FEATool Multiphysics
v1.17.0
Finite Element Analysis Toolbox
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EX_DIFFUSION1 2D Diffusion equation example on a unit square.
[ FEA, OUT ] = EX_DIFFUSION1( VARARGIN ) Diffusion equation on a unit square with exact solutions. Accepts the following property/value pairs.
Input Value/{Default} Description ----------------------------------------------------------------------------------- isol scalar {1} Exact solution 1 x*y 2 x^2-y^2 3 2*y/((1+x)^2+y^2) 4 2*y/((1+x)^2+y^2) 5 (sinh(pi*x)*sin(pi*y)+sinh(pi*y)* sin(pi*x))/sinh(pi) cd scalar {0.1} Diffusion coefficient igrid scalar 1/{0} Cell type (0=quadrilaterals, 1=triangles) hmax scalar {1/40} 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 ----------------------------------------------------------------------------------- fea struct Problem definition struct out struct Output struct
cOptDef = { ... 'isol', 1; ... 'cd', 0.1; ... 'igrid', 0; ... 'hmax', 1/40; ... 'sfun', 'sflag1'; ... 'iphys', 1; ... 'iplot', 1; ... 'fid', 1 }; [got,opt] = parseopt(cOptDef,varargin{:}); fid = opt.fid; switch opt.isol case 1 refsol = 'x*y'; case 2 refsol = 'x^2-y^2'; case 3 refsol = '2*y/((1+x)^2+y^2)'; case 4 refsol = 'exp(pi*x)*sin(pi*y)'; % pi can be substituted for any constant. case 5 refsol = '(sinh(pi*x)*sin(pi*y)+sinh(pi*y)*sin(pi*x))/sinh(pi)'; end % Geometry definition. gobj = gobj_rectangle(); fea.geom.objects = { gobj }; % Grid generation. switch opt.igrid case -1 fea.grid = rectgrid(round(1/opt.hmax)); fea.grid = quad2tri(fea.grid); case 0 fea.grid = rectgrid(round(1/opt.hmax)); 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,@convectiondiffusion); % Add convection and diffusion physics mode. fea.phys.cd.sfun = { opt.sfun }; % Set shape function. fea.phys.cd.eqn.coef{2,4} = { opt.cd }; % Set diffusion coefficient. fea.phys.cd.bdr.sel = [1 1 1 1]; fea.phys.cd.bdr.coef{1,end} = repmat({refsol},1,n_bdr); % Set Dirichlet boundary coefficient to reference solution. fea = parsephys(fea); % Check and parse physics modes. else fea.dvar = { 'c' }; % 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 = { opt.cd }; % Diffusion 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 = { 0 }; % Coefficient used in right hand side. % Define boundary conditions. fea.bdr.d = cell(1,n_bdr); [fea.bdr.d{:}] = deal(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); % Call to stationary solver. % Postprocessing. s_err = ['abs(',refsol,'-c)']; if ( opt.iplot>0 ) figure subplot(2,1,1) postplot(fea,'surfexpr','c') title('Solution c') subplot(2,1,2) postplot(fea,'surfexpr',s_err,'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('c',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.pass = out.err<0.01; if ( nargout==0 ) clear fea out end