## 4 Results of Comparison of Proposed MVD Algorithm with Existing Algorithms  
The proposed maximal vertical distance recursive algorithm (MVD), as illustrated in the Ramberg-Osgood model, is compared with two algorithms, one being the global Visvalingam Whyatt (VW) algorithm [31] and other one being an independent equal-intervals algorithm (EI) [29], both mentioned in Section 2.1. Adjustments were made to the original VW algorithm to ensure that it generates a simplified polyline with the same number of points as obtained from the MVD algorithm. This ensures a fair comparison, focusing solely on the accuracy of the resulting polylines in regard to the original material curve, eliminating the variable of a different number of points on a simplified polyline, which can result in different computing times when loading the multilinear curve to the finite element model. The VW algorithm computes the area of triangles formed by three successive points on the initial curve. This necessitates the discretization of the initial material curve, leading to potential variations in the position of points on the final simplified polyline based on the initial discretization. The VW algorithm requires user-input values, including the material constants of the used-material model, the number of initial discretization points, and the desired number of final points on the simplified polyline. Starting with the material constants, the material curve is initially discretized to a specified number of points. In each iteration, the algorithm calculates the minimal triangle area from the first to the last point. The middle point of the triangle with the minimal area is then removed, and the next iteration proceeds with all points except the removed one. The algorithm stops when it attains the desired number of points on a simplified polyline. Ultimately, the percentage of the stress deviation (\(SD_{\text{fin}}\)) of the constructed polyline is computed in relation to the original material curve. The EI algorithm, on the other hand, simply selects a fixed number of consecutive points with equal intervals based on strain and also maintains an equivalent number of points to the MVD algorithm for the sake of comparison. The comparative analysis of the MVD, VW, and EI algorithms is presented in Table 1 and is conducted on the plastic portion of Ramberg-Osgood material model with a starting strain value of 0.2% and final strain value of 30% for illustrative purposes. During the generation of simplified points through the MVD algorithm, two \(SD\) limits, 2% and 5%, were investigated. The resultant number of final simplified points obtained from the MVD algorithm was then employed as a stopping criterion for both the VW and EI algorithms. Figure 8: Proposed maximal vertical distance (MVD) recursive algorithm. For the VW algorithm, the initial plastic portion of the Ramberg-Osgood curve underwent discretization by generating equally spaced strain values between the starting and final strain values. To evaluate the VW algorithm's performance under different levels of discretization, the discretization of the initial Ramberg-Osgood curve was performed using different numbers of initial points: 500, 1000, and 10,000. Using the EI algorithm, stress values were also computed from the plastic portion of the Ramberg-Osgood curve at equal intervals of strain values, corresponding to the number of points obtained from the MVD algorithm. This systematic procedure was repeated for two distinct steel materials, each characterized by different Ramberg-Osgood monotonic material constants. These constants cover a wider range and are obtained from experimental tests from the available literature. The first studied material is characterized by \(K=549\) MPa and \(n=0.038\)[34], and the second material by \(K=1435.7\) MPa and \(n=0.276\)[35].