FEPy - Finite Element (Python)

Python project to build the Delaunay triangulation of 2D points and solve the Poisson and convection-diffusion equations in different study cases (Cavity and Oven).

Oven study case is inspired from a study case proposed during a class, which was initially solved using the open-source PDE solver FreeFem++.

The utlimate goal was to implement from scratch both a mesher and a PDE solver that could be used for this case. Along the way, additional features were developped such as time-dependant calculations and adaptive meshing.

Study case: Oven heating

We will note $\Omega$ the opening representing the oven. The oven is a square of 1 m side, and the room is a rectangle of 1 m by 0.4 m, in the center of the oven (to simplify, we will take the origin in the center of the square). We will note $C_i$ the resistors (modeled here as circles of radius 0.05), and placed at the points (±0.75, ±0.75) (initially, $N_r$ = 4). The upper edge of the furnace $\Gamma_u$ is assumed to be maintained at $T_u = 50 \degree C$, the lower edge $\Gamma_d$ at $T_d = 10 \degree C$, and the two lateral edges $\Gamma_l$ are isolated (zero heat flux). The temperature is thus a solution of the equation

$$ \begin{align*} -\text{div}(k \ \text{grad}T) & = f & \text{dans} \ \Omega \\ T & = T_u & \text{sur} \ \Gamma_u \\ T & = T_d & \text{sur} \ \Gamma_d \\ k\frac{\partial T}{\partial n} & = 0 & \text{sur} \ \Gamma_l \end{align*} $$

In this equation, $k$ is the thermal diffusion coefficient, which is variable in the oven: it is 1 in the room, and 10 in the rest of the oven. $f$ is the heat source, and will be of the form $f = \sum\limits_{i=1}^{N_r}\alpha_i{1}_{c_i}$, where the $\alpha_i$ coefficients are either chosen (in the straightforward problem), or to be determined (in the inverse problem).

Mesh

Delaunay triangulation

• Unstructured
• Constrained
• Maximum mesh size
• Minimum angle of 20°

PDE Solver

The variational formulation of the problem is as follows

$$ \begin{align*} T & = T_u & \text{sur} \ \Gamma_u \\ T & = T_d & \text{sur} \ \Gamma_d \\ \int_{\Omega} k \nabla(T).\nabla(v) & = \int_{\Omega} fv & \text{for all} \ v \in H^{1}(\Omega) \ \text{such that:} \ v=0 \ \text{sur} \ (\Gamma_u\cup\Gamma_d) \end{align*} $$