Numerical Analysis

An improved 4‐node quadrilateral assumed‐stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger–Reissner variational principle and the shape functions are formulated directly for the 4‐node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has 9 independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling and linear free vibration problems. The element is validated with standard test cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.

Efficient and optimal design of radar-based Advanced Driver Assistant Systems (ADAS) needs the evaluation of many different electromagnetic solutions for evaluating the impact of the radome on the electromagnetic wave propagation. Because of the very high frequency at which these devices operate, with the associated extremely small wavelength, very fine meshes are needed to accurately discretize the electromagnetic equations. Thus, the computational cost of each numerical solution for a given choice of the design or operation parameters, is high (CPU time consuming and needing significant computational resources) compromising the efficiency of standard optimization algorithms. In order to alleviate the just referred difficulties the present paper proposes an approach based on the use of reduced order modeling, in particular the construction of a parametric solution by employing a non-intrusive formulation of the Proper Generalized Decomposition, combined with a powerful phase-angle ...

SummaryComposite manufacturing processes usually proceed from preimpregnated preforms that are consolidated by simultaneously applying heat and pressure, so as to ensure a perfect contact compulsory for making molecular diffusion possible. However, in practice, the contact is rarely perfect. This results in a rough interface where air could remain entrapped, thus affecting the effective thermal conductivity. Moreover, the interfacial melted polymer is squeezed flowing in the rough gap created by the fibers located on the prepreg surfaces. Because of the typical dimensions of a composite prepreg, with thickness orders of magnitude smaller than its other in‐plane dimensions, and its surface roughness having a characteristic size orders of magnitude smaller than the prepreg thickness, high‐fidelity numerical simulations for elucidating the impact of surface and interface roughness remain today, despite the impressive advances in computational availabilities, unattainable. This work aim...

Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well-known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low-data limit. We provide examples on the performance of the technique in up to ten dimensions.

This paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems. It is shown that the Newton-Raphson (NR) method converges for any time step size, if the flux function is convex, concave, or linear, which is, in general, the case for CFD problems. In some problems, e.g., multiphase flow in porous media, the nonlinear flux function is S-shaped (not uniformly convex or concave); as a result, a standard NR iteration can diverge for large time steps, even if an implicit discretization scheme is used to solve the nonlinear system of equations. In practice, when such convergence difficulties are encountered, the current time step is cut, previous iterations are discarded, a smaller time step size is tried, and the NR process is repeated. The criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and/or NR iteration. Here, we propose a simple modification to the NR iteration scheme for conservation laws with S-shaped flux functions that converges for any time step size. The new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver. The proposed method can be implemented in an existing simulator, e.g., for CO 2 sequestration or reservoir flow modeling, quite easily. The numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media. The analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems.

A semianalytical model based on the method of eigenfunction expansions and domain decomposition is developed for Stokes shear flow over a grating composed of a periodic array of parallel slats, with finite slippage on solid surfaces and infinite slippage on the bottom of troughs mimicking a no-shear liquid-gas interface penetrating into the space between slats. The model gives the macroscopic slip lengths for flow parallel or normal to the slats in terms of the microscopic slip length of the liquid-solid interface, area fraction of the no-shear liquid-gas interface, and depth of the liquid-gas interface in the grooves. When the no-shear interface lies flat on the top of the slats, the macroscopic slip lengths are the maximum and can be estimated with reasonably good accuracy by simple formulas. However, the slip lengths, particularly the transverse one, are very sensitive to penetration of the no-shear interface into the grooves. They can be reduced by a large factor when the interface just slightly gets into the grooves. On comparing with some molecular-dynamics simulation measures, it is pointed out that the applied pressure, which has to be less than the capillary pressure in the superhydrophobic state, can be correlated with the penetration depth of the no-shear interface.

A Fourier-Chebyshev collocation spectral method is employed in this work to compute the Lagrangian drift or mass transport due to periodic surface pressure loading in a thin layer of non-Newtonian fluid mud, which is modeled as a power-law fluid. Because of the non-Newtonian rheology, these problems are nonlinear and must be solved numerically. On assuming that the solutions are of the same permanent waveform as the pressure loading, the governing equations are made time-independent by referring to a horizontal axis that moves at the same speed as the wave. The solutions are periodic in the horizontal direction, but are non-periodic in the vertical direction, and the computational domain is therefore discretized according to the Fourier-Chebyshev spectral collocation scheme. In this study, the spatial derivatives are computed with a differentiation matrix. In order to incorporate the boundary conditions, the matrix diagonalization technique is used to solve the matrix equation, and all the definite integrals in the vertical direction based on the collocation points are performed by the modified Clenshaw-Curtis quadrature rule. The developed method is applied to compute the first-and second-order motion of the mud. The comparison between the numerical results and the analytical solution in the Newtonian limit shows the good accuracy of the spectral method.

In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro‐differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.

A boundary element method (BEM) is utilized to find numerical solutions to boundary value problems of homogeneous media governed by as anisotropic-diffusion convection-reaction (DCR) equation. Some problems are considered. A FORTRAN script is developed for the computation of the solutions. The numerical solutions verify the validity of the analysis used to derive the boundary element method with accurate and consistent solutions. The computation shows that the BEM procedure elapses very efficient time in producing the solutions. In addition, results obtained for the considered examples show the effect of the anisotropy of the media on the solutions.

Supplier selection is one important decision factors in the supply chain management. Suppliers are necessary entities to any business, however wrong selection may affect the whole business processes; therefore the process of selecting suppliers is extremely important. The success of a supply chain is dependent on selection of better suppliers. Decision makers and managers always face challenges to select suppliers, for procurement of raw material and components for their manufacturing process. In this paper we develop a Mixed Integer Programming Model for Multi product-Multi supplier -Multi period inventory lot size problem (MMMILP) with supplier selection. The solution of the optimal mathematical model is obtained in terms of determining (i) which products should be ordered to which supplier. (ii) The optimal quantity to be ordered and the time of placing orders. We introduce budget constraint, storage capacity constraint and quantity discount approach. A numerical case study is so...

The goal of this paper is to discuss the role System Dynamics (SD) can play to enhance performance improvement in the public sector. It is remarked how SD can help decision makers to properly perceive the boundaries of the relevant system underlying observed phenomena. To this end, three real cases are analysed to show how SD can facilitate a better understanding of the relationships between the political and the organisational system in the public sector, and how to promptly attain efficiency and improve outcome, given the constraints that the institutional and political systems constitute. Copyright © 2010 John Wiley & Sons, Ltd.

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energyg. The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positiveh(the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameterhgoes to zero.

Modified Newton-Kantorovich method is developed to obtain an approximate solution for a system of nonlinear integral equations. The system of nonlinear integral equations is reduced to find the roots of nonlinear integral operator. This nonlinear integral operator is solved by the modified Newton-Kantorovich method with initial conditions and this procedure is continued by iteration method to find the unknown functions. The existence and uniqueness of the solutions of the system are also proven.

We present several ideas in direction of physical interpretation of qand f -oscillators as a nonlinear oscillators. First we show that an arbitrary one dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f -oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f -oscillator. By using dispersion relation for q-oscillator we solve the linear q-Schrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. Peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles we formulate the two circle theorem and construct solution as the q-periodic flow by non-symmetric q-calculus. Then we generalize this theorem to the flow in the wedge domain bounded by two arcs. This two circular-wedge theorem determines images of the flow by extension of q-calculus to two bases: the real one, corresponding to circular arcs and the complex one, with q as a primitive root of unity. As an application, the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f-oscillator is studied. Extending idea of q-oscillator to two bases with the golden ratio, we describe Fibonacci numbers as a special type of q-numbers with matrix Binet formula. We derive the corresponding golden quantum oscillator, nonlinear coherent states and Fock-Bargman representation. The spectrum of it satisfies the triple relations, while the energy levels relative difference approaches asymptotically to the golden ratio and has no classical limit.

On the basis of fifty morphological characters, including vegetative parts, flowers, fruits, seeds, pollen grains, and anatomical structure, a systematic study of 13 taxa belonging to genus Galium (Rubiaceae) from Egypt was conducted by means of numerical analysis. Four branches and clusters were distinguished. Representatives of these groups were clustered together according to characters with high factor loading in the principal coordinates analysis. The results showed congruence between the UPGMA clustering and principal coordinates analysis in suggesting four groups. There was some degree of similarity among the species of sect. Aparine (Kolgyda). The results indicated also that the sect. Leiogalium (G. mollugo) was a separate group, while Aparine (Kolgyda) was the most heterogeneous one.

The problem of combination between inertial sensors and CCD cameras is of paramount importance in various applications in robotics and autonomous navigation. In this paper we develop a totally geometric model for analysis of this problem, independently from a camera model and from the structure of the scene (landmarks etc.). This formulation can be used for data fusion in several inertial navigation problems. The estimation is then decoupled from the structure of the scene. We use it in the particular case of the estimation of the gyroscopes bias and we build a nonlinear observer which is easy to compute, provides an estimation of the biais, filters the image, and is by construction very robust to noise.

We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. We give an analogous expression for time-average events, which describe the steady-state behavior of reducible chains as well as the long-term behavior of irreducible chains.

As described in the classic works of Lee-Stewart and Short-Stewart, the numerical evaluation of linear stability of planar detonation waves is a computationally intensive problem of considerable interest in applications. Reexamining this problem from a modern numerical Evans function point of view, we derive a new algorithm for their stability analysis, related to a much older method of Erpenbeck, that, while equally simple and easy to implement as the standard method introduced by Lee-Stewart, appears to be potentially faster and more stable.

In Evans function computations of the spectra of asymptotically constant-coefficient linearized operators of large systems, a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices. In the case that the invariant subspace is spectrally separated from its complementary invariant subspace, we propose an efficient numerical implementation of a standard projection-based algorithm of Kato, for which the key step is the solution of an associated Sylvester problem. This may be recognized as the analytic cousin of a C k algorithm developed by Dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case. For a one-dimensional subspace, it reduces essentially to an algorithm of Bridges, Derks and Gottwald based on path-finding and continuation methods.

Arteriovenous graft (AVG) is artificially made with graft for hemodialysis in the patients with renal failure. Stenosis in the arterial or venous anastomosis of AVG results in its malfunction. Here, we made an AVG hemodynamic model with three different anastomotic angles (20˚, 30˚, 40˚) and analyzed hemodynamic parameters such as velocity vectors, WSS and OSI in the arterial and venous anastomosis to find what helps in developing new surgical techniques to reduce stenosis in the anastomosis. Recirculation flow, low WSS and high OSI in the venous anastomosis were demonstrated in 30˚ and 40˚ models, and recirculation flow, high WSS and high OSI in the arterial anastomosis were shown in all models. Conclusively, higher anastomosis angle in the venous anastomosis cause stenosis, but stenosis in the arterial anastomosis happens irregardless of anastomosis angle.

Reactive scalar mixing time scale have been investigated in direct numerical simulation data for turbulent premixed Bunsen flames with reduced methane-air chemistry. Previous conclusions from single step chemistry studies are confirmed regarding the role of dilatation and turbulence-chemistry interactions on the progress variable dissipation rate. Compared to the progress variable, the mixing rates of intermediate species can be several times greater. The variation of species mixing rates are explained with reference to the structure of one-dimensional flamelets. A new model is produced for the ratios of scalar mixing time scales which can be applied, for example, in transported probability density function simulations.

The phenomenological theory of continuous thickening of flocculated suspensions in an ideal cylindrical thickener is extended to vessels håving varying cross-section, including divergent or convergent conical vessels. The purpose of this contribution to draw attention to the corresponding mathematical model, whose key ingredient is a strongly degenerate parabolic partial differential equation. For ideal (non-floccuiated) suspensions, which do not form com pressible sediments, the mathematical model reduces to the kinematic approach by Anestis, who developed a method of construction of exact solution by the method of characteristics. The difficulty lies in the fact that characteristics and iso-concentration lines, unlike the conventional Kynch model for cylindrical vessels, do not coincide, and one has to resort to numerical methods to simulate the thickening process. A numerical algorithm is presented and employed for simu lations of continuous thickening. Implications of the mathematical model are also demonstrated by steady-state calculations, which lead to new possibilities in thickener design.

Continuously operated clarifier-thickener units can be modeled by a non-linear, scalar conservation law with a flux that involves two parameters that depend discontinuously on the space variable. This paper presents two numerical schemes for the solution of this equation that have formal second-order accuracy in both the time and space variable. One of the schemes is a standard total variation diminishing (TVD) method, while the other scheme, the so-called flux-TVD (FTVD) scheme, is based on the property that due to the presence of the discontinuous parameters, the flux of the solution (rather than the solution itself) has the TVD property. The FTVD property is enforced by a new nonlocal limiter algorithm. We prove that the FTVD scheme converges to a BVt solution of the conservation law with discontinuous flux. Numerical examples for both resulting schemes are presented. They produce comparable numerical errors, while the FTVD scheme is supported by convergence analysis. The accuracy of both schemes is superior to that of an available monotone first-order scheme. In the clarifier-thickener application there is interest in modelling sediment compressibility by an additional strongly degenerate diffusion term. Second-order schemes for this extended equation are obtained by combining either the TVD or the FTVD scheme with a Crank-Nicolson discretization of the degenerate diffusion term in a Strang-type operator splitting procedure. Numerical examples illustrate the resulting schemes.

Multiscale Hybrid-Mixed (MHM) finite element method have been recently developed for several operators, including hydro-dynamics and reaction-advection-diffussion models. The MHM method is a consequence of a hybridization procedure, and emerges as a method that naturally incorporates multiple scales while provides solutions with high-order precision. The computation of local problems is embedded in the upscaling procedure, which are completely independent and thus may be naturally obtained using parallel computation facilities. We conclude that the MHM method is naturally shaped to be used in parallel computing environments and appears to be a highly competitive option to handle realistic multiscale parabolic boundary value problems with precision on coarse meshes. Numerical experiments will also be shown in order to support the theoretical results.

A model is developed for predicting separation along interfaces of pressure sensitive adhesives. Many authors have used the cohesive zone approach to solve such problems but the parameter calibration of such models remains uncertain. This study reports a novel method for determining such parameters. In addition, it provides crucial evidence for the suitability of the cohesive zone model approach in modelling interface fractures. Peel tests were performed at various rates using specimens which consisted of a polyester backing membrane supporting an acrylic pressure-sensitive adhesive (PSA) adhered to a polyethylene substrate. Interfacial separation of the PSA from the polyethylene substrate was observed. Finite element (FE) peeling simulations were conducted which modeled the backing-membrane as an elasto-plastic power-law material, the adhesive as a viscoelastic material and the interfacial properties with a cohesive zone model (CZM). The material properties of the backing membrane and the pressure-sensitive adhesive were measured from tensile and stress relaxation experiments. The rate-dependent CZM parameters were measured directly from poker-chip probe-tack tests which were performed at pull-off speeds which corresponded to the rates employed for the peel tests. The effect of the PSA thickness and test rate on both tack and peel was investigated experimentally, as well as modeled numerically. Good agreement was found between the experimentally measured and numerically predicted peel forces for different peel angles, speeds and PSA thicknesses. In addition, it was proven that the rate dependence observed in the peel and probe-tack data was dominated by the rate dependence of the interface properties, i.e. the time dependence of the two CZM parameters of maximum stress and fracture energy, rather than the timedependent bulk viscoelasticity of the PSA peel arm.

Recently there have two different effective methods proposed by Kanzow et al. in [19] and [21], respectively, which commonly use the Fischer-Burmeister (FB) function to recast the mixed complementarity problem (MCP) as a constrained minimization problem and a nonlinear system of equations, respectively. They all remark that their algorithms may be improved if the FB function is replaced by other NCP functions. Accordingly, in this paper, we employ the generalized Fischer-Burmeister (GFB) where the 2-norm in the FB function is relaxed to a general p-norm (p > 1) for the two methods

Asymptotic and numerical methods are used to highlight different types of dynamical behaviors that occur for the motion of a localized spike-type solution to the singularly perturbed Gierer-Meinhardt and Schnakenberg reaction-diffusion models in a one-dimensional spatial domain. Depending on the parameter range in these models, there can either be a slow evolution of a spike toward the midpoint of the domain, a sudden oscillatory instability triggered by a Hopf bifurcation leading to an intricate temporal oscillation in the height of the spike, or a pulse-splitting instability leading to the creation of new spikes in the domain. Criteria for the onset of these oscillatory and pulse-splitting instabilities are obtained through asymptotic and numerical techniques. A moving-mesh numerical method is introduced to compute these different behaviors numerically, and results are compared with corresponding results computed using a method of lines based software package.

The paper describes a new approach used to simulate the vibration in high frequency (up to 4 kHz) of an hermetic compressor for domestic appliances. The described methodology has been applied to evaluate noise emission and optimize the compressor design at the first stage of its development. The approach can be applied to evaluate noise emission due to structural forces, considering contacts and impacts of mechanical components, and has been successfully applied to redesign a compressor with optimized noise emission.

The paper describes a model that can predict the noise emission of a compressor, taking into account both structure borne and air borne noise. Such a kind of model can be used to design shell and to uncouple sound sources and structural and cavity modes. The model was tested and validated on a production compressor. Each subsystem of the model has been validated by means of experimental modal analysis (shell and cavity). Finally the whole system was validated through measurement of the emitted sound power.

We present a new formulation of three-dimensional central finite volume methods on unstructured staggered grids for solving systems of hyperbolic equations. Based on the Lax-Friedrichs and Nessyahu-Tadmor one-dimensional central finite difference schemes, the numerical methods we propose involve a staggered grids in order to avoid solving Riemann problems at cell interfaces. The cells are barycentric, while those of the staggered grid are diamond-shaped. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in 3D, where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods and solve Euler equations. Our numerical results are in good agreement with corresponding results appearing in the literature.

MAURILO MONTEIRO TERRA ( 2,8 ), ERASMO JOSÉ PAIOLI PIRES ( 2 ). SÔNIA MARIA BONILHA MARCONDES COELHO ( 4 ) ILENE RIBEIRO DA SILVA PASSOS ( 2 ), RUI RIBEIRO DOS SANTOS ( 3 ), CELSO VALDEVINO POMMER ( 2,6 ), ANDRÉ CAMARGO PEREIRA DA SILVA ( 2 ) e IVAN JOSÉ ANTUNES RIBEIRO ( ) ( 1 ) Recebido para publicação em 19 de maio de 1989 e aceito em 17 de agosto da 1990.

In this paper we describe the syntax, semantics, and implementation of the constraint logic programming language CLP(F) and we prove that the implementation is sound. A CLP(F) constraint is a conjunction of equations and inequations in a first order theory of analytic univariate functions over the reals. The theory allows vector-valued functions over closed intervals to be constrained in several ways, including specifying functional equations (possibly involving the differentiation operator) that must hold at each point in the domain, arithmetic constraints on the value of a function at a particular point in its domain, and bounds on the range of a function over its domain. After describing the syntax and semantics of the constraint language for CLP(F) and giving several examples, we show how to convert these analytic constraints into a subclass of simpler functional constraints which involve neither differentiation nor evaluation of functions. We then present an algorithm for solving these latter constraints and prove that it is sound. This implies the soundness of the CLP(F) interpreter. We also provide some timing results from an implementation of CLP(F) based on GNU Prolog. The current implementation is able to solve a wide variety of analytic constraints, but on particular classes of constraints (such as initial value problems for autonomous ODES), it is not competitive with other non-constraint based, interval solvers such as Lohner's AWA system. CLP(F) should be viewed as a first step toward the long term goal of developing a practical, declarative, logic-based approach to numerical analysis.

W artykule przedstawiono główne zasady symulacji numerycznej nieliniowych drgań samowzbudnych obrabiarek przy toczeniu w układnie wielomodalnym o dwóch stopniach swobody. Uwzględniono zarówno wychodzenie narzędzia z materiału obrabianego, jak i przenikanie się powierzchni przyłoŜenia z powierzchnią skrawania. Opracowane wzory i algorytmy zaimplementowano w uniwersalnym i łatwym do rozbudowania o kolejne stopnie swobody programie komputerowym w środowisku LabVIEW. Dokładność uzyskiwanych symulacji zweryfikowano przez porównanie ich wyników z otrzymywanymi analitycznie bez nieliniowości.

New technologies and increased requirements for performances of digital systems require new mathematical theories and tools as a basis for future VLSI CAD systems. New or alternative mathematical approaches and concepts must be suitable to solve some concrete problems in VLSI and efficient algorithms for their efficient application should be provided. This paper is an attempt in this direction and relates with the recently renewed interest in arithmetic expressions for switching functions, instead representations in Boolean structures, and spectral techniques and differential operators in switching theory and applications. Logic derivatives are efficiently used in solving different tasks in logic design, as for example, fault detection, functional decomposition, detection of symmetries and co-symmetries of logic functions, etc. Their application is based on the property that by differential operators, we can measure the rate of change of a logic function. However, by logic derivativ...

Aux membres de mon jury, qui m'ont fait l'honneur de bien vouloir examiner mon travail. En premier lieu aux rapporteurs, pour l'attention bienveillante portée à mon mémoire. A Claude Marché, pour son soutien constant et actif dans la préparation de cette habilitation. A Eric Goubault, chef du laboratoire MEASI, qui a guidé mes premiers pas sur le sujet, et toujours encouragée depuis. Comment en quelques mots évoquer plus de 10 ans à bénéficier de son énergie et de sa curiosité scientifique? Plus généralement à mes collègues du CEA et de Digiteo; ce travail leur doit tout. Tout spécialement Karim Tekkal et Franck Védrine, grâce à qui l'aventure Fluctuat, avec ses hauts et ses bas, est restée un plaisir. A Khalil Ghorbal, mon premier étudiant en thèse, forcément unique.

A general model is presented for short-range hydrodynamic interactions and head-on particle-particle/wall collisions. The model has been embedded in two distinct numerical methods for fully resolved simulation of finite-size particles in a viscous fluid. It accounts for the material properties of the particles and lubrication effects prior to collision that cannot be fully resolved on a fixed grid. We demonstrate that the model is able to reproduce experimental data for the coefficient of restitution of particle-wall collisions over a wide range of Stokes number based on the particle impact velocity. The set of model parameters we selected and more generally the modelling approach we propose can be efficiently used for fully resolved simulations of moderately dense solid-liquid suspensions.

For any power equipment the control system and protection system plays an imperative role as the dependency on the power equipment in any industry will be very high. Thus for the Generator the Reactances and Time Constants becomes controlling parameter and hence protecting it is necessary for prolonged usage. In general reactance shall be defined as "non-resistive component of impedance in an AC circuit, arising from the effect of inductance or capacitance or both and causing the current to be out of phase with the electromotive force causing it". In other word it can also be defined as the imaginary part of the impedance in any power circuit. There are various Reactances and Time Constants which contributes during the selection of control systems and protection scheme for any Generator. Therefore practical evaluation of those parameters shall also be super critical same as those are by calculation. More accurate measurement or the calculation of reactance shall certainly give us more stabilized error free system and hence ensuring a healthy control system. As there are different tests or different methods to measure and calculate the Reactances and Time Constants, in this context we are considering Line to Line fault (L-L fault), Double line to Ground fault and Sudden three phase short circuit fault to determine the Negative Sequence reactance, Zero Sequence reactance, Sub-transient and Transient reactance and time constants respectively. The fault condition will be demonstrated with the 40MW, 11KV, 2624A, 4pole, 50 Hz synchronous generator and above mentioned parameters are calculated based on references available.