## 3 Strategy and Objective Function for Damage Identification
### Proposed Strategy  
In this section, a methodology for damage properties identification is described in three main steps. Figure 4 represents a complete flowchart of the proposed strategy for geometry damage properties identification.  
Figure 3: Relative error on natural frequencies versus COMSOL Multiphysics standard mesh parameters for 3D FE model of damaged beam. Each line corresponds to the eigenfrequency in increasing order.  
Step 1: A set of healthy state frequencies are identified from the experimental beam. 2D and 3D FE healthy model are built, and [PERSON]'s modulus of FE models are updated to closely match the frequencies with the healthy beam (the cost function is described later on). Then, in the 3D FE model, the damage is materialized as a rectangular discontinuity considering the position, width and depth of the damage. At the same time, the damage is materialized as a local reduction of bending stiffness in one element in the 2D FE model. Numerical correlation using FSC minimization between 2D and 3D FE models is performed, that is, for each set of physical parameter, a bending stiffness reduction is sought in the 2D model. At the end of this step, damage library is obtained, where 3D rectangular damage provides the width and depth, and 2D model gives bending stiffness reduction in percentage value of severity at particular damage position. More precisely, the damage library is a relation between physical parameters and bending stiffness reduction. Moreover, these parameters can be extracted in the form of one item from the damage library.  
Step 2: An updating of the 2D FE model of damaged beam is performed using FSC to identify location and severity of single or multiple damage. Typically, the basis for the model update is a set of frequencies issued from a 2D model, a 3D model or an experimental measurement. For a given updating procedure, four beam models with the same characteristics (healthy beam case, damaged beam case, 2D healthy model and 2D stiffness updated damaged model) are used. The stiffness reduction coefficient (or severity) and the localization of the reduction are updated during this procedure. The minimum value of the FSC is sought during the updating process. The argument of the minimum yield wanted location and severity. Practically, the minimization is performed using particle swarm optimization (PSO) [26] in MATLAB global optimization toolbox. If the damage is identified, then we can proceed to the next step for geometry investigation.  
Step 3: In this step, based on the results of **Step 2** and **Step 3**, the identification of damage physical parameters can be done. The estimated damage severity from **Step 2** is searched in the damage library. It corresponds to a (potentially nonunique) set of width and depth for a rectangular geometry. Note that curve fitting may be necessary if the library is not precise enough and that a tolerance can be introduced. The nonuniqueness of the solution and the way to go beyond will be discussed later on.  
Figure 4: Flowchart of proposed strategy for geometry damage properties identification  
Remark: The interest of using a damage library instead of directly correlating the tested structure to a 3D model is indeed to reduce the computational time.