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Rectangle, steady GF, single- and double-sum form.

The GF for the 2D rectangle satisfies:
$\displaystyle {\frac{\partial ^{2}G}{\partial x^{2}}}$ + $\displaystyle {\frac{\partial ^{2}G}{\partial y^{2}}}$ = - $\displaystyle \delta$(x - x$\scriptstyle \prime$)$\displaystyle \delta$(y - y$\scriptstyle \prime$) (8)
    0 < x < L;  0 < y < W  
ki$\displaystyle {\frac{\partial G}{\partial n_{i}}}$ + hiG = 0 for faces i = 1, 2,..., 4  

Each of the four faces of the rectangle may have 3 possible boundary condition types, so the above expression represents a total of 34 = 81 different GF for cases XIJYKL; for example umber X11Y11 represents a rectangle with all four faces of type 1. The 2D GF may be stated in any of three alternative forms.

Double-summation Form for the Rectangle
The GF may be found from separation of variables in both coordinate directions resulting in a double-summation form:
G2D(x, y $\displaystyle \left\vert\vphantom{ \,x^{\prime },y^{\prime }}\right.$ x$\scriptstyle \prime$, y$\scriptstyle \prime$$\displaystyle \left.\vphantom{ \,x^{\prime },y^{\prime }}\right.$) = $\displaystyle \sum_{m=0}^{\infty
}$$\displaystyle \sum_{n=0}^{\infty}$$\displaystyle {\frac{X_m(x^{\prime}) X_m(x)}{N_x(\lambda_m)}}$  
  x $\displaystyle {\frac{Y_{n}(y^{\prime })Y_{n}(y)}{N_{y}(\gamma_{n})}}$ x $\displaystyle \left\{\vphantom{
0; & m=n=0 \\
(\lambda_m^2 + \gamma_n^2)^{-1}; & \mbox{ otherwise}
\end{array}}\right.$$\displaystyle \begin{array}{cc}
0; & m=n=0 \\
(\lambda_m^2 + \gamma_n^2)^{-1}; & \mbox{ otherwise}
\end{array}$ $\displaystyle \left.\vphantom{
0; & m=n=0 \\
(\lambda_m^2 + \gamma_n^2)^{-1}; & \mbox{ otherwise}
\end{array}}\right.$ (9)

Here Nx($ \lambda_{m}^{}$) and Ny($ \gamma_{n}^{}$) denote the norms of the mth x-direction eigenfunction and the nth y-direction eigenfunction, respectively. Eigenfunctions Xm(x) and eigenvalues $ \lambda_{m}^{}$ are given in the Tables 2 and 3 above. Eigenfunctions Yn(y) and eigenvalues $ \gamma_{n}^{}$ may be found in the same tables by substituting y, W, and $ \gamma_{n}^{}$ in place of x, L, and $ \lambda_{m}^{}$, respectively. Generally the double-sum form of the 2D GF has poor convergence properties; the single-sum forms are recommended for calculating numerical values. The double-sum form can appear in temperature expressions involving the steady 3D GF or the transient 2D GF.

Single-Summation Form for the Rectangle
The GF for the rectangle has a single-summation form with previously described eigenfunction Yn and norm Ny, and with kernel function Pn, as follows:

G2D(x, y $\displaystyle \left\vert\vphantom{ \,x^{\prime },y^{\prime }}\right.$ x$\scriptstyle \prime$, y$\scriptstyle \prime$$\displaystyle \left.\vphantom{ \,x^{\prime },y^{\prime }}\right.$) = $\displaystyle \sum_{n=0}^{\infty}$$\displaystyle {\frac{Y_{n}(y^{\prime })Y_{n}(y)}{N_{y}(\gamma_{n})}}$Pn(x, x$\scriptstyle \prime$) (10)

The n = 0 term of the series is needed only when Y22 is part of the GF number (when zero is an eigenvalue). Special case X22Y22 is discussed later. Kernel function P0 is identical to the polynomial form of the 1D GF for the slab and is given in Table 1. That is, P0 = G1D(x, x$\scriptstyle \prime$). Kernel functions Pn for n $ \neq$ 0 are given by
Pn(x, x$\scriptstyle \prime$) = {S1-S2-exp[- $\displaystyle \gamma_{n}^{}$(2L - $\displaystyle \left\vert\vphantom{ x-x^{\prime} }\right.$x - x$\scriptstyle \prime$$\displaystyle \left.\vphantom{ x-x^{\prime} }\right\vert$)] (11)
    + S1+S2-exp[- $\displaystyle \gamma_{n}^{}$(2L - x - x$\scriptstyle \prime$)]  
    + S1+S2+exp[- $\displaystyle \gamma_{n}^{}$($\displaystyle \left\vert\vphantom{ x-x^{\prime} }\right.$x - x$\scriptstyle \prime$$\displaystyle \left.\vphantom{ x-x^{\prime} }\right\vert$)]  
    + S1-S2+exp[- $\displaystyle \gamma_{n}^{}$(x + x$\scriptstyle \prime$)]}  
  ÷ {2$\displaystyle \gamma_{n}^{}$[S1+S2+ - S1-S2-exp(- 2$\displaystyle \gamma_{n}^{}$L)]}  

where coefficients S1+, S1-, S2+, and S2- are given in Table 4. Kernel functions Pn for n $ \neq$ 0 have nine different forms, one for each of the boundary condition combinations XIJ for I,J = 1, 2, 3. (The kernel functions may also be stated with hyperbolic sines and cosines; however these can cause ``numerical overflow errors'' when evaluated on a computer).

Table 4. Coefficients for kernel functions.
geometry S1+ S1- S2+ S2-
X11 1 -1 1 -1
X12 1 -1 1 1
X13 1 -1 $ \gamma_{n}^{}$L + B2 $ \gamma_{n}^{}$L - B2
X21 1 1 1 -1
X22 1 1 1 1
X23 1 1 $ \gamma_{n}^{}$L + B2 $ \gamma_{n}^{}$L - B2
X31 $ \gamma_{n}^{}$L + B1 $ \gamma_{n}^{}$L - B1 1 -1
X32 $ \gamma_{n}^{}$L + B1 $ \gamma_{n}^{}$L - B1 1 1
X33 $ \gamma_{n}^{}$L + B1 $ \gamma_{n}^{}$L - B1 $ \gamma_{n}^{}$L + B2 $ \gamma_{n}^{}$L - B2

Alternative Single-sum GF for the Rectangle
In the above equation kernel functions Pn are placed along the x axis. An alternate GF may be constructed by placing the kernel functions along the y axis. The convergence behavior of the alternate single-sum forms of the rectangle GF are complementary. Where one converges slowly, the other generally converges rapidly, so that between them the GF may be computed anywhere in the rectangle.
next up previous
Next: Parallelepiped, steady GF, double- Up: Rectangular Coordinates. Finite Bodies, Previous: Plate, steady GF, polynomial
Kevin D. Cole