Finite Element Method - 2D Mesh Generator - Metfem2D
Transformation in local L-coordinates
Let us compute the determinant of the jacobian transformation between the global x, y and a local L1, L2, L3 coordinate frame. One notices immediately that the problem is degenerate. That is why, we introduce a new coordinate z as a linear combination of L1, L2, L3 i. e. z = L1 + L2 + L3. Note that z is not an independent coordinate and has a constant value equal 1. After taking into account relations Eq. (3) we find the jacobian matrix in the form
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J(L1, L2, L3) ≡
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∂(x, y, z)/∂(L1, L2, L3) =
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(
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∂x/∂L1 ∂x/∂L2 ∂x/∂L3
∂y/∂L1 ∂y/∂L2 ∂y/∂L3
∂z/∂L1 ∂z/∂L2 ∂z/∂L3
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)
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=
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(
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)
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Furthermore, we have the relation between the jacobian and an element area
det(J) ≡ 2 Δ,
where Δ denotes the area of a triangle which is based on vertices (x1, y1), (x2, y2), (x3, y3). And finally, we obtain the coordinates transformation rule
dxdy = 2 ΔdL1dL2, and L3 = 1 - L1 - L2.
The relation between the gradient operator Δ in cartesian and in new coordinates is given by:
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[∂/∂x, ∂/∂y] =
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[∂L1/∂x ∂/∂L1 + ∂L2/∂x ∂/∂L2 + ∂L3/∂x ∂/∂L3,
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∂L1/∂y ∂/∂L1 + ∂L2/∂y ∂/∂L2 + ∂L3/∂y ∂/∂L3]
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[∂/∂x, ∂/∂y] =
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1/(2Δ) [∂/∂L1, ∂/∂L2, ∂/∂L3]
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(
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)
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= [∂/∂L1, ∂/∂L2, ∂/∂L3]
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T
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where Lk = (akx + bky + ck)/(2Δ) (k = 1, 2, 3) and a1 = y2 - y3, b1 = x3 - x2, c1 = x2y3 - x3y2, the rest of coeficients is obtained by cyclic permutation of indices 1, 2 and 3.
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