Nonlinear finite elements/Newton method for finite elements

Newton's method for nonlinear finite elements

The finite element approximation of a nonlinear boundary value problem results in a system of nonlinear equations of the form

$K(u)u=f$

Here, K is a nonlinear function relating the nodal degrees of freedom and the internal nodal forces and f is the externally applied nodal forces. Now the problem is to find the values of the nodal degrees of freedom such that the above equation is satisfied. In general the above equation does not have a close form solution (or some algorithm to find the closed form solution), hence we resort to numerical root finding methods to find an approximate solution. Newtons method is commonly used solution procedure for the solution of system simultaneous nonlinear equations. We rewrite the above equation in the form $g(x)=0$

$g(u):=K(u)u-f$

Then, the Newton iteration formula can be written as

${\begin{aligned}u_{r+1}&=u_{r}-[{\cfrac {dg(u_{r})}{du}}]^{-1}g(u_{r})\\&=u_{r}-T^{-1}g(u_{r})\end{aligned}}$

The tangent stiffness is given by

$T={\cfrac {dg(u_{r})}{du}}$

The iterative procedure is terminated when either the residual is very small or the difference between successive solutions is less than a specified tolerance.

However, both the residual $\mathbf {r}$ and the solution $\mathbf {u}$ are vectors. We usually compare the $L_{2}$ (Euclidean) norm of the vectors with a tolerance $\epsilon$. In symbolic form, we check the norm of the residual using

$\lVert \mathbf {r} \rVert _{2}\leq \epsilon \qquad {\text{or}}\qquad {\sqrt {\mathbf {r} \cdot \mathbf {r} }}\leq \epsilon \qquad {\text{or}}\qquad {\sqrt {\sum _{i=1}^{n}r_{i}^{2}}}\leq \epsilon ~.$

For the difference between successive solutions we check

${\cfrac {\lVert \mathbf {u} _{r+1}-\mathbf {u} _{r}\rVert _{2}}{\lVert \mathbf {u} _{r+1}\rVert _{2}}}\leq \epsilon \qquad {\text{or}}\qquad {\cfrac {\sqrt {(\mathbf {u} _{r+1}-\mathbf {u} _{r})\cdot (\mathbf {u} _{r+1}-\mathbf {u} _{r})}}{\sqrt {\mathbf {u} _{r+1}\cdot \mathbf {u} _{r+1}}}}\leq \epsilon \qquad {\text{or}}\qquad {\cfrac {\sqrt {\sum _{i=1}^{n}(u_{i}^{r+1}-u_{i}^{r})^{2}}}{\sqrt {\sum _{i=1}^{n}(u_{i}^{r+1})^{2}}}}\leq \epsilon ~.$

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