaxation time with a Shepard interpolation which handles the discontinuity of elastic stress near the interface; and (c) a new mixed boundary treatment is proposed to treat the wall boundary in a viscoelastic flow. Moreover, a weighted color gradient scheme is adopted to handle the surface tension near the phase interface, and a particle shifting technique is used to regularize particle distributions and enhance numerical stability. Several two-phase benchmark problems are simulated to test the validity and reliability of the proposed CWCSPH-IT. Subsequently, the proposed particle scheme is adopted to simulate the droplet rising in viscous/viscoelastic matrix fluid with high density ratios. Further, the stable dimple shape of a polymer drop falling in a low-density matrix, and the complex overshooting, oscillation and breakup phenomena of a drop in a high-viscoelasticity shear fluid are numerically investigated. The comparison between the SPH and other reliable reference results demonstrates the ability of the present viscoelastic two-phase scheme in modeling complex viscous or viscoelastic multi-phase problems.

Slip flow in a rectangular cavity is studied. Using slip eigenfunctions, an exact series solution is found for a lid sliding longitudinally. For a transversely sliding lid, the stream function for Stokes flow is determined by a new set of slip eigenfunctions and supplemented by point match on the boundary. It is found that slip has considerable effect on the flow properties. Analyses of the region very close to the corner of the velocity discontinuity show the stress singularity of no-slip flow can be eliminated by slip for both longitudinal flow and transverse flow, except when the latter corner is a right angle.

This article aims to solve the nonsmooth resource allocation problem in the presence of a global network resource constraint and local set constraints in the framework of multiagent system optimization. It is assumed that multiagent systems are subject to some external disturbances, and the control inputs of the agents satisfy Lispchitz continuity. These two distinguished features render the existing distributed optimization algorithms, especially the subgradient-based algorithms inapplicable due to the employment of discontinuity of subgradients. To solve such a challenging resource allocation problem, a new kind of continuous-time proximal algorithm is designed with the aid of convex optimization theory and the internal-model technique. The proximal algorithm is further augmented by introducing an event-based communication scheme such that the continuous-time communication among the agents is avoided successfully. The theoretical analysis shows that the multiagent systems under the proposed algorithms can converge to the optimal solution of the considered problem, while the external disturbances are rejected. Besides, the Zeno behavior can be excluded for the proximal algorithm with event-based communication. Finally, the numerical simulations are given to verify the established theoretical results.

Abstract The determination of 3D particle shapes from the one-dimensional data on small-angle scattering from macromolecule solutions is ambiguous. In addition, due to the poor conditionality of the inverse problem, the determination is unstable and depends on the search algorithm parameters. Consequently, in practice, it is necessary not only to estimate the degree of stability of the solution of the problem but also to select the parameters of both the search method and the model itself. A search tactic is considered which consists in sequential determination of a set of models of particle shape represented in the form of a structure consisting of small close-packed spherical beads. The set of solutions is obtained by varying the relative contributions of objective function terms: the criterion for the deviation of the model scattering curve from the experimental one and the penalties for the looseness and discontinuity of the body structure, and the deviation of the average number of bead contacts from a specified value. Examples of the solution of model problems and the determination of molecular shape from the measurements available in the SASBDB bank of structures are presented.

In this study, an energy‐efficient speed trajectory planner is proposed for high‐speed trains traveling on tracks with steep gradients and speed limits, especially for situations in which the speed limit has been reached, which causes adjoint variable discontinuity during calculation. New optimal switching rules at points where the speed limit is reached on steep tracks are derived by analysing the jump condition of state‐constrained Pontryagin's maximum principle. Accordingly, a novel two‐step algorithm for high‐speed trains, including an optimal‐cruise minimum‐time algorithm and search‐substitution algorithm, is designed to solve dynamic train models considering time‐energy and space‐energy conversions, respectively. Practical case studies demonstrates that the proposed method can save energy by approximately 3% and 10% in comparison to the approximate‐optimal time‐satisfied and minimum running time strategies, respectively. Moreover, the proposed method approximately consumes 0.98% and 1.62% of the computation time taken by discrete dynamic programming and reinforcement learning, respectively.

Recurrent neural network (RNN) has become popular for human motion prediction thanks to its ability to capture temporal dependencies. However, it has limited capacity in modeling the complex spatial relationship in the human skeletal structure. In this work, we present a novel diffusion convolutional recurrent predictor for spatial and temporal movement forecasting, with multi-step random walks traversing bidirectionally along an adaptive graph to model interdependency among body joints. In the temporal domain, existing methods rely on a single forward predictor with the produced motion deflecting to the drift route, which leads to error accumulations over time. We propose to supplement the forward predictor with a forward discriminator to alleviate such motion drift in the long term under adversarial training. The solution is further enhanced by a backward predictor and a backward discriminator to effectively reduce the error, such that the system can also look into the past to improve the prediction at early frames. The two-way spatial diffusion convolutions and two-way temporal predictors together form a quadruple network.\ud \ud Furthermore, we train our framework by modeling the velocity from observed motion dynamics instead of static poses to predict future movements that effectively reduces the discontinuity problem at early prediction. Our method outperforms the state of the arts on both 3D and 2D datasets, including the Human3.6M, CMU Motion Capture and Penn Action datasets. The results also show that our method correctly predicts both high-dynamic and low-dynamic moving trends with less motion drift.

The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. An explicit Poincaré map is constructed with an implicit constraint on the switching time. The stability of the fixed points of the Poincaré map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle-type fixed point. The global dynamics of the system exhibits discontinuity induced bifurcations of the fixed points.

Magnetic Resonance Imaging (MRI) is a powerful imaging modality with highly specific applications in medicine. However, certain series and sequences of MRI do not produce images with good acuity. Widely-used sharpening algorithms like Shock Filter (SF) and Unsharp Masking (UM) do not provide output images with appreciable sharpness-to-noise ratio. Output images of UM are prone to discontinuity artefact. A noise-robust sharpening algorithm to enhance acuity of MR images termed as Nonlinear Amplification of Spatial Derivative (NASD) that does not have the issue of discontinuity artefact is introduced in this paper. In the NASD, the enhanced image is estimated from the amplified spatial derivative in an inverse way. To avoid the amplification of noise, the amplification factor at any pixel location is calculated from a nonlinear function of the mean of the absolute values of the gradients along eight different orientations at that location. On hundred test images, the sharpness-to-noise ratio exhibited by NASD, SF and UM are 0.0963 ± 0.0400, 0.1642 ± 0.0967 and 0.2851 ± 0.1610. The computation time (in seconds) of NASD, SF and UM are 0.0330 ± 0.0031, 0.0452 ± 0.0390 and 0.0554 ± 0.0112. The NASD exhibits higher values of sharpness-to-noise ratio than SF and UM. NASD is able to sharpen the edges in the input image without amplifying noise and it is computationally fast like SF and UM.

We extend the applicability of the popular interior penalty discontinuous Galerkin method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new H 1 − L 2 H^1-L_2 -type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.

Evaluating carbon’s candidacy as a light element in the Earth’s core is critical to constrain the budget and planet-scale distribution of this life-essential element. Here we use first principles molecular dynamics simulations to estimate the density and compressional wave velocity of liquid iron-carbon alloys with ~4-9 wt.% carbon at 0-360 gigapascals and 4000-7000 kelvin. We find that for an iron-carbon binary system, ~1-4 wt.% carbon can explain seismological compressional wave velocities. However, this is incompatible with the ~5-7 wt.% carbon that we find is required to explain the core’s density deficit. When we consider a ternary system including iron, carbon and another light element combined with additional constraints from iron meteorites and the density discontinuity at the inner-core boundary, we find that a carbon content of the outer core of 0.3-2.0 wt.%, is able to satisfy both properties. This could make the outer core the largest reservoir of terrestrial carbon. A carbon content in Earth’s outer core between 0.3 and 2.0 % by weight, along with at least two other light elements, is compatible with observational constraints, according to molecular dynamics simulations, and could make the core Earth’s largest carbon reservoir.

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%.

Modal network theory has proven to be a useful tool in analyzing waveguide discontinuities. Ports of a modal network correspond to waveguide eigenmodes on either side of the discontinuity. In this work, modal network theory is extended beyond conventional/geometrical waveguide discontinuities. We use the modal network formulation to analyze discontinuities resulting from metasurfaces. Specifically, we analyze azimuthally invariant cascaded sheet metasurfaces placed in a cylindrical waveguide excited by ${\text {TM}}_{0n}$ modes. The metasurface comprises multiple radially varying electric admittance sheets separated by dielectric spacers. We derive the modal representation of a single inhomogeneous, electric sheet admittance using two approaches: a conventional numerical integration approach and a discrete Hankel transform (DHT) approach. We show that the DHT approach is more efficient. Subsequently, the modal matrices of cascaded sheet metasurfaces are derived using the DHT. Finally, we propose an optimization-based procedure to synthesize transparent metasurface-based mode converters. The mode converters transform a set of incident modes on one side to another set of desired modes on the opposite side of the metasurface. Two designs are synthesized at 10 GHz: a single-mode converter, and a mode splitter. The designs are verified using the commercial finite element electromagnetic solvers.

Dynamic electromagnetic characteristics is significant for the design and operation of superconducting (SC) electrodynamic suspension (EDS) train. This paper established a three-dimensional transient numerical model in COMSOL. The current density excitation comes from the homogenized model of the superconducting magnet. In order to decrease the computing time, periodic boundary is introduced taking the place of finitely long route. Mapping operator is adopted to ensure the continuity of physical quantities in the case of geometric discontinuity in the process of motion. Induced current and magnetic force at different operation velocity and different structure are simulated. In general, this model could be applied to simulate long term operation. It is also suited for the simulation of propulsion systems for EDS systems. In addition, this model provides electromagnetic environment intuitively that would be helpful for the stability analysis and design of onboard SC magnets during long term operation.

Broadband impedance spectroscopy (BIS) provides a new method for cable fault location, however, this method has not been properly used in locating cable fault of cross-bonded (CB) cables due to the geometric discontinuity of impedance spectroscopy measurement loop. This study addresses the possibility of fault location by BIS in CB cables. First, this paper proposes a fault location method by transforming BIS in frequency domain into the location function in space domain with the aid of integral transformation (IT). The kernel function $\mathrm{e}^{j{2}\beta x}$ is used to obtain fault location function. Second, this paper demonstrates an impedance spectroscopy measurement loop in CB cables. BIS in CB cables is observed having a generalized orthogonality with the kernel function $\mathrm{e}^{j{2}\beta x}$ . Given this, the IT method is applicable for fault location of CB cables. Finally, this study carries out the location experiment of cable joints on a 110 kV CB cable in cable trench. The experiment shows that BIS in CB cables could be obtained by the loop proposed in this paper. Besides, the results verify the possibility of fault location in CB cables by IT method using BIS.

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.

This article focuses on the one of the many outcomes of the so-called rebranded philosophy of history, namely, the continuity-discontinuity issue. Eelco Runia’s, Noël Bonneuil’s and Paul A. Roth’s conceptions of historical time will be analyzed as representative of this subject in the landscape of the theory of history from 2010 on. The authors sampled not only provide the evidence that historical discontinuity remains alive as a theoretical and historiographical challenge, but they also disclose different arrays to think the relationship among past, present, and future, and historical transformation. The concepts of historical time analyzed recall Foucault’s discontinuously-base model of thinking historical time and add to it different varieties of historical discontinuity. Moreover, the continuity-discontinuity issue in the new backdrop involves operation of translating time into space (spatialization of time). As a result, the discontinuously-based model of historical time’s main characteristics will be summarized and its strength as a heuristic tool for further analyses of the concepts of historical time is outlined. Este artigo enfoca um dos muitos resultados da chamada filosofia da história redefinidas, qual seja, a questão da continuidade-descontinuidade. As concepções de tempo histórico de Eelco Runia, Noël Bonneuil e Paul A. Roth serão analisadas como representativas deste assunto no panorama da teoria da história a partir de 2010. Os autores estudados não apenas fornecem evidências de que a descontinuidade histórica permanece viva como um desafio teórico e historiográfico, mas também revelam diferentes arranjos para pensar a relação entre passado, presente e futuro, bem como a transformação histórica. Os conceitos de tempo histórico analisados ​​lembram o modelo de base descontínua de pensar o tempo histórico de Foucault, adicionando ainda a ele diferentes variedades de descontinuidade histórica. Além disso, a questão da continuidade-descontinuidade no novo pano de fundo envolve a operação de tradução do tempo no espaço (espacialização do tempo). Como resultado, as principais características do modelo baseado na descontinuidade do tempo histórico serão resumidas, assim como sua força como uma ferramenta heurística para análises posteriores dos conceitos de tempo histórico será ressaltada.

According to the existing driver model, the objective function is coupled with continuity factors and discontinuity factors, which makes it difficult to determine the weighting coefficients in multi-objective optimization, which will cause dangerous situations such as the vehicle rushing out of the road boundary; in response to this problem, this paper proposes a driver model adapted to the complex traffic environment, based on the mechanism modeling of continuity factors and rule modeling of discontinuity factors. In view of the difficulty of traditional optimization algorithms to find a balance between efficiency and accuracy, this paper proposes a grid optimization algorithm that takes into account both efficiency and accuracy. In order to reduce the amount of calculation in the preview decision-making process, this paper proposes a curve integral method based on the laws of vehicle kinematics to predict the position of the vehicle to judge whether a collision will occur. The driver model is established in the Simulink simulation environment, and the C-level prototype model in the vehicle dynamics simulation software CarSim is selected as the control object, the results show that the proposed the preview decision model effectively solves the problem of divergence in the optimization solution, and can also ensure safety and traffic rules in a complex traffic environment, improving the quality of the model.

This article explains that in the Grundrisse Marx considered that the discontinuity in his transformation procedure was no logical inconsistency, but a necessary feature of the disproportionate tra...

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.

In this work, the head-on collision and transmission with nonreciprocal properties of opposite propagating solitary waves are studied, in which the interface between different granular chains is considered. Due to the discontinuity of two periodic granular systems, the transmitted and reflected solitary waves are produced. The head-on collision appears at the interface and the reductive perturbation method is applied to derive the generated solitary waves. According to the derivation and numerical simulation, we can find that the transmitted and reflected solitary waves can propagate with the same speed when they locate at the same chain. Moreover, the influences of both the arrangement and prestress are discussed. It is found that the amplitude and velocity of solitary waves become larger because of a bigger prestress, which result in the nonreciprocal collision and transmission in the granular mechanical metamaterials. This study is expected to be helpful for the design and application of elastic wave metamaterials and mechanical diodes with nonlinear solitary waves.

In this paper, we extend the classical atmospheric Ekman equations to discontinuous differential equations of Ekman flows by considering the eddy viscosity with piecewise form. Local and global analysis for the corresponding discontinuous ODEs are studied respectively. In the section of local analysis, all the possible crossings of solutions are presented over the certain discontinuity linear subspaces associated with a curve. In the section of global analysis, crossings of solutions, bounded and unbounded solutions on dynamics are obtained.

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