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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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EX_HEATTRANSFER4 2D Heat transfer with convective cooling.
[ FEA, OUT ] = EX_HEATTRANSFER4( VARARGIN ) NAFEMS T4 benchmark example for two dimensional heat transfer with convective heat flux boundary conditions.
_ q_n=h*(T_amb-T)
^ +--------+
| | |
| q_n=0 | | q_n=h*(T_amb-T)
1m | |
| | T(0.6,0.2)?
| | |
v +--------+
T=100
|<-0.6m->|
A 0.6 by 1 m iron plate, with density 7850 kg/m^3, heat capacity 460 J/kgC, and thermal conductivity 52 W/mC, is prescribed a fixed temperature of T = 100 C at the bottom edge. The left side is insulated, and the right and top boundaries exposed to convective cooling with a heat transfer coefficient h = 750 W/m^2K. The steady state temperature at the point (0.6,0.2) is sought when the surrounding ambient temperature is T_amb = 0 C.
[1] Cameron AD, Casey JA, Simpson GB. Benchmark Tests for Thermal Analysis,
The National Agency for Finite Element Standards, UK, 1986.
Accepts the following property/value pairs.
Input Value/{Default} Description
-----------------------------------------------------------------------------------
hmax scalar {0.025} Grid cell size
igrid scalar {0}/1/2 Cell type (0=quadrilaterals, 1=triangles,
2=triangles converted from quadrilaterals)
sfun string {sflag1} Finite element shape function
solver string fenics/{} Use FEniCS or default solver
istat scalar {1}/0 Use stationary (=1), or time dependent solver
iplot scalar {1}/1 Plot solution (=1)
.
Output Value/(Size) Description
-----------------------------------------------------------------------------------
fea struct Problem definition struct
out struct Output struct
cOptDef = { 'hmax', 0.025;
'igrid', 0;
'sfun', 'sflag1';
'solver', '';
'istat', 1;
'iplot', 1;
'tol', 1e-2;
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
% Geometry definition.
gobj = gobj_rectangle( 0, 0.6, 0, 1 );
fea.geom.objects = { gobj };
% Grid generation.
switch opt.igrid
case 0
fea.grid = rectgrid( round(0.6/opt.hmax), round(1/opt.hmax), [0 0.6;0 1] );
case 1
fea.grid = gridgen( fea, 'hmax', opt.hmax, 'fid', opt.fid );
case 2
fea.grid = rectgrid( round(0.6/opt.hmax), round(1/opt.hmax), [0 0.6;0 1] );
fea.grid = quad2tri( fea.grid, 1 );
end
% Problem definition.
fea.sdim = { 'x', 'y' }; % Space coordinate name.
fea = addphys( fea, @heattransfer ); % Add heat transfer physics mode.
fea.phys.ht.sfun = { opt.sfun }; % Set shape function.
% Equation coefficients.
fea.phys.ht.eqn.coef{1,end} = 7850; % Density
fea.phys.ht.eqn.coef{2,end} = 460; % Heat capacity.
fea.phys.ht.eqn.coef{3,end} = 52; % Thermal conductivity.
fea.phys.ht.eqn.coef{7,end} = { 0 }; % Initial temperature.
% Boundary conditions.
fea.phys.ht.bdr.sel = [1 4 4 3];
fea.phys.ht.bdr.coef{1,end} = { 100 [] [] [] };
fea.phys.ht.bdr.coef{4,end}{2}{2} = 750;
fea.phys.ht.bdr.coef{4,end}{3}{2} = 750;
% Parse physics modes and problem struct.
fea = parsephys(fea);
fea = parseprob(fea);
% Compute solution.
if( strcmp(opt.solver,'fenics') )
fea = fenics( fea, 'fid', opt.fid, ...
'tstep', 100, 'tmax', 20000, 'ischeme', 2*(~opt.istat), 'nproc', 1 );
else
if( opt.istat )
fea.sol.u = solvestat( fea, 'fid', opt.fid, 'init', {'T0_ht'} );
else
[fea.sol.u, tlist] = solvetime( fea, 'fid', opt.fid, 'init', {'T0_ht'}, ...
'tmax', 20000, 'tstep', 100, 'toldef', 1e-4, 'maxnit', 5 );
end
end
% Postprocessing.
if( opt.iplot>0 )
postplot( fea, 'surfexpr', 'T', 'isoexpr', 'T' )
title('Temperature, T')
end
% Error checking.
T_sol = evalexpr( 'T', [0.6;0.2], fea );
T_ref = 18.3;
out.err = abs(T_sol-T_ref)/T_ref;
out.pass = out.err<opt.tol;
if( nargout==0 )
clear fea out
end