# Finite Element Simulation of HELICOIL(r) Inserts
## 2 Background  
In [2], the HELICOIL\({}^{\text{\textregisteredregistered}}\) insert was replaced by a contact with a cohesive zone setting for simulation purposes. The cohesive zone behaviour was defined by a multi-linear curve, which was identified manually. The follow-up step was to identify this curve using a gradient algorithm. During testing, it was found that the task occasionally converges to a sawtooth waveform (see Appendix A) in the initial part of the curve. This problem could be corrected manually; however, this is inappropriate for practical use. The second option was to find or design a suitable algorithm that would not lead to a sawtooth waveform. The last option was to design a mathematical model of the HELICOIL\({}^{\text{\textregistered}}\) insert that would clearly prescribe the shape, such that the problem would not occur. From our point of view, the second option was the most logical; that is, to test other known algorithms. During testing, it was found that the sawtooth waveform arises depending on:  
* The number and position of the points of the multi-linear curve;
* The settings of the algorithm used;
* The magnitude of the multi-linear curve update.  
Conditions were set that limited the formation of the saw curve shape. These conditions relate to the method of identifying the multi-linear curve, not the definition of the curve. The curve definition--that is, the mathematical model of the HELICOIL\({}^{\text{\textregistered}}\) insert--will be the next step, and multi-linear curves can be used for its analysis and validation. The associated conditions are as follows:  
1. Gradual adding of points to the curve;
2. Frequent updating of the gradient after a pre-defined number of cycles;
3. Multi-linear curve update size limit;
4. Reducing the effect of thread damage at the end of the measured time, when the HELICOIL\({}^{\text{\textregistered}}\) insert is gradually pulled out of the hole and the threads are already broken.  
The first point led to a proposal inspired by the interval halving method, which adds only one point to the interval found in the previous cycle. Next, we defined a partial objective function, which describes the influence of each point of the multi-linear curve on the value of the objective function, as well as a screening approach, which identifies the most appropriate place to add a new point. Thus, the curve could be identified successively with an increasing number of points, until the defined accuracy given by the objective function value is met. The points identified in previous cycles no longer change, in some cases, so a simple constraint based on the sign of the gradient was added to reduce the number of points identified.  
The second point added to the convergence check, allowing for checking of the maximum number of cycles per gradient calculation. The third point led to the use of a so-called normalised gradient, for which the magnitude of the added value can be controlled very well. The last point led to the correction of the values of the objective function with respect to time.  
These modifications were implemented into the algorithm and objective function using a penalty function. In the resulting algorithm, the formation of a sawtooth waveform no longer occurred in the test example. The conditions were not applied to the multi-linearcurve; which, in our opinion, would lead to the construction of a mathematical model for the HELICOIL(r) insert. Instead, the conditions were applied to the method of identifying the parameters of the multi-linear curve. The proposed identification method differs from standard procedures and, in our opinion, some of the modifications used may be useful in other cases.