## 3 Materials and Methods
### Defining the Maximal Vertical Distance Measure  
A maximal vertical distance measure is introduced and implemented to optimally select the points along a material curve for retention. The stopping criterion, defined as a percentage of stress deviation, \(SD\), will be demonstrated using a Ramberg-Osgood material model for illustrative purposes. However, with the necessary adjustments, the stopping criterion and accompanying algorithm can also be applied to other material models. The Ramberg-Osgood model presented in Equation (1) can be further simplified by segmenting the linear and nonlinear portions of the stress-strain curve as follows:  
\[\varepsilon=\left\{ \begin{array}{ll}\sigma/E&\quad\text{for }\sigma\leq R_{ \text{e}}\\ \frac{\sigma}{E}+\left(\frac{\sigma}{E}\right)^{\frac{1}{D}}&\quad\text{for } \sigma>R_{\text{e}}\end{array} \right.. \tag{5}\]  
This form is widely applied in finite element simulations, as it takes advantage of the linearity of the stress-strain relationship before reaching the yield point [33]. Materialsdisplay an elastic component of strain both before and after surpassing the yield stress, allowing for the further subdivision of the nonlinear portion of the complete stress-strain curve into elastic and plastic components. The plastic portion of the Ramberg-Osgood curve is a stress-strain power law given by the equation \(f(x)=Kx^{n}\), where \(x\) represents plastic strain values and \(f(x)\) represents stress values. The strain-hardening exponent, \(n\), has a value between 0 and 1. The strength coefficient, \(K\), is typically defined as the stress value corresponding to a plastic strain value of 100%. To visually demonstrate the maximal vertical distance measure applied to the chosen material model, the stress-strain power law model is presented in Figure 5. This figure shows a line denoted with a function \(g(x)=ax+b\) connecting two arbitrary points, \(A\) and \(B\), on the material model defined with power law \(f(x)\). As depicted in Figure 6, subtracting the lower line function \(g(x)\) from the upper material curve \(f(x)\) yields a new function labeled \(h(x)\): \(Kx^{n}-ax-b\)  
\[h(x)=f(x)-g(x)=Kx^{n}-ax-b. \tag{6}\]  
If a given function \(h(x)\) is differentiable on an interval \([a,b]\), then the maximum value of \(h(x)\) in that interval must occur either at \(x\in[a,b]\), where the first derivative is equal to zero, or at the endpoints \(a\) and \(b\) of the interval. Thus, the first derivate of this function is:  
\[h^{\prime}(x)=nKx^{n-1}-a. \tag{7}\]  
The \(x_{\max}\) value for which \(h(x)\) is maximized is then obtained by setting the derivative of \(h(x)\) to zero:  
\[nKx^{n-1}-a=0, \tag{8}\] \[x_{\max}=\left(\frac{a}{nK}\right)^{\frac{1}{n-1}}.\]  
Figure 5: Ramberg–Osgood example curve \(f(x)\) and line segment \(g(x)\). Figure 6: Maximal vertical distance measure I. The maximal vertical distance between the line segment \(g(x)\) and the material curve \(f(x)\) is calculated by substituting \(x_{\max}\) (Equation (8)) into the function \(h(x)\) (Equation (6)):  
\[ \begin{split} d_{\mathrm{v,max}}&=Kx_{\max}^{n}-ax_{ \max}-b,\\ d_{\mathrm{v,max}}&=K(\frac{a}{nK})^{\frac{1}{n-1}}-a( \frac{a}{nK})^{\frac{1}{n-1}}-b.\end{split}  \tag{9}\]  
Figure 7 illustrates the process of obtaining the maximal vertical distance between a continuous curve given by a power law and a line segment. The resulting maximum vertical measure (\(d_{\mathrm{v,max}}\)) is then normalized by the stress value corresponding to the strain value for the maximal vertical distance, denoted as \(\sigma_{\max}\). This yields a measure used as a stopping criterion in the proposed algorithm called the percentage of stress deviation (\(SD\)), with the formula defined as:  
\[SD=\frac{d_{\mathrm{v,max}}}{\sigma_{\max}}\cdot 100. \tag{10}\]  
When using an alternative material model defined by a different constitutive equation, it is essential to integrate the relevant parameters and functional expressions, along with their derivatives, into the presented framework.