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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_MULTIPHASE1 Static bubble example.
[ FEA, OUT ] = EX_MULTIPHASE1( VARARGIN ) Sets up and solves a stationary static bubble example where the pressure jump should be equal to sigma/r. Accepts the following property/value pairs.
Input Value/{Default} Description
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rho scalar {1e4} Density
miu scalar {1} Molecular/dynamic viscosity
sigma scalar {1} Coefficient of surface tension
r scalar {0.25} Bubble radius
lbi scalar 1/{0} Exact or transformed surface tension source term
igrid scalar 0/{1} Cell type (0=quadrilaterals, 1=triangles)
hmax scalar {1/30} Max grid cell size
ischeme scalar {3} Time stepping scheme
sf_u string {sflag2} Shape function for velocity
sf_p string {sflag1} Shape function for pressure
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', 1e4; ...
'miu', 1; ...
'sigma', 1; ...
'r', 0.25; ...
'lbi', 0; ...
'igrid', 1; ...
'hmax', 1/30; ...
'ischeme' 3; ...
'sf_u', 'sflag2'; ...
'sf_p', 'sflag1'; ...
'iplot', 1; ...
'tol', 0.3; ...
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Geometry and grid generation.
fea.sdim = { 'x' 'y' }; % Coordinate names.
fea.grid = rectgrid(round(1/opt.hmax),round(2/opt.hmax),[0 1;0 1]);
if ( opt.igrid==1 )
fea.grid = quad2tri( fea.grid );
end
n_bdr = max(fea.grid.b(3,:)) + 1; % Increment number of boundaries.
fea.grid.b(3,1) = n_bdr; % Set first boundary edge to boundary n_bdr.
% Problem definition.
phi = ['(sqrt((x-0.5)^2+(y-0.5)^2)-',num2str(opt.r),')']; % Level set function.
dh = num2str(1*opt.hmax); % Smoothing region width.
dw = ['(',phi,'/',dh,')'];
smhs = ['((0.5*(1+',dw,'+1/pi*sin(pi*',dw,')))*(',dw,'>-1)*(',dw,'<1)+(',dw,'>=1))']; % Smooth heaviside function.
smdel = ['0.5*(1+cos(pi*',dw,'))/',dh,'*(',dw,'>-1)*(',dw,'<1)']; % Smooth delta function.
sigma = num2str(opt.sigma);
nx = ['(2*x-1)/(2*sqrt((x-0.5)^2+(y-0.5)^2+eps))'];
ny = ['(2*y-1)/(2*sqrt((x-0.5)^2+(y-0.5)^2+eps))'];
% Surface tension force source term.
if ( ~opt.lbi ) % Exact curvature.
kappa = ['1/sqrt((x-0.5)^2+(y-0.5)^2+eps)'];
fx1 = ['-',sigma,'*',kappa,'*',nx,'*',smdel];
fy1 = ['-',sigma,'*',kappa,'*',ny,'*',smdel];
else % With Laplace-Beltrami transformation.
fx1 = ['-',sigma,'*(1-(',nx,')^2)*',smdel];
fx2 = ['-',sigma,'*(-(',nx,')*(',ny,'))*',smdel];
fy1 = ['-',sigma,'*(-(',ny,')*(',nx,'))*',smdel];
fy2 = ['-',sigma,'*(1-(',ny,')^2)*',smdel];
end
% Add Navier-Stokes equations physics mode.
fea = addphys(fea,@navierstokes);
fea.phys.ns.eqn.coef{1,end} = { opt.rho };
fea.phys.ns.eqn.coef{2,end} = { opt.miu };
fea.phys.ns.sfun = { opt.sf_u opt.sf_u opt.sf_p };
fea.phys.ns.eqn.coef{3,end} = { fx1 };
fea.phys.ns.eqn.coef{4,end} = { fy1 };
fea.phys.ns.bdr.sel(n_bdr) = 4; % Set pressure to zero on last boundary segment.
% Parse problem.
fea = parsephys(fea);
if ( opt.lbi )
fea.eqn.f.form{1} = [2 3];
fea.eqn.f.form{2} = [2 3];
fea.eqn.f.coef{1} = {fx1 fx2};
fea.eqn.f.coef{2} = {fy1 fy2};
end
fea = parseprob(fea);
% Call to time-dependent solver.
fea.sol.u = solvetime( fea, ...
'fid', fid, ...
'tmax', 0.3, ...
'tstep', 0.1, ...
'icub', 3, ...
'ischeme', opt.ischeme );
% Postprocessing.
if ( opt.iplot>0 )
postplot(fea,'surfexpr','p')
title('Pressure')
end
% Error checking.
ind_p = [fea.eqn.ndof(1)+fea.eqn.ndof(2)+1:fea.eqn.ndof(1)+fea.eqn.ndof(2)+fea.eqn.ndof(3)];
dp = max(fea.sol.u(ind_p,end)) - min(fea.sol.u(ind_p,end));
out.err = abs(opt.sigma/opt.r - dp)/(opt.sigma/opt.r);
out.pass = out.err<opt.tol;
if ( nargout==0 )
clear fea out
end