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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_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
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fea struct Problem definition struct
out struct Output struct
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 = 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.
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
%-----------------------------------------------