Fictitious boundary

The first term in the curved inner bracket is the energy of either the upper or lower half of the a/0 interface in Fig. 19.27, while the second represents half the energy of the area A. Their sum is therefore the total interfacial energy of the half-nucleus shape containing the fictitious boundary shown in Fig. 19.28. Using these results ... 

A closely related method is that of Boley (B8), who was concerned with aerodynamic ablation of a one-dimensional solid slab. The domain is extended to some fixed boundary, such as X(0), to which an unknown temperature is applied such that the conditions at the moving boundary are satisfied. This leads to two functional equations for the unknown boundary position and the fictitious boundary temperature, and would, therefore, appear to be more complicated for iterative solution than the Kolodner method. Boley considers two problems, the first of which is the ablation of a slab of finite thickness subjected on both faces to mixed boundary conditions (Newton s law of cooling). The one-dimensional heat equation is once again... 

Recognizing the intrinsic symmetry of the flow process or the repetitious nature of the process equipment can minimize the size and extent of the solution domain. If such a possibility exists, fictitious boundaries may be used to define the solution domain with special boundary conditions imposed on these such fictitious surfaces. Some of the commonly encountered cases are discussed below. 

T is a fictitious boundary to the assumed gas temperature profile. Since one of the profile boundary temperatures and the mean temperature are defined, the remaining profile boundary temperature, TT, is fixed but lot necessarily equal to the gas inlet temperature or saturation temperature. 

If the BVP contains a Neumann boundary condition, one way to solve the problem is to extend the original computational domain, i.e. from nhto (n + )h. This means introducing a fictitious boundary. This obviously requires one more grid point outside the original grid the additional one is atx +i, as shown in Example 6.8. By introducing the fictitious node, the accuracy OQ ) is maintained. 

Note that a Dirichlet boundary condition is specified for the left-hand side of the domain, i.e.0o = l.InordertoimplementtheNeumannboundary conditionontheright-handside of the domain (the no-heat-flux condition on the tip), we can extend the computational domain from =i x /i for i = 0,1, 2,..., n to i =0, 1, 2,..., n + 1. For this reason, tile dependent variable is also calculated at the fictitious boundary outside the domain, at fs. The Neuman boundary condition, i.e. the derivative at tiie right-hand side of the... 

Note that O5 is the solution at the fictitious boundary (outside the domain), it can be excluded when plotting the solution on the domain of interest. In order to reduce the error, the domain is simply discretized into smaller intervals. The solution to the problem using finite differences and a mesh with /z = 0.1 is shown in Figure 6.16. 

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Theory of the fictitious temperature field allows us to analyze the problems of residual stresses in glass using the mathematical apparatus of thermoelasticity. In this part we formulate the boundary-value problem for determining the internal stresses. We will Lheretore start from the Duhamel-Neuinan relations... 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

Which of the Equations [(4.13) or (4.14)] is determined to be the most important depends on which part of the boundary has the major resistance to the mass transport. If, as an example, the major resistance exists in the water film of the boundary, i.e., k2A < k1A, Equation (4.13) is the relevant description of the flux rate. The two equilibrium values of the molar fraction, xA and y A are fictitious, however, each is determined from Henry s law, i.e. ... 

For boundary conditions which require a prescribed value of the flux instead of concentration, we introduce what is usually called fictitious points. Let us assume that at x = 0, the condition is... 

We now get a fairly good feeling that handling boundary conditions in two dimensions is significantly more difficult than those of one-dimensional problems. Flux conditions can be applied to the boundaries using the method of fictitious points. 

It was assumed that the concentration boundary layer was thin relative to the shorter axis of the spheroid. The order of magnitude of the boundary layer thickness can be approximated by the thickness 5 of a fictitious film... 

The second example is the solidification of liquid enclosed between parallel boundaries. Here it is necessary to deal with two fictitious bodies of constant dimensions, one representing the liquid and the other the solid. For the solid phase one deals not with an initially finite body whose dimensions are shrinking, but with a body whose dimensions increase from zero. For this reason an imaginary initial temperature turns out to be more appropriate than an imaginary heat input. 

Recently, Attard [30] proposed a different approach which provides a variational formulation of the electrostatic potential in dielectric continua. His formulation of the free energy functional starts from Equation (1.77), which he justifies using a maximum entropy argument. He defines a fictitious surface charge, s, located on the cavity boundary. The charge s, which produces an electric field /, contributes together with the solute... 

A third possibility of approximating the mixed type boundary condition, widely used, implies considering a fictitious external gridpoint xo and the corresponding concentration eg. The centered-difference approximation of the boundary condition becomes ... 

A special treatment has to be applied to the central gridpoint n = 0. The singularity arising in the second term is overcome by imposing the natural boundary condition that the first order derivative vanishes at n = 0. By introducing a fictitious gridpoint r0 = ri - h, this condition may be approximated by the second order centered-differ-ence scheme ... 

We have thus far obtained the distribution functions for the incoming and reflected particles of reduced mass in different regions in terms of the unknown constants A and B and the unknown dimensionless surface temperature 8. The surface temperature was then related to the internal energy and the number density of the fictitious particle at the reflecting surface. Now, in order to determine the constants A and B, wc must specify the boundary conditions for the mass and the energy flux at the sphere of influence. 

Since the motion of the fictitious particle in Regions II and III is assumed to be collisionless, a reflected particle of reduced mass /Mr with kinetic cncigy less than the depth of the potential well 4>0 will be captured by the potential well. Therefore, the boundary condition for the mass flux is given by... 

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