Search results for: nonlinear systems of equations

Authors: Muhammad Danish Khan, Imran Naeem, Mudassar Imran

Abstract:

In this article, we study time-space Fractional Advection-Dispersion (FADE) equation and time-space Fractional Whitham-Broer-Kaup (FWBK) equation that have a significant role in hydrology. We introduce suitable transformations to convert fractional order derivatives to integer order derivatives and as a result these equations transform into Partial Differential Equations (PDEs). Then the Lie symmetries and corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations.

Keywords: modified Riemann-Liouville fractional derivative, lie-symmetries, optimal system, invariant solutions

Procedia PDF Downloads 433

Authors: Eugene Stepanov, Arcady Ponosov

Abstract:

To model gene regulatory networks one uses ordinary differential equations with switching nonlinearities, where the initial value problem is known to be well-posed if the trajectories cross the discontinuities transversally. Otherwise, the initial value problem is usually ill-posed, which lead to theoretical and numerical complications. In the presentation, it is proposed to apply the theory of hybrid dynamical systems, rather than switched ones, to regularize the problem. 'Hybridization' of the switched system means that one attaches a dynamic discrete component ('automaton'), which follows the trajectories of the original system and governs its dynamics at the points of ill-posedness of the initial value problem making it well-posed. The construction of the automaton is based on the classification of the attractors of the specially designed adjoint dynamical system. Several examples are provided in the presentation, which support the suggested analysis. The method can also be of interest in other applied fields, where differential equations contain switchings, e.g. in neural field models.

Keywords: hybrid dynamical systems, ill-posed problems, singular perturbation analysis, switching nonlinearities

Procedia PDF Downloads 187

Authors: Davit Shahnazaryan, Diogo Gomes, Mher Safaryan

Abstract:

Many symbolic computations and manipulations required in the analysis of partial differential equations (PDE) or systems of PDEs are tedious and error-prone. These computations arise when determining conservation laws, entropies or integral identities, which are essential tools for the study of PDEs. Here, we discuss a new Mathematica package for the symbolic analysis of PDEs that automate multiple tasks, saving time and effort. Methodologies: During the research, we have used concepts of linear algebra and partial differential equations. We have been working on creating algorithms based on theoretical mathematics to find results mentioned below. Major Findings: Our package provides the following functionalities; finding symmetry group of different PDE systems, generation of polynomials invariant with respect to different symmetry groups; simplification of integral quantities by integration by parts and null Lagrangian cleaning, computing general forms of expressions by integration by parts; finding equivalent forms of an integral expression that are simpler or more symmetric form; determining necessary and sufficient conditions on the coefficients for the positivity of a given symbolic expression. Conclusion: Using this package, we can simplify integral identities, find conserved and dissipated quantities of time-dependent PDE or system of PDEs. Some examples in the theory of mean-field games and semiconductor equations are discussed.

Keywords: partial differential equations, symbolic computation, conserved and dissipated quantities, mathematica

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Authors: Aliasghar Arab

Abstract:

Autonomous robots interacting with humans, as safety-critical nonlinear control systems, are complex closed-loop cyber-physical dynamical machines. Keeping these intelligent yet complicated systems safe and smooth during their operations is challenging. The aim of the safe predictive output feedback linearization control synthesis is to design a novel controller for smooth trajectory following while unsafe situations must be avoided. The controller design should obtain a linearized output for smoothness and invariance to a safety subset. Inspired by finite-horizon nonlinear model predictive control, the problem is formulated as constrained nonlinear dynamic programming. The safety constraints can be defined as control barrier functions. Avoiding unsafe maneuvers and performing smooth motions increases the predictability of the robot’s movement for humans when robots and people are working together. Our results demonstrate the proposed output linearization method obeys the safety constraints and, compared to existing safety-guaranteed methods, is smoother and performs better.

Keywords: robotics, collaborative robots, safety, autonomous robots

Procedia PDF Downloads 98

Authors: P. Selyshchev

Abstract:

Construction materials for nuclear facilities are operated under extreme thermal and radiation conditions. First of all, they are nuclear fuel, fuel assemblies, and reactor vessel. It places high demands on the control of their state, stability of their state, and their operating conditions. An irradiated material is a typical example of an open non-equilibrium system with nonlinear feedbacks between its elements. Fluxes of energy, matter and entropy maintain states which are far away from thermal equilibrium. The links that arise under irradiation are inherently nonlinear. They form the mechanisms of feed-backs that can lead to instability. Due to this instability the temperature of the sample, heat transfer, and the defect density can exceed the steady-state value in several times. This can lead to change of typical operation and an accident. Therefore, it is necessary to take into account the thermal instability to avoid the emergency situation. The point is that non-thermal energy can be accumulated in materials because irradiation produces defects (first of all these are vacancies and interstitial atoms), which are metastable. The stored energy is about energy of defect formation. Thus, an annealing of the defects is accompanied by releasing of non-thermal stored energy into thermal one. Temperature of the material grows. Increase of temperature results in acceleration of defect annealing. Density of the defects drops and temperature grows more and more quickly. The positive feed-back is formed and self-reinforcing annealing of radiation defects develops. To describe these phenomena a theoretical approach to thermal instability is developed via formalism of complex systems. We consider system of nonlinear differential equations for different components of microstructure and temperature. The qualitative analysis of this non-linear dynamical system is carried out. Conditions for development of instability have been obtained. Points of bifurcation have been found. Convenient way to represent obtained results is a set of phase portraits. It has been shown that different regimes of material state under irradiation can develop. Thus degradation of irradiated material can be limited by means of choice appropriate kind of evolution of materials under irradiation.

Keywords: irradiation, material, non-equilibrium state, nonlinear feed-back, thermal instability

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Authors: Esra Zengin, Sinan Akkar

Abstract:

Reliable and accurate prediction of nonlinear structural response requires specification of appropriate earthquake ground motions to be used in nonlinear time history analysis. The current research has mainly focused on selection and manipulation of real earthquake records that can be seen as the most critical step in the performance based seismic design and assessment of the structures. Utilizing amplitude scaled ground motions that matches with the target spectra is commonly used technique for the estimation of nonlinear structural response. Representative ground motion ensembles are selected to match target spectrum such as scenario-based spectrum derived from ground motion prediction equations, Uniform Hazard Spectrum (UHS), Conditional Mean Spectrum (CMS) or Conditional Spectrum (CS). Different sets of criteria exist among those developed methodologies to select and scale ground motions with the objective of obtaining robust estimation of the structural performance. This study presents ground motion selection and scaling procedure that considers the spectral variability at target demand with the level of ground motion dispersion. The proposed methodology provides a set of ground motions whose response spectra match target median and corresponding variance within a specified period interval. The efficient and simple algorithm is used to assemble the ground motion sets. The scaling stage is based on the minimization of the error between scaled median and the target spectra where the dispersion of the earthquake shaking is preserved along the period interval. The impact of the spectral variability on nonlinear response distribution is investigated at the level of inelastic single degree of freedom systems. In order to see the effect of different selection and scaling methodologies on fragility curve estimations, results are compared with those obtained by CMS-based scaling methodology. The variability in fragility curves due to the consideration of dispersion in ground motion selection process is also examined.

Keywords: ground motion selection, scaling, uncertainty, fragility curve

Procedia PDF Downloads 584

Authors: Gautam Kumar Saharia, Sagardeep Talukdar, Riki Dutta, Sudipta Nandy

Abstract:

The physics informed neural network (PINN) method opens up an approach for numerically solving nonlinear partial differential equations leveraging fast calculating speed and high precession of modern computing systems. We construct the PINN based on strong universal approximation theorem and apply the initial-boundary value data and residual collocation points to weekly impose initial and boundary condition to the neural network and choose the optimization algorithms adaptive moment estimation (ADAM) and Limited-memory Broyden-Fletcher-Golfard-Shanno (L-BFGS) algorithm to optimize learnable parameter of the neural network. Next, we improve the PINN with a weighted loss function to obtain both the bright and dark soliton solutions of Fokas-Lenells equation (FLE). We find the proposed scheme of adjustable weight coefficients into PINN has a better convergence rate and generalizability than the basic PINN algorithm. We believe that the PINN approach to solve the partial differential equation appearing in nonlinear optics would be useful to study various optical phenomena.

Keywords: deep learning, optical Soliton, neural network, partial differential equation

Procedia PDF Downloads 129

Authors: S.I.Mukhin, S. Seidov, A. Mukherjee

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The Dicke model is a key tool for the description of correlated states of quantum atomic systems, excited by resonant photon absorption and subsequently emitting spontaneous coherent radiation in the superradiant state. The Dicke Hamiltonian (DH) is successfully used for the description of the dynamics of the Josephson Junction (JJ) array in a resonant cavity under applied current. In this work, we have investigated a generalized model, which is described by DH with a frustrating interaction term. This frustrating interaction term is explicitly the infinite coordinated interaction between all the spin half in the system. In this work, we consider an array of N superconducting islands, each divided into two sub-islands by a Josephson Junction, taken in a charged qubit / Cooper Pair Box (CPB) condition. The array is placed inside the resonant cavity. One important aspect of the problem lies in the dynamical nature of the physical observables involved in the system, such as condensed electric field and dipole moment. It is important to understand how these quantities behave with time to define the quantum phase of the system. The Dicke model without frustrating term is solved to find the dynamical solutions of the physical observables in analytic form. We have used Heisenberg’s dynamical equations for the operators and on applying newly developed Rotating Holstein Primakoff (HP) transformation and DH we have arrived at the four coupled nonlinear dynamical differential equations for the momentum and spin component operators. It is possible to solve the system analytically using two-time scales. The analytical solutions are expressed in terms of Jacobi's elliptic functions for the metastable ‘bound luminosity’ dynamic state with the periodic coherent beating of the dipoles that connect the two double degenerate dipolar ordered phases discovered previously. In this work, we have proceeded the analysis with the extended DH with a frustrating interaction term. Inclusion of the frustrating term involves complexity in the system of differential equations and it gets difficult to solve analytically. We have solved semi-classical dynamic equations using the perturbation technique for small values of Josephson energy EJ. Because the Hamiltonian contains parity symmetry, thus phase transition can be found if this symmetry is broken. Introducing spontaneous symmetry breaking term in the DH, we have derived the solutions which show the occurrence of finite condensate, showing quantum phase transition. Our obtained result matches with the existing results in this scientific field.

Keywords: Dicke Model, nonlinear dynamics, perturbation theory, superconductivity

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Authors: Edward J. Kansa, Pavel Holoborodko

Abstract:

Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.

Keywords: partial differential equations, Meshfree radial basis functions, , no restrictions on spatial dimensions, Extended arithmetic precision.

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Authors: Ahmed Guerine, Ali El Hafidi, Bruno Martin, Philippe Leclaire

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This paper investigates the nonlinear dynamic response of a mechanical torque limiter which is used to protect drive parts from overload (helical transmission gears). The system is driven by four excitations: two external excitations (aerodynamics torque and force) and two internal excitations (two mesh stiffness fluctuations). In this work, we develop a dynamic model with lumped components and 28 degrees of freedom. We use the Runge Kutta step-by-step time integration numerical algorithm to solve the equations of motion obtained by Lagrange formalism. The numerical results have allowed us to identify the sources of vibration in the wind turbine. Also, they are useful to help the designer to make the right design and correctly choose the times for maintenance.

Keywords: two-stage helical gear, lumped model, dynamic response, torque-limiter

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Authors: Rachid Fermous, Rima Mebrek

Abstract:

Plasma expansion is an important physical process that takes place in laser interactions with solid targets. Within a self-similar model for the hydrodynamic multi-fluid equations, we investigated the expansion of dense plasma. The weakly relativistic electrons are produced by ultra-intense laser pulses, while ions are supposed to be in a non-relativistic regime. It is shown that dense plasma expansion is found to be governed mainly by quantum contributions in the fluid equations that originate from the degenerate pressure in addition to the nonlinear contributions from exchange and correlation potentials. The quantum degeneracy parameter profile provides clues to set the limit between under-dense and dense relativistic plasma expansions at a given density and temperature.

Keywords: plasma expansion, quantum degeneracy, weakly relativistic, under-dense plasma

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Authors: Yaping Zhao

Abstract:

In the present study, the exact solutions for the steady response of quasi-linear systems under non-white wide-band random excitation are considered by means of the stochastic averaging method. The non linearity of the systems contains the power-law damping and the cross-product term of the power-law damping and displacement. The drift and diffusion coefficients of the Fokker-Planck-Kolmogorov (FPK) equation after averaging are obtained by a succinct approach. After solving the averaged FPK equation, the joint probability density function and the marginal probability density function in steady state are attained. In the process of resolving, the eigenvalue problem of ordinary differential equation is handled by integral equation method. Some new results are acquired and the novel method to deal with the problems in nonlinear random vibration is proposed.

Keywords: random vibration, stochastic averaging method, FPK equation, transition probability density

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Authors: Martin Cermak, Stanislav Sysala

Abstract:

In this paper we present the efficient parallel implementation of elastoplastic problems based on the TFETI (Total Finite Element Tearing and Interconnecting) domain decomposition method. This approach allow us to use parallel solution and compute this nonlinear problem on the supercomputers and decrease the solution time and compute problems with millions of DOFs. In our approach we consider an associated elastoplastic model with the von Mises plastic criterion and the combination of linear isotropic-kinematic hardening law. This model is discretized by the implicit Euler method in time and by the finite element method in space. We consider the system of nonlinear equations with a strongly semismooth and strongly monotone operator. The semismooth Newton method is applied to solve this nonlinear system. Corresponding linearized problems arising in the Newton iterations are solved in parallel by the above mentioned TFETI. The implementation of this problem is realized in our in-house MatSol packages developed in MATLAB.

Keywords: isotropic-kinematic hardening, TFETI, domain decomposition, parallel solution

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Authors: Mohamed Omar

Abstract:

Steel bracing members are widely used in steel structures to reduce lateral displacement and dissipate energy during earthquake motions. Concentric steel bracing provide an excellent approach for strengthening and stiffening steel buildings. Using these braces the designer can hardly adjust the stiffness together with ductility as needed because of buckling of braces in compression. In this study the use of SMA bracing and steel bracing (Mega) utilized in steel frames are investigated. The effectiveness of these two systems in rehabilitating a mid-rise eight-storey steel frames were examined using time-history nonlinear analysis utilizing Seismo-Struct software. Results show that both systems improve the strength and stiffness of the original structure but due to excellent behavior of SMA in nonlinear phase and under compressive forces this system shows much better performance than the rehabilitation system of Mega bracing.

Keywords: finite element analysis, seismic response, shapes memory alloy, steel frame, mega bracing

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Authors: Khosrow Maleknejad, Yaser Rostami

Abstract:

In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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Authors: Karima Kimouche

Abstract:

In this paper, we consider the asymptotic problems in spectral analysis of stationary causal random fields. We impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear random fields. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given.

Keywords: spatial nonlinear processes, spectral estimators, GMC condition, bootstrap method

Procedia PDF Downloads 455

Authors: Meriem Mossaddek, El Mehdi Laadissi, El Mehdi Loualid, Chouaib Ennawaoui, Sohaib Bouzaid, Abdelowahed Hajjaji

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An electrochemical system is a subset of mechatronic systems that includes a wide variety of batteries and nickel-cadmium, lead-acid batteries, and lithium-ion. Those structures have several non-linear behaviors and uncertainties in their running range. This paper studies an effective technique for modeling Lithium-Ion (Li-Ion) batteries using a Nonlinear Auto-Regressive model with exogenous input (NARX). The Artificial Neural Network (ANN) is trained to employ the data collected from the battery testing process. The proposed model is implemented on a Li-Ion battery cell. Simulation of this model in MATLAB shows good accuracy of the proposed model.

Keywords: lithium-ion battery, neural network, energy storage, battery model, nonlinear models

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Authors: P.-W. Tsai, C.-Y. Chen, C.-W. Chen

Abstract:

In this paper, the stability analysis of a GA-Based adaptive fuzzy sliding model controller for a nonlinear system is discussed. First, a nonlinear plant is well-approximated and described with a reference model and a fuzzy model, both involving FLC rules. Then, FLC rules and the consequent parameter are decided on via an Evolved Bat Algorithm (EBA). After this, we guarantee a new tracking performance inequality for the control system. The tracking problem is characterized to solve an eigenvalue problem (EVP). Next, an adaptive fuzzy sliding model controller (AFSMC) is proposed to stabilize the system so as to achieve good control performance. Lyapunov’s direct method can be used to ensure the stability of the nonlinear system. It is shown that the stability analysis can reduce nonlinear systems into a linear matrix inequality (LMI) problem. Finally, a numerical simulation is provided to demonstrate the control methodology.

Keywords: adaptive fuzzy sliding mode control, Lyapunov direct method, swarm intelligence, evolved bat algorithm

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Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

Abstract:

An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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Authors: Gautam Kumar Saharia, Sagardeep Talukdar, Riki Dutta, Sudipta Nandy

Abstract:

The physics-informed neural network (PINN) method opens up an approach for numerically solving nonlinear partial differential equations leveraging fast calculating speed and high precession of modern computing systems. We construct the PINN based on a strong universal approximation theorem and apply the initial-boundary value data and residual collocation points to weekly impose initial and boundary conditions to the neural network and choose the optimization algorithms adaptive moment estimation (ADAM) and Limited-memory Broyden-Fletcher-Golfard-Shanno (L-BFGS) algorithm to optimize learnable parameter of the neural network. Next, we improve the PINN with a weighted loss function to obtain both the bright and dark soliton solutions of the Fokas-Lenells equation (FLE). We find the proposed scheme of adjustable weight coefficients into PINN has a better convergence rate and generalizability than the basic PINN algorithm. We believe that the PINN approach to solve the partial differential equation appearing in nonlinear optics would be useful in studying various optical phenomena.

Keywords: deep learning, optical soliton, physics informed neural network, partial differential equation

Procedia PDF Downloads 70

Authors: J.P. Henriques, E. R. Neto, G. Almeida, G. Ribeiro, J.V. Coutinho, A.B. Lugli

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Industry 4.0, or the Fourth Industrial Revolution, is transforming the relationship between people and machines. In this scenario, some technologies such as Cloud Computing, Internet of Things, Augmented Reality, Artificial Intelligence, Additive Manufacturing, among others, are making industries and devices increasingly intelligent. One of the most powerful technologies of this new revolution is the Digital Twin, which allows the virtualization of a real system or process. In this context, the present paper addresses the linear and nonlinear dynamic study of a didactic level plant using Digital Twin. In the first part of the work, the level plant is identified at a fixed point of operation, BY using the existing method of least squares means. The linearized model is embedded in a Digital Twin using Automation Studio® from Famous Technologies. Finally, in order to validate the usage of the Digital Twin in the linearized study of the plant, the dynamic response of the real system is compared to the Digital Twin. Furthermore, in order to develop the nonlinear model on a Digital Twin, the didactic level plant is identified by using the method proposed by Hammerstein. Different steps are applied to the plant, and from the Hammerstein algorithm, the nonlinear model is obtained for all operating ranges of the plant. As for the linear approach, the nonlinear model is embedded in the Digital Twin, and the dynamic response is compared to the real system in different points of operation. Finally, yet importantly, from the practical results obtained, one can conclude that the usage of Digital Twin to study the dynamic systems is extremely useful in the industrial environment, taking into account that it is possible to develop and tune controllers BY using the virtual model of the real systems.

Keywords: industry 4.0, digital twin, system identification, linear and nonlinear models

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Authors: Hassen M. Ouakad

Abstract:

In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.

Keywords: MEMS, NEMS, fringing-fields, mid-plane stretching, Galerkin

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Authors: Abtin Farokhipanah

Abstract:

In recent years, analysis of structures is based on ductility design in contradictory to strength design in surveying earthquake effects on structures. ASCE07-10 code offers to intensify relative drifts calculated from a linear analysis with Cd which is called (Deflection Amplification Factor) to obtain the real relative drifts which can be calculated using nonlinear analysis. This lateral drift should be limited to the code boundaries. Calculation of this amplification factor for different structures, comparing with ASCE07-10 code and offering the best coefficient are the purposes of this research. Following our target, short and tall building steel structures with various earthquake resistant systems in linear and nonlinear analysis should be surveyed, so these questions will be answered: 1. Does the Response Modification Coefficient (R) have a meaningful relation to Deflection Amplification Factor? 2. Does structure height, seismic zone, response spectrum and similar parameters have an effect on the conversion coefficient of linear analysis to real drift of structure? The procedure has used to conduct this research includes: (a) Study on earthquake resistant systems, (b) Selection of systems and modeling, (c) Analyzing modeled systems using linear and nonlinear methods, (d) Calculating conversion coefficient for each system and (e) Comparing conversion coefficients with the code offered ones and concluding results.

Keywords: ASCE07-10 code, deflection amplification factor, earthquake engineering, lateral displacement of structures, response modification coefficient

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Authors: A. Belmeguenai, K. Mansouri, R. Djemili

Abstract:

This work present a new algorithm based on the nonlinear filter generator for speech encryption and decryption. The proposed algorithm consists on the use a linear feedback shift register (LFSR) whose polynomial is primitive and nonlinear Boolean function. The purpose of this system is to construct Keystream with good statistical properties, but also easily computable on a machine with limited capacity calculated. This proposed speech encryption scheme is very simple, highly efficient, and fast to implement the speech encryption and decryption. We conclude the paper by showing that this system can resist certain known attacks.

Keywords: nonlinear filter generator, stream ciphers, speech encryption, security analysis

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Authors: A. Abdikian

Abstract:

Propagation of nonlinear acoustic wave in dense electron-positron (e-p) plasmas in the presence of an external magnetic field and stationary ions (to neutralize the plasma background) is studied. By means of the quantum hydrodynamics model and applying the reductive perturbation method, the Zakharov-Kuznetsov equation is derived. Using the bifurcation theory of planar dynamical systems, the compressive structure of electrostatic solitary wave and periodic travelling waves is found. The numerical results show how the ion density ratio, the ion cyclotron frequency, and the direction cosines of the wave vector affect the nonlinear electrostatic travelling waves. The obtained results may be useful to better understand the obliquely nonlinear electrostatic travelling wave of small amplitude localized structures in dense magnetized quantum e-p plasmas and may be applicable to study the particle and energy transport mechanism in compact stars such as the interior of massive white dwarfs etc.

Keywords: bifurcation theory, phase portrait, magnetized electron-positron plasma, the Zakharov-Kuznetsov equation

Procedia PDF Downloads 245

Authors: Qijiao He

Abstract:

MicroElectrical-Mechanical System (MEMS) is one of the most rapidly developing frontier research field both in theory study and applied technology. Micro-channel is a very important link component of MEMS. With the research and development of MEMS, the size of the micro-devices and the micro-channels becomes further smaller. Compared with the macroscale flow, the flow characteristics of gas in the micro-channel have changed, and the rarefaction effect appears obviously. However, for the rarefied gas and microscale flow, Navier-Stokes-Fourier (NSF) equations are no longer appropriate due to the breakup of the continuum hypothesis. A Nonlinear Coupled Constitutive Model (NCCM) has been derived from the Boltzmann equation to describe the characteristics of both continuum and rarefied gas flows. We apply the present scheme to simulate continuum and rarefied gas flows in a micro-channel structure. And for comparison, we apply other widely used methods which based on particle simulation or direct solution of distribution function, such as Direct simulation of Monte Carlo (DSMC), Unified Gas-Kinetic Scheme (UGKS) and Lattice Boltzmann Method (LBM), to simulate the flows. The results show that the present solution is in better agreement with the experimental data and the DSMC, UGKS and LBM results than the NSF results in rarefied cases but is in good agreement with the NSF results in continuum cases. And some characteristics of both continuum and rarefied gas flows are observed and analyzed.

Keywords: continuum and rarefied gas flows, discontinuous Galerkin method, generalized hydrodynamic equations, numerical simulation

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Authors: Nicolò Vaiana, Giorgio Serino

Abstract:

The nonlinear time history analysis of seismically base-isolated structures can require a significant computational effort when the behavior of each seismic isolator is predicted by adopting the widely used differential equation Bouc-Wen model. In this paper, a nonlinear exponential model, able to simulate the response of seismic isolation bearings within a relatively large displacements range, is described and adopted in order to reduce the numerical computations and speed up the nonlinear dynamic analysis. Compared to the Bouc-Wen model, the proposed one does not require the numerical solution of a nonlinear differential equation for each time step of the analysis. The seismic response of a 3d base-isolated structure with a lead rubber bearing system subjected to harmonic earthquake excitation is simulated by modeling each isolator using the proposed analytical model. The comparison of the numerical results and computational time with those obtained by modeling the lead rubber bearings using the Bouc-Wen model demonstrates the good accuracy of the proposed model and its capability to reduce significantly the computational effort of the analysis.

Keywords: base isolation, computational efficiency, nonlinear exponential model, nonlinear time history analysis

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Authors: Young Hee Geum

Abstract:

In this paper, we present the iterative method and determine the control parameters to converge cubically for solving nonlinear equations. In addition, we derive the asymptotic error constant.

Keywords: asymptotic error constant, iterative method, multiple root, root-finding, order of convergent

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Authors: Mohammad R. Jalali, Mohammad M. Jalali

Abstract:

The earliest theories of sloshing waves and solitary waves based on potential theory idealisations and irrotational flow have been extended to be applicable to more realistic domains. To this end, the computational fluid dynamics (CFD) methods are widely used. Three-dimensional CFD methods such as Navier-Stokes solvers with volume of fluid treatment of the free surface and Navier-Stokes solvers with mappings of the free surface inherently impose high computational expense; therefore, considerable effort has gone into developing depth-averaged approaches. Examples of such approaches include Green–Naghdi (GN) equations. In Cartesian system, GN velocity profile depends on horizontal directions, x-direction and y-direction. The effect of vertical direction (z-direction) is also taken into consideration by applying weighting function in approximation. GN theory considers the effect of vertical acceleration and the consequent non-hydrostatic pressure. Moreover, in GN theory, the flow is rotational. The present study illustrates the application of GN equations to propagation of sloshing waves and solitary waves. For this purpose, GN equations solver is verified for the benchmark tests of Gaussian hump sloshing and solitary wave propagation in shallow basins. Analysis of the free surface sloshing of even harmonic components of an initial Gaussian hump demonstrates that the GN model gives predictions in satisfactory agreement with the linear analytical solutions. Discrepancies between the GN predictions and the linear analytical solutions arise from the effect of wave nonlinearities arising from the wave amplitude itself and wave-wave interactions. Numerically predicted solitary wave propagation indicates that the GN model produces simulations in good agreement with the analytical solution of the linearised wave theory. Comparison between the GN model numerical prediction and the result from perturbation analysis confirms that nonlinear interaction between solitary wave and a solid wall is satisfactorilly modelled. Moreover, solitary wave propagation at an angle to the x-axis and the interaction of solitary waves with each other are conducted to validate the developed model.

Keywords: Green–Naghdi equations, nonlinearity, numerical prediction, sloshing waves, solitary waves

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Authors: Abhilash Sahu, Amit Priyadarshi

Abstract:

In this paper, we will discuss the existence of weak solutions of the Kirchhoff type boundary value problem on the Sierpinski gasket. Where S denotes the Sierpinski gasket in R² and S₀ is the intrinsic boundary of the Sierpinski gasket. M: R → R is a positive function and h: S × R → R is a suitable function which is a part of our main equation. ∆p denotes the p-Laplacian, where p > 1. First of all, we will define a weak solution for our problem and then we will show the existence of at least two solutions for the above problem under suitable conditions. There is no well-known concept of a generalized derivative of a function on a fractal domain. Recently, the notion of differential operators such as the Laplacian and the p-Laplacian on fractal domains has been defined. We recall the result first then we will address the above problem. In view of literature, Laplacian and p-Laplacian equations are studied extensively on regular domains (open connected domains) in contrast to fractal domains. In fractal domains, people have studied Laplacian equations more than p-Laplacian probably because in that case, the corresponding function space is reflexive and many minimax theorems which work for regular domains is applicable there which is not the case for the p-Laplacian. This motivates us to study equations involving p-Laplacian on the Sierpinski gasket. Problems on fractal domains lead to nonlinear models such as reaction-diffusion equations on fractals, problems on elastic fractal media and fluid flow through fractal regions etc. We have studied the above p-Laplacian equations on the Sierpinski gasket using fibering map technique on the Nehari manifold. Many authors have studied the Laplacian and p-Laplacian equations on regular domains using this Nehari manifold technique. In general Euler functional associated with such a problem is Frechet or Gateaux differentiable. So, a critical point becomes a solution to the problem. Also, the function space they consider is reflexive and hence we can extract a weakly convergent subsequence from a bounded sequence. But in our case neither the Euler functional is differentiable nor the function space is known to be reflexive. Overcoming these issues we are still able to prove the existence of at least two solutions of the given equation.

Keywords: Euler functional, p-Laplacian, p-energy, Sierpinski gasket, weak solution

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