Problem Statement   

      The form of DE for one-dimensional steady heat conduction is 

              on        0 < x < L 

where T is temperature, k is thermal conductivity and s is the source.  The boundary conditions (BCs) defining the analysis problem statement are 

                                   at         x = 0

                                                           at         x = L

 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