FEATool Multiphysics
v1.17.0
Finite Element Analysis Toolbox
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EX_POISSON2 2D Poisson equation example on a circle.
[ FEA, OUT ] = EX_POISSON2( VARARGIN ) Poisson equation on a circle with source term 1 and exact solution u=(1-r^2)/4. Accepts the following property/value pairs.
Input Value/{Default} Description ----------------------------------------------------------------------------------- radi scalar {1} Radius of circle 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) igrid scalar -1/{0} Cell type (0>=triangles, 0<quadrilaterals) iplot scalar 0/{1} Plot solution (=1) . Output Value/(Size) Description ----------------------------------------------------------------------------------- fea struct Problem definition struct out struct Output struct
cOptDef = { ... 'radi', 1; ... 'hmax', 0.1; ... 'refsol', '(1-(x^2+y^2))/4'; ... 'fsrc', '1'; ... 'sfun', 'sflag1'; ... 'iphys', 1; ... 'icub', 2; ... 'iplot', 1; ... 'igrid', 1; ... 'tol', 1e-1; 'fid', 1 }; [got,opt] = parseopt(cOptDef,varargin{:}); fid = opt.fid; % Geometry definition. gobj = gobj_circle( [0 0], opt.radi ); fea.geom.objects = { gobj }; % Grid generation. if( opt.igrid<0 ) fea.grid = circgrid(4,3); if( opt.igrid==-2 ) fea.grid = quad2tri( fea.grid ); end else 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' }; % 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} = { opt.fsrc }; % Set source term coefficient. fea.phys.poi.bdr.coef{1,end} = repmat({opt.refsol},1,n_bdr); % Set Dirichlet boundary coefficient to reference solution. % fea.phys.poi.eqn 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 = {{ opt.refsol opt.refsol opt.refsol opt.refsol ; '-x/2' '-x/2' '-x/2' '-x/2' ; '-y/2' '-y/2' '-y/2' '-y/2' }}; 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 = { opt.fsrc }; % 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 = {{ opt.refsol opt.refsol opt.refsol opt.refsol ; '-x/2' '-x/2' '-x/2' '-x/2' ; '-y/2' '-y/2' '-y/2' '-y/2' }}; else fea.bdr.d = cell(1,n_bdr); [fea.bdr.d{:}] = deal(opt.refsol); % Assign reference solution 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. s_err = ['abs(',opt.refsol,'-u)']; if ( opt.iplot>0 ) figure subplot(3,1,1) postplot(fea,'surfexpr','u','axequal','on') title('Solution u') subplot(3,1,2) postplot(fea,'surfexpr',opt.refsol,'axequal','on') title('Exact solution') subplot(3,1,3) postplot(fea,'surfexpr',s_err,'axequal','on','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('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