Finite Element Method - 3D Mesh Generator

FEM Approach

In turn, we can approximate the nodal electric potential

φ(x,y,z,t) =	∑	N_a(x,y,z) φ^a(t)
	a

and number of particles

n_i(x,y,z,t) =	∑	N_a(x,y,z) n_i^a(t)
	a

at a time t_n by

φ^a(t_n) ≈ φ^a_n
n_i^a(t_n) ≈ n^a_{i, n}

and evaluating

u_n =

[

n_{+, n}
n_{-, n}
φ_n

]

in the Taylor series we obtain a discrete approximation in time

u_n ≈ u_n-1 + Δt du_n-1/dt + β(Δt)² d²u_n-1 /dt² + O(Δt³)

where β takes values from [0, 1] and Δt denotes time step. After incorporating it into a general form of time-dependent equations A_{{1, 2}}(u)

K(u) + C du/dt = 0

where K(u) represents these parts of A_{1,2}(u) with a space-dependent operator we get time approximation for a given node

K(N_a Ι u^a_n) + C N_a Ι

{

1/(βΔt) (u^a_n - u^a_n-1) - (1 - β)/β du^a_n-1/dt

}

= 0.

When β = 1 an approximate solution to the semi - discrete equations at each time t_n is given by the Euler ,,backward'' scheme

K(Nu) + C N 1/(Δt) u_n = C N 1/(Δt) u_n-1

otherwise, the expression for u^a_n is as follows

C N_a Ι 1/(βΔt) u^a_n + K(N_a Ι u^a_n) - C N_a Ι 1/(βΔt) u^a_n-1 + (1 - β)/β K(N_a Ι u^a_n-1) = 0.