Boundary homogenization for a quasi-linear elliptic problem

We describe the asymptotic behaviour of the solution of a quasi-linear elliptic problem posed in a cylinder and with homogeneous Dirichlet boundary conditions imposed on narrow helicoidal strips ε-periodically distributed on the lateral boundary of this cylinder when the parameter ε goes to 0. We use epi-convergence arguments in order to establish the limit behaviour. We also compute the rate of convergence of the solution of the original problem to that of the limit one.     

Influence functions, integral formulas, and explicit solutions for thermoelastic spherical wedges

In this study, new exact Green’s functions and a new exact Green-type integral formula for a boundary value problem (BVP) in thermoelasticity for some spherical wedges with mixed homogeneous mechanical boundary conditions are derived. The thermoelastic displacements are subjected to a heat source applied in the inner points of the spherical wedges and to a mixed non-homogeneous boundary heat conditions. When the thermoelastic Green’s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by Dirac’s δ-function. All results are obtained in elementary functions that are formulated in a special theorem. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for spherical wedges also are included. In these particular BVPs, the thermoelastic displacements are subjected to a constant temperature (in the first particular BVP) or to a constant heat source (in the second particular BVP). In both BVPs, the constant temperature or the constant heat source is given on the segment of the radius of the quarter-space. On the boundary half-planes of the quarter-space zero temperature and zero heat flux are prescribed.     

On extra boundary condition in the stagnation point flow of a second grade fluid

The two dimensional stagnation point flow of a second grade fluid is considered. The flow is governed by a boundary value problem in which the order of differential equations is one more than the number of available boundary conditions. It is shown that without augmenting the boundary conditions at infinity it is possible to obtain a numerical solution of the problem for all values of K, where K is the dimensionless viscoelastic fluid parameter. The numerical results using the algorithm foreshadow an asymptotic behavior for large K. The asymptotic solution is derived up to terms of O(K⁻¹). Perturbation solutions are also obtained up to the terms of O(K²). Finally an approximate solution is developed, based on stretching of the independent variable and minimizing the residual of the differential equation in the least square sense. All these solutions are compared with the exact numerical solution and the appropriate conclusions are drawn.     

RBF meshless method for large deflection of thin plates with immovable edges

An efficient meshless formulation is presented for large deflection of thin plates with immovable edges. In this method, a fifth-order polynomial radial basis function (RBF) is used to approximate the solution variables. The governing equations are formulated in terms of the three displacement components u, v and w. The solution is obtained by satisfying three coupled partial differential equations and their boundary conditions inside the domain and over the boundary of the plate, respectively. The collocation procedure produces a system of coupled non-linear algebraic equations, which are solved using an incremental-iterative procedure. The numerical efficiency of the proposed method is illustrated through numerical examples.     

Analytical series solutions for three-dimensional supercritical flow over topography

An analytical series solution method for three-dimensional, supercritical flow over topography is presented. Steady, nonlinear solutions are calculated for a single layer of inviscid, constant-density fluid that flows irrotationally over an obstacle that varies significantly in the x-, y- and z-directions. Accurate series solutions for the free surface and a series of stream tubes throughout the flow region are calculated to demonstrate the three-dimensional properties of the problem. These solutions provide valuable insight into the three-dimensional interactions between the fluid and obstacle which is impossible to gain from any two-dimensional model. The model is described by a Laplacian free-boundary problem with fully nonlinear boundary conditions. The solution method consists of iteratively updating the location of the free surface (on top of the fluid) using a cost function which is derived from the Bernoulli equation. Root-mean-square errors in the boundary conditions are used as convergence criteria and a measure of the accuracy of the solution. This method has been used to solve the two-dimensional version of this problem in the past. Here, we detail the extensions required for three-dimensional flow.     

The elastostatic axisymmetric problem of a sphere containing a penny-shaped crack in a nonequatorial plane

The axisymmetric problem of a sphere containing a penny-shaped crack in a nonequitorial plane is solved with the use of Bousinesq stress functions. Two coordinate systems—oblate spheroidal for representing the crack surface and spherical polars for the spherical surface, translated along the z-axis with respect to each other—are used to satisfy boundary conditions. Integral representations and transformations of harmonic functions are used to relate stress functions in the two coordinate systems. This procedure-leads to a system of algebraic equations which is solved, for axisymmetric tractions on both the surfaces. Graphical results are presented for a specific loading case.     

Analytical solution of P3 approximation to radiative transfer equation for an infinite homogenous media and its validity

The P3 approximation of the spherical harmonics method for the radiative transfer equation whose homogenous solution is the spherical Bessel function is a nonlinear partial differential equation system. After breaking the spherical Bessel function into two exponential terms, we obtain its particular solution by the modified method of variation of parameters. The complete solution of this approximation is not uniquely determined by the Marshak or Mark boundary condition on the boundary (r = 0). Therefore, the constants of the full solution are determined by direct application of the energy conservation, and by the fact that the P3 approximation equals the P1 approximation when the absorption coefficient is much lower than the reduced scattering coefficient. With the analytic solution of the P3 approximation, the spherical harmonics method for the radiative transfer equation can predict accurately the optical property of a large absorption tissue with the ratio of the reduced scattering coefficient to absorption coefficient ranging over 3–10.     

Flow of a third grade fluid through a porous flat channel

The steady, laminar flow of a third grade fluid through a porous flat channel is considered, when the rate of injection of the fluid at one boundary is equal to the rate of suction at the other boundary. The flow is governed by a non-linear boundary value problem (BVP) in which the order of the differential equation is three, but only two boundary conditions are available. Two numerical schemes are developed to obtain the appropriate solution of the BVP. In the first scheme the dilemma is resolved by assuming that the solution is analytical in the neighborhood of K=0, where K is the non-dimensional viscoelastic fluid parameter. This scheme is practical to use only up to certain values of T, the third grade fluid parameter. The second scheme allows arbitrary values of T, but is restricted to small values of K and R, the cross-flow Reynolds number.A perturbation solution valid for small values of T is also derived. Finally two approximate solutions, based on Collatz’ iterative scheme, but with different starting trial solutions are obtained. A comparison is made of the results computed by using various methods and appropriate conclusions are drawn.     

Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory

A new higher order shear deformation theory for elastic composite/sandwich plates and shells is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The present theory accounts for an approximately parabolic distribution of the transverse shear strains through the shell thickness and tangential stress-free boundary conditions on the shell boundary surface. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved using Navier-type, closed form solutions. Static and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Shells and plates are subjected to bi-sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The accuracy of the present code is verified by comparing it with various available results in the literature.     

Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain

In the present article, a new method for the determination of the hardening law using the load displacement curve, F–h, of a spherical indentation test is developed. This method is based on the study of the error between an experimental indentation curve and a number of finite elements simulation curves. For the smaller values of these errors, the error distribution shape is a valley, which is defined with an analytic equation. Except for the fact that the identified hardening law is a Hollomon type, no assumption was made for the proposed identification method. A new representative strain of the spherical indentation, called “average representative strain,” ε _aR was defined in the proposed article. In the bottom of the valley, all the stress–strain curves that intersect at a point of abscissa ε _aR lead to very similar indentation curves. Thus, the average representative strain indicates the part of the hardening law that is the better identified from spherical indentation test. The results show that a unique material parameter set (yield stress σ _y, strain hardening exponent n) is identified when using a single spherical indentation curve. However, for the experimental cases, the experimental imprecision and the material heterogeneity lead to different indentation curves, which makes the uniqueness of solution impossible. Therefore, the identified solution is not a single curve but a domain that is called “solution domain” in the yield stress–work hardening exponent diagram, and “confidence domain” in the stress–strain diagram. The confidence domain gives clear answers to the question of uniqueness of the solution and on the sensitivity of the indentation test to the identified hardening laws parameters.     

A unified formulation and error estimation measure for the direct and the indirect boundary element methods in elasticity

In this paper, given a boundary value problem for a finite elastic body in two-dimensions, a problem of representing the solution by layers of point forces (F_j) and Somigliana dislocations (B_j) is considered in the infinite homogeneous body that contains the original finite body. In the boundary element method (BEM) solution, either the boundary displacement or traction component at each node is specified, but not both. This provides us a degree of freedom to arbitrarily specify the proportion of the densities F_j and B_j to be used in the direct and the indirect BEM formulations. The nature of the BEM solution errors is identified and a unified error estimation measure with a mesh refinement scheme for both formulations is proposed.     

Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method

In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale R_k in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of R_k are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale R_k can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.     

Interfacial Balance Equations for Diffusion Evaporation and Exact Solution for Weightless Drop

Introducing of additional terms into the balance equations to specify the conditions at the interface allows to study physical phenomena in the diffusion evaporation (condensation) of the liquid into the neutral gas. We have taken into account the vapour dynamic effects on evaporating liquid, as well as the waste of energy on deformation of the boundary, changing of the interfacial temperature (the interface has an internal energy and therefore heat capacity), to overcome the surface tension etc. This paper presents the balance conditions at the interface with the diffusion evaporation of the liquid into the neutral gas, for the case when the vapour is considered as an impurity in the gas phase. The analysis of the dimensionless criteria is carried out. The areas of parameters for which the effect of some physical factors take a place have been defined. The exact solution of the diffusion evaporation for a spherical drop at zero gravity conditions has been constructed. The explicit expression for the interfacial temperature and evaporation rate were derived. Solution for evaporation rate coincides with the solution obtained by Maxwell (1890).     

Time dependent model of gain saturation in microchannel plates and channel electron multipliers

We present and discuss a time dependent solution of the transmission line model of a channel electron multiplier, introduced in a previous paper and already solved in steady-state conditions. The model is applicable to all the situations in which the multiplier input current is sufficiently large so that the statistical variations of the gain for each electron can be ignored and it does not apply to photon counting detectors. By introducing the appropriate boundary conditions the time dependent non-linear equations of the model are reduced to an integral equation in implicit form, whose solution can be calculated numerically by a perturbative approach. In this way the multiplying current signal i(z,t) and the voltage V(z,t) are found as functions of the position z along the channel, and of the time t during the pulse itself, for any arbitrary shape of the input current waveform. The important case of the amplification of input current pulses with a short duration compared to the multiplier recovery time is investigated in detail, showing that the non-linear behavior can be entirely described by a general function of a conveniently defined saturation parameter and that this function is characteristic of any uniform channel multiplier. The model is then used to investigate the recovery of the multiplier after a saturating pulse, and it is found that the gain recovery from weak or moderate saturation levels is exponential to a very good approximation, but with a time constant different from the characteristic time constant RC of the multiplier. Finally the case of pulses of arbitrary shape and duration is considered and examples are given of the amplification of step pulses and of a regular sequence of identical pulses. A remarkable feature of the model is that the solution can be calculated from the time shape of the output pulse, rather than from the input. This makes possible to implement methods for pulse restoration, i.e. for recovering the original input pulse shape from a measured saturated output.     

Convection in superposed fluid and porous layers

This paper reports an analytical study of the stability and natural convection in a system consisting of a horizontal fluid layer over a layer of saturated porous medium. Neumann thermal boundary conditions are applied to the horizontal walls of the enclosure while the vertical walls are impermeable and adiabatic. At the interface between the fluid and the porous layers the empirical slip condition, suggested by Beavers and Joseph, is employed. An analytical solution is obtained using a parallel flow approximation, for constant-flux thermal boundary conditions, for which the onset of supercritical cellular convection occurs at a vanishingly small wavenumber and can thus be predicted by the present theory. The critical Rayleigh number, Ra _c , and Nusselt number, Nu, are found to depend on the depth ratio, η, the Darcy number, Da, the thermal conductivity ratio, γ and the slip parameter α. Results are presented for a wide range of each of the governing parameters. The results are compared with limiting cases of the problem and are found to be in agreement.     

Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition

This paper concerns the flow of an incompressible, viscous fluid past a porous spherical particle enclosing a solid core, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. No-slip and impenetrability boundary conditions on the solid spherical core have been used. The hydrodynamic drag force experienced by a porous spherical particle enclosing a solid core and permeability of membrane built up by solid particles with a porous shell are evaluated. It is found that the hydrodynamic drag force and dimensionless hydrodynamic permeability depends not only on the porous shell thickness, particle volume fraction γ and viscosities of porous and fluid medium, but also on the stress jump coefficient. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta–Morse’s condition). Some previous results for the hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Solution of laplace equation by the method of separation of variables

Abstract

The Laplace equation, which is used to describe the problem of two‐dimensional heat conduction with appropriate boundary conditions at steady state, is solved in this work by applying the method of separation of variables. The primary objective of this work involves discussing the effects of the constant value of the separation of variables (p) and the sequential order of substituting boundary conditions on the solution. Without appropriately arranging the sequential order of substituting the boundary conditions, the solution for non‐zero constant values of separation of variables (p) can not be obtained. For a zero value for the constant of the separation of variables, the solution obtained is trivial or does not exist. Solutions in different forms are obtained by using different values for the constant of the separation of variables (p) and for the sequential orders of substituting the boundary conditions.     

A simplified solution of the regenerator periodic problem: the case for air conditioning

This work presents an analytical solution for the periodic heat transfer problem of regenerators used in air conditioning, which are operating at low regeneration temperatures and mass flow rates. These types of regenerators are characterized by NTU/Cr ^?<1. The partial differential equations for hot and cold airflows as well as the regenerator matrix were solved using a successive transformation of variables. They were reduced to the ordinary Bessel differential equation of the type xf″+f′?xf=0. The conventional initial and reversal boundary conditions were used in this work. The solution gives a correlation for the prediction of the regenerator effectiveness. Besides the effectiveness, the solution facilitates the calculation of the matrix temperature distribution and exit airflow temperatures. The result is compared with the available numerical and analytical solutions from literature. The result of this analysis reveals that the consideration of a non-linear matrix temperature distribution as in some previous work for low temperature regenerators just complicates the solution procedure with no significant improvement in the accuracy within the parameter space typical for air conditioning applications.     

Meshfree natural vibration analysis of 2D structures

Determination of resonance frequencies and vibration modes of mechanical structures is one of the most important tasks in the product design procedure. The main goal of this paper is to describe a pioneering application of the solution structure method (SSM) to 2D structural natural vibration analysis problems and investigate the numerical properties of the method. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using analytical methods or traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward generalization to model vibrations of 3D structures.     

Non-similarity solution and correlation of transient heat transfer in laminar boundary layer flow over a wedge

With a comprehensive and rigorous method, this paper has successfully examined the transient heat transfer in a steady and two-dimensional (2D) laminar boundary layer flow on a wedge with sudden change of thermal boundary conditions of uniform wall temperature (UWT) and heat flux (UHF). Additionally, a correlation of unsteady forced convection was also formulated through an exact solution of transient heat conduction (ξ=0) and the similarity solutions of a steady forced convection on a wedge (ξ=1) in this study. Particularly, for the wedge with −0.198838?ξ?1, the deviation of the wall temperatures estimated by correlation is less than 7.5% within full time of 0?ξ?1 comparing with numerical results in the case of UHF ranging from Pr=10⁻⁴ to 10⁴.