The Lagrange polynomials p_{k}(x) are given by the general formula ^{[1,} ^{2]}
n | |||
p_{k}(x) = | ∏ | (x - x_{i})/(x_{k} - x_{i}) | |
i=1 | |||
i ≠ k |
for k = 1, …, n.
It is clearly seen from the above – given expression that for x = x_{k} p_{k}(x_{k}) = 1 and for x = x_{j} such that j ≠ k p_{k}(x_{j}) = 0. Between nodes values of p_{k}(x) vary according to the polynomial order i. e. n-1 which is the order of interpolation. Making use of these polynomials one can represent an arbitrary function φ(x) as
φ(x) = | ∑_{k} | p_{k}(x) φ_{k} |
On the other hand, when the interpolated function φ depends on two spacial coordinates one can define basis polynomials in the form
p_{m}(x, y) ≡ p_{IJ} (x, y) = p_{I}(x) p_{J}(y), |
where I J describe row and column number for the m-th node in a rectangular lattice (rows align along x and columns along y direction, respectively). And consequently, the set {p_{1}, …, p_{m}, …, p_{n}} is a basis of a n – dimensional functional space because each function p_{m} for m = 1, …, n equals 1 at the interpolation node (x_{m}, y_{m}) and 0 at others. It is easy to demonstrate that such functions are orthogonal^{[2]}. Instead of spacial coordinates any other coordinates can be considered. In the case of mesh elements the natural coordinates are the area coordinates L defined already in the Sec. The mathematical concept of FEM. On that basis the shape functions could be constructed as a composition of these three basis polynomials i. e. N_{m}(L_{1},L_{2},L_{3}) = p^{a}_{I}(L_{1})p^{b}_{J}(L_{2})p^{c}_{K}(L_{3}) where the values of a, b and c assign the polynomial order in each L_{k}-th coordinate and I, J and K denote the m-th node position in a triangular lattice (i. e. in the coordinates L_{1}, L_{2} and L_{3}, respectively).
In the ^{[1]} could be found a comprehensive description of various elements belonging to the triangular family starting from a linear through quadratic to cubic one. For simplicity, in the article only the linear case is looked on. It explicitly means that shape functions N_{k} = L_{k}(x, y), where k = 1, 2, 3, change between two nodes linearly (see Eq. (3)).