## 4 Mapping Test Model
### NSGA-II Optimization  
To ensure that the model outputs meet expectations across multiple sets of simulation trials, a multi-objective optimization algorithm is employed to optimize the model parameters. By minimizing the error between the output results of multiple data sets and the reference values, the optimal parameters are obtained. As shown in the Figure 8, NSGA-II [36; 37] loops through the interval to generate the parameters until the multiple objective function Y is minimized and the sum satisfies the condition as the optimal parameter of the mapping test model. NSGA-II is a multi-objective optimization algorithm, which is used to solve optimization problems with multiple variables and multiple objective functions. The steps include population initialization, non-dominant sorting and crowding allocation, selection operation, crossover and variation, recombination, and selection. Repeat the above steps, iterating over the parameters of the model until convergence reaches a satisfactory solution set. Three types of errors--delay error, amplitude error, and shape error--are considered as the three optimization objectives [38].  
The cross-correlation was used to obtain the delay error, which was originally used by [PERSON] to evaluate the error in the output of the MADYMO model [39; 40]. Suppose there are two signal sequences, \(a_{i}\), \(b_{i}\)\(i=1,2,3,...\), \(n\). The cross-correlation r at the delay d can be expressed as  
\[r(d)=\frac{\sum\limits_{i}{[a_{i}-\mu_{a}][b_{i-d}-\mu_{b}]}}{{\sqrt{\sum \limits_{i}{{{{(a_{i}-\mu_{a})}^{2}}}}}\sqrt{\sum\limits_{i}{{{{(b_{i}-\mu_{b})} ^{2}}}}}}} \tag{14}\]  
where \(\mu_{a}\) is the average value of \(a_{i}\), \(\mu_{b}\) is the average value of \(b_{i}\).  
When r is the largest, d is the delay of the signal, and the delay error is obtained by Formula (16):  
\[r(d_{0})=\max(r(d)) \tag{15}\]  
\[\varepsilon_{p}=d_{0}\cdot\Delta t \tag{16}\]  
\(\Delta t\) is the signal sampling period.  
Figure 8: NSGA-II flow.  
Figure 7: Mapping test model architecture.  
The amplitude error is defined as the relative error of the norms between the two sets of sequences, expressed by the following equation:  
\[\varepsilon_{m}=\frac{|a_{i}-b_{i}|}{|b_{i}|} \tag{17}\]  
The shape of the sequence is represented by the slope of the two sets of sequences, and the shape error is defined as  
\[\varepsilon_{s}=\frac{\left|\dot{a_{i}}-\dot{b_{i}}\right|}{\left|\dot{b_{i}} \right|} \tag{18}\]  
Before using NSGA-II to optimize the parameters, the search range of the parameters needs to be determined, and the model output range should cover the destination value, so the sum range is set as follows:  
\[\begin{split}\text{min}Y=(\varepsilon_{p}\varepsilon_{m} \varepsilon_{s})\\ \left\{\begin{array}{l}\text{max}(\sum\limits_{s=1}^{k}(y^{1}_{ i,j+s}.h^{11}_{s}+y^{2}_{i,j+s}.h^{21}_{s})+b^{1}_{j})\geq\text{max}(a^{FEM}_{ij})\\ \text{min}(\sum\limits_{s=1}^{k}(y^{1}_{i,j+s}.h^{11}_{s}+y^{2}_{i,j+s}.h^{21}_ {s})+b^{1}_{j})\leq\text{min}(a^{FEM}_{ij})\\ \text{max}(\sum\limits_{s=1}^{k}(y^{1}_{i,j+s}.h^{12}_{s}+y^{2}_{i,j+s}.h^{22} _{s})+b^{2}_{j})\geq\text{max}(F^{FEM}_{ij})\\ \text{min}(\sum\limits_{s=1}^{k}(y^{1}_{i,j+s}.h^{12}_{s}+y^{2}_{i,j+s}.h^{22} _{s})+b^{2}_{j})\leq\text{min}(F^{FEM}_{ij})\\ j=1,2,\ldots,n-k;\;s=1,2,\ldots,k\end{array}\right.\end{split} \tag{19}\]  
where \(a^{3}_{ijFEM}\) is the head acceleration calculated by the finite element model, and \(F^{FEM}_{ij}\) is the axial compression force of the neck calculated by the finite element model.  
Since it was observed that the response of Hybrid III 50 th FEM is delayed compared to the simple dummy, reversing the objective data in time can significantly reduce the convergence time.