FEATool Multiphysics  v1.16.4 Finite Element Analysis Toolbox
ex_diffusion1.m File Reference

## Description

EX_DIFFUSION1 2D Diffusion equation example on a unit square.

[ FEA, OUT ] = EX_DIFFUSION1( VARARGIN ) Diffusion equation on a unit square with exact solutions. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
isol        scalar {1}             Exact solution
1                x*y
2                x^2-y^2
3                2*y/((1+x)^2+y^2)
4                2*y/((1+x)^2+y^2)
5                (sinh(pi*x)*sin(pi*y)+sinh(pi*y)*
sin(pi*x))/sinh(pi)
cd          scalar {0.1}           Diffusion coefficient
igrid       scalar 1/{0}           Cell type (0=quadrilaterals, 1=triangles)
hmax        scalar {1/40}          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 = { ...
'isol',     1; ...
'cd',       0.1; ...
'igrid',    0; ...
'hmax',     1/40; ...
'sfun',     'sflag1'; ...
'iphys',    1; ...
'iplot',    1; ...
'fid',      1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid       = opt.fid;
switch opt.isol
case 1
refsol = 'x*y';
case 2
refsol = 'x^2-y^2';
case 3
refsol = '2*y/((1+x)^2+y^2)';
case 4
refsol = 'exp(pi*x)*sin(pi*y)';   % pi can be substituted for any constant.
case 5
refsol = '(sinh(pi*x)*sin(pi*y)+sinh(pi*y)*sin(pi*x))/sinh(pi)';
end

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

% Grid generation.
switch opt.igrid
case -1
fea.grid = rectgrid(round(1/opt.hmax));
case 0
fea.grid = rectgrid(round(1/opt.hmax));
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' };                % Coordinate names.
if ( opt.iphys==1 )

fea.phys.cd.sfun = { opt.sfun };           % Set shape function.
fea.phys.cd.eqn.coef{2,4}   = { opt.cd };  % Set diffusion coefficient.
fea.phys.cd.bdr.sel         = [1 1 1 1];
fea.phys.cd.bdr.coef{1,end} = repmat({refsol},1,n_bdr);   % Set Dirichlet boundary coefficient to reference solution.
fea = parsephys(fea);                 % Check and parse physics modes.

else

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

% Define equation system.
fea.eqn.a.form = { [2 3;2 3] };       % 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 = { opt.cd };          % Diffusion 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 = { 0 };               % Coefficient used in right hand side.

% Define boundary conditions.
fea.bdr.d     = cell(1,n_bdr);
[fea.bdr.d{:}] = deal(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);   % Call to stationary solver.

% Postprocessing.
s_err = ['abs(',refsol,'-c)'];
if ( opt.iplot>0 )
figure
subplot(2,1,1)
postplot(fea,'surfexpr','c')
title('Solution c')
subplot(2,1,2)
postplot(fea,'surfexpr',s_err,'evalstyle','exact')
title('Error')
end

% Error checking.
if ( size(fea.grid.c,1)==4 )
xi = [0;0];
else
xi = [1/3;1/3;1/3];
end
err = evalexpr0(s_err,xi,1,1:size(fea.grid.c,2),[],fea);
ref = evalexpr0('c',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.pass = out.err<0.01;
if ( nargout==0 )
clear fea out
end