Heat Toolbox

1. Models

2. Materials

Table 1. Materials properties defined in the heat toolbox (symbols are given without components suffix)
Name	Symbol	Shape	Description
thermal-conductivity	k	scalar or matrix	thermal conductivity
density	rho	scalar	density
thermal-expansion	beta	scalar	thermal expansion
specific-heat-capacity	Cp	scalar	specific heat capacity

3. Fields and symbols expressions

Table 2. Fields available
Name	Description	Shape
temperature	the temperature field	scalar

Table 3. Symbols expressions by considering `heat` as the keyword used with the toolbox
Symbol	Expression	Description
heat_T	\(T\)	evaluate the temperature
heat_grad_T_0	\(\frac{\partial T}{\partial x}\)	evaluate the first component of gradient of temperature
heat_grad_T_1	\(\frac{\partial T}{\partial y}\)	evaluate the second component of gradient of temperature
heat_grad_T_2	\(\frac{\partial T}{\partial z}\)	evaluate the third component of gradient of temperature
heat_dn_T	\(\nabla T \cdot \boldsymbol{n}\)	evaluate the normal derivative of temperature
heat_nflux	\(-k \nabla T \cdot \boldsymbol{n}\)	evaluate the normal flux

4. Boundary Conditions

4.1. Dirichlet

\[T = g \quad \text{ on } \Gamma\]

4.2. Neumann

\[-k \nabla T \cdot \boldsymbol{n} = g \quad \text{ on } \Gamma\]

4.3. Robin

\[-k \nabla T \cdot \boldsymbol{n} = h \left( T - g \right) \quad \text{ on } \Gamma\]

5. Initial Conditions

6. Post Processing

6.1. Exports

Table 4. Fields allowed to be exported in the `fields` section are:
Name	Description
temperature	the temperature field
velocity-convection	evaluate the gradient of temperature
pid	the mesh partitioning
all	all fields available

All materials properties given in the section Materials can be also exported by specifying the name in the fields entry.

6.2. Measures

6.2.1. Normal-Heat-Flux

Example of a Parameters section

"Normal-Heat-Flux": (1)
{
    "bottom": (2)
    {
        "markers":["bottom1,bottom2"], (3)
        "direction":"inward" (4)
     },
     "top": (5)
     {
         "direction":"inward"
     },

}

1	kind of measure
2	name of measure
3	surface markers
4	direction : `inward` or `outward`
5	another

6.2.2. Evaluation at points

temperature

7. Stabilization methods

7.1. GLS family