The Lagamine Inverse code allows the computation of parameter sensitivity through a semi-analytical calculation. This implementation does not fundamentally change the code, but the simulation with Lagamine Inverse will be made so as to obtain both a reference finite element curve and the derivative of this curve with respect to the specified parameters.
To compute the derivative of the curve with respect to parameters, Lagamine will run one reference simulation with the input parameters and 2 additional simulations for each optimized parameter: one with a positive and on with a negative perturbation applied to the parameter (see OPTIM Description).
The values of the reference curve and its derivatives are put in a *.sm file at each iteration. When the Lagamine Inverse is used, the parameters specified by the user in the *.ia file are not read from the *.lag file but from the *.op file.

In order to launch a simulation using the Lagamine Inverse, several things are necessary:

1. Modification of the IOPT parameter in the loading file *ex.dat;
2. Implementation in the Lagamine subroutines INVA.F and WSENSM.F to specify the finite element curves to use;
3. Creation of *.ia file and *.op file indicating the number of finite element curve, the parameters to be optimized, their initial values, …

In the *ex.dat file, the parameter IOPT (ninth parameter on the first line) must be set to 1 to activate the Lagamine Inverse simulation. It is also necessary to indicate an IDENT (5th parameter of first line) that allows the user to specify his curve in INVA.F and WSENSM.F (see figure 1).

Fig. 1: Example of *ex.dat file with activation of Lagamine Inverse

The user must implement in the INVA.F subroutine the finite element curves that will be compared with the experimental curves (for instance, a stress-strain curve). For each time step, subroutine INVA.F is called once for the reference simulation (ISENS=0), and twice for each optimized parameter (ISENS=1 for positive perturbation, ISENS=2 for negative perturbation). This allows to compute for every case the wanted curves REACTION (reference), REACTION_P (positive perturbation), REACTION_M (negative perturbation). The computation of the sensitivity is then done according to the following:
DUef_DX(I,ILI)=(REACTION_P(I)-REACTION_M(I))/(2*PREC_IA *XIAPAR(III,LLL))
Where:

• DUef_DX(I,ILI) is the derivatives of the Ith curve with respect to parameter n° ILI (ILI=1,NBPAR_OP)
• REACTION_P(I) is the value of the Ith curve with a positive perturbation PREC_IA*XIAPAR(III,LLL) on parameter n°ILI
• REACTION_M(I) is the value of the Ith curve with a negative perturbation PREC_IA*XIAPAR(III,LLL) on parameter n°ILI
• PREC_IA is the relative perturbation
• XIAPAR(III,LLL) is the value of parameter n°LLL of law n°III to be optimized