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### Plate, steady GF, polynomial and series form.

For the 1D plate, the steady GF satisfies:
 = - (x - x)  on domain  0 < x < L (4) ki + hiG = 0;  i = 1 or 2 (5)

There are two alternative forms of the 1D GF, polynomial and series.

Polynomial Form, 1-D Plate
The GF for the 1-D plate are given in Table 1 below. Note that these functions are piecewise linear with a jump in slope at x = x with the exception of special case X22 which is discussed later.

Table 1. Steady 1D Green's Function for cases XIJ.
Note B1 = h1L/k;  B2 = h2L/k.
 Case Range 1D steady GF; units are meters. X11 x > x x(1 - x/L) x < x x(1 - x/L) X12 x > x x x < x x X13 x > x x[1 - B2(x/L)/(1 + B2)] x < x x[1 - B2(x/L)/(1 + B2)] X21 x > x L - x x < x L - x X22 * x > x ( x2 + (x)2)/(2L) - x + L/3 x < x ( x2 + (x2)2)/(2L) - x + L/3 X23 x > x L(1 + 1/B2 - x/L) x < x L(1 + 1/B2 - x/L) X31 x > x (B1x - B1xx/L + L - x)/(1 + B1) x < x (B1x - B1xx/L + L - x)/(1 + B1) X32 x > x L(1/B1 + x/L) x < x L(1/B1 + x/L) X33 x > x B1B2x + B1x - B1B2xx/L - B2x + B2L + L ÷ (B1B2 + B1 + B2) x < x B1B2x + B1x - B1B2xx/L - B2x + B2L + L ÷ (B1B2 + B1 + B2)

* Special temperature solution needed with this pseudo GF.

Series Form, 1-D Plate.
The GF in the 1-D plate, cases XIJ (I and J = 1,2,3) may be expressed in series form as follows:

 G1D(x, x) = (6)

Here Nx() denotes the norm of the nth eigenfunction. (Strictly speaking, these are the squares of the norms of the eigenfunctions.) Eigenfunctions Xn(x) satisfy the following differential equation:

 Xn ' ' (x) + Xn(x) = 0. (7)

Table 2 contains eigenfunctions Xn(x) and norms and Table 3 contains the associated eigenconditions (and eigenvalues for simple cases). The 1D series form is important because it arises naturally as part of steady 2D and 3D temperature solutions. Whenever it appears, if you can recognize the 1D series form and replace it with the polynomial form, this will improve the numerical convergence.

Table 2. Eigenfunctions and inverse norm.
Index n = 1, 2,... for all cases except X22 with n = 0, 1, 2,... .
Case Xn(x) Nx-1
X11 sin(x) 2/L
X12 sin(x) 2/L
X13 sin(x) 2/L
X21 cos(x) 2/L
X22
 cos(x); 0 1; = 0
 2/L for  0 1/L for  = 0
X23 cos(x) 2/L
X31 sin((L - x)) 2/L
X32 cos((L - x)) 2/L
X33 Lcos(x) + (h1L/k)sin(x) 2/L
note: = (L)2 + (hiL/k)2 ÷ (L)2 + (hiL/k)2 + hiL/k
= ÷ (L)2 + (h1L/k)2 + (h1L/k)

Table 3. Eigenconditions and eigenvalues for Xn(x).
Index n = 1, 2,... for all cases except X22 with n = 0, 1, 2,....
 Case Eigencondition Eigenvalues X11 sin(L) = 0 , n = 1, 2,... X12 cos(L) = 0 , n = 1, 2,... X13 Lcot(L) = - h2L/k X21 cos(L) = 0 , n = 1, 2,... X22 sin(L) = 0 , n = 0, 1, 2,... X23 Ltan(L) = h2L/k X31 Lcot(L) = - h1L/k X32 Ltan(L) = h1L/k X33 tan(L) = [(h1 + h2)/k]/[ - h1h2k-2]

Next: Rectangle, steady GF, single- Up: Rectangular Coordinates. Finite Bodies, Previous: Rectangular Coordinates. Finite Bodies,
Kevin D. Cole
2002-12-31