Preface

Starting from the sixties of previous century, numerical simulation has become a significant, and at times, a crucial role in the progress of many areas of science. This book contains original and innovative research studies related to modeling and simulation of the physical phenomena in a very wide range of applications, including the macro- and micro-electrodynamics radiation and scattering, the fluid dynamics turbulence and emulsification, as well as the various industrial processes.

Recent numerical techniques, as well as the most accurate and advanced software are applied in order to perfectly explain the nature of the considered phenomena. The book can be useful for theoretical and applied researchers, who deal with numerical simulation in various areas of science.

The book chapters are divided into four sections according to the considered problems and corresponding areas of application. **Section 1** contains the latest advances in the bifurcation theory, network scheduling, epidemiology, physics of plasma, and mechanics.

**Chapter 1**, authored by Canan Çelik Karaaslanli, is aimed to study the bifurcation phenomena for the solution of ordinary differential equations and systems of such equations in conjunction with the analytical computation and numerical analysis. The governing equations of the bifurcation analysis are derived for the case of one dimension, for which the saddle-node, transcritical and pitchfork bifurcations are examined. For the two-dimensional case, the Hopf bifurcation theorem determines the condition providing the bifurcation of two types, namely, subcritical and supercritical bifurcation. It was substantiated and proved that Hopf bifurcation is possible for the systems consisting of two or more differential equations; such type of bifurcation is described in literature as Poincare-Andronov-Hopf bifurcation. The main theoretical result consists in center manifold reduction for Hopf bifurcation, this result allows to circumscribe the Hopf bifurcation for various types of the differential equation systems. The numerous exercises demonstrate initial equilibrium properties of the systems under consideration, as well as explain the difference between arising types of bifurcation.

A time-varying control approach designed for a nonlinear system is studied by Dong Bin Lee and C. Nataraj in **Chapter 2**. A cart-pole nonlinear dynamic model with the unknown system parameters is developed for the application of a proposed control algorithm and expressed in a state space form. The error signals are formulated from desired model-based reference system and applied for a trajectory tracking control. Based on the analysis of the developed time-varying error system, state transition matrix is given in a series form, and then a special form of this matrix is derived for the second-order error differential equation, which is used for obtaining the grammian matrix and the closed-loop controller. The control system is also applicable to reject disturbances via a projection-based adaptive control approach and modify laws for the parameter update. Results of numerical simulation demonstrate the validity of the proposed system. The developed approach can be extended to other nonlinear timevarying dynamic systems such as aerial, marine, or ground vehicles.

Preface XI

cylindrical plasma edge. The problem is formulated in coupled set of equations consisting of the normalized 2D Vlasov equation for ions distribution function and normalized 2D guiding-center equation for electron properties. The effective leapfrog scheme is proposed in order to split the equations in one-time step. The proposed scheme foresees three stages, in the first of them a set of homogeneous equations for determination of the distribution function and electron density profile is solved for half-time step. The second step gives the solution to homogeneous equations with regard to missing in the first step terms. The values of electron density, calculated in the previous step, are substituted in the equations which are solved in the first step in order to receive the solution with half-time step again in the third step. The analysis of modeling results provides with understanding a series of physical phenomena, what is the considerable contribution to the neoclassical Tokamak theory and its applications.

T. T. Truong and M. K. Nguyen use theoretical study and numerical simulation for describing the recent developments in the Compton scatter tomography (CST) in **Chapter 6**. The mathematical model foresees the calculation of the reconstructed electron density for the cases of internal and external scanning. The combination of the Radon transform along the respective curves in the conjunction with Chebyshev integral transform of initial equation for the reconstructed electron density is applied for this purpose. In the process of numerical simulation, the resulting expression for the reconstructed electron density is obtained using the explicit formula for curves of integration what leads to representation of integrals by discrete sums; the recurrence relations for arising integrals, as well as, a simple linear approximation of derivatives is used. The validation of approach is realized on the Shepp-Logan medical phantom investigation. It is observed that the quality of reconstruction is improved by passing from first to third CST modality, proposed by authors and adopted from the literature. The proposed approach can be used for research and development in the field of non-

Numerical simulations of the slip-stick elastic contact related to the contact mechanics and theory of elasticity is considered by Sergiu Spinu and Dumitru Amarandei in **Chapter 7**. The novelty of proposed approach is determined by utilization of the nonlinear theory of elasticity for study of the stress-strain state of materials with various dissimilarly elastic characteristics. The mathematical model of the slip-stick elastic contact problem in terms of normal and tangential forces is discussed. It is underlined that owing to the interdependency of the normal and tangential forces' components the model is essentially non-linear. Based on the two dimensional convolution product, the mathematical description of the discretized model is given in very clear and useful manner. The algorithm for solving the received disretized problem, involving the combination of the conjugate gradient method and uncoupled determination of normal and tangential forces is described very precisely and accurate. The additional taking into account of the torsional contact is validated by consideration of problem with spherical indenter and comparison of solution with

invasive imaging for biology, medicine and industry.

known analytical one for such problem.

Oscar Esquivel-Flores at al. discuss in **Chapter 3** the communication network control system based on scheduling strategy and applied for the real time controller design of 2-DOF helicopter prototype. The architecture and overview of the network control system is presented and explored. The True Time simulation tool based on the Matlab/Simulink software, presenting the kernel and network blocks, is scheduled in order to take into account the global network activity. This leads to considering a real time distributed system with three sets of frequency transmission parameters: minimum, real, and maximum frequency rates. The real frequency rate is determined as a solution to respective linear algebraic system with limitation to four sensor nodes. For the implementation purpose, the mathematical model of the 2-DOF helicopter prototype is studied, and the parameters for the network control are defined. The values of pitch and yew angles, as well as the values of their derivatives are the parameters under real time control. The network control flowchart integrated into closed control loop of the 2-DOF helicopter is constructed and applied for the dynamical adjustment of the helicopter position.

In **Chapter 4**, Sandile Motsa and Stanford Shateyi apply the nonlinear system of partial differential equations for SIR model which describes the temporal dynamics of a childhood disease in the presence of a preventive vaccine. The main stage of solving the problem consists of reduction of the obtained system to non interconnected equations. In spite of the fact that this system contains two initial unknown functions, the proposed quasilinearization method allows solving the respective equations related with one function only. Rearranging the unknown functions in right and left hand sides of equations, authors received such representation of the system which allows solving it by the effective iterative procedure. The numerical results are carried out for different cases of the DFE and EE phenomena. The approach is validated with calculations by the MATLAB initial value solver ode45. It is shown that for the case of absence of the disease eradication the scheme of zero order is sufficient to describe toward process adequately.

Magdi Shoucri in **Chapter 5** discusses the rigorous analytical description of the phenomena related to charge separation and electromagnetic field properties at the cylindrical plasma edge. The problem is formulated in coupled set of equations consisting of the normalized 2D Vlasov equation for ions distribution function and normalized 2D guiding-center equation for electron properties. The effective leapfrog scheme is proposed in order to split the equations in one-time step. The proposed scheme foresees three stages, in the first of them a set of homogeneous equations for determination of the distribution function and electron density profile is solved for half-time step. The second step gives the solution to homogeneous equations with regard to missing in the first step terms. The values of electron density, calculated in the previous step, are substituted in the equations which are solved in the first step in order to receive the solution with half-time step again in the third step. The analysis of modeling results provides with understanding a series of physical phenomena, what is the considerable contribution to the neoclassical Tokamak theory and its applications.

X Preface

unknown system parameters is developed for the application of a proposed control algorithm and expressed in a state space form. The error signals are formulated from desired model-based reference system and applied for a trajectory tracking control. Based on the analysis of the developed time-varying error system, state transition matrix is given in a series form, and then a special form of this matrix is derived for the second-order error differential equation, which is used for obtaining the grammian matrix and the closed-loop controller. The control system is also applicable to reject disturbances via a projection-based adaptive control approach and modify laws for the parameter update. Results of numerical simulation demonstrate the validity of the proposed system. The developed approach can be extended to other nonlinear time-

Oscar Esquivel-Flores at al. discuss in **Chapter 3** the communication network control system based on scheduling strategy and applied for the real time controller design of 2-DOF helicopter prototype. The architecture and overview of the network control system is presented and explored. The True Time simulation tool based on the Matlab/Simulink software, presenting the kernel and network blocks, is scheduled in order to take into account the global network activity. This leads to considering a real time distributed system with three sets of frequency transmission parameters: minimum, real, and maximum frequency rates. The real frequency rate is determined as a solution to respective linear algebraic system with limitation to four sensor nodes. For the implementation purpose, the mathematical model of the 2-DOF helicopter prototype is studied, and the parameters for the network control are defined. The values of pitch and yew angles, as well as the values of their derivatives are the parameters under real time control. The network control flowchart integrated into closed control loop of the 2-DOF helicopter is constructed and applied for the

In **Chapter 4**, Sandile Motsa and Stanford Shateyi apply the nonlinear system of partial differential equations for SIR model which describes the temporal dynamics of a childhood disease in the presence of a preventive vaccine. The main stage of solving the problem consists of reduction of the obtained system to non interconnected equations. In spite of the fact that this system contains two initial unknown functions, the proposed quasilinearization method allows solving the respective equations related with one function only. Rearranging the unknown functions in right and left hand sides of equations, authors received such representation of the system which allows solving it by the effective iterative procedure. The numerical results are carried out for different cases of the DFE and EE phenomena. The approach is validated with calculations by the MATLAB initial value solver ode45. It is shown that for the case of absence of the disease eradication the scheme of zero order is sufficient to describe

Magdi Shoucri in **Chapter 5** discusses the rigorous analytical description of the phenomena related to charge separation and electromagnetic field properties at the

varying dynamic systems such as aerial, marine, or ground vehicles.

dynamical adjustment of the helicopter position.

toward process adequately.

T. T. Truong and M. K. Nguyen use theoretical study and numerical simulation for describing the recent developments in the Compton scatter tomography (CST) in **Chapter 6**. The mathematical model foresees the calculation of the reconstructed electron density for the cases of internal and external scanning. The combination of the Radon transform along the respective curves in the conjunction with Chebyshev integral transform of initial equation for the reconstructed electron density is applied for this purpose. In the process of numerical simulation, the resulting expression for the reconstructed electron density is obtained using the explicit formula for curves of integration what leads to representation of integrals by discrete sums; the recurrence relations for arising integrals, as well as, a simple linear approximation of derivatives is used. The validation of approach is realized on the Shepp-Logan medical phantom investigation. It is observed that the quality of reconstruction is improved by passing from first to third CST modality, proposed by authors and adopted from the literature. The proposed approach can be used for research and development in the field of noninvasive imaging for biology, medicine and industry.

Numerical simulations of the slip-stick elastic contact related to the contact mechanics and theory of elasticity is considered by Sergiu Spinu and Dumitru Amarandei in **Chapter 7**. The novelty of proposed approach is determined by utilization of the nonlinear theory of elasticity for study of the stress-strain state of materials with various dissimilarly elastic characteristics. The mathematical model of the slip-stick elastic contact problem in terms of normal and tangential forces is discussed. It is underlined that owing to the interdependency of the normal and tangential forces' components the model is essentially non-linear. Based on the two dimensional convolution product, the mathematical description of the discretized model is given in very clear and useful manner. The algorithm for solving the received disretized problem, involving the combination of the conjugate gradient method and uncoupled determination of normal and tangential forces is described very precisely and accurate. The additional taking into account of the torsional contact is validated by consideration of problem with spherical indenter and comparison of solution with known analytical one for such problem.

#### XVI Preface

**Section 2** is devoted to numerical simulation in the areas of macro- and microelectrodynamics, as well as to the numerical investigation of the electromagnetic field characteristics in semiconductor devices of micro- and nano-electronics.

Preface XIII

charge waves, distribution of spectral components of the electric field, the spatial distribution of alternative part of electric field components, the components of electron drift velocity, and alternative part of the electron concentration are analyzed. The modeling results show a better performance of the investigated n-InP films in

The subject of **Chapter 11**, prepared by Takuichi Hirano at al., is electromagnetic simulation of the RF circuits S-parameters applied to structures on a Si CMOS substrate. The engineering features of method, the detailed description of the investigated ICs and de-embedded techniques, as well as the mathematical description of toward processes in three types of patterns are discussed. Electromagnetic modeling consists of calculation of transmission coefficients using the conventional HFSS software. The amplitude and phase of transmission coefficients are received for three types of lumped ports configuration for symmetric and asymmetric IC patterns. The comparison of accuracy between proposed variants of de-embedding techniques for the guided microstrip line is done for the attenuation and phase constant, as well as for the extracted characteristic impedance. A good agreement between the calculation and measurement results was obtained for the above characteristics. Summarizing the obtained results on the whole, one can conclude that the proposed de-embedded techniques on the base of numerical simulation and HFSS solver is effective instrument for investigation of transmission parameters of a wide class of RF circuits

Saadeddine Khemissi studies the conductive characteristics of the GaAs metalsemiconductor field effect transistors (MESFETs) within the framework of the elaboration of analytical model and numerical simulation in **Chapter 12**. Firstly, the respective Poisson's equation supplemented by proper boundary conditions is considered. The boundary conditions deciding the connections between the intrinsic gate-source, drain-source voltages, and electric field in the depletion-layer edges are specified in the second step of research. The Green's function approach and superposition technique are applied for extension of the initial model to the 2D one. The resulting depletion layer parameters are determined as a combination of initial values from 1D model and total tension function. Calculation of conductive characteristics consists of determination of the drain current for linear, pinch-off, and saturation regimes. In order to calculate the extrinsic characteristics, the effects of parasitic resistance are taken into account. Since they require the calculation of drain currents, which are also contained in the first terms of respective formulas, the

In **Chapter 13**, the comparative study of the Diamond SOI MOSFETs and the conventional ones is performed by Salvador Pinillos Gimenez and Marcello Bellodi using the numerical simulation. The ATLAS device simulator is applied for the comprehensive three-dimensional modeling of the devices under consideration. Lombardi's vertical and horizontal electric-field-depend mobility model which takes

iterative procedure for calculation of the drain currents is proposed.

comparison with the GaAs ones.

on chip.

In **Chapter 8**, Oleg A. Yurtsev and Grigory V. Ptashinsky apply numerical simulation for study of radiation characteristics of the dipole and loop antennas. The well approved Pocklington's integral equation is used for accurate formulation of the problem of determining the exciting current in the antenna elements. The respective linear algebraic system is derived and applied for numerical calculation of sought currents. The radiation and scattering characteristics, normalized cross section, input resistance, resonance frequency, and percentage bandwidth for the dipole antennas and Yagi-Uda antennas, dipole antennas with reduced dimensions, flat and convex dipole and Yagi arrays, linear antenna arrays with series excitation, are discussed during the numerical modeling. The high emphasis is placed onto investigation of various modifications of loop antennas. The constructions of such antennas, the interaction of array elements are studied in the terms of input resistance, voltagestanding wave ratio, gain, and matched bandwidth. The optimal parameters of antennas are determined for various geometries, and simulation results complete the existing data regarding the dipole antennas to a considerable degree.

Mykhaylo Andriychuk proposes the variational approach for solving the synthesis problems of antennas according to the given (desired) amplitude radiation pattern (RP) in **Chapter 9**. The absence of requirements to the phase RP is used as an additional possibility to improve the quality of approximation to the desired amplitude RP, but the incompleteness of initial data in the problem yields a class of essentially ill-posed problems, which are described by the nonlinear integral or matrix equations. The methods of nonlinear functional analysis which allow localization of the branching solutions are applied for investigation of solutions and determination of their number and qualitative characteristics. Depending on the restrictions which are imposed on the sought distribution of current or field in the antenna elements and type of antenna, the problems of amplitude-phase, amplitude, and phase synthesis are considered for the specific antennas. The methods of successive approximations are applied for solving the derived nonlinear integral equations; the direct optimization of the proposed functionals by the gradient methods is performed too.

The physical phenomena of amplification of the space charge waves in n-InP films are numerically studied by Abel García-Barrientos et al. in **Chapter 10.** The mathematical description of model is similar to the case of GaAs films. Since the 2D model of electron gas in the n-InP film is used, the respective Poisson's equation is reduced to the two dimensional case. The fast Fourier transform is used for solving this equation. The unknown coefficients in the series for potential and E-components are determined by satisfaction of boundary conditions what yields the respective linear algebraic system with three-diagonal matrix. A series of physical phenomena is studied in the framework of numerical simulation. The spatial increment of instability of the space charge waves, distribution of spectral components of the electric field, the spatial distribution of alternative part of electric field components, the components of electron drift velocity, and alternative part of the electron concentration are analyzed. The modeling results show a better performance of the investigated n-InP films in comparison with the GaAs ones.

XII Preface

**Section 2** is devoted to numerical simulation in the areas of macro- and microelectrodynamics, as well as to the numerical investigation of the electromagnetic field

In **Chapter 8**, Oleg A. Yurtsev and Grigory V. Ptashinsky apply numerical simulation for study of radiation characteristics of the dipole and loop antennas. The well approved Pocklington's integral equation is used for accurate formulation of the problem of determining the exciting current in the antenna elements. The respective linear algebraic system is derived and applied for numerical calculation of sought currents. The radiation and scattering characteristics, normalized cross section, input resistance, resonance frequency, and percentage bandwidth for the dipole antennas and Yagi-Uda antennas, dipole antennas with reduced dimensions, flat and convex dipole and Yagi arrays, linear antenna arrays with series excitation, are discussed during the numerical modeling. The high emphasis is placed onto investigation of various modifications of loop antennas. The constructions of such antennas, the interaction of array elements are studied in the terms of input resistance, voltagestanding wave ratio, gain, and matched bandwidth. The optimal parameters of antennas are determined for various geometries, and simulation results complete the

characteristics in semiconductor devices of micro- and nano-electronics.

existing data regarding the dipole antennas to a considerable degree.

the proposed functionals by the gradient methods is performed too.

Mykhaylo Andriychuk proposes the variational approach for solving the synthesis problems of antennas according to the given (desired) amplitude radiation pattern (RP) in **Chapter 9**. The absence of requirements to the phase RP is used as an additional possibility to improve the quality of approximation to the desired amplitude RP, but the incompleteness of initial data in the problem yields a class of essentially ill-posed problems, which are described by the nonlinear integral or matrix equations. The methods of nonlinear functional analysis which allow localization of the branching solutions are applied for investigation of solutions and determination of their number and qualitative characteristics. Depending on the restrictions which are imposed on the sought distribution of current or field in the antenna elements and type of antenna, the problems of amplitude-phase, amplitude, and phase synthesis are considered for the specific antennas. The methods of successive approximations are applied for solving the derived nonlinear integral equations; the direct optimization of

The physical phenomena of amplification of the space charge waves in n-InP films are numerically studied by Abel García-Barrientos et al. in **Chapter 10.** The mathematical description of model is similar to the case of GaAs films. Since the 2D model of electron gas in the n-InP film is used, the respective Poisson's equation is reduced to the two dimensional case. The fast Fourier transform is used for solving this equation. The unknown coefficients in the series for potential and E-components are determined by satisfaction of boundary conditions what yields the respective linear algebraic system with three-diagonal matrix. A series of physical phenomena is studied in the framework of numerical simulation. The spatial increment of instability of the space

The subject of **Chapter 11**, prepared by Takuichi Hirano at al., is electromagnetic simulation of the RF circuits S-parameters applied to structures on a Si CMOS substrate. The engineering features of method, the detailed description of the investigated ICs and de-embedded techniques, as well as the mathematical description of toward processes in three types of patterns are discussed. Electromagnetic modeling consists of calculation of transmission coefficients using the conventional HFSS software. The amplitude and phase of transmission coefficients are received for three types of lumped ports configuration for symmetric and asymmetric IC patterns. The comparison of accuracy between proposed variants of de-embedding techniques for the guided microstrip line is done for the attenuation and phase constant, as well as for the extracted characteristic impedance. A good agreement between the calculation and measurement results was obtained for the above characteristics. Summarizing the obtained results on the whole, one can conclude that the proposed de-embedded techniques on the base of numerical simulation and HFSS solver is effective instrument for investigation of transmission parameters of a wide class of RF circuits on chip.

Saadeddine Khemissi studies the conductive characteristics of the GaAs metalsemiconductor field effect transistors (MESFETs) within the framework of the elaboration of analytical model and numerical simulation in **Chapter 12**. Firstly, the respective Poisson's equation supplemented by proper boundary conditions is considered. The boundary conditions deciding the connections between the intrinsic gate-source, drain-source voltages, and electric field in the depletion-layer edges are specified in the second step of research. The Green's function approach and superposition technique are applied for extension of the initial model to the 2D one. The resulting depletion layer parameters are determined as a combination of initial values from 1D model and total tension function. Calculation of conductive characteristics consists of determination of the drain current for linear, pinch-off, and saturation regimes. In order to calculate the extrinsic characteristics, the effects of parasitic resistance are taken into account. Since they require the calculation of drain currents, which are also contained in the first terms of respective formulas, the iterative procedure for calculation of the drain currents is proposed.

In **Chapter 13**, the comparative study of the Diamond SOI MOSFETs and the conventional ones is performed by Salvador Pinillos Gimenez and Marcello Bellodi using the numerical simulation. The ATLAS device simulator is applied for the comprehensive three-dimensional modeling of the devices under consideration. Lombardi's vertical and horizontal electric-field-depend mobility model which takes into account the effects of low and high electric field is a theoretical background of the proposed approach. It is substantiated that the Diamond SOI nMOSFETs have more than two times larger drain current than conventional ICs, what can be used for increasing the fun-out and fun-in capability of the logic gates. The better performances regarding the transconductance (two times and more) can be used for improving the voltage gain and unit voltage gain frequency of the amplifiers used in the analog ICs. Comparative study of the electrical and physical behavior in the wide temperature range demonstrates also the advantages of the Diamond nMOSFETs. The results reported in Chapter give the important information regarding the electrical characterization of the novel Diamond SOI MOSFETs and comprehensive understanding of their electrical and physical internal behaviors.

Preface XV

The evolutionary processes caused by water wave propagation in the sea environment, the problems of instability of water waves, the fluid turbulent flows, and

In **Chapter 16**, the numerical model for prediction of beach changes in the various conditions of shoreline is elaborated and compared with experimental studies by Takaaki Uda et al. The proposed model is used for investigation of extension of sand spit on shallow seabed and formation of a cuspate foreland on step coast, a tidal flat facing on inland sea, and formation of slender sand bar in the case of depth seabed. The comprehensive numerical experiments regarding formation of barrier island on flat shallow seabed and formation of cuspate foreland on steep coast are carried out. The experiment foresees prescribing the initial data the adequate to the field conditions, accurate study of bathymetric phenomena, investigation of influence of change in wave field and sand transport flux. The results on change in longitudinal profiles of areas near shoreline present a specific practical importance. The numerical simulation is performed also for the case of formation of a slender sand bar in the condition of real coast. The results of Chapter are applicable to prediction of changes

Julien Touboul and Christian Kharif discuss the problem of propagation of the wave trains caused by the wind and dissipation effects in **Chapter 17**. The theoretical background of investigation is based on the potential flow theory, but new nonpotential effects caused by wind and viscosity of water waves can be taken into account through modification of the boundary conditions at the surface of waves. More exact, fully nonlinear approach upcoming in the framework of two-dimensional flows and based on the high-order spectral method is used to solve numerically the governing partial differential equations. In order to investigate the long time behavior of periodic water waves characterizing by the modulation instability, a special treatment is engaged. The main computational problem due to the instability phenomenon is calculation of vertical velocity of propagating waves. Prescribing the initial conditions for the propagating wave's trains is important for receiving the stable or non-stable solution. The numerical results testify that the propagation waves under action of wind and viscosity feature can not be considered likely a stationary and

Study and clarification of the physical properties of the compressible turbulent flows under rapid distortion theory consideration is the aim of **Chapter 18**, presented by Riahi Mohamed and Lili Taieb. The mathematical model of the physical processes consists of a set of equations describing the linear compressible flows in the absence of inertial effects. The liquid flow is interpreted as a homogeneous compressive turbulent shear flow described by a set of conventional equations. Using the Fourier transform, authors reduce the initial problem to a set of equations in the respective spectral space. The initial velocity components are determined by integration of additional double correlations which are contained in the velocity components. It is substantiated that

emulsification processes are simulated in **Section 3**.

in shoreline and creating the marine nature-conservative objects.

periodic.

Comparison of the output characteristics of the passively Q-switched solid state lasers obtained by the numerical simulation, based on the coupled differential equations system for photon distribution function, and the measurement data, is performed in **Chapter 14** authored by Ion Lancranjan et al. It is established that the accurate definition of photon density and population inversion constitutes a key role in the passive optical Q-switching solid state laser numerical simulation. The numerical simulation and comparison with experimental results are carried out for rod laser and slab laser oscillator/amplifier. For the first example, the good coincidence for the full width and half maximum characteristic, as well as the measured power density and predicated photon density curve is observed. In the case of slab laser oscillator/amplifier, a wide set of the output parameters was simulated and compared with experimental results for four types of prism samples. The analysis of the difference between the first passively Q-switched laser pulse and the next one, generated by a continuous and quasi-continuous wave pumped laser system, regarding the recovery condition to initial state of laser system, is reported firstly.

Jean-Luc Autran et al. in **Chapter 15** suggest a complete general purpose simulation platform for the numerical evaluation of the sensitivity of advanced semiconductor memories, such as static RAMs, subjected to natural radiation at ground level. It is substantiated that the atmospheric radiation environment and the telluric radiation sources are two major sources which cause soft error in the digital circuits. The structure of the TIARA-G4 Monte Carlo simulation code applied to the soft error rate evaluation of different SRAM CMOS built circuits of various technologies is discussed and explained from the utilization point of view. The simulation results illustrate the capabilities of the TIARA-G4 code trough studies on the simulation of 65 and 40 nm CMOS bulk SRAM circuits, subjected to different sources of atmospheric particles. The impact of thermal and low energy neutrons, the soft error rate estimation under high energy atmospheric neutrons, and effects of low-energy muons are investigated and various possible scenarios are discussed. In conclusion, it is accentuated that the novel TIARA-G4 simulation tool provides the researchers with the possibility of the comprehensive numerical evaluation of the sensitivity of advanced semiconductor memories, such as static RAMs.

The evolutionary processes caused by water wave propagation in the sea environment, the problems of instability of water waves, the fluid turbulent flows, and emulsification processes are simulated in **Section 3**.

XIV Preface

into account the effects of low and high electric field is a theoretical background of the proposed approach. It is substantiated that the Diamond SOI nMOSFETs have more than two times larger drain current than conventional ICs, what can be used for increasing the fun-out and fun-in capability of the logic gates. The better performances regarding the transconductance (two times and more) can be used for improving the voltage gain and unit voltage gain frequency of the amplifiers used in the analog ICs. Comparative study of the electrical and physical behavior in the wide temperature range demonstrates also the advantages of the Diamond nMOSFETs. The results reported in Chapter give the important information regarding the electrical characterization of the novel Diamond SOI MOSFETs and comprehensive

Comparison of the output characteristics of the passively Q-switched solid state lasers obtained by the numerical simulation, based on the coupled differential equations system for photon distribution function, and the measurement data, is performed in **Chapter 14** authored by Ion Lancranjan et al. It is established that the accurate definition of photon density and population inversion constitutes a key role in the passive optical Q-switching solid state laser numerical simulation. The numerical simulation and comparison with experimental results are carried out for rod laser and slab laser oscillator/amplifier. For the first example, the good coincidence for the full width and half maximum characteristic, as well as the measured power density and predicated photon density curve is observed. In the case of slab laser oscillator/amplifier, a wide set of the output parameters was simulated and compared with experimental results for four types of prism samples. The analysis of the difference between the first passively Q-switched laser pulse and the next one, generated by a continuous and quasi-continuous wave pumped laser system, regarding the recovery condition to initial state of laser system, is reported firstly.

Jean-Luc Autran et al. in **Chapter 15** suggest a complete general purpose simulation platform for the numerical evaluation of the sensitivity of advanced semiconductor memories, such as static RAMs, subjected to natural radiation at ground level. It is substantiated that the atmospheric radiation environment and the telluric radiation sources are two major sources which cause soft error in the digital circuits. The structure of the TIARA-G4 Monte Carlo simulation code applied to the soft error rate evaluation of different SRAM CMOS built circuits of various technologies is discussed and explained from the utilization point of view. The simulation results illustrate the capabilities of the TIARA-G4 code trough studies on the simulation of 65 and 40 nm CMOS bulk SRAM circuits, subjected to different sources of atmospheric particles. The impact of thermal and low energy neutrons, the soft error rate estimation under high energy atmospheric neutrons, and effects of low-energy muons are investigated and various possible scenarios are discussed. In conclusion, it is accentuated that the novel TIARA-G4 simulation tool provides the researchers with the possibility of the comprehensive numerical evaluation of the sensitivity of advanced semiconductor

memories, such as static RAMs.

understanding of their electrical and physical internal behaviors.

In **Chapter 16**, the numerical model for prediction of beach changes in the various conditions of shoreline is elaborated and compared with experimental studies by Takaaki Uda et al. The proposed model is used for investigation of extension of sand spit on shallow seabed and formation of a cuspate foreland on step coast, a tidal flat facing on inland sea, and formation of slender sand bar in the case of depth seabed. The comprehensive numerical experiments regarding formation of barrier island on flat shallow seabed and formation of cuspate foreland on steep coast are carried out. The experiment foresees prescribing the initial data the adequate to the field conditions, accurate study of bathymetric phenomena, investigation of influence of change in wave field and sand transport flux. The results on change in longitudinal profiles of areas near shoreline present a specific practical importance. The numerical simulation is performed also for the case of formation of a slender sand bar in the condition of real coast. The results of Chapter are applicable to prediction of changes in shoreline and creating the marine nature-conservative objects.

Julien Touboul and Christian Kharif discuss the problem of propagation of the wave trains caused by the wind and dissipation effects in **Chapter 17**. The theoretical background of investigation is based on the potential flow theory, but new nonpotential effects caused by wind and viscosity of water waves can be taken into account through modification of the boundary conditions at the surface of waves. More exact, fully nonlinear approach upcoming in the framework of two-dimensional flows and based on the high-order spectral method is used to solve numerically the governing partial differential equations. In order to investigate the long time behavior of periodic water waves characterizing by the modulation instability, a special treatment is engaged. The main computational problem due to the instability phenomenon is calculation of vertical velocity of propagating waves. Prescribing the initial conditions for the propagating wave's trains is important for receiving the stable or non-stable solution. The numerical results testify that the propagation waves under action of wind and viscosity feature can not be considered likely a stationary and periodic.

Study and clarification of the physical properties of the compressible turbulent flows under rapid distortion theory consideration is the aim of **Chapter 18**, presented by Riahi Mohamed and Lili Taieb. The mathematical model of the physical processes consists of a set of equations describing the linear compressible flows in the absence of inertial effects. The liquid flow is interpreted as a homogeneous compressive turbulent shear flow described by a set of conventional equations. Using the Fourier transform, authors reduce the initial problem to a set of equations in the respective spectral space. The initial velocity components are determined by integration of additional double correlations which are contained in the velocity components. It is substantiated that growth of initial gradient Mach number yields the different regimes of flow. The numerical results explain the physical effects arising in the process of the compressible turbulent flow propagation. Determination of dominant parameter in the turbulent kinetic energy budget is very important for better understanding the physics of flow. The limit values of Reynolds stress anisotropy tensor providing with the equilibrium state of homogeneous compressible turbulence are assessed.

Preface XVII

heat capacity characteristics, thermal conductivity, enthalpy balance, and adiabatic temperature of reaction are applied for rigorous description of process in whole. The numerical discretization scheme requires very precise arrangement because of the complexity of the mathematical model. The questions of dependence of the combustion front propagation, contribution of furnace temperature, ratio of released/absorbed energy by the boundary conditions, contribution of the cut-off temperature to the ignition and propagation, as well as analysis of double-front propagation are studied for a large set of the initial parameters of problem. The extension to 2D and 3D models provides with data corresponding to specific TiC components. The essential peculiarity of the proposed approach is the fact that it can be successfully extended to study of other

The process of combustion in porous media is studied numerically by Arash Mohammadi and Ali Jazayeri in **Chapter 22**. The background of combustion, including stability of flame, premixed and non-premixed mixture, reciprocating flow, hydrogen production, and the materials for porous media combustion, are introduced. The numerical step-by-step modeling foresees consideration of governing equations for the one-, two-, and three-dimensional models. The last model is considered for the cases of simple turbulence and radiation models, as well as for more complicate combustion model. The latter foresees the consideration of the chemical mechanisms: continuity, gas phase momentum, and gas phase energy equations, as well as the effects of turbulence. The finite volume equation formulation is applied for discretization of problem, the algorithms of discretization are explained in details. The numerical simulations are carried out on the example of thin cylinder domain. The distribution of gas and solid fractions, gas phase temperature, and temperature of solid phase are investigated in the temporal range. Obtained numerical results are explained from the physical point of view and they confirm a series of the known

In **Chapter 23**, the numerical simulation of slab broadening in a continuous casting of steel, validated by a series of experimental and factory results, is described by Jian-Xun Fu. The slab broadening in the process of casting is considered in both terms of the ratio of apparent shrinkage and the ratio of ultimate broadening. It is substantiated that the mechanical stresses, including the bending stress, straightening stress, rollermiss-alignment stress, the stress of rollers acting on the slab, and the static pressure of molten steel, determine the degree of broadening. An influence of the factory parameters, such as casting speed, width and thickness of slab, change of mold size, contraction of roll gap in the process of broadening, is carefully investigated and compared with the specific industrial data. The industrial experiments related to investigation of influence of the static pressure of molten steel on the broadening were performed and validated the proposed approach. General conclusion, that the higher casting speed, the lower intensity of secondary cooling, the thinner slab shell, the larger static pressure of molten steel, and the lower hardness of steel at high

ceramic materials, such as silicium carbide (SiC), and etc.

chemical and physical phenomena of combustion.

temperature increase slab broadening, summarizes the Chapter.

The phenomenon of droplet dynamics in membrane emulsification is studied in **Chapter 19** by Manabendra Pathak in order to determine the optimal geometrical parameters and physical properties for obtaining the stable condition of process. The conservation mass equation leading to continuity equation, the momentum or unsteady Navier-Stokes equation are specified in the framework of the volume of fluid method; the respective non-slip and impermeability boundary conditions are applied in order to rigorously describe the existing physical processes. Because of a large set of initial variables, the governing equations have been reduced in the dimensionless form. Two non-dimensional numbers (Reynolds' and Weber's ones) influence the emulsification process to the considerable extent. The characteristics of the growth and detachment of the droplets, effects of dispersed phase flow rate, velocity fields during droplets growth, effects of surface tension are investigated and optimal parameters of emulsification process are determined. It is substantiated that three important factors must be considered in order to obtain a high production rate of membrane emulsification, namely, a proper combination of continuous and dispersed phases, a proper distribution of pores in ground membrane, and sufficient cross-flow velocity.

**In Section 4**, the numerical modeling and computational approach are used for study of some industrial problems in the energetic, metallurgy and building.

In **Chapter 20**, Dumitru Toader et al. discuss the application of the numerical modeling for study of the transients in medium voltage networks. The mathematic model of electric circuits is based on the state-variable method and consists of a system of algebraic and differential equations. The first system is a result of applying the Kirchhoff's law to the circuit's topology, and the second one represents the temporal change of the circuit currents and voltages. As a result, the state-variable equations are obtained by elimination of the algebraic equations in the initial equation system. To obtain the transient characteristics, the conventional PSPICE tool is used. Simple ground and double grounding faults are analyzed for the different values of initial phase of voltage. It is concluded that the values of the initial phase of voltage does not exert a sufficient influence in the case of single phase-to-ground faults in the overcompensated and resonance regimes, but in the case of the double phase-toground faults this value influences the transient characteristics too much.

A. Aoufi and G. Damamme in **Chapter 21** deal with the numerical analysis of the processes of ignition and propagation of TiC combustion, described by the interconnected set of equations. The mathematical modeling takes into account the interaction between the all essential phenomena, namely, the equations describing the heat capacity characteristics, thermal conductivity, enthalpy balance, and adiabatic temperature of reaction are applied for rigorous description of process in whole. The numerical discretization scheme requires very precise arrangement because of the complexity of the mathematical model. The questions of dependence of the combustion front propagation, contribution of furnace temperature, ratio of released/absorbed energy by the boundary conditions, contribution of the cut-off temperature to the ignition and propagation, as well as analysis of double-front propagation are studied for a large set of the initial parameters of problem. The extension to 2D and 3D models provides with data corresponding to specific TiC components. The essential peculiarity of the proposed approach is the fact that it can be successfully extended to study of other ceramic materials, such as silicium carbide (SiC), and etc.

XVI Preface

growth of initial gradient Mach number yields the different regimes of flow. The numerical results explain the physical effects arising in the process of the compressible turbulent flow propagation. Determination of dominant parameter in the turbulent kinetic energy budget is very important for better understanding the physics of flow. The limit values of Reynolds stress anisotropy tensor providing with the equilibrium

The phenomenon of droplet dynamics in membrane emulsification is studied in **Chapter 19** by Manabendra Pathak in order to determine the optimal geometrical parameters and physical properties for obtaining the stable condition of process. The conservation mass equation leading to continuity equation, the momentum or unsteady Navier-Stokes equation are specified in the framework of the volume of fluid method; the respective non-slip and impermeability boundary conditions are applied in order to rigorously describe the existing physical processes. Because of a large set of initial variables, the governing equations have been reduced in the dimensionless form. Two non-dimensional numbers (Reynolds' and Weber's ones) influence the emulsification process to the considerable extent. The characteristics of the growth and detachment of the droplets, effects of dispersed phase flow rate, velocity fields during droplets growth, effects of surface tension are investigated and optimal parameters of emulsification process are determined. It is substantiated that three important factors must be considered in order to obtain a high production rate of membrane emulsification, namely, a proper combination of continuous and dispersed phases, a proper distribution of pores in ground membrane, and sufficient cross-flow velocity.

**In Section 4**, the numerical modeling and computational approach are used for study

In **Chapter 20**, Dumitru Toader et al. discuss the application of the numerical modeling for study of the transients in medium voltage networks. The mathematic model of electric circuits is based on the state-variable method and consists of a system of algebraic and differential equations. The first system is a result of applying the Kirchhoff's law to the circuit's topology, and the second one represents the temporal change of the circuit currents and voltages. As a result, the state-variable equations are obtained by elimination of the algebraic equations in the initial equation system. To obtain the transient characteristics, the conventional PSPICE tool is used. Simple ground and double grounding faults are analyzed for the different values of initial phase of voltage. It is concluded that the values of the initial phase of voltage does not exert a sufficient influence in the case of single phase-to-ground faults in the overcompensated and resonance regimes, but in the case of the double phase-to-

of some industrial problems in the energetic, metallurgy and building.

ground faults this value influences the transient characteristics too much.

A. Aoufi and G. Damamme in **Chapter 21** deal with the numerical analysis of the processes of ignition and propagation of TiC combustion, described by the interconnected set of equations. The mathematical modeling takes into account the interaction between the all essential phenomena, namely, the equations describing the

state of homogeneous compressible turbulence are assessed.

The process of combustion in porous media is studied numerically by Arash Mohammadi and Ali Jazayeri in **Chapter 22**. The background of combustion, including stability of flame, premixed and non-premixed mixture, reciprocating flow, hydrogen production, and the materials for porous media combustion, are introduced. The numerical step-by-step modeling foresees consideration of governing equations for the one-, two-, and three-dimensional models. The last model is considered for the cases of simple turbulence and radiation models, as well as for more complicate combustion model. The latter foresees the consideration of the chemical mechanisms: continuity, gas phase momentum, and gas phase energy equations, as well as the effects of turbulence. The finite volume equation formulation is applied for discretization of problem, the algorithms of discretization are explained in details. The numerical simulations are carried out on the example of thin cylinder domain. The distribution of gas and solid fractions, gas phase temperature, and temperature of solid phase are investigated in the temporal range. Obtained numerical results are explained from the physical point of view and they confirm a series of the known chemical and physical phenomena of combustion.

In **Chapter 23**, the numerical simulation of slab broadening in a continuous casting of steel, validated by a series of experimental and factory results, is described by Jian-Xun Fu. The slab broadening in the process of casting is considered in both terms of the ratio of apparent shrinkage and the ratio of ultimate broadening. It is substantiated that the mechanical stresses, including the bending stress, straightening stress, rollermiss-alignment stress, the stress of rollers acting on the slab, and the static pressure of molten steel, determine the degree of broadening. An influence of the factory parameters, such as casting speed, width and thickness of slab, change of mold size, contraction of roll gap in the process of broadening, is carefully investigated and compared with the specific industrial data. The industrial experiments related to investigation of influence of the static pressure of molten steel on the broadening were performed and validated the proposed approach. General conclusion, that the higher casting speed, the lower intensity of secondary cooling, the thinner slab shell, the larger static pressure of molten steel, and the lower hardness of steel at high temperature increase slab broadening, summarizes the Chapter.

XXII Preface

Syahroni Nur and Hidayat Mas in **Chapter 24** use 3D finite element simulation for study of residual stresses and distortions depending on the various welding sequences. The investigation of the thermomechanical effects of various welding sequences on the T-joint fillet weld is carried out. Theoretical background of approach foresees the study of basic mechanism of welding residual stresses, including the residual stresses due to mismatch and uneven distribution of non-elastic strains, as well as the analysis of welding distortions upon residual stresses. The existing thermal, stress and strain phenomena are examined in details and are evaluated from the engineering point of view. In the first part of procedure, welding simulation deals with determination of the temperature distributions for the different welding sequences. The temperature characteristics and peak temperature are determined. Using the above data the distribution of residual stresses and change of the welding structure are examined in the second part of investigations. The optimal welding sequences, resulting in the process with the smallest longitudinal and transverse residual stresses, as well as with the smallest angular distortion and difference, are revealed in the process of numerical simulation.

Hossein Jalalifar and Naj Aziz study the application of numerical simulation to prediction of the stressed-deformed state of composite materials in **Chapter 25**. The mathematical model foresees consideration of fully grouted rock bolt medium with bolt, grout, rock, and two interfaces under axial and lateral loading. Authors discuss the advantages of existing models from the points of adequate modeling and necessary computational resources. The mathematical modeling deals with description of finite elements for concrete, grout, and bolt; the specific finite elements are applied for the different areas in order to precisely describe the stresses and strains in each of material's component. The special attention is paid to modeling the bolts under lateral loading and bolt modeling under axial loading. The first part of investigations is devoted to obtaining the strain and stress development for the materials with various conditions and features, namely the different strength rock and pre-tension loads were examined. The conditions, at which the materials start to crush, are substantiated in the second part of research.

I would like to thank all authors for their responsibility, patience, and improvement of their chapters in the process of preparing the book. In conclusion, I express my sincere thanks to Ms. Mariana Jozipovich, who finished this project, as well as to Mr. Marijan Polic, who started as publishing manager for this book, for their professional assistance during the publishing process.

> **Mykhaylo I. Andriychuk**  Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, NASU, Lviv, Ukraine

**Simulation Background and Advances** 

**Chapter 0**

**Chapter 1**

**Bifurcation Analysis and Its Applications**

manifold theorem, including periodic normal forms for periodic orbits.

that is varied is known as the "bifurcation parameter".

Continuous dynamical systems that involve differential equations mostly contain parameters. It can happen that a slight variation in a parameter can have significant impact on the solution. The main questions of interest in this chapter are: How to continue equilibria and periodic orbits of dynamical systems with respect to a parameter? How to compute stability boundaries of equilibria and limit cycles in the parameter space? How to predict qualitative changes in system's behavior (bifurcations) occurring at these equilibrium points? This chapter will also cover the classification of bifurcations in terms of equilibria and periodic orbits. Especially it will present the specific bifurcation called "Hopf bifurcation" which refers to the development of periodic orbits from stable equilibrium point, as a bifurcation parameter crosses a critical value. Since the theory of bifurcation from equilibria based on center manifold reduction and Poincaré-Normal forms, the direction of bifurcations for the mathematical models will also be explained using this theory. Finally, by introducing several software packages and numerical methods this chapter will also cover the techniques to determine and continue in some control parameters all local bifurcations of periodic orbits of dynamical systems and relevant normal form computations combined with the center

In general, in a dynamical system, a parameter is allowed to vary, then the differential system may change. An equilibrium can become unstable and a periodic solution may appear or a new stable equilibrium may appear making the previous equilibrium unstable. The value of parameter at which these changes occur is known as "bifurcation value" and the parameter

In this chapter, we also discuss several types of bifurcations, saddle node, transcritical, pitchfork and Hopf bifurcation. Among these types, we especially focus on Hopf bifurcation. The first three types of bifurcation occur in scalar and in systems of differential equations. The fourth type called Hopf bifurcation does not occur in scalar differential equations because this type of bifurcation involves a change to a periodic solution. Scalar autonomous

and reproduction in any medium, provided the original work is properly cited.

©2012 Çelik,licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

Additional information is available at the end of the chapter

Canan Çelik Karaaslanlı

http://dx.doi.org/10.5772/50075

**1. Introduction**
