# Finite Element Simulation of HELICOIL(r) Inserts
## 4 Materials and Methods
### Screening
#### 4.4.4 Selection Function  
The key for any selection is the POF value (\(f_{i}\)). The parameters \(p_{i}\) with the lowest values of \(f_{i}\) have the lowest effect on \(f\), and it can be considered non-effective to further divide the corresponding intervals or to identify the values of parameters. Conversely, the parameters \(p_{i}\) with the highest value of \(f_{i}\) have a key effect on \(f\) and must, therefore, be selected preferentially for dividing the interval or for identifying parameter values. Several variants of the selection function (\(\mathbf{P}^{a+1}\leftarrow\ Re(\mathbf{P}^{*},\ldots)\)) were tested during algorithm debugging. The superscripts \(a\), \(a+1\), and \(*\) indicate two consecutive steps and the tested variant, respectively. The following rules can be used for the task:  
1. All points are used; it is not necessary to calculate the value of \(f_{i}\). This variant is not used in this article, but was used during debugging of the adding functions: \[\mathbf{P}=Re_{1}(\mathbf{P}^{*})=\mathbf{P}^{*}.\] (36) This option leads to a shortening of the time needed for screening, but usually increases the number of points and, thus, the time-consuming identification of corresponding parameter values.
2. All points from previous step are used. This rule was always used: \[\mathbf{P}=Re_{2}(\mathbf{P}^{a})=\mathbf{P}^{a}.\] (37) For Adding function 1, we consider it sufficient that \(\exists u_{i}^{*}\approx u_{j}^{a}\).
3. The parameters with the highest value of \(f_{i}^{*}\) are selected. The rule \(Re_{3}(\mathbf{P}^{*},f_{S})\) can be formulated with respect to the limit value \(f_{S}\), as follows: \[\mathbf{P}=Re_{3}(\mathbf{P}^{*},f_{S})=\ \mathbf{IF}\ f_{i}^{*}>f_{S}\ \mathbf{THEN}\ [u_{i}^{*},p_{i}^{*}]:i=1,\ldots,N^{*}.\] (38)
4. The parameters/intervals with the lowest value \(f_{i}\) are excluded from the selection. The procedure is similar to the previous case; therefore, it was no longer used.
5. For parameter identification, only one new parameter can be added to each interval. With respect to Adding function 2, a point lying approximately in the middle of the interval was added. The rule can be formulated as follows: \[\mathbf{P}=Re_{4}(\mathbf{P}^{*},\mathbf{P}^{a})=\ \mathbf{IF}\ \exists u_{j}^{*}\approx(u_{i}^{a}+u_{i+1}^{a})/2\ \mathbf{THEN}\ [u_{j}^{*},p_{j}^{*}]:i=0,1,\ldots,N^{a},\] (39) where \([u_{j}^{*},p_{j}^{*}]\in\mathbf{P}^{*}\). This rule greatly simplifies the identification of parameter values.  
When using Adding function 1 (\(Add_{1}(P^{a},N^{*})\)), the following combination of rules (40) was used, in order to determine a new set of parameters \(\mathbf{P}^{a+1}\):  
\[\mathbf{P}^{a+1}=S_{1}(\mathbf{P}^{a},N^{*},f_{S})=\begin{cases}Add_{1}(P^{a},N^{*})\rightarrow\mathbf{P}^{*},\\ Re_{2}(\mathbf{P}^{a})\cup(Re_{3}(\mathbf{P}^{*},f_{S})\cap Re_{4}(\mathbf{P}^{ *},\mathbf{P}^{a})).\end{cases} \tag{40}\]  
When using Adding function 2 (\(Add_{2}(P^{a},f_{S})\)), the following combination of rules (41) was used, in order to determine a new set of parameters \(\mathbf{P}^{a+1}\):  
\[\mathbf{P}^{a+1}=S_{2}(\mathbf{P}^{a},f_{S})=\begin{cases}Add_{2}(P^{a},f_{S} )\rightarrow\mathbf{P}^{*},\\ Re_{2}(\mathbf{P}^{a})\cup Re_{3}(\mathbf{P}^{*},f_{S}).\end{cases} \tag{41}\]