first of all, thanks for the help you guys provide! My question is the following: I have one power load $q^*$ [W/$m^2$] (a scalar) defined over a triangle $\mathcal{T}^*$ of surface $A^*$. My target is to compute the surface average over the set $\{\mathcal{T}_j\}_{j=1}^N$ which contains $\mathcal{T}^*$ itself and other $(N-1)$ neighboring triangles (neighbor = triangle which shares at least one vertex with the reference ($^*$) one), each one with its power load $q_j$. The result is then assigned to $\mathcal{T}^*$. To do so, I relied on the formula:
\begin{equation} \overline{q}^* = \frac{1}{A} \iint_{A}q(x,y,z) dA \end{equation}
where $q(x,y,z)$ is given by $\{q_j\}$ and $A$ by $\{A_j\}$. In this discrete case, it should read:
\begin{equation} \overline{q}^* = \frac{\sum_{j=1}^N q_jA_j}{\sum_{j=1}^NA_j} \end{equation}
With, in general, $q^*\neq \overline{q}^*$, the total power [W] on $A^*$ changes: $P^* = q^*A^*$ vs. $\overline{P}^*=\overline{q}^*A^*$. This operation is then to be carried out for a second triangle with its neighbors and so on. The total power $P_{tot}$ (no average) is simply given by the sum:
\begin{equation} P_{tot} = \sum_{i=1}^M P_i = \sum_{i=1}^M q_iA_i \end{equation}
with $M$ being the total number of triangles (in the mesh). Instead:
\begin{equation} \overline{P}_{tot} = \sum_{i=1}^M \overline{P}_i = \sum_{i=1}^M \overline{q}_iA_i \end{equation}
This operation does not seems to be energy (equivalent to power here...) conservative, i.e. $P_{tot} \neq \overline{P}_{tot}$. In particular, I applied this operation to different cases and I always end up with:
\begin{equation} P_{tot} < \overline{P}_{tot} \end{equation}
I would like to have an explanation for such an effect since, being tied to the energy conservation, is a relevant topic. It can also be put on a more general ground: given a discrete set of variables, find a suitable surface average which guarantees the conservation of a macroscopic quantity.
Thank you very much in advance,
Matteo