Contents Functions PDF Index |
Purpose
Formulation and material support for the heat equation.
Syntax
il = p_heat('default')
Description
This help starts by describing the main commands : p_heat Database and Dbval. Supported p_heat subtypes and their formats are then described. For theory see section 6.1.13.
Element properties are described by the row of an element property matrix or a data structure with an .il field containing this row (see section 7.4). Element property functions such as p_solid support graphical editing of properties and a database of standard properties.
p_heat database
il=p_heat('database');
Accepted commands for the database are
For fixed values, use p_heat('info').
Example of database property construction
il=p_heat([100 fe_mat('p_heat','SI',1) 0 -3 3],... 'dbval 101 d3 -3 2');
Element properties are described by the row of an element property matrix or a data structure with an .il field containing this row. Element property functions such as p_beam support graphical editing of properties and a database of standard properties.
[ProId fe_mat('p_heat','SI',1) CoordM Integ DIM]
ProID | element property identification number |
type | identifier obtained with fe_mat('p_beam','SI',1) |
Integ | is rule number in integrules |
DIM | is problem dimension 2 or 3 D |
[ProId fe_mat('p_heat','SI',2) CoordM Integ DIM]
ProID | element property identification number |
type | identifier obtained with fe_mat('p_beam','SI',2) |
Integ | is rule number in integrules |
DIM | is problem dimension 2 or 3 D |
This command can be used to define a surface exchange and optionally associated load. Surface exchange elements add a stiffness term to the stiffness matrix related to the exchange coefficient Hf defined in corresponding material property. One then should add a load corresponding to the exchange with the source temperature at T_{0} through a convection coefficient Hf which is Hf.T_0. If not defined, the exchange is done with source at temperature equal to 0.
model=p_heat('SetFace',model,SelElt,pl,il);
Command option -load T can be used to defined associated load, for exchange with fluid at temperature T. Note that if you modify Hf in surface exchange material property you have to update the load.
Following example defines a simple cube that exchanges with thermal source at 55 deg on the bottom face.
model=femesh('TestHexa8'); % Build simple cube model model.pl=m_heat('dbval 100 steel'); % define steel heat diffusion parameter model.il=p_heat('dbval 111 d3 -3 1'); % volume heat diffusion (1) model=p_heat('SetFace-load55',... % exchange at 55 deg model,... 'SelFace & InNode{z==0}',... % on the bottom face 100,... % keep same matid for exchange coef p_heat('dbval 1111 d3 -3 2')); % define 3d, integ-3, for surface exchange (2) cf=feplot(model); fecom colordatapro def=fe_simul('Static',model); % compute static thermal state mean(def.def)
Consider a bi-dimensional annular thick domain Ω with radii r_{e}=1 and r_{i}=0.5. The data are specified on the internal circle Γ_{i} and on the external circle Γ_{e}. The solid is made of homogeneous isotropic material, and its conductivity tensor thus reduces to a constant k. The steady state temperature distribution is then given by
− k Δθ(x,y) = f(x,y) in Ω. (9.13) |
The solid is subject to the following boundary conditions
| (x,y) = g(x,y) (9.14) |
θ(x,y)=θ_{ext}(x,y) (9.15) |
In above expressions, f is an internal heat source, θ_{ext} an external temperature at r=r_{e}, and g a function. All the variables depend on the variable x and y.
The OpenFEM model for this example can be found in ofdemos('AnnularHeat').
Numerical application : assuming k=1, f=0, Hf=1e^{−10}, θ_{ext}(x,y) = exp(x) cos(y) and g(x,y)= −exp(x)/r_{i} ( cos(y) x − sin(y) x ), the solution of the problem is given by
θ(x,y) = exp(x) cos(y) |
See also
section 6.1.13, section 4.5.1, fe_mat