Options for unsteady line search used by PTC in the startup phase, used by SolverOpts (diablo).
This is similar in principle but generally more useful than the steady LineSearch which can be activated in the Inexact Newton Phase.

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Use:
diablo%strtup_ls

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Description:
PTCLineSearch is the derived sub-type used by PTC in the startup phase

Unsteady line search algorithm, as implemented by D. Brown in Jetstream:
$\mathcal{R}_t\left(\mathbf{Q}^{\left(n+1\right)}\right)\equiv \mathcal{T}\left(\mathbf{Q}^{\left(n+1\right)}-\mathbf{Q}^{\left(n\right)}\right)+\mathcal{R}_s\left(\mathbf{Q}^{\left(n+1\right)}\right)$
$\mathcal{T}^{\left(n\right)}=\frac1{\Delta t}_i\mathcal{I}, \,\,\Delta t_i^{\left(n\right)}=T_i\zeta a\left(b\right)^{m\left\lfloor \frac{n}{m}\right\rfloor}, \,\, T_i=\frac1{1+J_i^{\frac13}}$
$\begin{align*} & \eta = \eta_0,\,\, \tilde{\mathbf{Q}}_\eta = \mathbf{Q}^{\left(n\right)}+\eta\Delta\mathbf{Q}^{\left(n\right)} \\ & \mathrm{\textbf{while}}\,\, \left\{\left(\left\|\mathcal{R}_t\left(\tilde{\mathbf{Q}}_\eta\right)\right\|>f_{\mathcal{R}}\left\|\mathcal{R}_t\left(\tilde{\mathbf{Q}}^{\left(n\right)}\right)\right\|\right) \,\&\, \left(\eta > \eta_{\mathrm{min}}\right)\right\} \,\, \mathrm{\textbf{do}}\\ & \,\,\,\,\,\,\,\, \eta\leftarrow f_{\eta}\eta,\,\, \tilde{\mathbf{Q}}_\eta=\mathbf{Q}^{\left(n\right)}+\eta\Delta\mathbf{Q}\\ & \mathrm{\textbf{done}}\\ & \mathrm{\textbf{if}}\,\, \left(\eta = \eta_0\right)\,\, \mathrm{\textbf{then}}\\ & \,\,\,\,\,\,\,\, \mathbf{Q}^{\left(n+1\right)} = \tilde{\mathbf{Q}}_\eta^{\left(n\right)} + \eta\Delta\mathbf{Q}\\ & \mathrm{\textbf{else}}\,\,\mathrm{\textbf{if}}\,\, \left(\eta > \eta_{\mathrm{min}}\right)\,\, \mathrm{\textbf{then}}\\ & \,\,\,\,\,\,\,\, \zeta \leftarrow f_{\zeta 1}\zeta,\,\, \mathbf{Q}^{\left(n+1\right)} = \tilde{\mathbf{Q}}_\eta^{\left(n\right)} + \eta\Delta\mathbf{Q}\\ & \mathrm{\textbf{else}}\\ & \,\,\,\,\,\,\,\, \zeta \leftarrow f_{\zeta 2}\zeta,\,\, \mathbf{Q}^{\left(n+1\right)}=\mathbf{Q}^{\left(n\right)}\\ & \mathrm{\textbf{end}}\,\,\mathrm{if} \end{align*}$

The algorithm generally works better if the steady line search is tracked instead of the unsteady residual (ls%steady = .true.). Recommended use is to just set diablo%strtup_ls%on=.true. and to use it with the default settings. If additional damping is desired, it is generally advised to match the time step relaxation with the solution update relaxation (ls%damp = ls%dt_red_fac).

A similar derived sub-type LineSearch can be used for steady line search in the Newton phase.

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ls% on

Description: Turn it on?
Parameter Type: logical

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ls% steady

Description: Track the steady residual? (Alternative is unsteady, as seen in the algorithm)
Parameter Type: logical

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ls% dt_red_fac(1:2)

Description: Damping factor on the time step update (f_{\zeta 1} and f_{\zeta 2} in the above algorithm)
Parameter Type: real
Range of Acceptable values: (0,1)

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ls% damp

Description: Damping factor on the solution update (f_{\eta} in the above algorithm)
Parameter Type: real
Range of Acceptable values: (0,1)

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ls% eta_in

Description: Initial value of \eta (\eta_0 in the above algorithm)
Parameter Type: real
Range of Acceptable values: (0,1]

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ls% eta_min

Description: Minimum value of \eta (\eta_{min} in the above algorithm)
Parameter Type: real
Range of Acceptable values: (0,ls%eta_in)

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ls% rfac_tol

Description: A factor that allows us to relax when to apply line search (f_R in the above algorithm)
Parameter Type: real
Range of Acceptable values: The intended use is f_R >= 1