Newton-Raphson method

The method is based on splitting of the loading process into series of subsequent incremental loading steps in which the incremental loading vector $\Delta$$f$ is applied. We are looking for the equilibrium at the end of loading step 2.1 using the iterative procedure outlined in 2.1. The algorithm is graphically outlined in Fig. 2.1.1 for a system with one unknown and summarized in Table 2.2.

Figure 2.1: Illustration of Newton-Raphson method

Table 2.1: Newton-Raphson method

Given
$f$$^{ext}_{n-1}$
$f$$^{ext}_{n}=$$f$$^{ext}_{n-1}+\delta$$f$$^{ext}_n$
$r$$^{0}_{n} =$   $r$$_{n-1}$
Looking for $r$$_n$, such that $f$$^{int}($$r$$_n) =$   $f$$^{ext}_n$
Solve for $i=1,2,\cdots$
$K$$^{i}\delta$$r$$^{i} =$   $f$$^{ext}_n-$$f$$^{int}_n($$r$$^{i-1}_{n})$
$r$$^i_{n} =$   $r$$^{i-1}_{n}+\delta$$r$$^i$
Until $\Vert$$f$$^{ext}_n-$$f$$^{int}_n($$r$$^{i-1}_{n})\Vert \le \varepsilon$

Based on update strategy for stiffness matrix, one can obtain different variants of the method. When the stiffness matrix $K$$^{i}$ is updated in each iteration, the full Newton-Raphson method is obtained. When stiffness matrix only every n-th iteration, one speaks about modified Newton-Raphson method. Finally, when the stiffness matrix is updated only at the beginning of the loading step, one obtains so called initial stiffness method. For the full Newton-Raphson method a quadratic convergence is obtained.

One can implement two blends of Newton-Raphson algorithm, where the loading can be driven by load control or by displacement control, where the prescribed increments of displacements are applied to selected DOFs.