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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_POISSON1 1D Poisson equation example.
[ FEA, OUT ] = EX_POISSON1( VARARGIN ) Poisson equation on a line with a constant source term equal to 1, homogenous boundary conditions, and exact solution (-x^2+x)/2. Accepts the following property/value pairs.
Input Value/{Default} Description
-----------------------------------------------------------------------------------
hmax scalar {1/10} Grid cell size
sfun string {sflag1} Finite element shape function
iphys scalar 0/{1} Use physics mode to define problem (=1)
or directly define fea.eqn/bdr fields (=0)
or use core assembly functions (<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 = { 'hmax', 1/10;
'sfun', 'sflag1';
'refsol', '(-x^2+x)/2';
'fsrc', '1';
'iphys', 1;
'icub', 2;
'iplot', 1;
'tol', 2e-2;
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Grid generation.
if( opt.hmax>0 )
nx = round( 1/opt.hmax );
fea.grid = linegrid( nx, 0, 1 );
else % Scrambled testing grid.
fea.grid.p = [ 0 1/10 4/10 1/3 1 1-1/3 ];
fea.grid.c = [ 1 4 2 6 3 ;
2 3 4 5 6 ];
fea.grid.a = [ 0 2 1 4 3 ;
2 4 3 0 5 ];
fea.grid.b = [ 1 1 1 -1 ;
4 2 2 1 ]';
fea.grid.s = ones(1,5);
end
n_bdr = 2; % Number of boundaries.
% Problem definition.
fea.sdim = { 'x' }; % Coordinate name.
switch opt.iphys
case 0 % Directly define fea.eqn/bdr fields.
fea.dvar = { 'u' }; % Dependent variable name.
fea.sfun = { opt.sfun }; % Shape function.
% Define equation system.
fea.eqn.a.form = { [2;2] }; % First row indicates test function space (2=x-derivative),
% second row indicates trial function space (2=x-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 = { opt.fsrc }; % Coefficient used in right hand side.
% Define boundary conditions.
if( strcmp(opt.sfun(end-1:end),'H3') ) % Prescribed derivatives at end points for Hermite elements.
fea.bdr.d = {{ 0 0 ; 1/2 -1/2 }};
else
fea.bdr.d = cell(1,n_bdr);
[fea.bdr.d{:}] = deal(0); % Assign zero to all boundaries (homogenous Dirichlet conditions).
end
fea.bdr.n = cell(1,n_bdr); % No Neumann boundaries ('fea.bdr.n' empty).
% 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.
case 1 % Use physics mode.
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} = { opt.fsrc }; % 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( strcmp(opt.sfun(end-1:end),'H3') ) % Prescribed derivatives at end points for Hermite elements.
fea.bdr.d = {{ 0 0 ; 1/2 -1/2 }};
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.
otherwise % Use core assembly functions.
fea.dvar = { 'u' }; % Dependent variable name.
fea.sfun = { opt.sfun }; % Shape function.
fea = parseprob(fea); % Check and parse problem struct.
% Assemble stiffness matrix.
form = [2;2];
sfun = {opt.sfun;opt.sfun};
coefa = 1;
sind = 1;
i_cub = opt.icub;
[vRowInds,vColInds,vAvals,n_rows,n_cols] = ...
assemblea(form,sfun,coefa,i_cub,fea.grid.p,fea.grid.c,fea.grid.a,fea.grid.s,[]);
A = sparse(vRowInds,vColInds,vAvals,n_rows,n_cols);
% Check and compare with finite difference stencil.
if (strcmp(opt.sfun,'sflag1'))
h = 1/nx;
n = nx+1;
e = ones(n,1);
A_ref = 1/h*spdiags([-e 2*e -e], -1:1, n, n);
A_ref(1) = A_ref(1)/2;
A_ref(end) = A_ref(end)/2;
err = norm(A(:)-A_ref(:));
if err>opt.tol
out.err = err;
out.pass = -1;
return
end
end
form = 1;
sfun = sfun{1};
coeff = 1;
f = assemblef(form,sfun,coeff,i_cub,fea.grid.p,fea.grid.c,fea.grid.a,fea.grid.s,[]);
% Check and compare with finite difference stencil.
if (strcmp(opt.sfun,'sflag1'))
f_ref = coeff*h*ones(n,1);
f_ref([1 end]) = coeff*1/2*h;
err = norm(f-f_ref);
if err>1e-6
out.err = err;
out.pass = -2;
return
end
end
% Set homogenous Dirichlet boundary conditions on first and last dof/node.
bind = [1 nx+1];
A = A'; %'
A(:,bind) = 0; % Zero out Dirichlet BC rows.
for i=1:length(bind) % Loop to set diagonal entry to 1.
i_a = bind(i);
A(i_a,i_a) = 1;
end
A = A'; %'
f(bind) = 0; % Set corresponding source term entries to Dirichlet BC values.
% Solve problem.
fea.sol.u = A\f;
end
% Postprocessing.
if ( opt.iplot>0 )
x = linspace( 0, 1, 41 );
u = evalexpr( 'u', x, fea )';
figure
subplot(3,1,1)
plot( x, u )
axis( [0 1 0 0.2])
grid on
title('Solution u')
subplot(3,1,2)
ux = (-x.^2+x)/2;
plot( x, ux )
axis( [0 1 0 0.2])
grid on
title('Exact solution')
subplot(3,1,3)
plot( x, abs(ux-u) )
title('Error')
end
% Error checking.
xi = [1/2; 1/2];
s_err = ['abs(',opt.refsol,'-u)'];
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: %e\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