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====== Heat Source/Flux ======
Heat source elements and heat flux boundary elements can be applied using special finite elements. As any finite element, their definition require an ''ElementProperty'' object and an ''Interaction'' (''HeatInteraction'') object. There are no ''Material'' objects associated to these elements.

''ElementProperty'' contains the type of element and all necessary information to compute the value of the heat flux/source at each Gauss point with a spatial distribution function. This spatial heat distribution function is expressed in a set of local coordinates {$x'$, $y'$, $z'$}, which is handled by the ''HeatInteraction''.

{{ doc:user:elements:boundaries:heat_localcoord.png?400 |Coordonnées locales du la source de chaleur}}

Using this set of local coordinates allows to easily define a moving/rotating heat source/flux, which is particularly useful for some applications (//e.g.// additive manufacturing).
===== Materials=====     

Since flux interactions are boundary conditions interactions, no materials must be associated to the element.

===== Element =====

Therefore, the first step consist in defining an ''[[doc:user:elements:general:def_element_properties|ElementProperties]]'', as

  prp = ElementProperties(typeEl)
  prp.put(param1, value1)
  prp.depend(param1, fct1, Lock1)) #optional
  ...

where
|''typeEl'' | desired element (for example ''Tm[2]HeatFlux[2|3]DElement'')|
|''param1''| name of the property associated to the element (for example ''HEATEL_VALUE'')|
|''value1''| value of the corresponding property |
|''fct1''| function which characterizes the dependency of the property (optional: no fct if no dependency) |
|''Lock1''| [[doc:user:general:locks|Lock]] which defines the dependency variable of the property (compulsory if there is a dependency) |
  ...

==== Tm[2]HeatFlux[2|3]DElement ====
=== Description ===
Thermal heat flux element in 2/3D, first or second order (thermal field of second order), that can be created on "boundary" geometries (//i.e.// curves in 2D and sides in 3D).

There are currently 4 different heat flux distributions types that are implemented for this element. These can be selected by using the ''HEATEL_TYPE'' parameter when defining the element properties.

== Constant Distribution (=default) ==
Heat flux at each Gauss point is equal to ''HEATEL_VALUE'' [W/m$^2$].
  prp.put(HEATEL_TYPE, HEATEL_CONSTANT)

== Rectangular Distribution ==
Heat flux at each Gauss point is equal to a uniform distribution of the total heat $Q_{src}$ within a rectangular surface centered on the local heat flux coordinates
$$
  q = \frac{Q_{src}}{4a b}~~~\text{if } x'\in [-a,~a],~~ y'\in [-b,~b],
$$
where $a$ and $b$ are the half lengths of the rectangle in the $x'$ and $y'$ local coordinate directions respectively.
  prp.put(HEATEL_TYPE, HEATEL_RECTANGULAR)

== Ellipsoid Distribution ==
Heat flux at each Gauss point is equal to an ellipsoid Gaussian distribution function of the total heat $Q_{src}$ centered on the local heat flux coordinates [Goldak //et. al.// 1986]
$$
q = \frac{Q_{src} 6\sqrt{3}}{ab \pi^\frac{3}{2}}~ e^{-3\left(\frac{x'}{a}\right)^2}~ e^{-3\left(\frac{y'}{b}\right)^2}
$$
where $a$ and $b$ are the semi-axes lengths of the ellipsoid in the $x'$ and $y'$ directions respectively.
  prp.put(HEATEL_TYPE, HEATEL_ELLIPSOID)

== Double Ellipsoid Distribution ==
Modification of the ellipsoid Gaussian distribution function to account for a different distribution at the front  ($x'>=0$) and at the rear ($x'<0$) of the heat flux [Goldak //et. al.// 1986]
$$
q_f = f \frac{Q_{src} 6\sqrt{3}}{ab \pi^\frac{3}{2}}~ e^{-3\left(\frac{x'}{a}\right)^2}~ e^{-3\left(\frac{y'}{b}\right)^2}, ~~~~~~~~x'>=0
$$
$$
q_r = (1-f) \frac{Q_{src} 6\sqrt{3}}{a_rb \pi^\frac{3}{2}}~ e^{-3\left(\frac{x'}{a_r}\right)^2}~ e^{-3\left(\frac{y'}{b}\right)^2}, x'<0
$$
where $a$ and $a_r$ are the front and rear semi-axes lengths in the $x'$ directions, $b$ is the semi-axis length in the $y'$ direction, and $f=\frac{ba}{a+a_r}$ is the balancing factor.
  prp.put(HEATEL_TYPE, HEATEL_DOUBLE_ELLIPSOID)

=== Parameters ===

^   Name                                          ^    Metafor Code      ^ Dependency  ^
| Type of surface distribution                                        |   ''HEATEL_TYPE''    |       -     |
| Total applied heat $Q_{src}$[W] \\ (heat per area for ''HEATEL_CONSTANT'')                              |  ''HEATEL_VALUE''  |    ''TO/TM''        |
| Semi-axis Length ($a$)                     |   ''HEATEL_A''   |     -        |
| Semi-axis Length ($b$)                     |   ''HEATEL_B''   |      -       |
| Semi-axis Length ($a_r$)                     |   ''HEATEL_AR''   |      -       |
| Number of integration points                  |   ''NIP''   |      -       |
| Material Stiffness  \\ (STIFF_ANALYTIC - STIFF_NUMERIC) \\ only if element Stiffness == STIFF_ANALYTIC | ''MATERIALSTIFFMETHOD''  |      -       |

==== Tm[2]HeatSource[2|3]DElement ====
=== Description ===
Thermal heat source element in 2/3D, first or second order (thermal field of second order), that can be created on "volume" geometries (//i.e.// sides in 2D and volumes in 3D).

There are currently 4 different types of heat source distributions that are implemented for this element. These can be selected by using the ''HEATEL_TYPE'' parameter when defining the element properties.

== Constant Distribution (=default) ==
Heat source at each Gauss point is equal to ''HEATEL_VALUE'' [W/m$^3$].
  prp.put(HEATEL_TYPE, HEATEL_CONSTANT)

== Rectangular Distribution ==
Heat source at each Gauss point is equal to a uniform distribution of the total heat $Q_{src}$ within a box volume centered on the local heat flux coordinates
$$
  q = \frac{Q_{src}}{8a b c}~~~\text{if } x'\in [-a,~a],~~ y'\in [-b,~b],~~ z'\in [-c,~c],
$$
where $a$, $b$ and $c$ are the half lengths of the rectangle in the $x'$, $y'$ and $z'$ local coordinate directions respectively.
  prp.put(HEATEL_TYPE, HEATEL_RECTANGULAR)

== Ellipsoid Distribution ==
Heat source at each Gauss point is equal to an ellipsoid Gaussian distribution function of the total heat $Q_{src}$ centered on the local heat flux coordinates [Goldak //et. al.// 1986]
$$
q = \frac{Q_{src} 12\sqrt{3}}{abc \pi^\frac{3}{2}}~ e^{-3\left(\frac{x'}{a}\right)^2}~ e^{-3\left(\frac{y'}{b}\right)^2}~ e^{-3\left(\frac{z'}{c}\right)^2}
$$
where $a$, $b$ and $c$ are the semi-axes lengths of the ellipsoid in the $x'$, $y'$ and $z'$ directions respectively.
  prp.put(HEATEL_TYPE, HEATEL_ELLIPSOID)

== Double Ellipsoid Distribution ==
Modification of the ellipsoid Gaussian distribution function to account for a different distribution at the front  ($x'>=0$) and at the rear ($x'<0$) of the heat flux [Goldak //et. al.// 1986]
$$
q_f = f \frac{Q_{src} 12\sqrt{3}}{ab \pi^\frac{3}{2}}~ e^{-3\left(\frac{x'}{a}\right)^2}~ e^{-3\left(\frac{y'}{b}\right)^2}~ e^{-3\left(\frac{z'}{c}\right)^2}, ~~~~~~~~x'>=0
$$
$$
q_r = (1-f) \frac{Q_{src} 12\sqrt{3}}{a_rb \pi^\frac{3}{2}}~ e^{-3\left(\frac{x'}{a_r}\right)^2}~ e^{-3\left(\frac{y'}{b}\right)^2}~ e^{-3\left(\frac{z'}{c}\right)^2}, x'<0
$$
where $a$ and $a_r$ are the front and rear semi-axes lengths in the $x'$ directions, $b$ and $c$ are the semi-axes lengths in the $y'$ and $z'$ direction, and $f=\frac{ba}{a+a_r}$ is the balancing factor.
  prp.put(HEATEL_TYPE, HEATEL_DOUBLE_ELLIPSOID)

=== Parameters ===

^   Name                                          ^    Metafor Code      ^ Dependency  ^
| Type of volume distribution                                        |   ''HEATEL_TYPE''    |       -     |
| Total applied heat $Q_{src}$[W]  \\ (heat per volume $q$[W/m$^3$] for ''HEATEL_CONSTANT'')                              |  ''HEATEL_VALUE''  |    ''TO/TM''        |
| Semi-axis Length ($a$)                     |   ''HEATEL_A''   |     -        |
| Semi-axis Length ($b$)                     |   ''HEATEL_B''   |      -       |
| Semi-axis Length ($c$)                   |   ''HEATEL_C''   |       -      |
| Semi-axis Length ($a_r$)                     |   ''HEATEL_AR''   |      -       |
| Number of integration points                  |   ''NIP''   |      -       |
| Material Stiffness  \\ (STIFF_ANALYTIC - STIFF_NUMERIC) \\ only if element Stiffness == STIFF_ANALYTIC | ''MATERIALSTIFFMETHOD''  |      -       |

==== Tm[2]ConvectionHeatFlux[2|3]DElement ====
<note important> **Metafor version >= 3545** </note>
=== Description ===
Thermal convection heat flux element in 2/3D, first or second order (thermal field of second order), that can be created on "boundary" geometries (//i.e.// curves in 2D and sides in 3D). These elements are particularly suited to model moving hot gas torches.

These elements are similar to ''Tm[2]HeatFlux[2|3]DElement'' except that heat flux is computed as a convection boundary
$$
q = h(x',y')~(T_s - T_f),
$$
with the surface temperature $T_s$ and fluid temperature $T_f$ (''TEMP_FLUIDE''). The convection coefficient $h$ is a space-dependent quantity (in the local coordinates) which writes
$$
h(x',y')=A~\mathcal{f}(x',y'),
$$
where $A$ is the amplitude of the convection coefficient (''CONV_COEF'') and $\mathcal{f}\in [0,~1]$ is a spatial distribution function which can be selected using the ''HEATEL_TYPE'' parameter.

== Rectangular Distribution ==
Convection coefficient is equal to a constant value within a rectangular surface centered in the local coordinates axes and 0 outside the surface
$$
h(x',y') =\begin{cases}
 A~~\text{if}~~~ x'\in [-k_x,~k_x] ~~ \text{and} ~~ y'\in [-k_y,~k_y],\\
 0~~~\text{else}.
\end{cases}
$$
where $k_x$ and $k_y$ are the half lengths of the rectangle in the $x'$ and $y'$ directions respectively. 
   prp.put(HEATEL_TYPE, CONVHEATEL_RECTANGULAR)

== Gaussian Distribution ==
Convection coefficient is distributed with a Gaussian distribution as defined by [Zacherl //et. al.// 2023] centered on the local coordinates 
$$
h(x',y') = A~e^{(-\left[ \frac{x'}{k_x} \right]^2 -\left[ \frac{y'}{k_y} \right]^2)},
$$
where $k_x$ and $k_y$ are concentration coefficients which define the slope of the curve in the $x'$ and $y'$ directions respectively.
   prp.put(HEATEL_TYPE, CONVHEATEL_GAUSSIAN)

== Modified Log-Normal  Distribution ==
Convection coefficient is distributed with a modified log-normal distribution as defined by [Zacherl //et. al.// 2023] centered on the local coordinates 
$$
h(x',y') = A~e^{(-[\text{ln}\left( \frac{|x'|}{k_x}+1 \right)]^2 -[\text{ln}\left( \frac{|y'|}{k_y}+1 \right)]^2)},
$$
where $k_x$ and $k_y$ are concentration coefficients which define the slope of the curve in the $x'$ and $y'$ directions respectively.
   prp.put(HEATEL_TYPE, CONVHEATEL_LOGNORM)

== Combined Distribution ==
Allows to choose between a Gaussian or log-normal distribution in the front ($x' \geq 0$), rear ($x'<0$) and $y'$ directions. Convection coefficient is distributed as
$$
h(x',y') =\begin{cases}
 A~\mathcal{f}_{xf}(x',k_x)~\mathcal{f}_{y}(y',k_y) ~~~~~ x' \geq 0\\
 A~\mathcal{f}_{xr}(x',k_{xr})~\mathcal{f}_{y}(y',k_y) ~~~~ x' < 0.
\end{cases}
$$
where distribution function $\mathcal{f}$ is either a Gaussian distribution (''CONVHEATEL_GAUSSIAN'')
$$
\mathcal{f_i} = e^{-\left[ \frac{i}{k_i} \right]^2},
$$
or a modified log-normal distribution (''CONVHEATEL_LOGNORM'')
$$
\mathcal{f_i} = e^{-[\text{ln}\left( \frac{|i|}{k_i}+1 \right)]^2}.
$$
Difference between these 2 distribution types is highlighted below for $k_i=1$.

{{ doc:user:elements:boundaries:heat_GaussLogNorm.png?400 |Distribution Gaussienne v.s. Log-Normale avec k=1}}

__Example:__ \\
Modelling of an inclined hot gas torch in Automated Fiber Placement process (AFP).
Convection heat flux element is modelled using a modified log-normal distribution at the rear and Gaussian distributions at the front and along $y'$.
    # convection heat source (LogNorm - Gauss - Gauss)
    prpHeat = ElementProperties(TmConvectionHeatFlux3DElement)
    prpHeat.put(        HEATEL_TYPE,  CONVHEATEL_COMBINE)
    prpHeat.put(        TEMP_FLUIDE,          p['T_HGT'])
    prpHeat.put(          CONV_COEF,          p['HGT_A'])
    prpHeat.put( CONVHEATEL_TYPE_XF,  CONVHEATEL_LOGNORM) #front
    prpHeat.put(      CONVHEATEL_KX,         p['HGT_kf'])
    prpHeat.put( CONVHEATEL_TYPE_XR, CONVHEATEL_GAUSSIAN) #rear
    prpHeat.put(     CONVHEATEL_KXR,         p['HGT_kr'])
    prpHeat.put(  CONVHEATEL_TYPE_Y, CONVHEATEL_GAUSSIAN) #y
    prpHeat.put(      CONVHEATEL_KY,         p['HGT_ky'])

=== Parameters ===

^   Name                                          ^    Metafor Code      ^ Dependency  ^
| Type of surface distribution                                        |   ''HEATEL_TYPE''    |       -     |
| Distribution along $x'>=0$ \\ (only for ''HEATEL_TYPE'' = ''CONVHEATEL_COMBINE'')                                      |   ''CONVHEATEL_TYPE_XF''    |       -     |
| Distribution along $x'<0$ \\ (only for ''HEATEL_TYPE'' = ''CONVHEATEL_COMBINE'')                                      |   ''CONVHEATEL_TYPE_XR''    |       -     |
| Distribution along $y'$ \\ (only for ''HEATEL_TYPE'' = ''CONVHEATEL_COMBINE'')                                      |   ''CONVHEATEL_TYPE_Y''    |       -     |
| Fluid temperature $T_f$                             |  ''TEMP_FLUIDE''  |    ''TO/TM''        |
| Amplitude of the convection coefficient $A$                             |  ''CONV_COEF''  |    ''TO/TM''        |
| Concentration factor ($k_x$)                    |   ''CONVHEATEL_KX''   |     -        |
| Concentration factor ($k_y$)                     |   ''CONVHEATEL_KY''   |      -       |
| Concentration factor ($k_{xr}$)                   |   ''CONVHEATEL_KXR''   |       -      |
| Number of integration points                  |   ''NIP''   |      -       |
| Material Stiffness  \\ (STIFF_ANALYTIC - STIFF_NUMERIC) \\ only if element Stiffness == STIFF_ANALYTIC | ''MATERIALSTIFFMETHOD''  |      -       |

===== Interaction =====
The interaction is defined as:

  load = HeatInteraction(no)
  load.push(gObject1)
  load.push(gObject2)
  ...
  load.setAxes(Ox, Oz)
  load.useRescale(bool)
  load.addProperty(prp)
  interactionset.add(load)

where

| ''no''       | number of the ''Interaction''  | 
| ''gObject1'', ''gObject2''  | mesh geometric entity where the boundary conditions are applied  | 
| ''prp''      | [[doc:user:elements:general:def_element_properties|Properties]] of [[#Element|boundary condition elements]] to generate |
| ''Ox'', ''Oz''  | Curve entities that define the local coordinates $x'$ and $z'$ for the heat distribution function (not necessary for ''HEATEL_CONSTANT'')\\ [[doc:user:conditions:displacements]] and/or [[doc:user:conditions:rotations]] can be applied on these curves to obtain a moving heat source | 
| ''useRescale(bool)''  | Rescaling of the heat flux \\ = False (default): do nothing \\ = True: allows the rescaling of all heat source at each beginning of time-step to obtain the exact value of total applied ''HEATEL_VALUE'' \\ :!: not for ''HEATEL_CONSTANT'' and ''Tm[2]ConvectionHeatFlux[2|3]DElement'' |