Problem Statement
The form of DE for
one-dimensional steady heat conduction is
where T is temperature, k is thermal conductivity and s
is the source. The boundary
conditions (BCs) defining the analysis problem statement are
where fn
is an imposed boundary heat flux. This linear problem possesses an
analytical solution for comparison to the FEA results. The exact solution
happens to be of the form
defined for any approximation TN(x) , i.e.,
for N = 3. Because of this problem statement’s simplicity, the FE
solution on any mesh Wh
will produce nodally exact temperatures. However, the FE solution Th
exhibits quantifiable error as discussed in the lecture, hence is only an
approximation.
The FE solution is generated via discrete implementation of the
Galerkin weak statement GWSN. Symbolically, GWSN Þ GWSh = Se {WS}e = 0 The FE algorithm template
pseudo-code for WSe is {WS}e = (constant) (avg.data)e {dist.data}e (metric)e [FE matrix] {Q or data}e {WS}e = (COND) ( ) { } (-1) [A211L]e {Q}e -
( ) ( ) { } (1) [A200L]e {SRC}e -
(FN) ( ) { } ( ) [ONE] { } When assembled, GWSh
takes the matrix form
{GWSh} Þ
[MATRIX] {Q} – {b(data)} = {0} As illustrated in the
lecture, after modifying the last row in [MATRIX] to read QM+1
= Tb, this algebraic equation was directly solved by
moving {b(data)} across the equal sign.
However, successive labs deals with non-linear problem statements,
which require a matrix iterative solution procedure, the FEmPSE
Matlab toolbox is organized for the Newton algorithm statement.
[JAC]{dQ}p+1
= -{GWSh} For iteration index p
= 0, 1, 2, …, the nodal DOF array is updated as {Q}p+1 = {Q}p + {dQ}p+1
and the iteration is
stopped when
max |dQ|
< e where e is the convergence requirement, a user input. For a linear problem statement, the Newton algorithm will converge in one iteration to machine zero as follows. By definition
and since no unknown temperature nodal DOF are known a priori {Q}0 º {0} Thereby, for p = 0, [JAC]{dQ}1 = -{GWSh} becomes [MATRIX]{dQ}1 = - (-{b(data)}) the solution of which is obviously {dQ}1 = {Q}. The estimator of error in Th uses the available asymptotic convergence theory expressed in terms of the energy norm || × ||E, specifically
|| errorh ||E ≤
Cle2
|| data ||W
È
¶W Under uniform mesh refinement, i.e., sequentially doubling the number M of equal-length elements spanning Wh, the estimate for the FE solution error on mesh Wh/2 is || errorh/2 || = D || Th ||E / 3 You have assurance of a quality FE solution when the error data plots according to the theory, i.e., parallel to a straight line of slope 2 on log-log scales. The computation of the FE solution energy norm is a post processing operation. The definition is
The slope computation for the generated solution data is
|