Let us define the set of mesh triangles Ω = { T_{j}, j = 1, 2, …, M } and a set T^{i} of triangle mesh elements to which a node p_{i} belongs. The closest neighbours C(p_{i}) of the mesh point p_{i} are defined as a subset of mesh points p_{j} ∈ P
∀ p_{i} ∃ T^{i} ⊂ Ω: p_{i} ∈ T^{i} C(p_{i}) = { p_{j}: p_{j} ∈ T^{i} where p_{j} ≠ p_{i} }. |
Note, that the closest region is not the same what the Voronoi region ^{[1]}. Presented definition is needed to proceed with the Metropolis algorithm ^{[2]} which will be applied in order to adjust triangle's area to the desired size given by the element size h.
In turn, a proper triangulation is the essence of the finite element method as it is stated in the Sec. 2. Let us divide the whole problem into two different tasks. The first one focuses on finding an optimization for mesh elements being the internal elements whereas the second one is developed for so – called the edge elements. They are the elements for which one triangle's bar belongs to the boundary Γ of the domain Ω. It is assumed that a proper triangulation gives a discrete set of triangles T_{j} which approximates the domain Ω well.