FEATool Multiphysics  v1.16.4 Finite Element Analysis Toolbox
ex_poisson6.m File Reference

## 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.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

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