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1
Song, Xiao, Yaofei Ma, Wei Zhang, and Jiangyun Wang. "Quantized State Based Simulation of Time Invariant and Time Varying Continuous Systems." Mathematical Problems in Engineering 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/141607.
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Continuous system can be discretized for computer simulation. Quantized state systems (QSS) method has been used to discretize time invariant systems based on the discretization of the state space. A HLA based QSS method is proposed in this paper to address issues of real-time advancements in simulation and an aircraft control example was introduced to illustrate our method. Moreover, to simulate time varying systems, a novel approach is also proposed and exemplified with a practical case.
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Jiang, Xiefu, Zongming Yin, and Jinjing Wu. "Stability Analysis of Linear Systems under Time-Varying Samplings by a Non-Standard Discretization Method." Electronics 7, no. 11 (October 27, 2018): 278. http://dx.doi.org/10.3390/electronics7110278.
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This paper is concerned with the stability of linear systems under time-varying sampling. First, the closed-loop sampled-data system under study is represented by a discrete-time system using a non-standard discretization method. Second, by introducing a new sampled-date-based integral inequality, the sufficient condition on stability is formulated by using a simple Lyapunov function. The stability criterion has lower computational complexity, while having less conservatism compared with those obtained by a classical input delay approach. Third, when the system is subject to parameter uncertainties, a robust stability criterion is derived for uncertain systems under time-varying sampling. Finally, three examples are given to show the effectiveness of the proposed method.
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Camlibel, Kanat, and Aneel Tanwani. "A discretization algorithm for time-varying composite gradient flow dynamics." IFAC-PapersOnLine 54, no. 9 (2021): 558–63. http://dx.doi.org/10.1016/j.ifacol.2021.06.116.
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GRÜNE, LARS. "ROBUST ASYMPTOTIC CONTROLLABILITY UNDER TIME-VARYING PERTURBATIONS." Stochastics and Dynamics 04, no. 03 (September 2004): 297–316. http://dx.doi.org/10.1142/s0219493704001085.
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We investigate the effect of time-varying perturbations on the dynamical behavior of nonlinear control systems. More specifically, we study the effect of such perturbations on the controlled equivalent of asymptotically stable sets, i.e. asymptotically controllable sets. In the first part of this paper we illustrate by a simple example how different types of perturbations affect this dynamical behavior and use concepts from dynamical game theory in order to identify classes of perturbations which allow to model the effects of numerical discretization errors both in time and space. In the second part we introduce appropriate robustness properties and prove that these are inherent properties for asymptotically controllable sets under these classes of perturbations.
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Okoniewski, Piotr, and Jacek Piskorowski. "Short Transient Parameter-Varying IIR Filter Based on Analog Oscillatory System." Applied Sciences 9, no. 10 (May 16, 2019): 2013. http://dx.doi.org/10.3390/app9102013.
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This paper presents a concept for digital infinite impulse response (IIR) lowpass filter with reduced transient response. The proposed digital filtering structure is based on an analog oscillatory system. In order to design the considered digital filter, the analog prototype is subjected to a discretization process and, then, the parameters describing the dynamical properties of the oscillatory system are temporarily varied in time, so as to suppress the transient response of the designed filter. An optimization method, aimed at reducing the settling time by proper parameter manipulation, is presented. Simulation results, along with a real-life application proving the usefulness of the proposed concept, are also shown and discussed.
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KO, Jeong-Wan, and PooGyeon PARK. "Delay-Dependent Stability Criteria for Systems with Time-Varying Delays: State Discretization Approach." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E92-A, no. 4 (2009): 1136–41. http://dx.doi.org/10.1587/transfun.e92.a.1136.
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Shao, Chenhui, Shuang Li, Haitao Li, and Jie Sheng. "Control for Time‐Varying Delay Systems by Integrating Semi‐Discretization and Hysteresis‐Based Switching." Asian Journal of Control 21, no. 5 (August 8, 2018): 2181–92. http://dx.doi.org/10.1002/asjc.1829.
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De la Sen, M. "About a Class of Positive Hybrid Dynamic Linear Systems and an Associate Extended Kalman-Yakubovich-Popov Lemma." Discrete Dynamics in Nature and Society 2017 (2017): 1–22. http://dx.doi.org/10.1155/2017/3928970.
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This paper formulates an “ad hoc” robust version under parametrical disturbances of the discrete version of the Kalman-Yakubovich-Popov Lemma for a class of positive hybrid dynamic linear systems which consist of a continuous-time system coupled with a discrete-time or a digital one. An extended discrete system, whose state vector contains both the digital one and the discretization of the continuous-time one at sampling instants, is a key analysis element in the formulation. The hyperstability and asymptotic hyperstability properties of the studied class of positive hybrid systems under feedback from any member of a nonlinear (and, eventually, time-varying) class of controllers, which satisfies a Popov’s-type inequality, are also investigated as linked to the positive realness of the associated transfer matrices.
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WANG, JINJIE, GUOHUA CAO, YANDONG WANG, and RONGHUA WU. "A NOVEL DRIVING STRATEGY FOR DYNAMIC SIMULATION OF HOISTING ROPE WITH TIME-VARYING LENGTH." International Journal of Modeling, Simulation, and Scientific Computing 04, no. 03 (August 19, 2013): 1350009. http://dx.doi.org/10.1142/s1793962313500098.
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A simulation method for investigating the vibration behavior of hoisting rope with time-varying length is improved. By previously creating markers in the MSC.ADAMS software package, the parametric model of the rope wound along helix is established based on the concentrated-mass theory with multi-degree of freedom (multi-DOF). A novel driving strategy, cooperating fixed joints with angle sensors under the control of driving script, is proposed to substitute conventional contact force. Researching on the hoisting rope in the sinking winch mechanism, an equivalent discretization model is obtained with complicated boundary conditions considered. The differential equations of motion of the hoisting system are formulated employing Lagrange's equation and numerically solved using Runge–Kutta method. The simulation indicates that the horizontal swing is decreased in principle and the simulation with 800 discrete ropes is not performed more than 61 min. Therefore, this feasible strategy could not only guarantee the accuracy but also promote simulation efficiency and stability. The motion curves exported from ADAMS simulation coincide with one in numerical simulation, which validates both the numerical model and the driving strategy.
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Bürger, Raimund, Julio Careaga, and Stefan Diehl. "A method-of-lines formulation for a model of reactive settling in tanks with varying cross-sectional area." IMA Journal of Applied Mathematics 86, no. 3 (May 20, 2021): 514–46. http://dx.doi.org/10.1093/imamat/hxab012.
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Abstract Reactive settling denotes the process of sedimentation of small solid particles dispersed in a viscous fluid with simultaneous reactions between the components that constitute the solid and liquid phases. This process is of particular importance for the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as wastewater treatment plants. A spatially 1D model of reactive settling in an SST is formulated by combining a mechanistic model of sedimentation with compression with a model of biokinetic reactions. In addition, the cross-sectional area of the tank is allowed to vary as a function of height. The final model is a system of strongly degenerate parabolic, nonlinear partial differential equations that include discontinuous coefficients to describe the feed, underflow and overflow mechanisms, as well as singular source terms that model the feed mechanism. A finite difference scheme for the final model is developed by first deriving a method-of-lines formulation (discrete in space, continuous in time) and then passing to a fully discrete scheme by a time discretization. The advantage of this formulation is its compatibility with common practice in development of software for WRRFs. The main mathematical result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in SSTs in WRRFs illustrate the model and its discretization.
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Wiechetek, Katarzyna, and Jacek Piskorowski. "A Concept of the Non-Stationary Filtering Network with Reduced Transient Response." Applied Sciences 9, no. 21 (October 28, 2019): 4570. http://dx.doi.org/10.3390/app9214570.
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This paper presents a concept of the non-stationary filtering network with reduced transient response consisting of the first-order digital elements with time-varying parameters. The digital filter section is based on the analog system. In order to design the filtering network, the analog prototype was subjected to the discretization process. The time constant and the gain factor were then temporarily varied in time in order to suppress the transient response of the designed filtering structure. The optimization method, based on the Particle Swarm Optimization (PSO) algorithm which is aimed at reducing the settling time by a proper parameter manipulation, is presented. Simulation results proving the usefulness of the proposed concept are also shown and discussed.
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Attaullah, Şuayip Yüzbaşı, Sultan Alyobi, Mansour F. Yassen, and Wajaree Weera. "A Higher-Order Galerkin Time Discretization and Numerical Comparisons for Two Models of HIV Infection." Computational and Mathematical Methods in Medicine 2022 (November 9, 2022): 1–24. http://dx.doi.org/10.1155/2022/3599827.
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Human immunodeficiency virus (HIV) infection affects the immune system, particularly white blood cells known as CD4+ T-cells. HIV destroys CD4+ T-cells and significantly reduces a human’s resistance to viral infectious diseases as well as severe bacterial infections, which can lead to certain illnesses. The HIV framework is defined as a system of nonlinear first-order ordinary differential equations, and the innovative Galerkin technique is used to approximate the solutions of the model. To validate the findings, solve the model employing the Runge-Kutta (RK) technique of order four. The findings of the suggested techniques are compared with the results obtained from conventional schemes such as MuHPM, MVIM, and HPM that exist in the literature. Furthermore, the simulations are performed with different time step sizes, and the accuracy is measured at various time intervals. The numerical computations clearly demonstrate that the Galerkin scheme, in contrast to the Runge-Kutta scheme, provides incredibly precise solutions at relatively large time step sizes. A comparison of the solutions reveals that the obtained results through the Galerkin scheme are in fairly good agreement with the RK4 scheme in a given time interval as compared to other conventional schemes. Moreover, having performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes, it is observed that the Galerkin scheme is noticeably slower than the Runge-Kutta scheme. On the other hand, this work is also concerned with the path tracking and damped oscillatory behaviour of the model with a variable supply rate for the generation of new CD4+ T-cells (based on viral load concentration) and the HIV infection incidence rate. Additionally, we investigate the influence of various physical characteristics by varying their values and analysing them using graphs. The investigations indicate that the lateral system ensured more accurate predictions than the previous model.
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Ahmad, Alalyani. "On the Solution of a Nonlinear Fractional-Order Glucose-Insulin System Incorporating β -cells Compartment." Malaysian Journal of Mathematical Sciences 17, no. 1 (March 27, 2023): 1–12. http://dx.doi.org/10.47836/mjms.17.1.01.
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In this work, we are interested in studying variations in plasma glucose and insulin levels over time using a fractional-order version of a mathematical model. Applying the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy β-cells. The main aim is to obtain the numerical solution of the proposed model in order to show variations in plasma glucose and insulin levels over time, by applying the generalized Euler method. The local stability analysis of the proposed (discretization) Caputo fractional-order model was discussed. To check the feasibility of our analysis, we have investigated some numerical simulations for various fractional orders by varying values of the parameters with help of Mathematica. Numerical simulations were in good agreement with the theoretical findings. Three specific numerical examples are given as applications of the main results.
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Manin, L., and D. Play. "Thermal Behavior of Power Gearing Transmission, Numerical Prediction, and Influence of Design Parameters." Journal of Tribology 121, no. 4 (October 1, 1999): 693–702. http://dx.doi.org/10.1115/1.2834125.
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Thermal behavior prediction of Power Gearing Transmissions during preliminary design becomes necessary in order to optimize all the parts of mechanical systems (lubrication, cooling device dimensioning, static and dynamic stress resistance, etc.). It also reduces prototype tests. The thermal network method is used to model each technological element as a thermal finite element and a nonlinear system of equations is obtained. Geometric discretization is adapted to the scale of phenomenological observation and result requirements. As assumptions have to be made for the modeling of convection heat transfer and oil flow, experimental verifications are made. The influence of gear parameters, the effect of time varying running conditions, and oil flow defaults are then studied and discussed.
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Alqaraghuli, Hasan, Abdul Rashid Husain, Nik Rumzi Bin Nik Idris, Waqas Anjum, and Muhammad Abbas Abbasi. "A new method for controlling an induction motor using a hybrid discretization model predictive field orientated control." PLOS ONE 17, no. 6 (June 16, 2022): e0267459. http://dx.doi.org/10.1371/journal.pone.0267459.
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The dynamic performance of the Model Predictive Control (MPC) of an Induction Motor (IM) relies on the accuracy and computational efficiency of the Discretisation Technique (DT). If the discretisation process is inaccurate or slow approximation, the MPC will exhibit high torque ripple and lower load handling capabilities. Traditionally, Euler’s method is used to discretise the MPC, which merely relies on the predictor to yield a fast, but less accurate system approximation. In contrast, Heun’s method uses a combination of predictor and corrector at alternate sampling intervals to improve the discretisation accuracy; however, the controller response becomes slow due to increased computational intensity of the algorithm. In this study, a new Hybrid Discretisation Technique (HDT) for Model Predictive Field Oriented Control (MPFOC) for IM control systems is presented to achieve robust discretisation with improved accuracy. In the proposed approach, Euler’s method is used to discretise the system at the first nine samples, followed by the predictor-corrector at the tenth sampling interval, accomplishing the desired speed and accuracy of discretisation. This newly proposed HDT in MPFOC is verified with Processor-In-Loop (PIL) for a three-phase IM with bi-directional rotation under varying load conditions. The results indicate that the IM torque ripple is reduced by up to 20%, whereas, the load handling capability is increased by up to 10%. Moreover, the controller gives 20% and 23% improvement in rise time and settling time, respectively, under high loading conditions, as compared to traditional Euler and Heun methods.
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Debeljkovic, Dragutin, Mica Jovanovic, and Nemanja Visnjic. "Singular system theory applied to the evaporator dynamics of a once - through subcritical steam generator: The differential discrete mathematical modeling based approach." Chemical Industry and Chemical Engineering Quarterly 13, no. 1 (2007): 27–32. http://dx.doi.org/10.2298/ciceq0701027d.
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A dynamic thermal-hydraulic mathematical model of the evaporator dynamics of a once - through sub critical steam generator was derived and presented. This model allows the investigation of evaporator dynamics including its transient responses. The evaporator was considered as part of a three-section (economizer, evaporator and super-heater) model with time varying phase boundaries and was described by a set of linearized discrete - difference equations which, with some other algebraic equations, constituted a closed system of equations possible for exact computer solution. This model was derived using the fundamental equations of mass, energy and momentum balance. For the first time, a discrete differential approach was applied in order to investigate such complex, two phase processes. Namely, this approach allows one to escape from the model of this process usually described by a set of partial differential equations and enables one, using this method, to simulate evaporator dynamics in an extraordinarily simple way. In the current literature this approach is sometimes called physical discretization.
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Ortiz, Adolfo, Jesús Seoane, J. Yang, and Miguel Sanjuán. "Dynamics of the helmholtz oscillator with fractional order damping." Acta Universitaria 23 (December 1, 2013): 12–15. http://dx.doi.org/10.15174/au.2013.584.
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The dynamics of the nonlinear Helmholtz Oscillator with fractional order damping are studied in detail. The discretization of differential equations according to the Grünwald-Letnikov fractional derivative definition in order to get numerical simulations is reported. Comparison between solutions obtained through a fourth-order Runge-Kutta method and the fractional damping system are comparable when the fractional derivative of the damping term a is fixed at 1. That proves the good performance of the numerical scheme. The effect of taking the fractional derivative on the system dynamics is investigated using phase diagrams varying a from 0.5 to 1.75 with zero initial conditions. Periodic motions of the system are obtained at certain ranges of the damping term. On the other hand, escape of the trajectories from a potential well result at a certain critical value of the fractional derivative. The history of the displacement as a function of time is shown also for every a selected.
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Xinhua Long and B. Balachandran. "Stability of Up-milling and Down-milling Operations with Variable Spindle Speed." Journal of Vibration and Control 16, no. 7-8 (June 2010): 1151–68. http://dx.doi.org/10.1177/1077546309341131.
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In this article, a stability treatment is presented for up-milling and down-milling processes with a variable spindle speed (VSS). This speed variation is introduced by superimposing a sinusoidal modulation on a nominal spindle speed. The VSS milling dynamics is described by a set of delay differential equations with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite-dimensional transition matrix is reduced to a finite-dimensional matrix over this period. The eigenvalues of this finite-dimensional matrix provide information on VSS milling stability with respect to control parameters, such as the axial depth of cut and the nominal spindle speed. The stability charts obtained for VSS milling operations are compared with those obtained for constant spindle speed milling operations, and the benefits of VSS milling operations are discussed.
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Attaullah, Mansour F. Yassen, Sultan Alyobi, Fuad S. Al-Duais, and Wajaree Weera. "On the comparative performance of fourth order Runge-Kutta and the Galerkin-Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology." AIMS Mathematics 8, no. 2 (2022): 3699–729. http://dx.doi.org/10.3934/math.2023185.
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<abstract><p>Anti-viral medication is comparably incredibly beneficial for individuals who are infected with numerous viruses. Mathematical modeling is crucial for comprehending the various relationships involving viruses, immune responses and health in general. This study concerns the implementation of a <italic>continuous</italic> Galerkin-Petrov time discretization scheme with mathematical models that consist of nonlinear ordinary differential equations for the hepatitis B virus, the Chen system and HIV infection. For the Galerkin scheme, we have two unknowns on each time interval which have to be computed by solving a $ 2 \times 2 $ block system. The proposed method is accurate to order 3 in the whole time interval and shows even super convergence of order 4 in the discrete time points. The study presents the accurate solutions achieved by means of the aforementioned schemes, presented numerically and graphically. Further, we implemented the classical fourth-order Runge-Kutta scheme accurately and performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes. The performances of the fourth order Runge-Kutta and the Galerkin-Petrov time discretization approaches for solving nonlinear ordinary differential equations were compared, with applications towards certain mathematical models in epidemiology. Several simulations were carried out with varying time step sizes, and the efficiency of the Galerkin and Runge Kutta schemes was evaluated at various time points. A detailed analysis of the outcomes obtained by the Galerkin scheme and the Runge-Kutta technique indicates that the results presented are in excellent agreement with each other despite having distinct computational costs in terms of time. It is observed that the Galerkin scheme is noticeably slower and requires more time in comparison to the Runge Kutta scheme. The numerical computations demonstrate that the Galerkin scheme provides highly precise solutions at relatively large time step sizes as compared to the Runge-Kutta scheme.</p></abstract>
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Duchaud, Jean-Laurent, Cyril Voyant, Alexis Fouilloy, Gilles Notton, and Marie-Laure Nivet. "Trade-Off between Precision and Resolution of a Solar Power Forecasting Algorithm for Micro-Grid Optimal Control." Energies 13, no. 14 (July 10, 2020): 3565. http://dx.doi.org/10.3390/en13143565.
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With the development of micro-grids including PV production and storage, the need for efficient energy management strategies arises. One of their key components is the forecast of the energy production from very short to long term. The forecast time-step is an important parameter affecting not only its accuracy but also the optimal control time discretization, hence its efficiency and computational burden. To quantify this trade-off, four machine learning forecast models are tested on two geographical locations for time-steps varying from 2 to 60 min and horizons from 10 min to 6 h, on global irradiance horizontal and tilted when data was available. The results are similar for all the models and indicate that the error metric can be reduced up to 0.8% per minute on the time-step for forecasts below one hour and up to 1.7% per ten minutes for forecasts between one and six hours. In addition, it is shown that for short term horizons, it may be advantageous to forecast with a high resolution then average the results at the time-step needed by the energy management system.
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Ide, Satoshi, Minoru Takeo, and Yasuhiro Yoshida. "Source process of the 1995 Kobe earthquake: Determination of spatio-temporal slip distribution by Bayesian modeling." Bulletin of the Seismological Society of America 86, no. 3 (June 1, 1996): 547–66. http://dx.doi.org/10.1785/bssa0860030547.
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Abstract The Kobe (Hyogo-ken Nanbu) earthquake (MS 6.8) occurred on 16 January 1995, in western Japan, bringing about a tremendous disaster. We constructed its source model by determining a spatio-temporal slip distribution using strong-motion seismograms recorded at a number of stations near the hypocenter. The source area is expressed using both a rough and a detailed subfault discretization system and a source-time function expanded with a finite number of basis functions at each subfault is determined. To perform stable inversions with many parameters, we construct a Bayesian model with unknown hyperparameters, incorporating prior constraints on the smoothness of spatio-temporal slip distribution in the observation equation. Using Akaike's Bayesian Information Criterion (ABIC), the optimal set of hyperparameters are objectively determined from observed data, which enables us to uniquely determine a spatio-temporal slip distribution. After rough analysis in which the actual rupture area is estimated as 45 × 20 km, we carried out a detailed analysis with a finer subfault discretization system (5 × 5 km). In our final source model, most of the slip exists beneath Awaji Island, and the rupture below the city of Kobe, where the damage was most serious, is relatively smaller. The large slips of about 2 m in the shallowest subfaults located southwest of the hypocenter correspond to the surface offsets observed along the Nojima fault. The overall focal mechanism of the source is right-lateral strike slip with a small thrust component, while total seismic moment is 2.0 × 1019 Nm. The source-time functions at shallow subfaults show a longer duration than those at deeper ones, being hypothesized to represent a possible combined effect of low rupture propagation velocity and varying friction behavior in shallow regions of crust.
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Basarab, M. A., B. S. Lunin, and I. P. Ivanov. "Analytical Solution of the Dynamics Equations for a Wave Solid-State Gyroscope Using the Angular Rate Linear Approximation." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 5 (98) (October 2021): 17–32. http://dx.doi.org/10.18698/1812-3368-2021-5-17-32.
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The exact solution is provided of the dynamics equation for an elastic inextensible ring being the basic model of a wave solid-state gyroscope with the linear law of the base angular rotation rate alteration. This solution is presented in terms of the parabolic cylinder functions (Weber function). Asymptotic approximations are used in the device certain operating modes. On the basis of the solution obtained, the analytical solution to the equation of the ring dynamics in case of piecewise linear approximation of the angular rate arbitrary profile on a time grid is derived. This significantly expands the class of angular rate dependences, for which the solution could be written down analytically. Earlier, in addition to the simplest case of constant angular rate, solutions were obtained for angular rate varying according to the square root law with time (Airy function), as well as according to the harmonic law (Mathieu function). Error dependence of such approximation on the discretization step in time is estimated numerically. Results obtained make it possible to reduce the number of operations, when it is necessary to study long-term evolutions of the dynamic system oscillations, as well as to quantitatively and qualitatively control convergence of finite-difference schemes in solving dynamics equations for a wave solid-state gyroscope with the ring resonator
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Ding, Hu, Qiao-Yun Yan, and Jean W. Zu. "Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime." International Journal of Bifurcation and Chaos 24, no. 05 (May 2014): 1450062. http://dx.doi.org/10.1142/s021812741450062x.
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This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.
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Perez-Salesa, Irene, Rodrigo Aldana-Lopez, and Carlos Sagues. "Precise Dynamic Consensus under Event-Triggered Communication." Machines 11, no. 2 (January 17, 2023): 128. http://dx.doi.org/10.3390/machines11020128.
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This work addresses the problem of dynamic consensus, which consists of estimating the dynamic average of a set of time-varying signals distributed across a communication network of multiple agents. This problem has many applications in robotics, with formation control and target tracking being some of the most prominent ones. In this work, we propose a consensus algorithm to estimate the dynamic average in a distributed fashion, where discrete sampling and event-triggered communication are adopted to reduce the communication burden. Compared to other linear methods in the state of the art, our proposal can obtain exact convergence under continuous communication even when the dynamic average signal is persistently varying. Contrary to other sliding-mode approaches, our method reduces chattering in the discrete-time setting. The proposal is based on the discretization of established exact dynamic consensus results that use high-order sliding modes. The convergence of the protocol is verified through formal analysis, based on homogeneity properties, as well as through several numerical experiments. Concretely, we numerically show that an advantageous trade-off exists between the maximum steady-state consensus error and the communication rate. As a result, our proposal can outperform other state-of-the-art approaches, even when event-triggered communication is used in our protocol.
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Zernin, M. V., A. V. Matyuhin, and N. N. Rybkin. "Methods for calculation and experimental evaluation of nucleation and development of the fatigue damages in tin based babbitt and babbitt lining of plain bearings." Industrial laboratory. Diagnostics of materials 87, no. 11 (November 21, 2021): 43–54. http://dx.doi.org/10.26896/1028-6861-2021-87-11-43-54.
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Fatigue damage to babbitt layers of plain bearings is often manifested during operation. The goal of the study is to develop a model for accumulation of the fatigue damage and destruction of antifriction materials and layers of plain bearings. A generalized fatigue diagram of tin-based babbitts including the main stages of fatigue damage and a diagram of the fatigue damage development in the antifriction layer of plain bearings are presented. The generalized model of V. V. Bolotin for damage accumulation and destruction is modified with regard to antifriction materials containing rather large structural elements. An explicit (direct) modeling of damage processes appeared possible for such materials. The model describes dissipated accumulation of microcracks (interpreted as destruction of the elements of the material structure), initiation and development of a system of short cracks, initiation and development of macro-cracks up to the limit state of the object. The model suggests discretization of the volume into sections with constant levels of complex stress state and discretization of the time axis into the intervals (blocks of loading cycles). The problem of identifying the parameters of a multistage model of the fatigue damage accumulation in the alloy is solved proceeding from the analysis of the results of testing babbitt specimens. We used the simplest optimization procedure, i.e., the method of deformable polyhedron. The parameters of the power function in the dependence of the rate of microdamage accumulation on the level of stresses are obtained. The parameters of the initiation and development of the crack system in the babbitt layer are obtained from the analysis of experimental results of studying steel-babbitt samples. The problem of calculating the durability of antifriction babbitt layers required the development of a new software. The program is examined by comparing calculated and experimental values of the durability of fatigue-tested bearing specimens forced against a rotating shaft by varying cyclic load. The calculated values of the durability match the experimental which confirms the performance of the calculated model.
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Wanjing, Luo, and Tang Changfu. "A Semianalytical Solution of a Vertical Fractured Well With Varying Conductivity Under Non-Darcy-Flow Condition." SPE Journal 20, no. 05 (October 20, 2015): 1028–40. http://dx.doi.org/10.2118/178423-pa.
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Summary Fracture distributions (simple or complex fractures), fracture-conductivity heterogeneity (uniform or varying conductivity along the fracture), and flow regimes inside the fracture (Darcy or non-Darcy flow) are the three main issues that have been widely investigated for transient-pressure analysis of vertical fracture systems. In this study, we focus on the latter two issues by proposing a semianalytical solution to discuss the transient-pressure behaviors of a varying-conductivity fracture under non-Darcy-flow condition. First, a general fracture-flow equation is established for the uniform-/varying-conductivity fracture under Darcy/non-Darcy flow. Second, for the case of a varying-conductivity fracture, a dimension transformation and an unequal-length-discretization model are proposed to obtain the pressure solution. Then, the transient-pressure response for the case of non-Darcy flow in the fracture can be also obtained by use of an iterative procedure in each timestep in the Laplace domain. It is shown that results from our solutions agree very well with those reported in the literature (Guppy et al. 1982; Poe et al. 1992). Third, the transient-pressure behaviors of the varying-conductivity fracture under Darcy- and non-Darcy-flow condition are discussed in detail. Results show that non-Darcy flow in the fracture mainly reduces the effective conductivity and the transient-pressure curve follows the curve of an equivalently constant conductivity except for the case of extremely small conductivities. The pressure behaviors of varying-conductivity fractures depend on the value of average conductivity, the distribution of conductivity along the fracture, and the maximum-minimum-conductivity ratio. The presence of the varying conductivity not only affects the effective conductivity in the early and late times, but also changes the shape of the pressure curve, especially for the high-conductivity fracture in the early time. It is very difficult to accurately estimate the fracture parameters by well test for most of the cases of varying conductivities under non-Darcy flow in the fracture.
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Chen, Dekun, Kun Li, Nianli Lu, and Peng Lan. "A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency." Machines 11, no. 4 (March 29, 2023): 433. http://dx.doi.org/10.3390/machines11040433.
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In this paper, a space-time absolute nodal coordinate formulation cable (SAC) element forming technique based on the Lagrange family of shape functions is proposed. Two distinct SAC elements, each with a distinct spatial shape function, have been generated by this method. Moreover, the external forces such as the bending moment and the air resistance formula have been accounted for. The Lagrange multiplier method, along with the concepts of replacement constraint and supplementary constraint, has been employed to provide a solution for the dynamics of constrained mechanical systems. Additionally, a constraint conversion strategy has been suggested. The solver has been constructed through Hamilton’s law of varying action. The space-time finite element method is used to solve dynamic problems, employing the Newton algorithm and quasi-Newton algorithm. The accuracy and efficiency of the solution has been verified by three simulations and one experiment. The circle-bending static simulation and the double-ended velocity impact dynamic simulation demonstrate the accuracy of the two elements. The correlation between statics and dynamics has been studied for different discretization methods and different solvers’ calculation accuracy and efficiency. Different modeling methods, time steps, order and the application of the quasi-Newton method all have a bearing on the efficiency of the solution. Finally, a comparison with an experiment in the free-pendulum simulation reveals the capability of this model to simulate dynamic problems with air resistance.
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Iatkliang, Thitthita, Supaporn Kaewta, Nguyen Minh Tuan, and Sekson Sirisubtawee. "Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method." WSEAS TRANSACTIONS ON MATHEMATICS 22 (June 2, 2023): 432–50. http://dx.doi.org/10.37394/23206.2023.22.50.
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The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The exact explicit solutions to the problems obtained with the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simulations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very important for understanding nonlinear phenomena in beta partial differential equation models for systems involving the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion relations.
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Zhang, X., R. S. Padgett, and O. A. Basaran. "Nonlinear deformation and breakup of stretching liquid bridges." Journal of Fluid Mechanics 329 (December 25, 1996): 207–45. http://dx.doi.org/10.1017/s0022112096008907.
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In this paper, the nonlinear dynamics of an axisymmetric liquid bridge held captive between two coaxial, circular, solid disks that are separated at a constant velocity are considered. As the disks are continuously pulled apart, the bridge deforms and ultimately breaks when its length attains a limiting value, producing two drops that are supported on the two disks. The evolution in time of the bridge shape and the rupture of the interface are investigated theoretically and experimentally to quantitatively probe the influence of physical and geometrical parameters on the dynamics. In the computations, a one-dimensional model that is based on the slender jet approximation is used to simulate the dynamic response of the bridge to the continuous uniaxial stretching. The governing system of nonlinear, time-dependent equations is solved numerically by a method of lines that uses the Galerkin/finite element method for discretization in space and an adaptive, implicit finite difference technique for discretization in time. In order to verify the model and computational results, extensive experiments are performed by using an ultra-high-speed video system to monitor the dynamics of liquid bridges with a time resolution of 1/12 th of a millisecond. The computational and experimental results show that as the importance of the inertial force – most easily changed in experiments by changing the stretching velocity – relative to the surface tension force increases but does not become too large and the importance of the viscous force – most easily changed by changing liquid viscosity – relative to the surface tension force increases, the limiting length that a liquid bridge is able to attain before breaking increases. By contrast, increasing the gravitational force – most readily controlled by varying disk radius or liquid density – relative to the surface tension force reduces the limiting bridge length at breakup. Moreover, the manner in which the bridge volume is partitioned between the pendant and sessile drops that result upon breakup is strongly influenced by the magnitudes of viscous, inertial, and gravitational forces relative to surface tension ones. Attention is also paid here to the dynamics of the liquid thread that connects the two portions of the bridge liquid that are pendant from the top moving rod and sessile on the lower stationary rod because the manner in which the thread evolves in time and breaks has important implications for the closely related problem of drop formation from a capillary. Reassuringly, the computations and the experimental measurements are shown to agree well with one another.
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Min, Minghui, Jiayang Xiao, Peng Zhang, Jinling Song, and Shiyin Li. "Learning-Based IRS-Assisted Secure Transmission for Mine IoTs." Sensors 23, no. 14 (July 12, 2023): 6321. http://dx.doi.org/10.3390/s23146321.
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Mine Internet of Things (MIoT) devices in intelligent mines often face substantial signal attenuation due to challenging operating conditions. The openness of wireless communication also makes it susceptible to smart attackers, such as active eavesdroppers. The attackers can disrupt equipment operations, compromise production safety, and exfiltrate sensitive environmental data. To address these challenges, we propose an intelligent reflecting surface (IRS)-assisted secure transmission system for an MIoT device which enhances the security and reliability of wireless communication in challenging mining environments. We develop a joint optimization problem for the IRS phase shifts and transmit power, with the goal of enhancing legitimate transmission while suppressing eavesdropping. To accommodate time-varying channel conditions, we propose a reinforcement learning (RL)-based IRS-assisted secure transmission scheme that enables MIoT device to optimize both the IRS reflecting coefficients and transmit power for optimal transmission policy in dynamic environments. We adopt the deep deterministic policy gradient (DDPG) algorithm to explore the optimal transmission policy in continuous space. This can reduce the discretization error caused by traditional RL methods. The simulation results indicate that our proposed scheme achieves superior system utility compared with both the IRS-free (IF) scheme and the IRS randomly configured (IRC) scheme. These results demonstrate the effectiveness and practical relevance of our contributions, proving that implementing IRS in MIoT wireless communication can enhance safety, security, and efficiency in the mining industry.
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Hao, Junbo, Zedong Wang, Wenwu Yi, Yan Chen, and Jiyao Chen. "Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine." Applied Sciences 11, no. 19 (September 24, 2021): 8876. http://dx.doi.org/10.3390/app11198876.
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The finite element discretization of a tower system based on the two-node Euler-Bernoulli beam is carried out by taking the cubic Hermite polynomial as the form function of the beam unit, calculating the structural characteristic matrix of the tower system, and establishing the wind turbine-nacelle-tower multi-degree-of-freedom finite element numerical model. The equation for calculating the aerodynamic load for any nacelle attitude angle is derived. The effect of the flexible tower vibration feedback on the aerodynamic load of the wind turbine is studied. The results show that, when the stiffness of the tower is large, the effect of having tower vibration feedback or not on the aeroelastic load of the wind turbine is small. For the more flexible tower system, wind-induced vibration time-varying feedback will cause larger aeroelastic load variations, especially the top of the tower overturning moment, thus causing a larger impact on the dynamic behavior of the tower downwind and crosswind. As the flexibility of the tower system increases, the interaction between tower vibration and pneumatic load is also gradually increasing. Taking into account the influence of flexible towers on the aeroelastic load of a wind turbine can help predict the pneumatic load of a wind turbine more accurately and improve the efficiency of wind energy utilization on the one hand and analyze the dynamic behavior of the flexible structure of a wind turbine more accurately on the other hand, which is extremely beneficial to the structural optimization of wind turbine.
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NOLTE, BODO, INGO SCHÄFER, JAN EHRLICH, MARTIN OCHMANN, RALF BURGSCHWEIGER, and STEFFEN MARBURG. "NUMERICAL METHODS FOR WAVE SCATTERING PHENOMENA BY MEANS OF DIFFERENT BOUNDARY INTEGRAL FORMULATIONS." Journal of Computational Acoustics 15, no. 04 (December 2007): 495–529. http://dx.doi.org/10.1142/s0218396x07003536.
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Different numerical approaches for the physical phenomena of scattering waves from an obstacle are presented. They are based on different integral formulations. Fluid structure interaction effects are numerically treatable as well. We use the Boundary Element Method (BEM) in different approaches because the inherently satisfied Sommerfeld radiation condition makes sure that no reflecting waves from boundaries at infinity occur. One of the biggest disadvantages of numerical methods like BEM is the fact that they have difficulties with handling the high frequency range. For the high frequency range approximations like the Kirchhoff–Helmholtz integral equation have to be used. With varying assumptions of the reflecting behavior of the structure different approaches for the higher frequency range are obtained, where the explicit solving of a system of equations is not necessary. Another high frequency approach is the plane wave approximation which is compared with the Kirchhoff approach of the first kind. Additionally a modified Kirchhoff approach is introduced. Because the incident pressure on the scatterer's surface is known the integral is evaluated analytically on triangular patches. The discretization is no longer frequency dependent and the size of the patches only depends on the curvature of the structure. Large planar parts can be discretized with one element only. This leads to a substantial advantage in terms of calculation time over the traditional Kirchhoff approach. Like the traditional approach this procedure is valid under the assumption of high frequency or far field conditions.
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Chen, Zhiwei, Xiaopeng Li, and Xiaobo Qu. "A Continuous Model for Designing Corridor Systems with Modular Autonomous Vehicles Enabling Station-wise Docking." Transportation Science 56, no. 1 (January 2022): 1–30. http://dx.doi.org/10.1287/trsc.2021.1085.
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The “asymmetry” between spatiotemporally varying passenger demand and fixed-capacity transportation supply has been a long-standing problem in urban mass transportation (UMT) systems around the world. The emerging modular autonomous vehicle (MAV) technology offers us an opportunity to close the substantial gap between passenger demand and vehicle capacity through station-wise docking and undocking operations. However, there still lacks an appropriate approach that can solve the operational design problem for UMT corridor systems with MAVs efficiently. To bridge this methodological gap, this paper proposes a continuum approximation (CA) model that can offer near-optimal solutions to the operational design for MAV-based transit corridors very efficiently. We investigate the theoretical properties of the optimal solutions to the investigated problem in a certain (yet not uncommon) case. These theoretical properties allow us to estimate the seat demand of each time neighborhood with the arrival demand curves, which recover the “local impact” property of the investigated problem. With the property, a CA model is properly formulated to decompose the original problem into a finite number of subproblems that can be analytically solved. A discretization heuristic is then proposed to convert the analytical solution from the CA model to feasible solutions to the original problem. With two sets of numerical experiments, we show that the proposed CA model can achieve near-optimal solutions (with gaps less than 4% for most cases) to the investigated problem in almost no time (less than 10 ms) for large-scale instances with a wide range of parameter settings (a commercial solver may even not obtain a feasible solution in several hours). The theoretical properties are verified, and managerial insights regarding how input parameters affect system performance are provided through these numerical results. Additionally, results also reveal that, although the CA model does not incorporate vehicle repositioning decisions, the timetabling decisions obtained by solving the CA model can be easily applied to obtain near-optimal repositioning decisions (with gaps less than 5% in most instances) very efficiently (within 10 ms). Thus, the proposed CA model provides a foundation for developing solution approaches for other problems (e.g., MAV repositioning) with more complex system operation constraints whose exact optimal solution can hardly be found with discrete modeling methods.
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Best Mckay, Maricela, Brittany A. Erickson, and Jeremy E. Kozdon. "A computational method for earthquake cycles within anisotropic media." Geophysical Journal International 219, no. 2 (July 15, 2019): 816–33. http://dx.doi.org/10.1093/gji/ggz320.
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SUMMARY We present a numerical method for the simulation of earthquake cycles on a 1-D fault interface embedded in a 2-D homogeneous, anisotropic elastic solid. The fault is governed by an experimentally motivated friction law known as rate-and-state friction which furnishes a set of ordinary differential equations which couple the interface to the surrounding volume. Time enters the problem through the evolution of the ordinary differential equations along the fault and provides boundary conditions for the volume, which is governed by quasi-static elasticity. We develop a time-stepping method which accounts for the interface/volume coupling and requires solving an elliptic partial differential equation for the volume response at each time step. The 2-D volume is discretized with a second-order accurate finite difference method satisfying the summation-by-parts property, with boundary and fault interface conditions enforced weakly. This framework leads to a provably stable semi-discretization. To mimic slow tectonic loading, the remote side-boundaries are displaced at a slow rate, which eventually leads to earthquake nucleation at the fault. Time stepping is based on an adaptive, fourth-order Runge–Kutta method and captures the highly varying timescales present. The method is verified with convergence tests for both the orthotropic and fully anisotropic cases. An initial parameter study reveals regions of parameter space where the systems experience a bifurcation from period one to period two behaviour. Additionally, we find that anisotropy influences the recurrence interval between earthquakes, as well as the emergence of aseismic transients and the nucleation zone size and depth of earthquakes.
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Praß, Julian, Jörg Franke, and Stefan Becker. "Investigation of Drag Reduction due to Dimpled Surfaces in Narrow Channels by Means of Flow Simulations." Applied Mechanics and Materials 871 (October 2017): 244–51. http://dx.doi.org/10.4028/www.scientific.net/amm.871.244.
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Reduction of drag and flow resistance in systems containing moving fluids is a prominenttool to increase energy efficiency. Besides active flow control – such as moving surfaces or boundarylayer suction – passive techniques such as surface patterning by means of dimples are promising sinceno additional energy consumer is introduced into the system. Even though the effect of drag reductiondue to dimples has often been observed, the physical principals responsible for this effect are not yetunderstood. Most of the research concerning dimples and drag reduction published so far has beencarried out experimentally, not many numerical investigations on this topic have been done. The mainreason for this is that the tiny, transient flow structures generated in direct vicinity of dimples can noteasily be resolved in simulations. Even in case of time dependent numerical investigations it is notclear, whether and with which method of sub-grid scale modeling Large Eddy Simulations are capableof modeling these structures sufficiently. In this work we investigated different surfaces with dimpledepth to diameter ratios h/D reaching from 0.01 to 0.1 in channels of height H = 0.417D at Reynoldsnumbers ReD 5 830 and ReD 11 650 using steady state simulations with a k-omega-SST turbulencemodel. Drag reductions were observed for all setups h/D < 0.08 compared to the smooth channel.The best results were obtained with dimple depths of 4-5 % of diameter showing a slight dependenceof Re which is in good agreement with literature. As the experimental investigation of the flow overdimpled surfaces is limited in spacial and temporal resolution we could demonstrate that numericalinvestigations give the possibility to overcome this drawback. However the solution of simulationsstrongly depends on numerous factors such as the discretization scheme, the numerical models andthe grid used to obtain results which might be a reason for slightly varying results of such simulationsfound in literature.
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Delis, Anargiros I., Maria Kazolea, and Maria Gaitani. "On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids." Water 14, no. 21 (November 7, 2022): 3584. http://dx.doi.org/10.3390/w14213584.
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This work aims to supplement the realization and validation of a higher-order well-balanced unstructured finite volume (FV) scheme, that has been relatively recently presented, for numerically simulating weakly non-linear weakly dispersive water waves over varying bathymetries. We investigate and develop solution strategies for the sparse linear system that appears during this FV discretisation of a set of extended Boussinesq-type equations on unstructured meshes. The resultant linear system of equations must be solved at each discrete time step as to recover the actual velocity field of the flow and advance in time. The system’s coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the undisturbed water depth and the mesh topology. To this end, we investigate the application of different well-known iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. The iiterative methods considered are the GMRES and the BiCGSTAB, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is finally proposed which combines the use of the BiCGSTAB method with the ILUT preconditioner and the Reverse Cuthill–McKee reordering.
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McLay, Laura A., and David E. Goldberg. "Efficient Genetic Algorithms Using Discretization Scheduling." Evolutionary Computation 13, no. 3 (September 2005): 353–85. http://dx.doi.org/10.1162/1063656054794752.
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In many applications of genetic algorithms, there is a tradeoff between speed and accuracy in fitness evaluations when evaluations use numerical methods with varying discretization. In these types of applications, the cost and accuracy vary from discretization errors when implicit or explicit quadrature is used to estimate the function evaluations. This paper examines discretization scheduling, or how to vary the discretization within the genetic algorithm in order to use the least amount of computation time for a solution of a desired quality. The effectiveness of discretization scheduling can be determined by comparing its computation time to the computation time of a GA using a constant discretization. There are three ingredients for the discretization scheduling: population sizing, estimated time for each function evaluation and predicted convergence time analysis. Idealized one- and two-dimensional experiments and an inverse groundwater application illustrate the computational savings to be achieved from using discretization scheduling.
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Kularatne, Dhanushka, Subhrajit Bhattacharya, and M. Ani Hsieh. "Optimal Path Planning in Time-Varying Flows Using Adaptive Discretization." IEEE Robotics and Automation Letters 3, no. 1 (January 2018): 458–65. http://dx.doi.org/10.1109/lra.2017.2761939.
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Suris, Yuri B. "Time Discretization of F. Calogero’s “Goldfish” System." Journal of Nonlinear Mathematical Physics 12, sup1 (January 2005): 633–47. http://dx.doi.org/10.2991/jnmp.2005.12.s1.49.
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Xuan, Hejun, Xinhui Zhu, Jian Li, Huaping Guo, and Yanling Li. "General Third-Order-Accuracy Formulas for Time Discretization Applied to Time-Varying Optimization." IEEE Access 8 (2020): 224235–45. http://dx.doi.org/10.1109/access.2020.3045185.
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Fritsch, H. "Time discretization of a setvalued stochastic dynamic system." Optimization 35, no. 4 (January 1995): 391–400. http://dx.doi.org/10.1080/02331939508844158.
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Borri, M., M. Lanz, and P. Mantegazza. "Comment on "Time Finite Element Discretization of Hamilton's Law of Varying Action''." AIAA Journal 23, no. 9 (September 1985): 1457–58. http://dx.doi.org/10.2514/3.48615.
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Mergel, Janine C., Roger A. Sauer, and Sina Ober-Blöbaum. "C1-continuous space-time discretization based on Hamilton's law of varying action." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 97, no. 4 (November 7, 2016): 433–57. http://dx.doi.org/10.1002/zamm.201600062.
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Kazantzis, Nikolaos, K. T. Chong, J. H. Park, and Alexander G. Parlos. "Control-Relevant Discretization of Nonlinear Systems With Time-Delay Using Taylor-Lie Series." Journal of Dynamic Systems, Measurement, and Control 127, no. 1 (April 9, 2004): 153–59. http://dx.doi.org/10.1115/1.1870046.
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A new time-discretization method for the development of a sampled-data representation of a nonlinear continuous-time input-driven system with time delay is proposed. It is based on the Taylor-Lie series expansion method and zero-order hold assumption. The mathematical structure of the new discretization scheme is explored and characterized as useful for establishing concrete connections between numerical and system-theoretic properties. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. The resulting time-discretization provides a finite-dimensional representation for nonlinear control systems with time-delay enabling the application of existing controller design techniques. The performance of the proposed discretization procedure is evaluated using the case study of a two-degree-of-freedom mechanical system that exhibits nonlinear behavior. Various sampling rates and time-delay values are considered.
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Zhang, Zheng, Yuanliang Zhang, Young Won Chang, and Kil To Chong. "Time discretization of nonlinear time-delay system using matrix exponential method." International Journal of Control, Automation and Systems 9, no. 6 (December 2011): 1219–26. http://dx.doi.org/10.1007/s12555-011-0624-2.
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Sun, Min, Maoying Tian, and Yiju Wang. "Discrete-Time Zhang Neural Networks for Time-Varying Nonlinear Optimization." Discrete Dynamics in Nature and Society 2019 (April 8, 2019): 1–14. http://dx.doi.org/10.1155/2019/4745759.
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As a special kind of recurrent neural networks, Zhang neural network (ZNN) has been successfully applied to various time-variant problems solving. In this paper, we present three Zhang et al. discretization (ZeaD) formulas, including a special two-step ZeaD formula, a general two-step ZeaD formula, and a general five-step ZeaD formula, and prove that the special and general two-step ZeaD formulas are convergent while the general five-step ZeaD formula is not zero-stable and thus is divergent. Then, to solve the time-varying nonlinear optimization (TVNO) in real time, based on the Taylor series expansion and the above two convergent two-step ZeaD formulas, we discrete the continuous-time ZNN (CTZNN) model of TVNO and thus get a special two-step discrete-time ZNN (DTZNN) model and a general two-step DTZNN model. Theoretical analyses indicate that the sequence generated by the first DTZNN model is divergent, while the sequence generated by the second DTZNN model is convergent. Furthermore, for the step-size of the second DTZNN model, its tight upper bound and the optimal step-size are also discussed. Finally, some numerical results and comparisons are provided and analyzed to substantiate the efficacy of the proposed DTZNN models.
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Bartosiewicz, Zbigniew. "Invariance under discretization for positive systems." Mathematics of Control, Signals, and Systems 33, no. 2 (March 27, 2021): 315–29. http://dx.doi.org/10.1007/s00498-021-00283-1.
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AbstractPositive dynamical or control systems have all their variables nonnegative. Euler discretization transforms a continuous-time system into a system on a discrete time scale. Some structural properties of the system may be preserved by discretization, while other may be lost. Four fundamental properties of positive systems are studied in the context of discretization: positivity, positive stability, positive reachability and positive observability. Both linear and nonlinear systems are investigated.
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Li, Yuezhe, Tiffany Jann, and Paola Vera-Licona. "Benchmarking time-series data discretization on inference methods." Bioinformatics 35, no. 17 (January 18, 2019): 3102–9. http://dx.doi.org/10.1093/bioinformatics/btz036.
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AbstractSummaryThe rapid development in quantitatively measuring DNA, RNA and protein has generated a great interest in the development of reverse-engineering methods, that is, data-driven approaches to infer the network structure or dynamical model of the system. Many reverse-engineering methods require discrete quantitative data as input, while many experimental data are continuous. Some studies have started to reveal the impact that the choice of data discretization has on the performance of reverse-engineering methods. However, more comprehensive studies are still greatly needed to systematically and quantitatively understand the impact that discretization methods have on inference methods. Furthermore, there is an urgent need for systematic comparative methods that can help select between discretization methods. In this work, we consider four published intracellular networks inferred with their respective time-series datasets. We discretized the data using different discretization methods. Across all datasets, changing the data discretization to a more appropriate one improved the reverse-engineering methods’ performance. We observed no universal best discretization method across different time-series datasets. Thus, we propose DiscreeTest, a two-step evaluation metric for ranking discretization methods for time-series data. The underlying assumption of DiscreeTest is that an optimal discretization method should preserve the dynamic patterns observed in the original data across all variables. We used the same datasets and networks to show that DiscreeTest is able to identify an appropriate discretization among several candidate methods. To our knowledge, this is the first time that a method for benchmarking and selecting an appropriate discretization method for time-series data has been proposed.Availability and implementationAll the datasets, reverse-engineering methods and source code used in this paper are available in Vera-Licona’s lab Github repository: https://github.com/VeraLiconaResearchGroup/Benchmarking_TSDiscretizations.Supplementary informationSupplementary data are available at Bioinformatics online.
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Dong, Haoyu, Changna Lu, and Hongwei Yang. "The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations." Mathematics 6, no. 10 (October 18, 2018): 211. http://dx.doi.org/10.3390/math6100211.
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We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.
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Lelis, Levi, Sandra Zilles, and Robert Holte. "Time Complexity of Iterative-Deepening A*: The Informativeness Pathology (Abstract)." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (August 4, 2011): 1800–1801. http://dx.doi.org/10.1609/aaai.v25i1.8053.
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Korf, Reid, and Edelkamp launched a line of research aimed at predicting how many nodes IDA* will expand with a given depth bound. This paper advances this line of research in three ways. First, we identify a source of prediction error that has hitherto been overlooked. We call it the "discretization effect." Second, we disprove the intuitively appealing idea that a "more informed" prediction system cannot make worse predictions than a ``less informed'' one. More informed systems are more susceptible to the discretization effect, and in our experiments the more informed system makes poorer predictions. Our third contribution is a method, called "Epsilon-truncation," which makes a prediction system less informed, in a carefully chosen way, so as to improve its predictions by reducing the discretization effect. In our experiments Epsilon-truncation improved predictions substantially.