Description

EX_HEATTRANSFER6 2D axisymmetric heat conduction.

[ FEA, OUT ] = EX_HEATTRANSFER6( VARARGIN ) NAFEMS benchmark example for heating of a solid cylider with an internal hole.

  _              T=T_amb
  ^            +---------+
  |    q_n=0   |         |
  |            :         |
0.14m  q_n=5e5 |         | T=T_amb
  |            :         |
  |    q_n=0   |         |
  v            +---------+
                 T=T_amb
               r=0.02
               |<-0.08m->|

The geometry can be considered axisymmetric and the solid has a thermal conductivity of 52 W/mC, the middle part of the inside of the cylider is heated by 5e5 W/mK. The steady temperature at the point (0.04,0.04) is sought when the surrounding ambient temperature is T_amb = 0 C.

Reference

  [1] Cameron AD, Casey JA, Simpson GB. Benchmark Tests for Thermal Analysis,
      The National Agency for Finite Element Standards, UK, 1986.

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
hmax        scalar {0.005}         Grid cell size
sfun        string {sflag1}        Finite element shape function
solver      string fenics/{}       Use FEniCS or default solver
iplot       scalar {1}/0           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { 'hmax',     0.005;
             'sfun',     'sflag1';
             'solver',   '';
             'iplot',    1;
             'tol',      1e-2;
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});


% Geometry definition.
 gobj = gobj_polygon( [ 0.02 0.1 0.1  0.02 0.02 0.02 ;
                        0    0   0.14 0.14 0.1  0.04 ]' );
 fea.geom.objects = { gobj };


% Grid generation.
 fea.grid = gridgen( fea, 'hmax', opt.hmax, 'fid', opt.fid );


% Problem definition.
 fea.sdim  = { 'r', 'z' };             % Space coordinate name.
 fea = addphys( fea, @heattransfer );  % Add heat transfer physics mode.
 fea.phys.ht.sfun = { opt.sfun };      % Set shape function.

% Equation coefficients.
 fea.phys.ht.eqn.seqn = '- r*k_ht*(Tr_r + Tz_z) = 0';

 fea.phys.ht.eqn.coef{3,end} =    52;       % Thermal conductivity.
 fea.phys.ht.eqn.coef{7,end} = { 273.15 };  % Initial temperature.

% Boundary conditions.
 fea.phys.ht.bdr.sel = [1 1 1 3 4 3];
 fea.phys.ht.bdr.coef{1,end}   = { 273.15 273.15 273.15 [] [] [] };
 fea.phys.ht.bdr.coef{4,end}{5}{1} = 'r*5e5';


% Parse physics modes and problem struct.
 fea = parsephys(fea);
 fea = parseprob(fea);


% Compute solution.
 if( strcmp(opt.solver,'fenics') )
   fea = fenics( fea, 'fid', opt.fid );
 else
   fea.sol.u = solvestat( fea, 'fid', opt.fid, 'init', {'T0_ht'} );
 end


% Postprocessing.
 if( opt.iplot>0 )
   postplot( fea, 'surfexpr', 'T', 'isoexpr', 'T' )
   title('Temperature, T')
 end


% Error checking.
 T_sol = evalexpr( 'T', [0.04;0.04], fea );
 T_ref = 332.97;
 out.err  = abs(T_sol-T_ref)/T_ref;
 out.pass = out.err<opt.tol;


 if( nargout==0 )
   clear fea out
 end