3. Implementation: implicit integration

Implicit integration consists in finding the increments of the internal variables \(\Delta {V}_{i}=(\Delta P,\Delta {\eta }_{i},\Delta {P}_{i},\Delta g,\Delta {\underline{\varepsilon }}_{e})\) on a discrete time increment \(\Delta t\) [ref]. The discretization of the equations in the table leads to the following system of nonlinear equations:

\({\underline{R}}_{e}=\Delta {\underline{\varepsilon }}_{e}+\Delta p\underline{n}+\Delta {p}_{i}\underline{n}+\Delta g\underline{1}-\Delta {\underline{\varepsilon }}_{m}=0\)

\({R}_{p}={\sigma }_{\text{eq}}-{\sigma }^{f}=0\)

\({R}_{\eta }=\Delta {\eta }_{i}-{\zeta }_{f}{\sigma }_{\text{eq}}\Delta \varphi =0\)

\({R}_{i}=\Delta {P}_{i}-H({\eta }_{i}-{\eta }_{i}^{s}){A}_{i}{\sigma }_{\text{eq}}\Delta \varphi =0\)

\({R}_{g}=\Delta g-{A}_{g}\Delta \Phi =0\)

Table 3.1: system of discretized equations

Where \(H\) is the Heavyside function and \(\Delta \Phi =\phi \Delta t\) is the fluence increment. By asking \(R=({\underline{R}}_{e},{R}_{p},{R}_{\eta },{R}_{i},{R}_{g})\), we are therefore trying to solve the system: \(R(\Delta {V}_{i})=0\). Internal variables that appear directly or indirectly (for example \({\sigma }_{\text{eq}}\) can be expressed in terms of \({\underline{\varepsilon }}_{e}\)) are expressed as: \({\text{v}}_{i}={\text{v}}_{i}^{0}+\theta \Delta {\text{v}}_{i}\), where \({\text{v}}_{i}^{0}\) represents the values of the variables at the beginning of the increment. \(\theta\) is a parameter that varies between 0 and 1. For \(\theta =0\), we get an explicit Euler schema (to be avoided); for \(\theta =1\) we get a totally implicit schema.

The system of equations in the table is solved by using a Newton-Raphson method that requires the calculation of the Jacobian: \(J=\partial R/\partial \Delta {\text{v}}_{i}\). This can be calculated block by block (zero terms are omitted). Note \(\underline{\underline{N}}=\partial \underline{n}/\partial \underline{\sigma }\) and \({\sigma }_{,p}^{f}=\partial {\sigma }^{f}/\partial p\).

Derivatives of \({\underline{R}}_{e}\):

\(\frac{\partial {\underline{R}}_{e}}{\partial \Delta {\underline{\varepsilon }}_{e}}=\underline{\underline{1}}+\theta \cdot \Delta p\cdot \underline{\underline{N}}:\underline{\underline{E}}+\theta \cdot \Delta {p}_{i}\cdot \underline{\underline{N}}:\underline{\underline{E}}\) \(\frac{\partial {\underline{R}}_{e}}{\partial \Delta p}=\underline{n}\)

\(\frac{\partial {\underline{R}}_{e}}{\partial \Delta {p}_{i}}=\underline{n}\) \(\frac{\partial {\underline{R}}_{e}}{\partial \Delta g}=\underline{1}\)

Derivatives of \({R}_{p}\):

\(\frac{\partial {R}_{p}}{\partial \Delta {\underline{\varepsilon }}_{e}}=\theta \underline{n}:\underline{\underline{E}}\) \(\frac{\partial {R}_{p}}{\partial \Delta p}=-\theta {\sigma }_{,p}^{f}\)

Derivatives of \({R}_{\eta }\):

\(\frac{\partial {R}_{\eta }}{\partial \Delta {\underline{\varepsilon }}_{e}}=-{\zeta }_{f}\theta \cdot \Delta \varphi \cdot \underline{n}:\underline{\underline{E}}\) \(\frac{\partial {R}_{\eta }}{\partial \Delta {\eta }_{i}}=1\)

Derivatives of \({R}_{i}\):

\(\frac{\partial {R}_{i}}{\partial \Delta {\underline{\varepsilon }}_{e}}=-{A}_{i}\cdot \theta \cdot \Delta \varphi \cdot \underline{n}:\underline{\underline{E}}\) \(\frac{\partial {R}_{i}}{\partial \Delta {p}_{i}}=1\)

Derivatives of \({R}_{g}\):

\(\frac{\partial {R}_{g}}{\partial \Delta g}=1\)