Finite Element Method - 2D Mesh Generator - Metfem2D
The Lagrange polynomials.
The Lagrange polynomials pk(x) are given by the general formula [1,
2]
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n
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pk(x) =
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∏
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(x - xi)/(xk - xi)
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i=1
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i ≠ k
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for k = 1, …, n.
It is clearly seen from the above – given expression that for x = xk pk(xk) = 1 and for x = xj such that j ≠ k pk(xj) = 0. Between nodes values of pk(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
On the other hand, when the interpolated function φ depends on two spacial coordinates one can define basis polynomials in the form
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pm(x, y) ≡ pIJ (x, y) = pI(x) pJ(y),
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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 {p1, …, pm, …, pn} is a basis of a n – dimensional functional space because each function pm for m = 1, …, n equals 1 at the interpolation node (xm, ym) 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. Nm(L1,L2,L3) = paI(L1)pbJ(L2)pcK(L3) where the values of a, b and c assign the polynomial order in each Lk-th coordinate and I, J and K denote the m-th node position in a triangular lattice (i. e. in the coordinates L1, L2 and L3, 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 Nk = Lk(x, y), where k = 1, 2, 3, change between two nodes linearly (see Eq. (3)).
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