Description

EX_HEATTRANSFER9 1D Transient heat diffusion with a point source.

[ FEA, OUT ] = EX_HEATTRANSFER9( VARARGIN ) Transient heat diffusion problem with a point source at one end and analytic solution.

       +---------- L=20m ----------+ dT/dn = 0
     Q(x=0) = 1   T(t=0) = 0

References

[1] Carslaw HS, Jaeger JC.Conduction of heat in solids, 2ndEd., Oxford at the Clarendon Press, 1959.

[2] Strang G, Fix G. An analysis of the Finite Element Method, 2nd Ed., Wellesley-Cambridge Press, 2008.

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
nel         scalar {100}           Number of grid cells
sfun        string {sflag1}        Finite element shape function
solver      string fenics/{}       Use FEniCS or default solver
ischeme     scalar {2}/1/3         Time stepping scheme
imass       scalar {2}/1/3/4       Mass matrix lumping
tmax        scalar {0.2}           Maximum time
tstep       scalar {0.01}          Time step size
iplot       scalar {1}/0           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { 'nel',      100;
             'sfun',     'sflag1';
             'solver',   '';
             'ischeme',  2;
             'imass',    2;
             'tmax',     1;
             'tstep',    0.01;
             'iplot',    1;
             'tol',      1e-2;
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});


% Geometry and grid.
 L = 20;

 fea.sdim = { 'x' };
 fea.geom.objects = { gobj_line(0,L) };
 fea.grid = linegrid( opt.nel, 0, L );


% Physics definition.
 rho = 1;
 cp  = 1;
 k   = 1;
 Q   = 0;
 T0  = 0;

 fea = addphys( fea, @heattransfer );
 fea.phys.ht.eqn.coef{1,end} = { rho };
 fea.phys.ht.eqn.coef{2,end} = {  cp };
 fea.phys.ht.eqn.coef{3,end} = {   k };
 fea.phys.ht.eqn.coef{5,end} = {   Q };
 fea.phys.ht.eqn.coef{6,end} = {  T0 };
 fea.phys.ht.sfun = { opt.sfun };

% Point source.
 fea.pnt.type  = 'source';
 fea.pnt.index = 0;
 fea.pnt.dvar  = 1;
 fea.pnt.expr  = 1;

 fea = parsephys( fea );
 fea = parseprob( fea );


% Compute solution.
 if( strcmp(opt.solver,'fenics') )
   fea = fenics( fea, 'fid', opt.fid, ...
                 'tstep', opt.tstep, 'tmax', opt.tmax, 'ischeme', opt.ischeme, 'nproc', 1 );
 else
   [fea.sol.u,fea.sol.t] = solvetime( fea, ...
                                      'tstep',   opt.tstep, ...
                                      'tmax',    opt.tmax, ...
                                      'ischeme', opt.ischeme, ...
                                      'imass',   opt.imass, ...
                                      'init',    { 'T0_ht' }, ...
                                      'fid', opt.fid );
 end

% Analytical solution.
 x = fea.grid.p;
 t = fea.sol.t;
 for i=1:length(t)   % Loop over time
   T_ref = 2*(t(i)/pi)^(1/2)*((exp((-x.^2./(4*t(i))))-0.5.*x.*(sqrt(pi/t(i)).*erfc((x./(2*sqrt(t(i))))))));
 end


% Error checking.
 T_sol = evalexpr( 'T', x, fea );
 out.err = norm( T_ref(:) - T_sol(:) );
 out.pass = out.err < opt.tol;


% Postprocessing.
 if( opt.iplot )
   figure, hold on
   plot( x, T_sol, 'b-',  'linewidth', 2 )
   plot( x, T_ref, 'r--', 'linewidth', 1.5 )

   grid on
   axis normal
   ax = axis;
   ax(1:2) = [0,2];
   axis( ax )

   title( sprintf('Solution at time %g',fea.sol.t(end)) )
   ylabel('T')
   xlabel('x')
   legend( 'Computed solution', 'Analytical solution' )
 end


 if( nargout==0 )
   clear fea out
 end