Fictitious points

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. 

The node x0 is the graph GM input and fictitious points xt are its outputs. The Mason formula for this graph takes the form... 

We are now ready to apply the discretisations, but must decide on the vector of unknown concentrations at all the grid points in Fig. 12.3. It is convenient to include even the boundary points (but not those at j = —1, which serve only as fictitious points), setting these to known values in the large linear system to be generated. Thus we note that the total number N of unknowns is given by... 

As can be deduced, for m > 2, expression (2.67) leads to cross derivatives by x and y, whose evaluation is rather cumbersome. To alleviate this difficulty, only one fictitious point can be considered at each side of the interface and hence only the zero- and first-order jump conditions are implemented. While this notion gives reliable solutions, an alternative quasi-fourth-order strategy has been presented in [28] for the consideration of higher order conditions and crossderivative computation. A fairly interesting feature of the derivative matching method is that it encompasses various schemes with different orders that permit its hybridization with other high-accuracy time-domain approaches. 

Let us now analyze the more complex shielding application of Figure 7.12(a), which illustrates a rectangular enclosure partitioned by two equal horizontal PEC walls. In the front plane, there is a centered (20 x 5) cm horizontal aperture. The dimensions are a = b = 60 cm, d = 120 cm, / = 70 cm, w = 2 cm, and the excitation is launched by a vertical coaxially fed monopole. Due to the nonstandard operators of (3.43), the domain is discretized into the coarse grid of30 x 60 x 30 cells with Ax = Ay = Az = 2 cm and At = 30.567 ps. In the area of the aperture, spatial derivatives are computed by the fictitious-point technique of Section 2.4.5, whereas the DRP schemes of Section 2.5.3 are also utilized. Figure 7.12(b) displays the shielding efficiency defined as the ratio of the electric field amplitude evaluated in front of... 

Convective transport in the primary flow direction is evalnated as a second-order-correct first derivative at the fictitious point, x, yj, and z +i/2> midway between Zk and Zk+i ... 

Note that the point is a fictitious point, since it is outside the domain of interest. 

To summarize the finite difference method, all we have to do is to replace all derivatives in the equation to be solved by their appropriate approximations to yield a finite difference equation. Next, we deal with boundary conditions. If the boundary condition involves the specification of the variable y, we simply use its value in the finite difference equation. However, if the boundary condition involves a derivative, we need to use the fictitious point which is outside the domain to effect the approximation of the derivative as we did in the above example at x =. The final equations obtained will form a set of algebraic equations which are amenable to analysis by methods such as those in Appendix A. If the starting equation is linear, the finite difference equation will be in the form of tridiagonal matrix and can be solved by the Thomas algorithm presented in the next section. 

As the based point is varied across the spatial domain, a set of simultaneous linear equations is generated which may be represented in diagonal band matrix form. However, at the base points (z, 1) and i,N), it is difficult to write the analogs since they require points outside the solution domain, that is, at (z,0), (i-l-1,0), i,N+ ), and (i-l-1, N+ ). This problem is handled by writing the analogs using the fictitious points, and then the points outside the spatial domain are eliminated via use of the boundary conditions. In this way, the boundary conditions are incorporated in the solution. 

When the boundary conditions are finite-differenced, solved for the fictitious points, and substituted into the energy and material balance equations, the following results, written about base point (z, 1) ... 

The radius of gyration can also be defined as the distance that separates the rotational axis of an object from the fictitious point where all its mass would be... 

Replacing the partial derivative in Eq. (6.35) with a forward difference does not require the use of fictitious points. However, it is important to use the forward difference formula with the same accuracy as the other equations. In this case, Eq. (4.41) should be used for evaluation of the partial derivative at x = 0 (/ = 0) ... 

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