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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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EX_POISSON5 3D Poisson equation example on a unit sphere.
[ FEA, OUT ] = EX_POISSON5( VARARGIN ) Poisson equation on a unit sphere with source term f=1 and homogenous (zero) Dirichlet boundary conditions on the sphere surface. The exact solution to this problem is u_ref=(1-r^2)/6 where r is the radius from the origin. Accepts the following property/value pairs.
Input Value/{Default} Description
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igrid scalar 1/{0} Cell type (0=quadrilaterals, 1=triangles)
hmax scalar {0.35} Max grid cell size
sfun string {sflag1} Shape function
iplot scalar 0/{1} Plot solution (=1)
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Output Value/(Size) Description
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fea struct Problem definition struct
out struct Output struct
cOptDef = { ...
'igrid', 0; ...
'hmax', 0.35; ...
'refsol', '(1-(x^2+y^2+z^2))/6'; ...
'sfun', 'sflag1'; ...
'iphys', 1; ...
'icub', 2; ...
'iplot', 1; ...
'tol', 0.2;
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Geometry definition.
gobj = gobj_sphere();
fea.geom.objects = { gobj };
% Grid generation.
switch opt.igrid
case -2
fea.grid = blockgrid( 1, 1, 1, [-1 1;-1 1;-1 1] );
case -1
fea.grid = spheregrid(round(1/opt.hmax*4/7),round(1/opt.hmax*3/7));
fea.grid = hex2tet(fea.grid);
case 0
fea.grid = spheregrid(max(2,5*round(1/opt.hmax*4/7)),max(1,5*round(1/opt.hmax*3/7)));
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({opt.refsol},1,n_bdr); % Assign reference solution to all boundaries (Dirichlet).
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(opt.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,'icub',opt.icub); % Call to stationary solver.
% Postprocessing.
if ( opt.iplot>0 )
figure
postplot(fea,'surfexpr','u','selexpr','(y>0)','axequal','on')
end
% Error checking.
s_err = ['abs(',opt.refsol,'-u)'];
if ( size(fea.grid.c,1)==8 )
xi = [0;0;0];
else
xi = [1/4;1/4;1/4;1/4];
end
err = evalexpr0(s_err,xi,1,1:size(fea.grid.c,2),[],fea);
ref = evalexpr0('u',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.tol = opt.tol;
out.pass = out.err<out.tol;
if ( nargout==0 )
clear fea out
end