Bounds on the Volume Fraction of the Two-Phase Shallow Shell Using One Measurement

We study the size estimate problem for the two-phase shallow shell equations in this paper. Our aim is to derive bounds on the volume fraction of each phase assuming that the material properties of the two phases are given. The approach in this paper is based on the translation method. One of the key steps is to connect the shallow shell equations to the thin plate equation.     

On the momentum equations in dispersed two-phase systems

For so-called dispersed two-phase flow systems the equations of motion are derived for each phase separately. The equations are developed using a minimum of mathematics by applying mass and momentum balances respectively over a small cubic volume element. Special attention is paid to those particles which are cut by the boundaries of this control volume element. In fact these particles are treated by means of two methods, both methods giving the same final result.     

An efficient implicit mesh‐less method for compressible flow calculations

A dual‐time implicit mesh‐less scheme is presented for calculation of compressible inviscid flow equations. The Taylor series least‐square method is used for approximation of spatial derivatives at each node which leads to a central difference discretization. Several convergence acceleration techniques such as local time stepping, enthalpy damping and residual smoothing are adopted in this approach. The capabilities of the method are demonstrated by flow computations around single and multi‐element airfoils at subsonic, transonic and supersonic flow conditions. Results are presented which indicate good agreements with other reliable finite‐volume results. The computational time is considerably reduced when using the proposed mesh‐less method compared with the explicit mesh‐less and finite‐volume schemes using the same point distributions. Copyright © 2010 John Wiley & Sons, Ltd.     

Derivation of equations of state for ideal crystals of simple structure

We consider an approach to the derivation of thermodynamic equations of state by averaging the dynamic equations of particles of the crystal lattice. Microscopic analogs of macroscopic variables such as pressure, volume, and thermal energy are introduced. An analysis of the introduced variables together with the equations of motion permits obtaining the equation of state. Earlier, this approach was used to obtain the equation of state in the Mie-Grüneisen form for a one-dimensional lattice. The aim of this paper is to develop and generalize this approach to the three-dimensional case. As a result, we obtain the dependence of the Grüneisen function on the volume, which is compared with the computations performed according to well-known models with experimental data taken into account. It is proved that the Grüneisen coefficient substantially depends on the form of the strain state. Moreover, we refine the equation of state; namely, we show that the Grüneisen coefficient depends on the thermal energy, but this dependence in the three-dimensional case is much weaker than in the one-dimensional case. A refined equation of state containing a nonlinear dependence on the thermal energy is obtained     

A new approach to the group analysis of one-dimensional stochastic differential equations

Stochastic evolution equations are investigated using a new approach to the group analysis of stochastic differential equations. It is shown that the proposed approach reduces the problem of group analysis for this type of equations to the same problem of group analysis for evolution equations of special form without stochastic integrals.     

Imperfection sensitivity in the nonlinear vibration of functionally graded plates

In this paper, the nonlinear partial differential equations of nonlinear vibration for an imperfect functionally graded plate (FGP) in a general state of arbitrary initial stresses are presented. The derived equations include the effects of initial stresses and initial imperfections size. The material properties of a functionally graded plate are graded continuously in the direction of thickness. The variation of the properties follows a simple power-law distribution in terms of the volume fractions of the constituents. Using these derived governing equations, the nonlinear vibration of initially stressed FGPs with geometric imperfection was studied. Present approach employed perturbation technique, Galerkin method and Runge–Kutta method. The perturbation technique was used to derive the nonlinear governing equations. The equations of motion of the imperfect FGPs was obtained using Galerkin method and then solved by Runge–Kutta method. Numerical solutions are presented for the performances of perfect and imperfect FGPs. The nonlinear vibration of a simply supported ceramic/metal FGP was solved. It is found that the initial stress, geometric imperfection and volume fraction index greatly affect the behaviors of nonlinear vibration.     

Global and bifurcation analysis of a structure with cyclic symmetry

This article combines the application of a global analysis approach and the more classical continuation, bifurcation and stability analysis approach of a cyclic symmetric system. A solid disc with four blades, linearly coupled, but with an intrinsic non-linear cubic stiffness is at stake. Dynamic equations are turned into a set of non-linear algebraic equations using the harmonic balance method. Then periodic solutions are sought using a recursive application of a global analysis method for various pulsation values. This exhibits disconnected branches in both the free undamped case (non-linear normal modes, NNMs) and in a forced case which shows the link between NNMs and forced response. For each case, a full bifurcation diagram is provided and commented using tools devoted to continuation, bifurcation and stability analysis.     

Averaged topological equations for dispersed two-phase flows

A general relationship between the volume fraction and the specific interfacial area for averaged dispersed two-phase flows is proposed. This relationship, expressed as a basic set of two scalar evolution equations and two vectorial non-uniformity state equations, is an analytical result obtained by a systematic approach using the derivatives of some generalized functions and a local volume-averaging technique. The proposed set of equations was expressed for measurable macroscopic parameters of the system and has the same generality as the averaged transport equations of two-phase flows. By combination of the basic set of equations, called the averaged topological equations (ATEs), second-order ATEs for the volume fraction were found. The second-order ATEs were expressed both by a Lagrangian formulation and by a Eulerian formulation. The importance and physical meaning of the ATEs developed in this study were clarified within the framework of the theory of kinematic waves.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.     

Analysis and thermal optimization of an ecological ventilated self-weighted wood panel for roofs by FVM

This work is to study the performance of an ecological ventilated self-weighted wood panel used on roofs in order to get a high energy savings. With this aim, we have carried out a convective heat transfer analysis of the panel by the finite volume method (FVM). Pure conduction is found in the wood panel through their thermal properties. Heat transfer by convection is always accompanied by conduction, that is to say, among the external air and the upper internal surface of the panel, and the internal air and the inner lower surface of the panel taking into account the heat conduction of the internal ribs. The finite volume method (FVM) is a method for representing and evaluating partial differential equations as algebraic equations. Similar to the finite difference method, values are calculated at discrete places on a meshed geometry. ‘Finite volume’ refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. One advantage of the finite volume method over the finite difference method is that it does not require a structured mesh, although a structured mesh can be used. The FVM can solve problems on irregular geometries. Furthermore, one advantage of the finite volume method over the finite element method is that it can conserve the variables on a coarse mesh easily. This is an important characteristic for fluid problems just as in this case. Finally, conclusions of this study are exposed.     

A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.     

Linear and Nonlinear Analyses of the Onset of Buoyancy-Induced Instability in an Unbounded Porous Medium Saturated by Miscible Fluids

This study analyzes the stability of an initially sharp interface between two miscible fluids in a porous medium. Linear stability equations are first derived using the similarity variable of the basic state, and then transformed into a system of ordinary differential equations using a spectral expansion with and without quasi-steady-state approximation (QSSA). These transformed equations are solved using the eigenanalysis and initial value problem approach. The initial growth rate analysis shows that initially the system is unconditionally stable. The stability characteristics obtained under the present QSSA are quantitatively same as those obtained without the QSSA. To support these theoretical results, numerical simulations are conducted using the Fourier-spectral method. The results of theoretical linear stability analyses and the numerical simulations validate to each other.     

Mechanical buckling of a functionally graded cylindrical shell with axial and circumferential stiffeners using the third-order shear deformation theory

This paper deals with an analytical approach of the buckling behavior of a functionally graded circular cylindrical shell under axial pressure with external axial and circumferential stiffeners. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations and equilibrium and stability equations are derived using the third-order shear deformation theory. The resulting equations are employed to obtain the closed-form solution for the critical buckling loads. A simply supported boundary condition is considered for both edges of the shell. The comparison of the results of this study with those in the literature validates the present analysis. The effects of material composition (volume fraction exponent), of the number of stiffeners and of shell geometry parameters on the characteristics of the critical buckling load are described. The analytical results are compared and validated using the finite-element method. The results show that the inhomogeneity parameter, the geometry of the shell and the number of stiffeners considerably affect the critical buckling loads.     

A staggered control volume scheme for unstructured triangular grids

The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D/axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier–Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure–velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. © 1997 John Wiley & Sons, Ltd.     

A finite volume method for multicomponent gas transport in a porous fuel cell electrode

We present a mathematical model for multicomponent gas transport in an anisotropic fuel cell electrode.The model couples the Maxwell–Stefan equations for multicomponent diffusion along with Darcy's law for flow in a porous medium. The equations are discretized using a finite volume approach with the method of lines, and the resulting non‐linear system of differential equations is integrated in time using a stiff ODE solver. Numerical simulations are performed to validate the model and to investigate the effect of various parameters on fuel cell performance. Copyright © 2003 John Wiley & Sons, Ltd.     

Partitioned solution to fluid–structure interaction problem in application to free-surface flows

In this work we discuss a way to compute the impact of free-surface flow on nonlinear structures. The approach chosen relies on a partitioned strategy that allows us to solve the strongly coupled fluid–structure interaction problem. It is then possible to re-use the existing and validated strategy for each sub-problem. The structure is formulated in a Lagrangian way and solved by the finite element method. The free-surface flow approach considers a Volume-Of-Fluid (VOF) strategy formulated in an Arbitrary Lagrangian–Eulerian (ALE) framework, and the finite volume is used to discrete and solve this problem. The software coupling is ensured in an efficient way using the Communication Template Library (CTL). Numerical examples presented herein concern the 2D validation case but also 3D problems with a large number of equations to be solved.     

A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two‐dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi‐discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo‐time step requires the solution of a sparse linear system for the flow variables. In this study, a non‐overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson‐type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes. Copyright © 2001 John Wiley & Sons, Ltd.     

A finite element method study of Eulerian droplets impingement models

To compute droplet impingement on airfoils, an Eulerian model for air flows containing water droplets is proposed as an alternative to the traditional Lagrangian particle tracking approach. Appropriate boundary conditions are presented for the droplets equations, with a stability analysis of the solution near the airfoil surface. Several finite element formulations are proposed to solve the droplets equations, based on conservative and non‐conservative forms and using different stabilization terms. Numerical results on single and multi‐elements airfoils for droplets of mean volume diameter, as well as for a Langmuir distribution of diameters, are presented and validated against measured values. Copyright © 1999 John Wiley & Sons, Ltd.     

Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids

In this paper, based on three-dimensional linear generalized thermoelasticity, an exact analysis of free vibration of a simply supported homogeneous isotropic, thermally conducting, cylindrical panel with voids initially at uniform temperature and undeformed state has been presented. Three displacement potential functions are introduced for solving the equations of motion, heat conduction and volume fraction field. The purely transverse wave gets decoupled from rest of motion and is not affected by thermal and volume fraction (voids) fields. After expanding the displacement potentials, volume fraction and temperature functions with orthogonal series, the equations of the considered vibration problem are reduced to five-second order coupled ordinary differential equations whose formal solution can be expressed by using Bessel functions with complex arguments. The corresponding results for thermoelastic panel without voids, elastic panel with and without voids have been deduced as special cases from the present analysis. In order to illustrate the analytical results, the numerical solutions of various relations and equations have been obtained to compute the lowest frequency as function of different cylindrical panel parameters. The computer simulated results have been presented graphically.     

On the Computer Formulations of the Wheel/Rail Contact Problem

In this investigation, four nonlinear dynamic formulations that can be used in the analysis of the wheel/rail contact are presented, compared and their performance is evaluated. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail (constraint approach), while in the other two formulations the contact force is modeled using a compliant force element (elastic approach). The goal of the four formulations is to provide accurate nonlinear modeling of the contact between the wheel and the rail, which is crucial to the success of any computational algorithm used in the dynamic analysis of railroad vehicle systems. In the formulations based on the elastic approach, the wheel has six degrees of freedom with respect to the rail, and the normal contact forces are defined as function of the penetration using Hertzs contact theory or using assumed stiffness and damping coefficients. The first elastic method is based on a search for the contact locations using discrete nodal points. As previously presented in the literature, this method can lead to impulsive forces due to the abrupt change in the location of the contact point from one time step to the next. This difficulty is avoided in the second elastic approach in which the contact points are determined by solving a set of algebraic equations. In the formulations based on the constraint approach, on the other hand, the case of a non-conformal contact is assumed, and nonlinear kinematic contact constraint equations are used to impose the contact conditions at the position, velocity and acceleration levels. This approach leads to a model, in which the wheel has five degrees of freedom with respect to the rail. In the constraint approach, the wheel penetration and lift are not permitted, and the normal contact forces are calculated using the technique of Lagrange multipliers and the augmented form of the system dynamic equations. Two equivalent constraint formulations that employ two different solution procedures are discussed in this investigation. The first method leads to a larger system of equations by augmenting all the contact constraint equations to the dynamic equations of motion, while in the second method an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated. Numerical results are presented in order to examine the performance of various methods discussed in this study.