Finite Element Method - 3D Mesh Generator - Metfem3D
System of Nonlinear Equations
Newton's method
The above written set of equations is of a nonlinear type. Let us denote all of them as
F(n1, …, nM) = 0
Thus to solve it we have to employ the iterative Newton's method[1]. It means that we have to start from an initial guess of {ni0}Mi=1 values.
And during next iterations for k = 0, 1, …
|
nk+1 = nk -
|
{
|
F'(n1k, …, nMk)
|
}
|
-1 F(n1k, …, nMk)
|
the solution should be achieved. F'(n1k, …, nMk) denotes the following matrix of partial derivatives:
|
F'(n1k, …, nMk) =
|
[
|
| ∂F1 / ∂n1k |
… |
∂F1 / ∂nMk |
| … |
… |
… |
| ∂FM / ∂n1k |
… |
∂FM / ∂nMk |
|
]
|
where
∂Fb/∂nak = ki|zi|e / (ε0ε) {∑e K*b,ec}-1 ∑e Kb,ec nck ∑e Kb,eac + D ∑e K*b,ea
+ 1/(Δt) ∑e Kb,ea + 2ki|zi|e / (ε0ε) {∑e K*b,ec}-1 ∑e Kb,ec nak ∑e Kb,eac
where c ≠ a.
References
