## 3 Physical and Computational Model  
For the simulation of the liner deformation, the complex three-dimensional thermal effects, as well as the local stresses and strains, have to be regarded. This can be done with Finite Element Methods (FEM) [21]. The main challenge for FEM is the computational requirement and capacity, namely computational power, memory, and computational time. Therefore, it is important to work with sufficiently simple yet accurate models. The current study was done with the ANSYS mechanical software, which is able to study liner deformation [22; 23; 24; 25; 26]. It is attractive due to its capability of handling dynamic, thermal, material, geometric, and contact nonlinearities in the finite element model.  
The simulated engine in this work was based on a NISSAN CA18 gasoline engine, which had a piston stroke of 83.6 mm, cylinder bore of 83 mm, and total displacement of 1809 cm\({}^{3}\). The liner thickness was 1.6 mm, with a clearance of 80 \(\upmu\)m from the engine block. More details about this engine dimensions and specifications have been described by the author of [27]. The cylinder block and the cylinder liner were made of cast iron, while the cylinder head was made of an aluminum alloy. The physical material properties for the grey cast iron are shown in Table 1. The deformation of this cylinder liner was measured at 4000 rpm under full load [25]. The measured temperature at this operation points varied between 123 \({}^{\circ}\)C to 185 \({}^{\circ}\)C at the engine block and between 150 \({}^{\circ}\)C to 197 \({}^{\circ}\)C at the cylinder liner. The coolant temperature was 80 \({}^{\circ}\)C, and the used bolt pretension was 80,000 N. The top deck was restrained from the movement in the axial direction. Further details about the physical model have been described by the authors of [14].  
The numerical study was done on the terminal cylinder located at the end of the four-cylinder engine. The deformation of the terminal cylinder is different from the other intermediate cylinders due to uneven supply of the combustion heat in the front-rear direction. This leads to a third-order deformation pattern for the terminal cylinder in addition to the first-, second-, and fourth-order  
\begin{table}
\begin{tabular}{c c} \hline Density & 7200 kg/m\({}^{3}\) \\ \hline Thermal expansion coefficient (20–200 \({}^{\circ}\)C) & 11.7 \(\upmu\)m/mK \\ \hline Heat conductivity (200 \({}^{\circ}\)C) & 47.5 W/mK \\ \hline Elastic modulus & 103–118 GPa \\ \hline Tensile strength & 250–350 MPa \\ \hline \end{tabular}
\end{table}
Table 1: Physical properties for grey cast iron [28].  
deformations that also appear in the intermediate cylinders. One and a half cylinders were simulated to include the effects of the thermal load from the adjacent cylinder and the surrounding (see Figure 1). The simulated deformations were found to be in the range of up to 30 \(\upmu\)m for the fired conditions. Both the trends and the values of the deformations have good agreement with experimental data described by the authors of [25]. The detailed model configuration, boundary conditions, temperature distributions, mesh properties, grid independence test, circumferential position dependent deformation, and model validation have been explained by the authors of [14].