# Finite Element Method - 3D Mesh Generator - Metfem3D

## FEM Approach

### Finite Element Method. Discrete Approximation In Time.

In turn, we can approximate the nodal electric potential

 φ(x,y,z,t) = ∑ Na(x,y,z) φa(t) a

and number of particles

 ni(x,y,z,t) = ∑ Na(x,y,z) nia(t) a

at a time tn by

φa(tn) ≈ φan
nia(tn) ≈ nai, n

and evaluating

 un = [ n+, n n-, n φn ]

in the Taylor series we obtain a discrete approximation in time

un ≈ un-1 + Δt dun-1/dt + β(Δt)2 d2un-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(Na Ι uan) + C Na Ι { 1/(βΔt) (uan - uan-1) - (1 - β)/β duan-1/dt } = 0.

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

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

otherwise, the expression for uan is as follows

C Na Ι 1/(βΔt) uan + K(Na Ι uan) - C Na Ι 1/(βΔt) uan-1 + (1 - β)/β K(Na Ι uan-1) = 0.

### References

1. ^ O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Sixth edition., Elsevier 2005