# Finite Element Simulation of HELICOIL(r) Inserts
## 4 Materials and Methods
### Objective Functions  
The vector of data \(\hat{\mathbf{Y}}\) and the vector of results \(\mathbf{Y}\) are used as inputs to the objective functions. It is obvious that the \(y_{k}\) values are obtained from FEA, while the \(\hat{y}_{k}\) values are obtained from an experiment. The dimension and the physical meaning of both is the same. We assume deformation loading in one axis (i.e., \(n=1\)) in this article. The results contain a vector \([t_{k},F_{k}]\) (\(m=1\)), where \(t_{k}\) and \(F_{k}\) are the \(k^{\text{th}}\) time point [s] and the \(k^{\text{th}}\) force [N], respectively. The time points are the same for vectors \(\mathbf{R}\), \(\mathbf{Y}\), and \(\hat{\mathbf{Y}}\).  
Based on the least squares method, COF \(g(\mathbf{Y},\hat{\mathbf{Y}})\) is defined by the following norm:  
\[\|\hat{\mathbf{Y}}-\mathbf{Y}\|_{C}=\frac{1}{N_{T}}\sum_{k=1}^{N_{T}}{(F_{k}^ {\text{EXP}}-F_{k}^{\text{FEA}})^{2}},[N^{2}], \tag{9}\]  
where the superscript FEA denotes an element of the vector \(\mathbf{Y}\) and the superscript EXP denotes an element of the vector \(\hat{\mathbf{Y}}\).  
For the use of some methods, it may be advantageous to divide the OF into several parts. We want individual POFs to indicate the influence of individual parameters on the result. Therefore, there are the same number of POFs as the searched parameters. The parameter point is described by its position \(u_{i}\) and by the value of the parameter itself \(p_{i}\), as defined in the previous section. Suppose a POF \(f_{i}(\mathbf{Y},\hat{\mathbf{Y}})\) is defined in the following form:  
\[\|\hat{\mathbf{Y}}-\mathbf{Y}\|_{i}=\frac{1}{N_{T}}\sum_{k=1}^{N_{T}}{(F_{k}^ {\text{EXP}}-F_{k}^{\text{FEA}})^{2}}h_{t}(t_{k})h_{p}(t_{k},\ldots),[N^{2}], \tag{10}\]  
where \(h_{t}(t_{k})\) and \(h_{p}(t_{k},\ldots)\) are correction functions.  
We define a time correction function \(h_{t}(t_{k})\), which reduces the effect of data differences with increasing time. The time correction function is defined as follows:  
\[h_{t}(t_{k})=a_{1}+a_{2}t_{k}:0\leq h_{t}(t_{k})\leq 1,[-], \tag{11}\]  
where \(a_{1}\) and \(a_{2}\) are parameters of the time correction function.  
A parameter correction function \(h_{p}(t_{k},p_{i})\) is used to include the effect of the \(i^{\text{th}}\) parameter. To determine \(h_{p}(t_{k},p_{i})\), we start from the classical objective function and attempt to identify the effect of the parameter, as follows:  
\[\frac{\partial\|\hat{\mathbf{Y}}-\Gamma(\mathbf{P},\mathbf{R})\|_{C}}{ \partial p_{i}}=\frac{\partial}{\partial p_{i}}(\frac{1}{N_{T}}\sum_{k=1}^{N_ {T}}{(F_{k}^{\text{EXP}}-F_{k}^{\text{FEA}})^{2}})=\frac{2}{N_{T}}\sum_{k=1}^ {N_{T}}{(F_{k}^{\text{EXP}}-F_{k}^{\text{FEA}})\frac{\partial\Gamma(\mathbf{ P},\mathbf{R})}{\partial p_{i}}}.\]  
We use the forward difference formula to solve the partial derivative, as follows:  
\[\frac{\partial\Gamma(\mathbf{P},\mathbf{R})}{\partial p_{i}}\approx\frac{ \Gamma(\{\ldots[u_{i},p_{i}+\Delta p],\ldots\},\mathbf{R})-\Gamma(\{\ldots[u_ {i},p_{i}],\ldots\},\mathbf{R})}{\Delta p}. \tag{12}\]  
Here, the value \(\Delta p\) is the search step size, and we use the same value for all parameters. Based on Equations (1), (9), and (12), we assume that the parameter correction function will correspond to a difference of the forces \(\Delta F_{i,k}^{\text{FEA}}\), as follows:  
\[\Delta F_{i,k}^{\text{FEA}}=F_{k}^{\text{FEA}}-F_{k}^{\text{FEA}},[N], \tag{13}\]where \(k\) is the subscript of the time point, \(i\) is the subscript of the parameter \((p_{i}+\Delta p)\), \(F_{k}^{\text{FEM}}\) is a reference value for the force, and \(F_{i,k}^{\text{FEM}}\) is a modified value for the force. The reference value is taken from the current best solution. The parameter correction function is designed as follows:  
\[h_{p}(t_{k},p_{i})=\frac{\left|\Delta F_{i,k}^{\text{FEM}}\right|}{\|\Delta F _{i,k}^{\text{FEM}}\|_{\infty,j}}:0\leq h_{p}(t_{k},p_{i})\leq 1,[-], \tag{14}\]  
where \(\|\Delta F_{i,k}^{\text{FEM}}\|_{\infty,i}\) is an infinite norm.