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$            RIGID FORMAT No. 3 (APP HEAT), Nonlinear Heat Conduction
$               Nonlinear Heat Transfer in an Infinite Slab (3-5-1)
$ 
$ A. Description
$ 
$ This problem demonstrates NASTRAN's capability to solve nonlinear steady state
$ heat conduction problems. The infinite slab is subjected to uniform heat
$ addition per unit volume. There is no heat flux on one face and the other face
$ is kept at zero degrees. The conductivity is temperature dependent. This is a
$ one dimensional problem, since there is no temperature gradient parallel to
$ the surfaces of the slab.
$ 
$ B. Input
$ 
$ Linear elements BAR, CONROD, ROD, and TUBE with areas of pi square units and
$ boundary condition element HBDY (POINT) are used. The heat addition is
$ specified on a QVOL card and is referenced in Case Control by a LOAD card. The
$ area factor for the HBDY is given on the PHBDY card and heat flux is zero. The
$ initial temperatures are given on a TEMPD card and referenced in Case Control
$ by a TEMP (MATERIAL) card. The conductlvity is specified on a MAT4 card and is
$ made temperature dependent by the MATT4 card referencing table TABLEM3. The
$ convergence parameter, the maximum number of iterations, and an option to have
$ the residual vector output are specified on PARAM cards. The temperature at
$ the outer surface is specified by an SPC card. Temperature output is punched
$ on TEMP bulk data cards for future use in static analysis.
$ 
$ C. Theory
$ 
$ The conductivity, k, is defined by
$ 
$    k(T) = 1 + T/l00                                                        (1)
$ 
$ where T is the temperature.
$ 
$ The heat flow per area, q, is
$ 
$               dT                 dT
$    q(x) = - k -- = - (1 + T/100) --                                        (2)
$               dx                 dx
$ 
$ The heat input per volume, q , affects the heat flow by the equatIon
$                             v
$ 
$    dq(x)
$    ----- = q                                                               (3)
$      dx     v
$ 
$ A convenient substitution of variables in Equations (2) and (3) is
$ 
$                                     2
$    u = - integral of q(x)dx = (T + T /200)                                 (4)
$ 
$ Differentiation and substitution for q in Equation (3) results in the second-
$ order equation in u:
$ 
$     2
$    d u
$    --- = -q                                                                (5)
$      2     v
$    dx
$ 
$ From the following boundary conditions
$ 
$    u = 0 at x = l
$ 
$ and
$ 
$    du
$    -- = 0 at x = 0
$    dx
$ 
$ the solution to Equation (5) is
$ 
$        q
$         v    2    2
$    u = -- ( l  - x  )                                                      (6)
$        2
$ 
$ Therefore the solution for the temperature is
$ 
$                               2    2       1/2
$    T = 100 [ -1 +/- (1 + q  (l  - x  )/100)   ]                            (7)
$                           v
$ 
$ Since heat is flowing into the system, the positive temperature solution will
$ occur.
$ 
$ D. Results
$ 
$ A comparison with NASTRAN results is shown in Table 1.
$ 
$            Table 1. Comparison of Theoretical and NASTRAN Temperatures
$                 for Nonlinear Heat Conduction in an Infinite Slab
$                 ----------------------------------
$                 Grid     Theoretical      NASTRAN
$                 Point    Temperature      Solution
$                 ----------------------------------
$                   1          73.20         73.13
$                   2          69.56         69.53
$                   3          58.11         58.11
$                   4          36.93         36.93
$                   5           0.00          0.00
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