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ex_poisson5.m File Reference

Description

EX_POISSON5 3D Poisson equation example on a unit sphere.

[ FEA, OUT ] = EX_POISSON5( VARARGIN ) Poisson equation on a unit sphere with source term f=1 and homogenous (zero) Dirichlet boundary conditions on the sphere surface. The exact solution to this problem is u_ref=(1-r^2)/6 where r is the radius from the origin. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
igrid       scalar 1/{0}           Cell type (0=quadrilaterals, 1=triangles)
hmax        scalar {0.35}          Max grid cell size
sfun        string {sflag1}        Shape function
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output stuct

Code listing

 cOptDef = { ...
   'igrid',    0; ...
   'hmax',     0.35; ...
   'refsol',   '(1-(x^2+y^2+z^2))/6'; ...
   'sfun',     'sflag1'; ...
   'iphys',    1; ...
   'icub',     2; ...
   'iplot',    1; ...
   'tol',      0.2;
   'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


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


% Grid generation.
 switch opt.igrid
   case -2
     fea.grid = blockgrid( 1, 1, 1, [-1 1;-1 1;-1 1] );
   case -1
     fea.grid = spheregrid(round(1/opt.hmax*4/7),round(1/opt.hmax*3/7));
     fea.grid = hex2tet(fea.grid);
   case 0
     fea.grid = spheregrid(max(2,5*round(1/opt.hmax*4/7)),max(1,5*round(1/opt.hmax*3/7)));
   case 1
     fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid);
 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({opt.refsol},1,n_bdr);   % Assign reference solution to all boundaries (Dirichlet).
   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(opt.refsol);     % Assign reference solution 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 )
   figure
   postplot(fea,'surfexpr','u','selexpr','(y>0)','axequal','on')
 end


% Error checking.
 s_err = ['abs(',opt.refsol,'-u)'];
 if ( size(fea.grid.c,1)==8 )
   xi = [0;0;0];
 else
   xi = [1/4;1/4;1/4;1/4];
 end
 err = evalexpr0(s_err,xi,1,1:size(fea.grid.c,2),[],fea);
 ref = evalexpr0('u',xi,1,1:size(fea.grid.c,2),[],fea);
 err = sqrt(sum(err.^2)/sum(ref.^2));

 if( ~isempty(fid) )
   fprintf(fid,'\nL2 Error: %f\n',err)
   fprintf(fid,'\n\n')
 end


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