Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires Part II. Thermal modeling

A numerical procedure to determine the temperature rise in aircraft tires under free rolling conditions is presented in this article. Energy dissipation from cyclic inelastic deformation is considered the main heat generation source. This modeling considers the deformation process of the tire to be a steady-state problem, where all concurrent cycles are assumed to be the same as the first. The inelastic energy is determined by imposing a phase lag between the strain and the stress fields. The phase lag is assumed to be frequency independent in the range of interest, in keeping with the experimental observations in aircraft tire materials. It is further assumed that the inelastic energy is completely converted into volumetric heat input for a transient thermal conduction analysis. A conduction model is described and results are compared against thermocouple data recorded by Clark and Dodge [1].     

Numerical evaluation of the temperature field of steady-state rolling tires

Rubber is the main component of pneumatic tires. The tire heating is caused by the hysteresis effects due to the deformation of the rubber during operation. Tire temperatures can depend on many factors, including tire geometry, inflation pressure, vehicle load and speed, road type and temperature and environmental conditions. The focus of this study is to develop a finite element approach to computationally evaluate the temperature field of a steady-state rolling tire. For simplicity, the tire is assumed to be composed of rubber and body-ply. The nonlinear mechanical behavior of the rubber is characterized by a Mooney–Rivlin model while the body-ply is assumed to be linear elastic material. The coupled effects of the inflation pressure and vehicle loading are investigated. The influences of body-ply stiffness are studied as well. The simulation results show that loading is the main factor to determine the temperature field. The stiffer body-ply causes less deformation of rubber and consequently decreases the temperature.     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Numerical analysis and modeling of large deformation and necking behavior of tensile specimens

The present paper is concerned with numerical simulations of large deformation and necking behavior of axisymmetric tensile specimens. In particular, an efficient framework for the numerical analysis of finite deformation behavior of elastic-rate-independent plastic problems is summarized which is based on a plastic predictor method. Furthermore, numerical modeling of conventional tensile tests as well as their finite deformation and localization phenomena are discussed in some detail, and the results will be compared to those obtained by simplified numerical simulations.     

Numerical investigation of Maxwell–Vlasov equations – Part I: Basic physics and algorithms

We present a numerical scheme to solve the fully 3D Maxwell–Vlasov equations. The aim is to investigate the interaction of a laser pulse with a plasma and the acceleration of electrons and ions. We choose the approach based on numerical particles to approximate the distribution function interpreted, according to the Klimontovich formalism, as a probability distribution function. The key points are a high order implicit scheme to approximate the space differential operators and a time evolution scheme of adequate accuracy.     

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.     

A new element for analyzing large deformation of thin Naghdi shell model. Part 1: Elastic

One of the best approaches for modeling large deformation of shells is the Cosserat surface. However, the finite-element implementation of this model suffers from membrane and shear locking, especially for very thin shells. The basic assumption of this theory is that the mid-surface of the shell is regarded as a Cosserat surface with one inextensible director. In this paper, it is shown that by constraining the director vector normal to the mid-surface, besides very good and accurate results, shear locking is also eliminated. This constraint is in fact a limiting analysis of the Cosserat theory in which Kirichhoff’s hypothesis is enforced. Numerical solution is performed using nine-node isoparametric element. The principal of virtual work is used to obtain the weak form of the governing differential equations and the material and geometric stiffness matrices are derived through a linearization process. The validity and the accuracy of the method are illustrated by numerical examples.     

Numerical investigation of Maxwell–Vlasov equations. Part II: Preliminary tests and validation

We show that with an eighth order scheme the dispersion relation is very accurately reproduced and the numerical Cherenkov effect is small so that a good isotropy is obtained for the phase velocity. The comparison with an exact solution for the Gaussian wave packet confirms the accuracy of the method. The evolution of particles in the plane wave field is considered and an analysis of the first integrals of motion confirm the accuracy of the characteristics of the distribution function.     

Adaptive numerical solution of thick plates using first‐order shear deformation theory. Part II: Adaptivity

This is the second in a pair of articles concerned with the adaptive finite element solution of Riessner‐Mindlin thick plates modeled using first‐order shear deformation theory. This article is concerned with enhancing the a posteriori energy‐error estimators developed in Part I in order to accomodate transition elements in the finite element mesh. The resulting estimators are then used in an adaptive finite element model employing transition elements and the subsequent results discussed and compared with those in Part I. A major part of the article is devoted to identifying a novel patch assembly node algorithm for using the ZZ recovery‐type estimator with transition elements. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 227–253, 2003.     

Upper bound of the constant in the strengthened C.B.S. inequality for FEM 2D elasticity equations

The basic theory of the strengthened Cauchy–Buniakowskii–Schwarz (C.B.S.) inequality is the main tool in the convergence analysis of the recently proposed algebraic multilevel iterative methods. An upper bound of the constant γ in the strengthened C.B.S. inequality for the case of the finite element solution of 2D elasticity problems is obtained. It is assumed that linear triangle finite elements are used, the initial mesh consisting of right isosceles triangles and the mesh refinement procedure being uniform. For the resulting linear algebraic systems we have proved that γ²<0.75 uniformly on the mesh parameter and on Poisson's ratio ν ? (0, 1/2). Furthermore, the presented numerical tests show that the same relation holds for arbitrary initial right triangulations, even in the case of degeneracy of triangles. The theoretical results obtained are practically important for successful implementation of the finite element method to large-scale modeling of complicated structures. They allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real–life elasticity problems.     

Numerical analysis for factorial data analysis. Part I: Numerical software — the package INDA for microcomputers

A large collection of factorial data analysis methods can be characterized by the following matrices: X , the k x n matrix of data, and A, B the symmetric positive definite matrices of size n, k which represent the chosen norms of ?ⁿ, ?^k, respectively. All methods amount to computing the largest eigenvalues of U = XAX^TB or the largest singular values of E = B^1/2XA^1/2 . In Part I of this paper we begin by a geometrical and probabilistic interpretation of the various methods, showing how U and E are defined in each case. We then define the computational kernel for factorial data analysis. We conclude by devising the numerical aspects of software implementation for this kernel on microcomputers and presenting the package INDA.     

Numerical modeling of hydrodynamic and hydrothermal characteristics in subtropical alpine lake

A three-dimensional, time-dependent hydrodynamic and hydrothermal model was performed and applied to the subtropical alpine Yuan-Yang Lake (YYL) in northeastern region of Taiwan. The model was driven with discharge inflow, heat, and wind stress to simulate the hydrodynamic and hydrothermal in the lake. The model was validated with measured water surface elevation, current, and temperature in 2008. The overall model simulation results are in quantitative agreement with the available field data. The validated model was then used to investigate wind-driven current, mean circulation, and residence time in the YYL. The modeling results reveal that the velocity field along the wind axis present the variations over depth with return current where the velocity at the surface layer is along the wind direction while it is opposite near 1 m below water surface. The simulated mean current indicates that the surface currents flow towards the southwest direction and form a clock-wise rotation. The calculated residence time is strongly dependent on the inflows and wind effects. Regression analysis of model results reveals that an exponential regression equation can be employed to correlate the residence time to change of discharge input. The residence time without wind stress is higher than that with wind effect, indicating that wind plays an important role in lake mixing. The calculated residence time is approximately 2-2.5 days under low inflow with wind effect.     

Numerical study of dynamic thermal conductivity of nanofluid in the forced convective heat transfer

In this work, forced convective heat transfer of nanofluid in the developing laminar flow (entrance region) in a circular tube is considered. The nanofluid thermal conductivity, as an important parameter, is considered as two parts: static and dynamic part. Simulated results show that the dynamic part of nanofluid thermal conductivity due to the Brownian motion has a minor effect on the heat transfer coefficients, on the other hand, static part of thermal conductivity including nanolayer around nanoparticle has an important role in heat transfer.     

Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation

Complex analytical structure of Stokes wave for two‐dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave prop agating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with radians angle on the crest. A conformal map is used that maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half‐plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half‐plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase in the numerical precision and subsequent increase in the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one square‐root branch point per horizontal spatial period λ of Stokes wave located at the distance from the real line. The increase in the scaled wave height from the linear limit to the critical value marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called the Stokes wave of the greatest height). Here, H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10^?26. The number of poles in tables increases from a few for near‐linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with      

A new computer approach to the modeling of atoms,ions and molecules: Part I — Hydrogen

A computer approach to atomic and molecular dynamics is developed which utilizes known experimental values of ionization energies. In this first paper, applications are made to the hydrogen atom, the hydrogen ion, and the hydrogen molecule. A technique for the approximation of bond lengths is developed and illustrative examples of electron motions and, where significant, proton motions are described and discussed.     

Adaptive numerical solution of thick plates using first‐order shear deformation theory. Part I: Error estimates

A posteriori error estimation employing both a residual based estimator and a recovery based estimator is discussed. Interest is focused upon the application to Reissner‐Mindlin type thick plates modeled using first‐order shear deformation theory, and our investigation is limited to uniform meshes of bilinear quadrilateral elements. Numerical results for selected test problems are presented for the resulting error estimators and discussed. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 44–66, 2003     

Characteristic features of the deformation process on creep and secondary creep of polymers under conditions of monaxial tensioning. Part I

Experiments carried out in the field of linear and nonlinear creep of a filled polyethylene and Teflon resin, including two complete cycles of creep and deformation recovery at monaxial tensioning, show that the bulk deformations (t) change nonmonotonically, and relationships computed by summation of terms containing a function of time cannot be used to approximate the deformation.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 399–405, May–June, 1974.     

Numerical modeling of the effect of heat and mass transfer in porous low-temperature heat insulation in composite material structures on the magnitude of stresses which develop

The stressed state of multilayer low-temperature heat insulation for a cryogenic fuel tank is considered. Account is taken of heat and mass transfer in foam plastic (the main heat insulation material) occurring at cryogenic temperatures. A method is developed for solving a set of differential equations and boundary conditions. Numerical studies of the main features of these processes are performed. It is established that below 200 K the stresses which arise in foam plastic markedly exceed the ultimate strength for this material. Stresses develop as a result of both a reduction in temperature and a drop in pressure in the foam plastic pores connected with material cooling. On the basis of the results obtained it is established that the combination of thermophysical processes which occur in foam plastic during cooling to cryogenic temperatures leads to changes in the stress-strained state of structure, which should be considered in planning aerospace technology.Scientific Research Institute of Applied Mathematics and Mechanics, Tomsk State University, Tomsk, Russia. Translated from Mekhanika Kompozitmykh Materialov, Vol. 33, No. 3, pp. 384–393, May–June, 1997.     

Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals

A new finite difference (FD) method, referred to as "Cartesian cut-stencil FD", is introduced to obtain the numerical solution of partial differential equations on any arbitrary irregular shaped domain. The 2^nd-order accurate two-dimensional Cartesian cut-stencil FD method utilizes a 5-point stencil and relies on the construction of a unique mapping of each physical stencil, rather than a cell, in any arbitrary domain to a generic uniform computational stencil. The treatment of boundary conditions and quantification of the solution accuracy using the local truncation error are discussed. Numerical solutions of the steady convection-diffusion equation on sample complex domains have been obtained and the results have been compared to exact solutions for manufactured partial differential equations (PDEs) and other numerical solutions.     

On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D (the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.