Arc-length method

We start by assuming the parametrized loading, in which the total external load vector is expressed as

$f$$$^{ext}(\lambda) = +\lambda$$$f$$$^{ext}_p$$

where $f$$_p$ is proportional, reference load vector, and $\lambda$ is load scaling parameter, The arc-length method is based on idea of controlling the length passed along the loading path. For the differential length of loading path we can write

$$\Delta l = \sqrt{\Delta\mbox{\boldmath$r$}^T\Delta\mbox{\boldmath$r$}+(c^2\Delta\lambda^2\mbox{\boldmath$f$}^T_p\mbox{\boldmath$f$}_p)}$$ (2.3)

where $c$ is coefficient of generalized metrics used to define $\Delta l$ (taking into account different units of displacement and load). For selected increment of loading path length $\Delta l$, we are looking for the equilibrium, where the unknowns are nodal displacements $r$ and the load scaling parameter $\lambda$. We have the equilibrium equation and additional scalar equation 2.3:

$$\mbox{\boldmath$f$} ^{int}( \mbox{\boldmath$r$} _n) = \mbox{\boldmath$f$} ^{ext}(\lambda_n \mbox{\boldmath$f$} _p)$$ (2.4)

$$\Delta l_n^2 = \Delta \mbox{\boldmath$r$} _n^T\Delta \mbox{\boldmath$r$} _n+c^2\Delta\lambda^2 \mbox{\boldmath$f$} ^T_p \mbox{\boldmath$f$} _p$$ (2.5)

Figure 2.2: Illustration of Acr-length method

At the end of n-th loading step and i-th iteration the displacement vector can be written as $r$$^i_n =$   $r$$^{i-1}_n+\delta$$r$ and similarly the load scaling parameter as $\lambda_n^i = \lambda_n^{i-1}+\delta\lambda$. Substituting this into equilibrium equation 2.4 we get

$f$$$^{int}($$$r$$$_n^{i-1}+\delta$$$r$$$) =$$$f$$$^{ext}((\lambda_n^{i-1}+\delta\lambda)$$$f$$$_p)$$

By linearization of $F$$^{int}$ around known state $r$$_n^{i-1}$ we get

$f$$$^{int}_n($$$r$$$_n^{i-1})+$$$K$$$_n^{i-1}\delta$$$r$$$=$$   $f$$$^{ext,i-1}+\delta\lambda$$$f$$$_p$$

and finally for unknown $\delta$$r$

$$\delta \mbox{\boldmath$r$} = \underbrace{(\mbox{\boldmath$K$}_n^{i-1})^{-1}(\mbox{\boldmath$... ...ath$K$}_n^{i-1})^{-1}\mbox{\boldmath$f$}_p}_{\delta\mbox{\boldmath$r$}_\lambda}$$ (2.6)

Note that the vectors $\delta$$r$$_r$ and $\delta$$r$$_\lambda$ can be computed and the only unknown remaining is the incremental change of loading parameter $\delta\lambda$, which could be determined from 2.5

$$\Delta l_n^2 = (\Delta \mbox{\boldmath$r$} _n^{i-1}+\delta \mbox{\boldmath$r$} _r+\delta\lambda\delta \mbox{\boldmath$r$} _\lambda)^T(\Delta \mbox{\boldmath$r$} _n^{i-1}+\delta \mbox{\boldmath$r$} _r+\delta\lambda\delta \mbox{\boldmath$r$} _\lambda) + c^2(\Delta\lambda_n^{i-1}+\delta\lambda)^2 \mbox{\boldmath$f$} ^T_p \mbox{\boldmath$f$} _p$$ (2.7)

This finally yields a quadratic equation for unknown increment of loading parameter $\delta\lambda$. The algorithm is summarized in Table .

Table 2.2: Newton-Raphson method

Given
$f$$^{ext}_{n-1},$   $f$$_p$
$r$$^{0}_{n} =$   $r$$_{n-1}$
Evaluate
$\;\;\delta$$r$$_\lambda = ($$K$$_n)^{-1}$$f$$_p$
$\;\;\Delta\lambda^0=\pm{\Delta l}/{\sqrt{\delta\mbox{\boldmath $r$}_\lambda^T\... ...mbox{\boldmath $r$}_\lambda+c^2\mbox{\boldmath $f$}_p^T\mbox{\boldmath $f$}_p}}$
$\;\;\Delta$$r$$_n^0 = ($$K$$_n)^{-1}(\Delta\lambda$$f$$_p) = ($$K$$_n)^{-1}((\lambda_n+\Delta\lambda^0)$$f$$_p -$   $f$$^{int,0}_n)$
Repeat for $i=1,2,\cdots$
$\;\;\delta$$r$$_\lambda = ($$K$$_n^{i-1})^{-1}$$f$$_p$
$\;\;\delta$$r$$_r = ($$K$$_n^{i-1})^{-1}(\lambda_n^{i-1}$$f$$_p -$$f$$^{int}($$r$$_n^{i-1}))$
$\;\;$Solve quadratic equation 2.7 for $\delta\lambda$
$\;\;\delta$$r$$^i = \delta$$r$$_r+\delta\lambda\delta$$r$$_\lambda$
$\;\;\Delta$$r$$_n^i = \Delta$$r$$^{i-1}+\delta$$r$$^i,\$   $r$$_n^i =$   $r$$_n^{i-1}+\delta$$r$$^i$
$\;\;\lambda^i = \lambda^{i-1}+\delta\lambda,\ \Delta\Lambda_n^i = \Delta\Lambda_n^{i-1}+\delta\lambda$
Until convergence reached