# Parameter Identification Methods for Hyperelastic and Hyper-Viscoelastic Models
## 2 Parameter Identification Method for the Hyperelastic Model
### The Identified Parameters and the Numerical Verification  
In this paper, the uniaxial tension, biaxial tension, and planar tension data from Treloar are taken as the experiment data Treloar (2007), and the Ogden model is chosen as the hyperelastic model owing to its favorable performances under large deformations. In order to solve Equation (11), the PSA is employed as the first step for its remarkable ability in searching for the global optimum results. Then, the L-M algorithm is chosen to further optimize the results from the first step due to the probable poor efficiency of the PSA when approaching the optimum results. All the procedures are realized in MATLAB MATLAB (2008). The number of terms considered for the Ogden model are respectively set as three and four, and seven groups of different initial values of undetermined parameters for each one are iterated in parallel. For the limited space and the similarity between the results of different groups, only the ones with the least iterations and its corresponding deviation \(S_{1}\) are chosen and listed in Table 1. The estimated parameters presented by Treloar are presented as well Treloar (2007). The unit of \(\mu_{i}\) in Table 1 is Mpa, and the item 'PSA 169000+' means more than 169 thousand iterations in PSA, and the 'LMA 48' means 48 iterations in LMA after the previous PSA procedure. Moreover, the computation time for the whole procedure with an i7 CPU is about 6 h.  
As the Table 1 shows, these three groups of parameters are all capable of fitting the experiment data very well, and the last group is the best owing to its least deviation 0.012., However, despite thousands of iterations of the PSA and LMA, the two sets of the identified parameters in Table 1 may not be the optimal results of Equation (12), because there may be local minimum points extremely close to the global minimum one, and these local optimal points have not been excluded by the PSA for their relatively excellent, not merely most excellent, competitions and they are identified by the LMA. Even so, these identified parameters, being optimal or near-optimal, are still very useful due to enough accuracy and controlled computation cost. Additionally, It should be mentioned that the PSA takes more than 200 thousand iterations to reduce the \(S_{1}\) from 0.1 to 0.0598, while the LMA only takes 65 iterations to reduce the deviation from 0.06 to 0.012, which strongly reflects the inefficiency of the PSA when approaching the global optimal point, and taking the LMA as a succeeding procedure to further optimize results is practical and efficient. In the last iteration of the LMA, the sensitivity values for the items \(\partial S/\partial u_{i}\) and \(\partial S/\partial\alpha_{i}\) are all more than \(10^{9}\), which means accurate estimations of all the parameters. For the purpose of verifying the estimated parameters, the last group of the identified parameters is taken as the example to predict the response of rubbers in different types of experiments in ANSYS. The results of the numerical simulation, the fitted data obtained by the use of the proceeding algorithm, along with the experiment data from Treloar are all plotted in Figure 1.  
As illustrated in the Figure 1, the results identified by the PSA and LMA fit the experiment data very well. Besides, in the numerical simulation tests, the results of uniaxial tension and planar tension coincide well with the experiment data (when the principal stretch is larger than 5, the FE analysis of uniaxial tension tests cannot converge due to some uncertain reasons). However, the results of biaxial tension tests differ a bit from the experiment data, the reason for the errors lies in that the boundary conditions of the biaxial tension experiment cannot be accurately satisfied when the principal stretch ranges from 1.5 to 1.9 in the simulation test. The FE initial models of uniaxial tension, planar tension, and biaxial tension experiment and their deformed shapes with the stretch being 5, 6.14, and 5.04 are presented respectively in Figure 2.