# Finite Element Simulation of HELICOIL(r) Inserts
## 4 Materials and Methods
### Minimisation Method  
Many methods described in the literature can be used to solve minimisation problems (6). We first assumed that we are able to find a local/global minimum. However, in the task, we solve the problem (6) repeatedly with a different number of parameters, and assume a gradually increasing number of parameters. Some parameters may reach a local minimum earlier if parameter values are read from the previous minimisation phase. Other parameters may reach theirs later, if they were added in the previous screening phase. The solution method was designed with this problem in mind. Our update strategy is based on the following iterative formula for a parameter value:  
\[p_{i}^{a+1}=p_{i}^{a}+\Delta p_{i}^{a}=p_{i}^{a}+d_{i}^{a}\Delta p, \tag{15}\]  
where \(\Delta p\), \(\Delta p_{i}^{a}\), and \(d_{i}^{a}\) are the search step size, step size, and search direction, respectively, and \(a\) and \(a+1\) again represent two consecutive steps. For simplicity, we chose the same search step size for all parameters. For two consecutive steps, we require:  
\[f(\Gamma(\mathbf{P}^{a+1},\mathbf{R}),\hat{\mathbf{Y}})-f(\Gamma(\mathbf{P}^{ a},\mathbf{R}),\hat{\mathbf{Y}})<0. \tag{16}\]  
From (5) and the Taylor polynomial of the first degree of a function of \(N\) variables, the following applies (\(f_{j}(\Gamma(\mathbf{P}^{a},\mathbf{R}),\hat{\mathbf{Y}})=f_{j}^{a}\)):  
\[\sum_{j=1}^{N}(\sum_{i=1}^{N}\frac{\partial f_{j}^{a}}{\partial p_{i}}\Delta p _{i}^{a}+r_{j}^{a})<0, \tag{17}\]  
where \(r_{j}^{a}\) is a residual value of the \(j^{\text{th}}\) POF. We can use the forward differences, as follows:  
\[\frac{\partial f_{j}^{a}}{\partial p_{i}}\approx\frac{f_{j}(\Gamma([\ldots,[u _{i},p_{i}^{a}+\Delta p_{i}^{*}],\ldots],\mathbf{R}),\hat{\mathbf{Y}})-f_{j}( \Gamma([\ldots,[u_{i},p_{i}^{a}],\ldots],\mathbf{R}),\hat{\mathbf{Y}})}{\Delta p _{i}^{*}}=\frac{\Delta f_{j,\Delta i}^{a}}{\Delta p_{i}^{*}},\]  
\[\frac{\partial f^{a}}{\partial p_{i}}\approx\frac{f(\Gamma([\ldots,[u_{i},p_{i }^{a}+\Delta p_{i}^{*}],\ldots],\mathbf{R}),\hat{\mathbf{Y}})-f(\Gamma([ \ldots,[u_{i},p_{i}^{a}],\ldots],\mathbf{R}),\hat{\mathbf{Y}})}{\Delta p_{i} ^{*}}=\frac{\Delta f_{\Delta i}^{a}}{\Delta p_{i}^{*}},\]  
where \(\Delta f_{j,\Delta i}^{a}\) is the difference for the \(j^{\text{th}}\) POF, \(\Delta f_{\Delta i}^{a}\) is the difference for the OF, and \(\Delta p_{i}^{*}\) is the parameter change. The Equation (17) changes as follows:  
\[\sum_{j=1}^{N}(\sum_{i=1}^{N}\frac{\Delta f_{j,\Delta i}^{a}}{\Delta p_{i}^{*} }\Delta p_{i}^{a}+r_{j}^{a})<0, \tag{18}\]  
and can also be modified as follows:  
\[\sum_{i=1}^{N}\frac{\Delta p_{i}^{a}}{\Delta p_{i}^{*}}(\sum_{j=1}^{N}\Delta f _{j,\Delta i}^{a}+r_{i}^{a})=\sum_{i=1}^{N}\frac{\Delta p_{i}^{a}}{\Delta p_{i} ^{*}}(\Delta f_{\Delta i}^{a})<0, \tag{19}\]where \(r_{i}^{a}\) is a residual value of \(i^{\text{th}}\) parameter, and we assume that \(r_{i}^{a}\approx 0\). \(\Delta p_{i}^{a}\) and \(\Delta p_{i}^{*}\) can generally have different sizes and signs. Therefore, we introduce the following condition:  
\[sig(\Delta p_{i}^{a})=sig(\Delta p_{i}^{*}). \tag{20}\]  
The condition (20) must be fulfilled, which ensures that a sign vector is obtained. The initial values of the sign vector are:  
\[K^{0}=[x_{i}^{0}=1:\ i=1,2,\ldots,N]. \tag{21}\]  
The elements of the sign vector are calculated as follows:  
\[\kappa_{i}^{a+1}=\ \text{IF}\ \Delta f_{\Delta i}^{a}>-r_{min}\approx 0\left\{ \begin{array}{l}\text{\bf THEN}\ -\kappa_{i}^{a},\\ \text{\bf ELSE}\ \kappa_{i}^{a},\end{array} \right. \tag{22}\]  
where \(r_{min}\) is a residual value. A penalisation function \(\phi_{i}^{a}\) for the search direction (15) is defined as follows:  
\[\phi_{i}^{a}=\ \text{IF}\ (\Delta f_{\Delta i}^{a}>0)\left\{ \begin{array}{l} \text{\bf THEN}\ 0,\\ \text{\bf ELSE}\ 1.\end{array} \right. \tag{23}\]  
For all parameters, we define \(\Phi^{s}=[\phi_{i}^{s}:\ i=1,2,\ldots,N]\) as a penalty vector.