@CalcGradF

Description

This method is used for generalized midpoint implicit integration. The routine will be required to furnish the residual of incremental variable evolution equations, and the Jacobian matrix of all the residual equations with respect to the internal variables.

\[\Delta\chi = fn\left( \Delta\bf g_{tot}, {\cal A}\left(\chi_\theta\right) \right)\]

\[\chi_\theta = \chi_{ini} + \theta\Delta\chi\]

\[{\cal R}^k = {\cal F}^k(\chi_\theta) - {\cal F}_0\]

\[{\cal F}(\chi_\theta, \Delta\chi) = {\cal F}_0\]

\[\Delta\chi^{k+1}= \Delta\chi^{k} + [\nabla {\cal F}_\theta^k]^{-1}{\cal R}^k\]

CODE	DESCRIPTION
theta	the \(\theta\) value used for the midpoint
f_0	vector of imposed variable increments
f_vec	vector for the variable residual
f_grad	Jacobian matrix storage
chi_vec	variables calculated at \(\theta\)
d_chi	increment of integrated variables

CODE	DESCRIPTION
f_vec_ \(vname_i\)	residual space for variable \(vname_i\)
d\(vname_i\)_d\(vname_j\)	Jacobian space for the residual equation of \(vname_i\) with respect to the variable \(vname_j\)