1. Introduction#

Before actually addressing the problem of resetting, it is useful to recall a few elements on the identification of parameters. Suppose we want to identify \(n\) parameters from a given mechanical test. In the context of this identification, the quantities are defined:

  • \(\mathrm{c}\), the vector of the \(n\) parameters to be identified, belonging to \(O\), closed convex of \({ℝ}^{n}\).

  • \(\mathrm{d}\), the vector of the quantities calculated during a simulation of the test using the parameters \(c\), as opposed to \({d}^{\text{exp}}\), the vector of the quantities measured during an experimental test. Both belong to space \(L\) of observable quantities. The simulation of the experimental test, parameterized by the vector \(c\), can be carried out by various methods: finite differences, finite elements, border elements,… This is what we will call the direct problem.

The aim of the identification is to determine the set of parameters \(c\) reducing the difference between measured and experimental quantities (strongly hoping that the reduction of this difference is sufficient to obtain the desired set of parameters…). We therefore introduce a functional cost noted \(J\) depending on \(c\) and measuring the distance between \(d\) and \({d}^{\text{exp}}\).

\(J(c)=\parallel d-{d}^{\text{exp}}\parallel\) eq 1-1

where \(\parallel \mathrm{.}\parallel\) refers to a standard over \(L\).

Identification is therefore expressed in the form of the following minimization problem:

Determine \({c}^{\text{*}}\in O\) such as \(J({c}^{\text{*}})=\underset{c\in O}{\text{Min}}J(c)\)

Finally, realignment is defined as the minimization of a particular type of functional function called « least squares » which are expressed in the form:

\(J(c)=\sum _{n=1}^{N}{j}_{n}^{2}(c)\) eq 1-2

where \({j}_{n}(c)\) represents the \(n\) component of the difference between the vector of calculated and experimental quantities.

It is commonly accepted that among deterministic minimization algorithms, the most effective for this type of functional is the Levenberg-Marquardt algorithm. It is the latter that was historically the first to be implemented in the command MACR_RECAL of Code_Aster and that we present below.