## 1 Introduction
### The Constants Identification Methods
#### 1.2.2 Numerical Methods  
In this method, the computations continue until some geometrical parameters such as the final profile of the specimen predicted by numerical simulation converges to that obtained from an experiment such as tensile test. Plain or notched specimens can be used in the experiments. Determination of the constants of Gurson damage model through direct measurement and experimental testing is very difficult. On the other hand, the response of materials in numerical simulation definitely depends on these constants. Consequently, the constants can be determined by comparing material response in different loading states in numerical simulation with that of experimental measurements. Various researchers utilized this idea to find the damage models constants. [PERSON] et al. [9; 10] obtained the variation of fracture strain versus stress triaxiality coefficient for notched steel and copper specimens of different notch radii. The results were used for determining the constants \(D_{1}\) to \(D_{3}\) in Johnson-Cook damage model. They also obtained the variation of fracture strain versus Ln \(\hat{\varepsilon}\) under dynamic test conditions. The results were employed for determining the constant \(D_{4}\) in Johnson-Cook damage model. [PERSON] et al. [11] performed plain specimen tensile test and measured the fracture displacement of the specimen. Then, they simulated the same tests using a finite element code. The constants \(q_{1}\), \(S_{n}\), \(\varepsilon_{n}\), \(f_{0}\) and \(f_{n}\) were obtained from the literature. For determining \(f_{c}\) and \(f_{f}\), simulation was performed for different sets of \(f_{c}\) and \(f_{f}\) and for each simulation the fracture displacement was recorded. The next step was to define the difference between fracture displacement of the specimens obtained from experiment and that of simulation as an objective function. By minimizing this objective function, \(f_{c}\) and \(f_{f}\) were identified. They used the computed constants for numerical simulation of automobile component under crash loading and found a good compatibility with experimental results. In another study, [PERSON] et al. [12] predicted the \(f_{c}\) and \(f_{f}\) constants of Gurson model for aluminum die cast alloy and high strength steel, using minimization of the difference between component displacement parameter obtained from numerical simulation and the one measured in tensile test of the plain sample. They applied the same constants in a numerical simulation of the tensile test of notched samples. Moreover, they used these constants for automobile components in crash tests. The results obtained were found to agree well with the experimental results. [PERSON] and [PERSON] [13] determined the constants of Gurson damage model using the displacement-load curves obtained from experiment and a nonlinear optimization method. They defined the difference between the load measured in experiment and predicted by simulation at some points of displacement-load diagram as objective function for optimization. This objective function is defined as:  
\[\phi(p)=\frac{1}{2}\sum_{i=1}^{n_{l}}\Bigl{[}f_{i}(p)-\overline{f}_{i}(p) \Bigr{]}^{2}, \tag{17}\]  
where, \(n_{l}\) is the number of points on displacement-load diagram, \(\phi(p)\) is the objective function, \(f_{i}(p)\) is the force obtained from simulation and \(\overline{f}_{i}(p)\) is the force measured from experiment. The optimization of this objective function was accomplished by an iterative nonlinear method. [PERSON] et al. [14] used digital photography method for obtaining the specimen's profile in each loading increment. By collecting the profiles for each loading increment, they acquired the data to determine the damage and material models constants. [PERSON] and [PERSON] [15] employed local deformation measurements to determine GTN damage model constants.