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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_NAVIERSTOKES1 2D Example for incompressible stationary flow in a channel.
[ FEA, OUT ] = EX_NAVIERSTOKES1( VARARGIN ) Sets up and solves stationary Poiseuille flow in a rectangular channel. The inflow profile is constant and the outflow should assume a parabolic profile ( u(y)=U_max*4/h^2*y*(h-y) ). Accepts the following property/value pairs.
Input Value/{Default} Description
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rho scalar {1} Density
miu scalar {0.001} Molecular/dynamic viscosity
umax scalar {0.3} Maximum magnitude of inlet velocity
h scalar {0.5} Channel height
l scalar {2.5} Channel length
igrid scalar 1/{0} Cell type (0=quadrilaterals, 1=triangles)
hmax scalar {0.04} Max grid cell size
sf_u string {sflag1} Shape function for velocity
sf_p string {sflag1} Shape function for pressure
iphys scalar 0/{1} Use physics mode to define problem (=1)
solver string openfoam/su2/{} Use OpenFOAM, SU2, FEniCS, or default solver
ischeme scalar {0} Time stepping scheme (0 = stationary)
iplot scalar 0/{1} Plot solution and error (=1)
.
Output Value/(Size) Description
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fea struct Problem definition struct
out struct Output struct
cOptDef = { ...
'rho', 1;
'miu', 1e-3;
'umax', 0.3;
'h', 0.5;
'l', 2.5;
'igrid', 1;
'hmax', 0.04;
'sf_u', 'sflag1';
'sf_p', 'sflag1';
'iphys', 1;
'solver', '';
'ischeme', 0;
'iplot', 1;
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Model parameters.
rho = opt.rho; % Density.
miu = opt.miu; % Molecular/dynamic viscosity.
umax = opt.umax; % Maximum magnitude of inlet velocity.
% Geometry and grid parameters.
h = opt.h; % Height of rectangular domain.
l = opt.l; % Length of rectangular domain.
% Discretization parameters.
sf_u = opt.sf_u; % FEM shape function type for velocity.
sf_p = opt.sf_p; % FEM shape function type for pressure.
% Geometry definition.
gobj = gobj_rectangle( 0, l, 0, h );
fea.geom.objects = { gobj };
fea.sdim = { 'x' 'y' }; % Coordinate names.
% Grid generation.
if ( opt.igrid==1 )
fea.grid = gridgen(fea,'hmax',opt.hmax,'fid',fid);
else
fea.grid = rectgrid(round(l/opt.hmax),round(h/opt.hmax),[0 l;0 h]);
if( opt.igrid<0 )
fea.grid = quad2tri( fea.grid );
end
end
n_bdr = max(fea.grid.b(3,:)); % Number of boundaries.
% Boundary conditions.
dtol = opt.hmax;
i_inflow = findbdr( fea, ['x<',num2str(dtol)] ); % Inflow boundary number.
i_outflow = findbdr( fea, ['x>',num2str(l-dtol)] ); % Outflow boundary number.
s_inflow = ['2/3*',num2str(umax)]; % Definition of inflow profile.
s_refsol = ['4*',num2str(umax),'*(y*(',num2str(h),'-y))/',num2str(h),'^2']; % Definition of velocity profile.
% Problem definition.
if ( opt.iphys==1 )
fea = addphys(fea,@navierstokes); % Add Navier-Stokes equations physics mode.
fea.phys.ns.eqn.coef{1,end} = { rho };
fea.phys.ns.eqn.coef{2,end} = { miu };
fea.phys.ns.eqn.coef{5,end} = { s_inflow };
if( any(strcmp(opt.solver,{'openfoam','su2'})) )
fea.phys.ns.sfun = { 'sflag1', 'sflag1', 'sflag1' };
else
fea.phys.ns.sfun = { sf_u sf_u sf_p }; % Set shape functions.
end
fea.phys.ns.bdr.sel(i_inflow) = 2;
fea.phys.ns.bdr.sel(i_outflow) = 4;
fea.phys.ns.bdr.coef{2,end}{1,i_inflow} = s_inflow; % Set inflow profile.
fea = parsephys(fea); % Check and parse physics modes.
else
fea.dvar = { 'u' 'v' 'p' }; % Dependent variable name.
fea.sfun = { sf_u sf_u sf_p }; % Shape function.
% Define equation system.
cvelx = [num2str(rho),'*',fea.dvar{1}]; % Convection velocity in x-direction.
cvely = [num2str(rho),'*',fea.dvar{2}]; % Convection velocity in y-direction.
fea.eqn.a.form = { [2 3 2 3;2 3 1 1] [2;3] [1;2];
[3;2] [2 3 2 3;2 3 1 1] [1;3];
[2;1] [3;1] [] };
fea.eqn.a.coef = { {2*miu miu cvelx cvely} miu -1;
miu {miu 2*miu cvelx cvely} -1;
1 1 [] };
fea.eqn.f.form = { 1 1 1 };
fea.eqn.f.coef = { 0 0 0 };
% Define boundary conditions.
fea.bdr.d = cell(3,n_bdr);
[fea.bdr.d{1:2,:}] = deal( 0 );
fea.bdr.d{1,i_inflow} = s_inflow;
[fea.bdr.d{:,i_outflow }] = deal([]);
% fea.bdr.d{end,i_outflow} = 0; % Set pressure to zero on outflow boundary.
fea.bdr.n = cell(3,n_bdr);
end
% Parse and solve problem.
fea = parseprob(fea); % Check and parse problem struct.
if( opt.iphys==1 && strcmp(opt.solver,'fenics') )
fea = fenics( fea, 'fid', fid, 'ischeme', opt.ischeme, 'tmax', 10 );
elseif( opt.iphys==1 && strcmp(opt.solver,'openfoam') )
if( opt.ischeme==0 )
dt = 1.0;
tstop = 1000;
ddtSchemes = 'steadyState';
elseif( opt.ischeme==1 )
dt = 0.1;
tstop = 100;
ddtSchemes = 'backward';
elseif( opt.ischeme>=2 )
dt = 0.1;
tstop = 100;
ddtSchemes = 'CrankNicolson 0.9';
end
logfid = fid; if( ~got.fid ), fid = []; end
fea.sol.u = openfoam( fea, 'fid', fid, 'logfid', logfid, 'ddtSchemes', ddtSchemes, 'deltaT', dt, 'endTime', tstop, 'nproc', 1 );
fid = logfid;
elseif( opt.iphys==1 && strcmp(opt.solver,'su2') )
logfid = fid; if( ~got.fid ), fid = []; end
fea.sol.u = su2( fea, 'fid', fid, 'logfid', logfid, 'ischeme', opt.ischeme, 'tstep', 0.5, 'tmax', 20+30*(opt.ischeme==1) );
fid = logfid;
else
if( opt.ischeme==0 )
jac.form = {[1;1] [1;1] [];[1;1] [1;1] []; [] [] []};
jac.coef = {[num2str(rho),'*ux'] [num2str(rho),'*uy'] []; [num2str(rho),'*vx'] [num2str(rho),'*vy'] []; [] [] []};
fea.sol.u = solvestat( fea, 'fid', fid, 'nsolve', 2, 'jac', jac ); % Call to stationary solver.
else
fea.sol.u = solvetime( fea, 'fid', fid, 'ischeme', opt.ischeme, 'tmax', 10 );
end
end
fea.sol.u = fea.sol.u(:,end);
% Postprocessing.
s_velm = 'sqrt(u^2+v^2)';
s_err = ['abs(sqrt((',s_refsol,')^2)-(',s_velm,'))'];
s_len = ['(x>',num2str(3/4*l),')'];
if ( opt.iplot>0 )
figure
subplot(3,1,1)
postplot(fea,'surfexpr',s_velm,'evaltype','exact')
title('Velocity field')
subplot(3,1,2)
postplot(fea,'surfexpr','p','evaltype','exact')
title('Pressure')
subplot(3,1,3)
postplot(fea,'surfexpr',[s_err,'*',s_len],'evaltype','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
c_ind = find(evalexpr0(s_len,xi,1,1:size(fea.grid.c,2),[],fea))';
err = evalexpr0(s_err,xi,1,c_ind,[],fea);
ref = evalexpr0(['sqrt((',s_refsol,')^2)'],xi,1,c_ind,[],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 = err<0.06;
if ( nargout==0 )
clear fea out
end