Finite Element Method - 3D Mesh Generator - Metfem3D

FEM Approach

Tetrahedral elements

For tetrahedral linear elements shape functions N_i can be assumed as equal area coordinates L_i given by the formula

L_i = (a_i + b_i x + c_i y + d_i z)/(6V^e), i = 1, 2, 3, 4

where V^e is a volume of tetrahedron. The following integration formula can be useful

∫	L^α₁L^β₂L^μ₃L^ν₄ dxdydz = (α!β!μ!ν!)/(α + β + μ + ν + 3)! 6V^e

in order to calculate integrals K^b,e_ac, K*^b,e_a and K^b,e_a. Shape functions for linear elements are N_a = L_a for a = 1, 2, 3, 4. This gives

K^b,e_a,c =

∫

(∇ L^b)^T L_a∇ L_c dΩ^e = 1/(4!6V^e)[b^b, c^b, d^b] [b_c, c_c, d_c]^T =

1/(4!6V^e)(b^bb_c + c^bc_c + d^bd_c)

K*^b,e_a =

∫

(∇ L^b)^T∇ L_a dΩ^e = 1/(36V^e) (b^bb_a + c^bc_a + d^bd_a)

K^b,e_a =

∫

L_a L^b dΩ^e = 6V^e/5!

when a ≠ b

K^a,e_a =

∫

L_a L^a dΩ^e = 12V^e/5!.

Finally, we have

{

k₊

(

∑_e K^b,e_ac

)

φ^c + D₊

∑_e

K*^b,e_a + 1/(Δt) ∑_e K^b,e_a

}

n^+,a = f^+,b,

{

k_-

(

∑_e K^b,e_ac

)

φ^c + D_- ∑_e K*^b,e_a + 1/(Δt) ∑_e K^b,e_a

}

n^-,a = f^-,b,

ε₀εφ^a ∑_eK^b,e_a = |z_i|e ∑_e K^b,e_a (n^+,a - n^-,a)

where a, b, c = 1, …, M

where f^i,b = 1/(Δt)(∑_e K^b,e_a)n^{i, a}_n-1 with i = {+, -} and only these nodes a, b and c that participate in the particular element e can give a non-zero contribution to the sums of the general type ∑_e K^e.
Let us assume that all values of n^a are known at a time t_n. Then we can solve the third equation in equation obtaining the result

φ^a = (|z_i|e)/(ε₀ε) {∑_eK*^b,e_a}^-1 (∑_e K^b,e_a) (n^+,a - n^-,a)

where {∑_eK*^b,e_a}^-1 denotes elements of the inverse matrix to K*. After substitution the solution for φ to equation we get

n^i,a {(k_i |z_i|e)/(ε₀ε {∑_eK*^b,e_c}^-1 (∑_eK^b,e_c) (n^+,c - n^-,c) (∑_eK^b,e_ac)

+ D_i ∑_e K*^b,e_a + 1/(Δt) ∑_e K^b,e_a} - f^i,b = 0.

where i={+,-}

<< >>

Finite Element Method - 3D Mesh Generator - Metfem3D

FEM Approach

Tetrahedral elements

References