Description

EX_POISSON6 3D Poisson equation example on a unit cube.

[ FEA, OUT ] = EX_POISSON6( VARARGIN ) Poisson equation on a [0..1]^3 unit cube.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
igrid       scalar 1/{0}           Cell type (0=hexahedra, 1=tetrahedra)
hmax        scalar {1/9}           Max grid cell size
sfun        string {sflag1}        Shape function
iphys       scalar 0/{1}           Use physics mode to define problem    (=1)
                                   or directly define fea.eqn/bdr fields (=0)
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { ...
   'igrid',    0; ...
   'hmax',     1/9; ...
   'sfun',     'sflag1'; ...
   'iphys',    1; ...
   'icub',     2; ...
   'iplot',    1; ...
   'tol',      0.05; ...
   'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Geometry definition.
 gobj = gobj_block();
 fea.geom.objects = { gobj };


% Grid generation.
 switch opt.igrid
   case -1
     fea.grid = blockgrid(round(1/opt.hmax));
     fea.grid = hex2tet(fea.grid);
   case 0
     fea.grid = blockgrid(round(1/opt.hmax));
   case 1
     fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid,'dprim',false);
 end
 n_bdr = max(fea.grid.b(3,:));           % Number of boundaries.


% Problem definition.
 fea.sdim  = { 'x' 'y' 'z' };            % Coordinate names.
 if ( opt.iphys==1 )

   fea = addphys(fea,@poisson);          % Add Poisson equation physics mode.
   fea.phys.poi.sfun = { opt.sfun };     % Set shape function.
   fea.phys.poi.eqn.coef{3,4} = { 1 };   % Set source term coefficient.
   fea.phys.poi.bdr.coef{1,end} = repmat({0},1,n_bdr);   % Set Dirichlet boundary coefficient to zero.
   fea = parsephys(fea);                 % Check and parse physics modes.

 else

   fea.dvar  = { 'u' };                  % Dependent variable name.
   fea.sfun  = { opt.sfun  };            % Shape function.

% Define equation system.
   fea.eqn.a.form = { [2 3 4;2 3 4] };   % First row indicates test function space   (2=x-derivative + 3=y-derivative),
% second row indicates trial function space (2=x-derivative + 3=y-derivative).
   fea.eqn.a.coef = { 1 };               % Coefficient used in assembling stiffness matrix.

   fea.eqn.f.form = { 1 };               % Test function space to evaluate in right hand side (1=function values).
   fea.eqn.f.coef = { 1 };               % Coefficient used in right hand side.

% Define boundary conditions.
   fea.bdr.d     = cell(1,n_bdr);
  [fea.bdr.d{:}] = deal(0);              % Assign zero to all boundaries (Dirichlet).

   fea.bdr.n     = cell(1,n_bdr);        % No Neumann boundaries ('fea.bdr.n' empty).

 end


% Parse and solve problem.
 fea       = parseprob(fea);             % Check and parse problem struct.
 fea.sol.u = solvestat(fea,'fid',fid,'icub',opt.icub);   % Call to stationary solver.


% Postprocessing.
 if ( opt.iplot>0 )
   p      = l_xgrid(fea.grid,opt.sfun);
   u_ref  = refsol_poi3dcube(p(:,1),p(:,2),p(:,3),3);
   figure
   subplot(1,2,1)
   postplot(fea,'surfexpr','u','selexpr','(y>0.5)','axequal','on')
   title('Solution u')
   subplot(1,2,2)
   postplot(fea,u_ref(1:size(fea.grid.p,2)),'surfexpr','u','selexpr','(y>0.5)','axequal','on')
   title('Reference Solution')
 end


% Error checking.
 x = linspace( 0+sqrt(eps), 1-sqrt(eps), 11 );
 [x,y,z] = ndgrid(x,x,x);
 u = evalexpr( 'u', [x(:) y(:) z(:)]', fea );
 u_ref = refsol_poi3dcube( x(:), y(:), z(:), 6 );
 err = sqrt(sum((u-u_ref).^2)/sum(u_ref.^2));
 if( ~isempty(fid) )
   fprintf(fid,'\nL2 Error: %f\n',err)
   fprintf(fid,'\n\n')
 end


 out.err  = err;
 out.pass = out.err<opt.tol;
 if ( nargout==0 )
   clear fea out
 end


%-----------------------------------------------