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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_POISSON3 2D Poisson equation example on a unit square.
[ FEA, OUT ] = EX_POISSON3( VARARGIN ) Poisson equation on a [0..1]^2 unit square.
Input Value/{Default} Description
-----------------------------------------------------------------------------------
igrid scalar 1/{0} Cell type (0=quadrilaterals, 1=triangles)
hmax scalar {0.1} 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 = { ...
'igrid', 0; ...
'hmax', 0.1; ...
'sfun', 'sflag1'; ...
'iphys', 1; ...
'icub', 2; ...
'iplot', 1; ...
'tol', 0.01;
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Geometry definition.
fea.geom.objects = { gobj_rectangle() };
% Grid generation.
switch opt.igrid
case -2
n = round(1/opt.hmax);
fea.grid = rectgrid(n,n,[0 1;0 1]);
ix = setdiff( 1:(n+1)^2, [1:(n+1) (n+1):(n+1):(n+1)^2 (n+1)^2:-1:(n+1)^2-(n+1)+1 (n+1)^2-(n+1)+1:-(n+1):1 ] );
h = 0.3*1/n;
fea.grid.p(1,ix) = fea.grid.p(1,ix) + h*(2*rand(1,numel(ix)) - 1);
fea.grid.p(2,ix) = fea.grid.p(2,ix) + h*(2*rand(1,numel(ix)) - 1);
case -1
fea.grid = rectgrid(round(1/opt.hmax),round(1/opt.hmax),[0 1;0 1]);
fea.grid = quad2tri(fea.grid);
case 0
fea.grid = rectgrid(round(1/opt.hmax),round(1/opt.hmax),[0 1;0 1]);
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,@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.
if( any(strcmp(opt.sfun,{'sf_tri_H3','sf_quad_H3'})) ) % Prescribed derivatives at end points for Hermite elements.
fea.bdr.d = {{ 0 0 0 0 ;
0 '-ex_poisson3_derivative(y)' 0 'ex_poisson3_derivative(y)' ;
'ex_poisson3_derivative(x)' 0 '-ex_poisson3_derivative(x)' 0 }};
end
else
fea.dvar = { 'u' }; % 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 = { 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.
if( any(strcmp(opt.sfun,{'sf_tri_H3','sf_quad_H3'})) ) % Prescribed derivatives at end points for Hermite elements.
fea.bdr.d = {{ 0 0 0 0 ;
0 '-ex_poisson3_derivative(y)' 0 'ex_poisson3_derivative(y)' ;
'ex_poisson3_derivative(x)' 0 '-ex_poisson3_derivative(x)' 0 }};
else
fea.bdr.d = cell(1,n_bdr);
[fea.bdr.d{:}] = deal(0); % Assign zero to all boundaries (Dirichlet).
end
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
g = rectgrid( 20 );
u = evalexpr( 'u', g.p, fea );
fv.faces = g.c';
fv.vertices = [g.p' u];
fv.facevertexcdata = u;
fv.facecolor = 'interp';
patch( fv )
grid on, axis normal, view(3)
xlabel( 'x' )
ylabel( 'y' )
title('Solution u')
end
% Error checking.
x = linspace( 0, 1, 11 );
[x,y] = meshgrid(x,x);
u = evalexpr( 'u', [x(:) y(:)]', fea );
u_ref = l_poisol2( x(:), y(:), 6 );
u_diff = u - u_ref;
err = norm(u_diff);
if( ~isempty(fid) )
fprintf(fid,'\nL2 Error: %f\n',err)
fprintf(fid,'\nL00 Error: %f\n',max(abs(u_diff)))
fprintf(fid,'\n\n')
end
out.err = err;
out.tol = opt.tol;
out.pass = out.err<opt.tol;
if ( nargout==0 )
clear fea out
end
% -----------------------------------------------------------------------------