# Finite Element Method - 3D Mesh Generator - Metfem3D

## FEM Approach

Next, they can be solved numerically using the Finite Element Method where the problem is represented as

 ∫ vT A(u) dΩ = ∫ [v1A1(u) + v2A2(u) + v3A3(u)] dΩ = 0 Ω Ω ∫ rT B(u) dΓ = ∫ [r1B1(u) + r2B2(u) + r3B3(u)] dΓ = 0 Γ Γ

where

 u = [ n+ n- φ ]

and v and r are sets of arbitrary functions equal in number to the number of equations (or components of u) involved. A1(u), A2(u) and A3(u) are given by the following formulas, respectively

 A1(u) = ∇T ( -k+ uT [ 1 0 0 ] ∇ uT [ 0 0 1 ] - D+ ∇ uT [ 1 0 0 ] ) + [1, 0, 0] ∂/∂t u
 A2(u) = ∇T ( -k- uT [ 0 1 0 ] ∇ uT [ 0 0 1 ] - D- ∇ uT [ 0 1 0 ] ) + [0, 1, 0] ∂/∂t u
 A3(u) = ε0ε ∇T∇ uT [ 0 0 1 ] + [1, -1, 0] ze u.

where ki = Dizie / (kBT), i = {+, -}.
An expression B(u) gives the boundary conditions on Γ, however, we choose a forced type of boundary conditions on Γ i. e.

 φ - φboun = 0 ni - ni, boun = 0.

Let us approximate unknown u functions by the expansion

 u ≈ ∑ Na Ι [ n+a n-a φa ] = N ubar a

where Ι is the unit matrix. In place of any function v we put a set of approximate functions

 v ≈ ∑ wb Ι δubarb b

where δubarb are arbitrary parameters. Putting wb = Nb we end up with the Galerkin's formulation of the problem. The original basis (shape) functions are used as weighting ones.

### References

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