Perturbed Lagrange-multiplier method

Functional Form

In the previous sections we introduced the strong form of general mixed formulation for small and finite strain and by considering test functions \((\bm v, q)\) as derived in (153) and (164). Alternatively, we can derive the mixed \(\bm u- p\) weak formulation based on minimization of two fields functional \(\Pi(\bm u, p)\), which it is known as the perturbed Lagrangian approach. For the mixed linear case, we can write

(211)\[ \begin{aligned} \Pi (\bm u, p) &= \int_{\Omega} \left[ \mu \, \bm \varepsilon_{\text{dev}} \tcolon \bm \varepsilon_{\text{dev}} - p\, \trace \bm \varepsilon + \frac{\bulk_p}{2} \left(\trace \bm \varepsilon \right)^2 - \frac{p^2}{2 (\bulk-\bulk_p)} \right] \, dv - \Pi_{\text{ext}} (\bm u) \\ \Pi_{\text{ext}} (\bm u) &= \int_{\Omega}{\bm{u} \cdot \rho \bm{g}} \, dv + \int_{\partial \Omega}{\bm{u} \cdot \bar{\bm{t}}} \, da \end{aligned} \]

and by invoking the stationarity of \(\Pi\) with respect to \(\bm u\) and \(p\), we obtain

(212)\[ \begin{aligned} \int_{\Omega} \nabla \delta \bm u \tcolon \left[ 2 \mu \, \bm \varepsilon_{\text{dev}} + \left(\bulk_p \trace \bm \varepsilon - p \right) \bm{I} \right] \, dv - \bm L_{\text{ext}}(\delta \bm u) &=0, \\ \int_{\Omega} \delta p \left( -\trace \bm \varepsilon - \frac{p}{\bulk-\bulk_p} \right) \, dv &= 0, \end{aligned} \]

where \(\bm L_{\text{ext}}(\delta \bm u) = \int_{\Omega}{\delta \bm{u} \cdot \rho \bm{g}} \, dv + \int_{\partial \Omega}{\delta \bm{u} \cdot \bar{\bm{t}}} \, da\), and \(\delta \bm u\) and \(\delta p\) are virtual displacement and pressure and can be seen as the test functions \(\bm v, q\), i.e., \(\delta \bm u = \bm v, \, \delta p = q\). It is clear that the weak form (212) agrees with what we obtained in (153). However, the hyperelastic weak form (164), can not be derived by minimizing any functional and its linearization is not symmetric.

To write the mixed functional \(\Pi(\bm u, p)\) for hyperelastic, we need to consider the strain energy of the form

(213)\[ \psi \left(\bm{C} \right) = \psi_{\text{vol}}(J) + \psi_{\text{iso}}(\bar{\bm{C}}) = \frac{\bulk}{2} \left(U(J)\right)^2 + \psi_{\text{iso}}(\bar{\bm{C}}) \]

we can write a two fields energy functional as

(214)\[ \begin{aligned} \Pi (\bm u, p) &= \int_{\Omega_0} \left[ \psi_{\text{iso}}(\bar{\bm{C}}) - p \, U(J) - \frac{1}{2}\frac{p^2}{\bulk - \bulk_p} + \frac{\bulk_p}{2} U^2\right] \, dV - \Pi_{\text{ext}} (\bm u) \\ \Pi_{\text{ext}} (\bm u) &= \int_{\Omega_0}{\bm{u} \cdot \rho_0 \bm{g}} \, dV + \int_{\partial \Omega_0}{\bm{u} \cdot \bar{\bm{t}}} \, dA \end{aligned} \]

Finding the stationary conditions with respect to \(\bm{u}\) and \(p\) by taking Gateaux derivative gives the weak form

(215)\[ \begin{aligned} \int_{\Omega_0} \bm F \underbrace{\left( \bm{S}_{\text{iso}} + (\bulk_p U - p) \, J \, U' \, \bm{C}^{-1} \right)}_{\bm S} \tcolon \nabla_X \bm v \, dV &= L_{\text{ext}} (\bm v) \\ \int_{\Omega_0} \left(- U(J) - \frac{p}{\bulk - \bulk_p} \right) q \, dV &= 0 \end{aligned} \]

where, \(L_{\text{ext}} (\bm v) = \int_{\Omega_0}{\bm{v} \cdot \rho_0 \bm{g}} \, dV + \int_{\partial \Omega_0}{\bm{v} \cdot \bar{\bm{t}}} \, dA \) and we have used

\[ \begin{aligned} \frac{\partial \psi_{\text{iso}}}{\partial \bm u} \cdot {\delta \bm u} &= \frac{\partial \psi_{\text{iso}}}{\partial \bm E} \tcolon \frac{\partial \bm E}{ \partial \bm u}{\delta \bm u} = \bm{S}_{\text{iso}} \tcolon {\delta \bm E} = \bm{S}_{\text{iso}} \tcolon \text{sym} \left(\bm F^T \delta \bm F \right) \\ \frac{\partial U}{\partial \bm u} \cdot {\delta \bm u} &= \frac{\partial U}{\partial J} \frac{\partial J}{\partial \bm u} {\delta \bm u} = U' {\delta J} = J \, U' \, \bm{C}^{-1} \tcolon \tcolon {\delta \bm E} = J \, U' \, \bm{C}^{-1} \tcolon \text{sym} \left(\bm F^T \delta \bm F \right) \end{aligned} \]

where \(\delta \bm{F} = \nabla_X \delta \bm u = \nabla_X \bm v\). The Jacobian for problem (215) can be written as

(216)\[ \begin{aligned} \int_{\Omega_0} \nabla_X \bm{v} \tcolon \left(\bm F \diff \bm{S} + \diff \bm{F} \bm{S} \right) dV &= -R_u^{PL}, \\ \int_{\Omega_0} q \left( -\diff U - \frac{\diff p}{\bulk - \bulk_p} \right) dV &= -R_p^{PL}, \end{aligned} \]

where

(217)\[ \begin{aligned} \diff \bm{S} &= \diff \bm{S}_{\text{iso}} + \diff \bm{S}_{\text{vol}}^u + \diff \bm{S}_{\text{vol}}^p, \\ \diff \bm{S}_{\text{vol}}^u &= \left(\bulk_p U' \diff J \right) J U' \bm{C}^{-1} + \left(\bulk_p U - p \right) \left(\diff J \, U' \bm{C}^{-1} + J\, U^{''} \diff J \bm{C}^{-1} + J U' \diff \bm{C}^{-1} \right), \\ &= \left[\bulk_p \left( J U' \right)^2 + \left(\bulk_p U - p \right) \left( J U' + J^2 U^{''} \right) \right] \left( \bm{C}^{-1} \tcolon \diff \bm E \right) \bm{C}^{-1} \\ &+ \left(\bulk_p U - p \right) J U' \, \diff \bm{C}^{-1}\\ \diff \bm{S}_{\text{vol}}^p &= - dp J U' \, \bm{C}^{-1} \\ \diff U &= U' \diff J = J \, U' \, \bm{C}^{-1} \tcolon \diff \bm E, \end{aligned} \]

where \(\diff \bm{S}_{\text{iso}}\) is derived for neo-Hookean, Mooney-Rivlin and Ogden in (171), (186) and (203). To complete the derivation we only need \(U(J)\) function with condition

\[ U(J) = 0 \quad \text{if and only if} \quad J = 1. \]

For the volumetric strain energy function of the form given in (154), if we choose \(V(J) = \frac{1}{4} \left(J^2 - 1 - 2 \log J \right)\), from \(\bulk V(J) = \frac{\bulk}{2} U^2(J)\), we will have

\[ U(J) = \pm \sqrt{2 V} = \frac{\sign(J-1)}{\sqrt{2}} ( \underbrace{J^2 - 1 - 2 \log J }_{A})^{1/2} \]

where the derivatives are

\[ \begin{aligned} U' &= \frac{\partial U}{\partial J} = \sign(J-1) \frac{J^2 - 1}{J \sqrt{2}} A^{-1/2}, \\ U{''} &= \frac{\partial^2 U}{\partial J^2} = \sign(J-1) \frac{1}{J^2 \sqrt{2}} \left((J^2 + 1) A^{-1/2} - (J^2 - 1)^2 A^{-3/2} \right). \end{aligned} \]

Command-line interface

To enable the perturbed Lagrange model, use the model option -model elasticity-mixed-neo-hookean-PL-current or -model elasticity-mixed-neo-hookean-PL-initial and set the material parameters listed in Perturbed Lagrange-multiplier model options. Any parameter without a default option is required.

Table 28 Perturbed Lagrange-multiplier model options

Option	Description	Default value
-model elasticity-mixed-neo-hookean-PL-current | elasticity-mixed-neo-hookean-PL-initial	Required to enable the perturbed Lagrange model, | denotes an option for the user.
-E [real]	Young’s modulus, \(E > 0\)
-nu [real]	Poisson’s ratio, \(\nu \leq 0.5\).
-nu_primal [real]	Primal part of the Poisson’s ratio for Jacobian operator, \(-1 \leq \nu_p < \nu \leq 0.5\). This parameter may improve solver performance and should not affect the solution.	-1.0
-nu_primal_pc [real]	Primal part of the Poisson’s ratio for preconditioner operator, \(-1 \leq \nu_{pc} < \nu \leq 0.5\). This parameter may improve solver performance and should not affect the solution.	-nu_primal value

An example using the pertrubed Lagrange model can be run via

$ ./bin/ratel-quasistatic -options_file examples/ymls/ex01-static-elasticity-mixed-neo-hookean-PL-current-pcjacobi.yml