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p_heat

• [Database,Dbval] ...
• Heat equation element properties
• 1 : Volume element for heat diffusion (dimension DIM)
• 2 : Surface element for heat exchange (dimension DIM-1)
• 1 : Heat equation material
• 2D validation

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.

[Database,Dbval] ...

p_heat database

 il=p_heat('database');

Heat equation element properties

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_beam support graphical editing of properties and a database of standard properties.

For a tutorial on material/element property handling see section 4.2.1. For a programmers reference on formats used to describe element properties see section 7.4.

1 : Volume element for heat diffusion (dimension DIM)

  [ProId fe_mat('p_heat','SI',1) CordM Integ DIM]

2 : Surface element for heat exchange (dimension DIM-1)

   [ProId fe_mat('p_heat','SI',2) CordM Integ DIM] 

1 : Heat equation material

   [MatId fe_mat('m_heat','SI',2) k rho C alpha]

2D validation

Consider a bi-dimentinal annular thick domain Ω with radii r_e=1 and r_i=0.5. The data are specified on the internal circle Γ_i ans 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

The solid is subject to the following boundary conditions

• Γ_i  (r=r_i) : Neumann condition
  

  ∂ θ ∂ n

  (x,y) = g(x,y)     (8.14)

• Γ_e  (r=r_e) : Dirichlet condition
  

  θ(x,y)=θext(x,y)     (8.15)

In above expressions, f is an internal heat source, θ_ext an external temperature at r=r_e, and g a fonction. 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, α=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

See also

Section 4.2.1, section 7.4, fe_mat