Search results for: generalized linear mixed model (GLMM)

Authors: Yohei Saika

Abstract:

On the basis of Bayesian inference using the maximizer of the posterior marginal estimate, we carry out phase unwrapping using multiple interferograms via generalized mean-field theory. Numerical calculations for a typical wave-front in remote sensing using the synthetic aperture radar interferometry, phase diagram in hyper-parameter space clarifies that the present method succeeds in phase unwrapping perfectly under the constraint of surface- consistency condition, if the interferograms are not corrupted by any noises. Also, we find that prior is useful for extending a phase in which phase unwrapping under the constraint of the surface-consistency condition. These results are quantitatively confirmed by the Monte Carlo simulation.

Keywords: Bayesian inference, generalized mean-field theory, phase unwrapping, statistical mechanics.

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Authors: Tomohiro Hachino, Kenji Shimoda, Hitoshi Takata

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This paper presents a method of model selection and identification of Hammerstein systems by hybridization of the genetic algorithm (GA) and particle swarm optimization (PSO). An unknown nonlinear static part to be estimated is approximately represented by an automatic choosing function (ACF) model. The weighting parameters of the ACF and the system parameters of the linear dynamic part are estimated by the linear least-squares method. On the other hand, the adjusting parameters of the ACF model structure are properly selected by the hybrid algorithm of the GA and PSO, where the Akaike information criterion is utilized as the evaluation value function. Simulation results are shown to demonstrate the effectiveness of the proposed hybrid algorithm.

Keywords: Hammerstein system, identification, automatic choosing function model, genetic algorithm, particle swarm optimization.

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Authors: Lenka Ševelová, Aleš Florian

Abstract:

Appropriate and progressive tool for analyzing behavior of low volume roads are probabilistic models used in reliability analyses. The necessary part of the probabilistic model is the deterministic model of structural behavior. The FE model of low volume roads is created in the ANSYS software. It is able to determine the state of stress and deformation in any point of the structure and thus generate data required for the reliability analysis. The paper compares two material constitutive models used for modeling of unbound non-homogenous materials used in low volume roads. The first model is linear elastic model according to Hook theory (H model), the second one is nonlinear elastic-plastic Drucker-Prager model (D-P model).

Keywords: FEA, FEM, geotechnical materials, low volume roads, material constitutive models, pavement.

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Authors: Andre A. Keller

Abstract:

The optimal control is one of the possible controllers for a dynamic system, having a linear quadratic regulator and using the Pontryagin-s principle or the dynamic programming method . Stochastic disturbances may affect the coefficients (multiplicative disturbances) or the equations (additive disturbances), provided that the shocks are not too great . Nevertheless, this approach encounters difficulties when uncertainties are very important or when the probability calculus is of no help with very imprecise data. The fuzzy logic contributes to a pragmatic solution of such a problem since it operates on fuzzy numbers. A fuzzy controller acts as an artificial decision maker that operates in a closed-loop system in real time. This contribution seeks to explore the tracking problem and control of dynamic macroeconomic models using a fuzzy learning algorithm. A two inputs - single output (TISO) fuzzy model is applied to the linear fluctuation model of Phillips and to the nonlinear growth model of Goodwin.

Keywords: fuzzy control, macroeconomic model, multiplier - accelerator, nonlinear accelerator, stabilization policy.

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Authors: M. A. Omoloye, M. I. Yusuff, O. K. S. Emiola

Abstract:

The use of mathematical models for solving biological problems varies from simple to complex analyses, depending on the nature of the research problems and applicability of the models. The method is more common nowadays. Many complex models become impractical when transmitted analytically. However, alternative approach such as numerical method can be employed. It appropriateness in solving linear and non-linear model equation in Differential Transformation Method (DTM) which depends on Taylor series make it applicable. Hence this study investigates the application of DTM to solve dynamic transmission of Lassa fever model in a population. The mathematical model was formulated using first order differential equation. Firstly, existence and uniqueness of the solution was determined to establish that the model is mathematically well posed for the application of DTM. Numerically, simulations were conducted to compare the results obtained by DTM and that of fourth-order Runge-Kutta method. As shown, DTM is very effective in predicting the solution of epidemics of Lassa fever model.

Keywords: Differential Transform Method, Existence and uniqueness, Lassa fever, Runge-Kutta Method.

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Authors: R. B. Ogunrinde, C. C. Jibunoh

Abstract:

In this paper, a spectral decomposition method is developed for the direct integration of stiff and nonstiff homogeneous linear (ODE) systems with linear, constant, or zero right hand sides (RHSs). The method does not require iteration but obtains solutions at any random points of t, by direct evaluation, in the interval of integration. All the numerical solutions obtained for the class of systems coincide with the exact theoretical solutions. In particular, solutions of homogeneous linear systems, i.e. with zero RHS, conform to the exact analytical solutions of the systems in terms of t.

Keywords: Spectral decomposition, eigenvalues of the Jacobian, linear RHS, homogeneous linear systems.

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Authors: Radouane Iqdour, Abdelouhab Zeroual

Abstract:

The Multi-Layered Perceptron (MLP) Neural networks have been very successful in a number of signal processing applications. In this work we have studied the possibilities and the met difficulties in the application of the MLP neural networks for the prediction of daily solar radiation data. We have used the Polack-Ribière algorithm for training the neural networks. A comparison, in term of the statistical indicators, with a linear model most used in literature, is also performed, and the obtained results show that the neural networks are more efficient and gave the best results.

Keywords: Daily solar radiation, Prediction, MLP neural networks, linear model

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Authors: G. Zahedi, H. Yaghoobi

Abstract:

Modeling of a heterogeneous industrial fixed bed reactor for selective dehydrogenation of heavy paraffin with Pt-Sn- Al2O3 catalyst has been the subject of current study. By applying mass balance, momentum balance for appropriate element of reactor and using pressure drop, rate and deactivation equations, a detailed model of the reactor has been obtained. Mass balance equations have been written for five different components. In order to estimate reactor production by the passage of time, the reactor model which is a set of partial differential equations, ordinary differential equations and algebraic equations has been solved numerically. Paraffins, olefins, dienes, aromatics and hydrogen mole percent as a function of time and reactor radius have been found by numerical solution of the model. Results of model have been compared with industrial reactor data at different operation times. The comparison successfully confirms validity of proposed model.

Keywords: Dehydrogenation, fixed bed reactor, modeling, linear alkyl benzene.

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Authors: Ashu Ahuja, Shiv Narayan, Jagdish Kumar

Abstract:

This paper realized the 2-DOF controller structure for first order with time delay systems. The co-prime factorization is used to design observer based controller K(s), representing one degree of freedom. The problem is based on H∞ norm of mixed sensitivity and aims to achieve stability, robustness and disturbance rejection. Then, the other degree of freedom, prefilter F(s), is formulated as fixed structure polynomial controller to meet open loop processing of reference model. This model matching problem is solved by minimizing integral square error between reference model and proposed model. The feedback controller and prefilter designs are posed as optimization problem and solved using Particle Swarm Optimization (PSO). To show the efficiency of the designed approach different variety of processes are taken and compared for analysis.

Keywords: 2-DOF, integral square error, mixed sensitivity function, observer based controller, particle swarm optimization, prefilter.

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Authors: Parham Azimi, Hamid Reza Charmchi

Abstract:

In this research, we have developed a new efficient heuristic algorithm for the dynamic facility layout problem with budget constraint (DFLPB). This heuristic algorithm combines two mathematical programming methods such as discrete event simulation and linear integer programming (IP) to obtain a near optimum solution. In the proposed algorithm, the non-linear model of the DFLP has been changed to a pure integer programming (PIP) model. Then, the optimal solution of the PIP model has been used in a simulation model that has been designed in a similar manner as the DFLP for determining the probability of assigning a facility to a location. After a sufficient number of runs, the simulation model obtains near optimum solutions. Finally, to verify the performance of the algorithm, several test problems have been solved. The results show that the proposed algorithm is more efficient in terms of speed and accuracy than other heuristic algorithms presented in previous works found in the literature.

Keywords: Budget constraint, Dynamic facility layout problem, Integer programming, Simulation

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Authors: Monika Garg, Lillie Dewan

Abstract:

Three novel and significant contributions are made in this paper Firstly, non-recursive formulation of Haar connection coefficients, pioneered by the present authors is presented, which can be computed very efficiently and avoid stack and memory overflows. Secondly, the generalized approach for state analysis of singular bilinear time-invariant (TI) and time-varying (TV) systems is presented; vis-˜a-vis diversified and complex works reported by different authors. Thirdly, a generalized approach for parameter estimation of bilinear TI and TV systems is also proposed. The unified framework of the proposed method is very significant in that the digital hardware once-designed can be used to perform the complex tasks of state analysis and parameter estimation of different types of bilinear systems single-handedly. The simplicity, effectiveness and generalized nature of the proposed method is established by applying it to different types of bilinear systems for the two tasks.

Keywords: Bilinear Systems, Haar Wavelet, Haar ConnectionCoefficients, Parameter Estimation, Singular Bilinear Systems, StateAnalysis.

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Authors: Joabe Silva, Ginalber Serra

Abstract:

This paper proposes the analysis and design of robust fuzzy control to Stochastic Parametrics Uncertaint Linear systems. This system type to be controlled is partitioned into several linear sub-models, in terms of transfer function, forming a convex polytope, similar to LPV (Linear Parameters Varying) system. Once defined the linear sub-models of the plant, these are organized into fuzzy Takagi- Sugeno (TS) structure. From the Parallel Distributed Compensation (PDC) strategy, a mathematical formulation is defined in the frequency domain, based on the gain and phase margins specifications, to obtain robust PI sub-controllers in accordance to the Takagi- Sugeno fuzzy model of the plant. The main results of the paper are based on the robust stability conditions with the proposal of one Axiom and two Theorems.

Keywords: Fuzzy Systems; Robust Stability, Stochastic Control, Stochastic Process

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Authors: Aryanejad Mir Bahador Goli, Zahra Honarmand Shah Zileh

Abstract:

In most study fields, a phenomenon may not be studied directly but it will be examined indirectly by phenomenon model. Making an accurate model of system, there is attained new information from modeled phenomenon without any charge, danger, etc... there have been developed more solutions for describing and analyzing the recent complicated systems but few of them have analyzed the performance in the range of system description. Petri nets are of limited solutions which may make such union. Petri nets are being applied in problems related to modeling and designing the systems. Theory of Petri nets allow a system to model mathematically by a Petri net and analyzing the Petri net can then determine main information of modeled system-s structure and dynamic. This information can be used for assessing the performance of systems and suggesting corrections in the system. In this paper, beside the introduction of Petri nets, a real case study will be studied in order to show the application of generalized stochastic Petri nets in modeling a resource sharing production system and evaluating the efficiency of its machines and robots. The modeling tool used here is SHARP software which calculates specific indicators helping to make decision.

Keywords: Flexible manufacturing system, generalizedstochastic Petri nets, Markov chain, performance evaluation.

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Authors: Chompunut Jantarasorn, Chutima Prommak

Abstract:

The important issue considered in the widespread deployment of Wireless Sensor Networks (WSNs) is an efficiency of the energy consumption. In this paper, we present a study of the optimal relay station planning problems using Binary Integer Linear Programming (BILP) model to minimize the energy consumption in WSNs. Our key contribution is that the proposed model not only ensures the required network lifetime but also guarantees the radio connectivity at high level of communication quality. Specially, we take into account effects of noise, signal quality limitation and bit error rate characteristics. Numerical experiments were conducted in various network scenarios. We analyzed the effects of different sensor node densities and distribution on the energy consumption.

Keywords: Binary Integer Linear Programming, BILP, Energy consumption, Optimal node placement and Wireless sensor networks.

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Authors: Z.A. Jaffery, Vinay Kumar Chandna, Sunil Kumar Chaudhary

Abstract:

This paper present a MATLAB-SIMULINK model of a single phase 2.5 KVA, 240V RMS controlled PV VSI (Photovoltaic Voltage Source Inverter) inverter using IGBTs (Insulated Gate Bipolar Transistor). The behavior of output voltage, output current, and the total harmonic distortion (THD), with the variation in input dc blocking capacitor (Cdc), for linear and non-linear load has been analyzed. The values of Cdc as suggested by the other authors in their papers are not clearly defined and it poses difficulty in selecting the proper value. As the dc power stored in Cdc, (generally placed parallel with battery) is used as input to the VSI inverter. The simulation results shows the variation in the output voltage and current with different values of Cdc for linear and non-linear load connected at the output side of PV VSI inverter and suggest the selection of suitable value of Cdc.

Keywords: DC Blocking capacitor, IGBTs, PV VSI, THD.

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Authors: Kourosh Parand, Jamal Amani Rad

Abstract:

In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.

Keywords: Exp-function method, generalized Pochhammer- Chree equation, solitary wave solution, ODE's.

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Authors: Kaushalendra K. Khadanga, Lee Hee Hyol

Abstract:

Passenger comfort has been paramount in the design of suspension systems of high speed cars. To analyze the effect of vibration on vehicle ride quality, a vertical model of a six degree of freedom railway passenger vehicle, with front and rear suspension, is built. It includes car body flexible effects and vertical rigid modes. A second order linear shaping filter is constructed to model Gaussian white noise into random rail excitation. The temporal correlation between the front and rear wheels is given by a second order Pade approximation. The complete track and the vehicle model are then designed. An active secondary suspension system based on a Linear Quadratic Gaussian (LQG) optimal control method is designed. The results show that the LQG control method reduces the vertical acceleration, pitching acceleration and vertical bending vibration of the car body as compared to the passive system.

Keywords: Active suspension, bending vibration, railway vehicle, vibration control.

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Authors: Mingyu Xie, Mietek Brdys

Abstract:

The paper develops a Non-Linear Model Predictive Control (NMPC) of water quality in Drinking Water Distribution Systems (DWDS) based on the advanced non-linear quality dynamics model including disinfections by-products (DBPs). A special attention is paid to the analysis of an impact of the flow trajectories prescribed by an upper control level of the recently developed two-time scale architecture of an integrated quality and quantity control in DWDS. The new quality controller is to operate within this architecture in the fast time scale as the lower level quality controller. The controller performance is validated by a comprehensive simulation study based on an example case study DWDS.

Keywords: Model predictive control, hierarchical control structure, genetic algorithm, water quality with DBPs objectives.

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Authors: Yun Jong Choi, Nam Woong, PooGyeon Park

Abstract:

Variable Structure Control (VSC) is one of the most useful tools handling the practical system with uncertainties and disturbances. Up to now, unfortunately, not enough studies on the input-saturated system with linear-growth-bound disturbances via VSC have been presented. Therefore, this paper proposes an asymp¬totic stability condition for the system via VSC. The designed VSC controller consists of two control parts. The linear control part plays a role in stabilizing the system, and simultaneously, the nonlinear control part in rejecting the linear-growth-bound disturbances perfectly. All conditions derived in this paper are expressed with Linear Matrices Inequalities (LMIs), which can be easily solved with an LMI toolbox in MATLAB.

Keywords: Input saturation, linear-growth bounded disturbances, linear matrix inequality (LMI), variable structure control

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Authors: Rampal Singh, S. Balasundaram

Abstract:

In this paper, we study the application of Extreme Learning Machine (ELM) algorithm for single layered feedforward neural networks to non-linear chaotic time series problems. In this algorithm the input weights and the hidden layer bias are randomly chosen. The ELM formulation leads to solving a system of linear equations in terms of the unknown weights connecting the hidden layer to the output layer. The solution of this general system of linear equations will be obtained using Moore-Penrose generalized pseudo inverse. For the study of the application of the method we consider the time series generated by the Mackey Glass delay differential equation with different time delays, Santa Fe A and UCR heart beat rate ECG time series. For the choice of sigmoid, sin and hardlim activation functions the optimal values for the memory order and the number of hidden neurons which give the best prediction performance in terms of root mean square error are determined. It is observed that the results obtained are in close agreement with the exact solution of the problems considered which clearly shows that ELM is a very promising alternative method for time series prediction.

Keywords: Chaotic time series, Extreme learning machine, Generalization performance.

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Authors: S. K. Pillai, M. K. Jeyakumar

Abstract:

Estimation of model parameters is necessary to predict the behavior of a system. Model parameters are estimated using optimization criteria. Most algorithms use historical data to estimate model parameters. The known target values (actual) and the output produced by the model are compared. The differences between the two form the basis to estimate the parameters. In order to compare different models developed using the same data different criteria are used. The data obtained for short scale projects are used here. We consider software effort estimation problem using radial basis function network. The accuracy comparison is made using various existing criteria for one and two predictors. Then, we propose a new criterion based on linear least squares for evaluation and compared the results of one and two predictors. We have considered another data set and evaluated prediction accuracy using the new criterion. The new criterion is easy to comprehend compared to single statistic. Although software effort estimation is considered, this method is applicable for any modeling and prediction.

Keywords: Software effort estimation, accuracy, Radial Basis Function, linear least squares.

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Authors: Zilong Feng, Hong Li, Yang Liu, Siriguleng He

Abstract:

In this article, an adaptive least-squares mixed finite element method is studied for pseudo-parabolic integro-differential equations. The solutions of least-squares mixed weak formulation and mixed finite element are proved. A posteriori error estimator is constructed based on the least-squares functional and the posteriori errors are obtained.

Keywords: Pseudo-parabolic integro-differential equation, least squares mixed finite element method, adaptive method, a posteriori error estimates.

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Authors: I. Abasi, L. Fata, M. Sadeghi, S. Banihashemi, A. Mohammadee

Abstract:

Background: Dimensional and transdiagnostic approaches as a result of high comorbidity among mental disorders have captured researchers and clinicians interests for exploring the latent factors to development and maintenance of some psychological disorders. The goal of present study is comparing some of these common factors between generalized anxiety disorder and unipolar mood disorder. Methods: 27 patients with generalized anxiety disorder, 29 patients with depression disorder were recruited by using SCID-I and 69 non-clinical populations were selected by using GHQ cut off point. MANCOVA was used for analyzing data. Results: The results show that worry, rumination, intolerance of uncertainty, maladaptive metacognitive beliefs, and experiential avoidance were all significantly different between GAD and unipolar mood disorder groups. However, there weren’t any significant differences in difficulties in emotion regulation and neuroticism between GAD and unipolar mood disorder groups. Discussion: Results indicate that although there are some transdiagnostic and common factors in GAD and unipolar mood disorder, there may be some specific vulnerability factors for each disorder. Further study is needed for answering these questions.

Keywords: Depression, emotion regulation, generalized anxiety disorder, transdiagnostic.

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Authors: Ratanaporn Leesing, Rattanaporn Baojungharn, Thidarat Papone

Abstract:

Compared to oil production from microorganisms, little work has been performed for mixed culture of microalgae and yeast. In this article it is aimed to show high oil accumulation potential of mixed culture of microalgae Chlorella sp. KKU-S2 and oleaginous yeast Torulaspora maleeae Y30 using sugarcane molasses as substrate. The monoculture of T. maleeae Y30 grew faster than that of microalgae Chlorella sp. KKU-S2. In monoculture of yeast, a biomass of 6.4g/L with specific growth rate (m) of 0.265 (1/d) and lipid yield of 0.466g/L were obtained, while 2.53g/L of biomass with m of 0.133 (1/d) and lipid yield of 0.132g/L were obtained for monoculture of Chlorella sp. KKU-S2. The biomass concentration in the mixed culture of T. maleeae Y30 with Chlorella sp. KKU-S2 increased faster and was higher compared with that in the monoculture and mixed culture of microalgae. In mixed culture of microalgae Chlorella sp. KKU-S2 and C. vulgaris TISTR8580, a biomass of 3.47g/L and lipid yield of 0.123 g/L were obtained. In mixed culture of T. maleeae Y30 with Chlorella sp. KKU-S2, a maximum biomass of 7.33 g/L and lipid yield of 0.808g/L were obtained. Maximum cell yield coefficient (YX/S, 0.229g/L), specific yield of lipid (YP/X, 0.11g lipid/g cells) and volumetric lipid production rate (QP, 0.115 g/L/d) were obtained in mixed culture of yeast and microalgae. Clearly, T. maleeae Y30 and Chlorella sp. KKU-S2 use sugarcane molasses as organic nutrients efficiently in mixed culture under mixotrophic growth. The biomass productivity and lipid yield are notably enhanced in comparison with monoculture.

Keywords: Microbial oil, Chlorella sp. KKU-S2, Chlorella vulgaris, Torulaspora maleeae Y30, mixed culture, biodiesel.

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Authors: S. Kavitha, Nirmala P. Ratchagar

Abstract:

This paper presents the novel deterministic dynamic programming approach for solving optimization problem with quadratic objective function with linear equality and inequality constraints. The proposed method employs backward recursion in which computations proceeds from last stage to first stage in a multi-stage decision problem. A generalized recursive equation which gives the exact solution of an optimization problem is derived in this paper. The method is purely analytical and avoids the usage of initial solution. The feasibility of the proposed method is demonstrated with a practical example. The numerical results show that the proposed method provides global optimum solution with negligible computation time.

Keywords: Backward recursion, Dynamic programming, Multi-stage decision problem, Quadratic objective function.

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Authors: Imad Chaddad, Andrei A. Kolyshkin

Abstract:

Linear and weakly nonlinear analysis of shallow wake flows is presented in the present paper. The evolution of the most unstable linear mode is described by the complex Ginzburg-Landau equation (CGLE). The coefficients of the CGLE are calculated numerically from the solution of the corresponding linear stability problem for a one-parametric family of shallow wake flows. It is shown that the coefficients of the CGLE are not so sensitive to the variation of the base flow profile.

Keywords: Ginzburg-Landau equation, shallow wake flow, weakly nonlinear theory.

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Authors: Alpana Tiwari, N. K. Gaur

Abstract:

We have incorporated the translational rotational (TR) coupling effects in the framework of three body force shell model (TSM) to develop an extended TSM (ETSM). The dynamical matrix of ETSM has been applied to compute the phonon frequencies of orientationally disordered mixed crystal (ND4Br)x(KBr)1-x in (q00), (qq0) and (qqq) symmetry directions for compositions 0.10≤x≤0.50 at T=300K.These frequencies are plotted as a function of wave vector k. An unusual acoustic mode softening is found along symmetry directions (q00) and (qq0) as a result of translation-rotation coupling.

Keywords: Orientational glass, phonons, TR-coupling.

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Authors: V. S. Klimash, Ye Min Thu

Abstract:

Thyristor rectifiers, inverters grid-tied, and AC voltage regulators are widely used in industry, and on electrified transport, they have a lot in common both in the power circuit and in the control system. They have a common mathematical structure and switching processes. At the same time, the rectifier, but the inverter units and thyristor regulators of alternating voltage are considered separately both theoretically and practically. They are written about in different books as completely different devices. The aim of this work is to combine them into one class based on the unity of the equations describing electromagnetic processes, and then, to show this unity on the mathematical model and experimental setup. Based on research from mathematics to the product, a conclusion is made about the methodology for the rapid conduct of research and experimental design work, preparation for production and serial production of converters with a unified bundle. In recent years, there has been a transition from thyristor circuits and transistor in modular design. Showing the example of thyristor rectifiers and AC voltage regulators, we can conclude that there is a unity of mathematical structures and grid-tied thyristor converters.

Keywords: Direct current, alternating current, rectifier, AC voltage regulator, generalized mathematical model.

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Authors: Tamás Mankovits, Dávid Huri, Imre Kállai, Imre Kocsis, Tamás Szabó

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Non-linear FEM calculations are indispensable when important technical information like operating performance of a rubber component is desired. Rubber bumpers built into air-spring structures may undergo large deformations under load, which in itself shows non-linear behavior. The changing contact range between the parts and the incompressibility of the rubber increases this non-linear behavior further. The material characterization of an elastomeric component is also a demanding engineering task. In this paper a comprehensive investigation is introduced including laboratory measurements, mesh density analysis and complex finite element simulations to obtain the load-displacement curve of the chosen rubber bumper. Contact and friction effects are also taken into consideration. The aim of this research is to elaborate a FEM model which is accurate and competitive for a future shape optimization task.

Keywords: Rubber bumper, finite element analysis, compression test, Mooney-Rivlin material model.

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Authors: Delfim Soares Jr.

Abstract:

In this work, a new family of time marching procedures based on Green’s function matrices is presented. The formulation is based on the development of new recurrence relationships, which employ time integral terms to treat initial condition values. These integral terms are numerically evaluated taking into account Newton-Cotes formulas. The Green’s matrices of the model are also numerically computed, taking into account the generalized-α method and subcycling techniques. As it is discussed and illustrated along the text, the proposed procedure is efficient and accurate, providing a very attractive time marching technique. 

Keywords: Dynamics, Time-Marching, Green’s Function, Generalized-α Method, Subcycling.

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