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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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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
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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
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fea struct Problem definition struct
out struct Output struct
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));
fea.grid = quad2tri(fea.grid);
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 = addphys(fea,@convectiondiffusion); % Add convection and diffusion physics mode.
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