Finite Element Method  2D Mesh Generator  Metfem2D
Generator Optimization via the Metropolis method
Boundary mesh elements
The Metropolis algorithm^{[1}^{, 2}^{]} applied to boundary nodes slightly differs from the above– described case and could be summarize in the following steps:

Find all the boundary or edge nodes i. e. nodes for which p_{k, edge} ∈ Γ.

Find triangles in the closest neighbourhood of the considered p_{k,edge} node. Then calculate an area of each triangle A_{l,edge}.

Calculate the force acting on each boundary node except the constant nodes and coming from only two boundary nodes connected to it. This imposes the following constrain on the motion of the kth node in order to keep it in the boundary Γ

F_{i} = 

∑
_{j=1}

F_{j} δ r_{ji}


where δr_{jk} is defined as previously. The force is tangential to the boundary Γ.

Similarly, find an area of each triangle A^{new}_{l,edge} after shifting p_{k,edge} → p^{new}_{k,edge} according to the force F_{k}.

Adopt the Metropolis energetic condition to the boundary case i.e.

δE =

∑
_{l}

(
(A^{new}_{l, edge}  A)^{2}  (A_{l, edge}  A)^{2}
)

l = 1, 2, …, L.

If
e^{δE⁄T} > r
the new configuration is accepted otherwise is rejected. T denotes temperature and a random number r ∈ U(0; 1) as previously.

The main point of this part is to ensure that the boundary nodes are moved just along the boundary Γ.
References