## 5 Geometric Deformation Indicators  
To check if the use of the conical liner affected the roundness of the liner's cross sections, the roundness errors were calculated at different elevations. The roundness was evaluated with the least squares circle method [14]. Briefly, the roundness error is the difference between the maximum and minimum circles that confine all the data points in the cross section (see Figure 6a). Furthermore, the straightness error is calculated as the minimum distance between two straight lines that contain all the points as explained in Figure 6b. The straightness error represents the change of the radius over the length, while the roundness error represents the change of the radius over the circumference. The parallelism error equals the difference between maximum and minimum distances between points at identical elevations in two lines facing each other, as explained in Figure 6c and Equation (5). \[\textit{Parallelism error}=d_{max}-d_{min} \tag{5}\]  
The maximum value of the parallelism error is the summation of the straightness error of these two lines. The parallelism represents the change of the diameter over length, and thus shows the change in the bore tightness in a certain direction. This error is critical for both engine friction and PRCL performance. The reason for the use of both straightness and parallelism error is to predict the closeness of the liner's wall to the required straight parallel shape, since the parallelism error alone cannot predict the existence of parallel curved walls. The results are summarized in Tables 2-4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Cold State**} & \multicolumn{3}{c}{**Fixed State**} \\ \hline
**Shape** & **Side** & **Thrust-Antithrust** & **Front-Rear** & **Thrust-Antithrust** & **Front-Rear** \\ \hline Straight circle (ref.) & 0.0 & 0.0 & 11.6 & 13.5 \\ \hline Conical circle & 9.8 & 9.8 & 4.4 & 5.4 \\ \hline Straight ellipse & 0.0 & 0.0 & 12.3 & 13.5 \\ \hline Conical ellipse & 9.8 & 9.8 & 4.4 & 4.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parallelism deviation in the axial direction for the liner in both the cold state and under typical fired operational conditions. (Numbers are given in micrometers). Figure 6: (a) Roundness error; (b) Straightness error; (c) Parallelism error. Tables 2 and 3 show that both the parallelism and straightness of the liner increased in the fired state when a conical liner was used. The parallelism deviation decreased by a factor of three from values of around 12 \(\upmu\)m to values around 4 \(\upmu\)m. Simultaneously, the straightness error decreased from values of around 6 \(\upmu\)m to values around 3 \(\upmu\)m, which indicates that the walls of the liner became straighter and more parallel. They also showed almost the same values for both elliptical and circular cross sections, which indicates that the parallelism and straightness were independent of the liner cross section. In the thrust-antithrust direction, the liner was almost straight and parallel. An identical deviation of two micrometers out of a straight line was found at the top region of the liner on both sides. Here, the reason was given essentially from the pre-tightening load [25]. It can be assumed that this deviation could also be reduced if the longitudinal bore shape of the liner in its cold state was processed with a detailed nonlinear shape. Then, this shape could be derived from the reverse deviation of the fired state deformation. For the front-rear direction, a deviation from a straight shape occurred in the lower part in the rear side due to an uneven supply for the combustion heat from the terminal side and from the side of the adjacent cylinders. This led to a certain eccentricity in the lower part of the liner. From Table 4, it can be seen that the roundness error increased significantly in the fired state if a circular shape was used in the cold state. This holds true for both the straight and the conical circular liner.