## 1 Introduction  
The piston ring-cylinder liner (PRCL) assembly contributes to 20% to 50% of the total mechanical energy loss of internal combustion engine (ICE) [1]. In passenger cars, approximately 4%-7.5% of fuel energy is used to overcome the frictional losses by the piston assembly, and the oil control ring alone corresponds to 0.23% to 2.8% of the fuel energy in the diesel engine [2]. Piston rings need to be conformed to the bore shape to prevent lubricant oil consumption and corresponding emission of particular and hydrocarbons. However, high piston ring tension leads to increased friction losses, which means more fuel consumption and greenhouse gas (GHG) emissions [3; 4]. This implies that decreasing the friction in this contact without altering the sealing function can both increase the efficiency of the engine and reduce GHG emissions, which is a major challenge for the automotive industry [5; 6]. Many researchers have discussed the tribological performance of the PRCL friction pair. Most of them have focused on the honing texture, roughness, and orientation [7; 8; 9; 10; 11]. Only few publishedworks have discussed the enhancement of the PRCL conformation through the increase of the shape accuracy in the fired state. Increasing the roundness of the deformed liner in the fired state provides a significant opportunity for improving the PRCL performance [12; 13]. The authors of [14] studied the performance of initially noncircular liners numerically. In this study, it was shown that an engine with noncircular liners in the cold state can increase the roundness of the liners in its fired state significantly. The authors of [15] studied the effects of liner bore shape on the friction using a floating liner research engine. They found that using a larger bore diameter in the bottom of the liner decreased the friction. There are many reasons which cause the liner to deform from its ideal circular shape. This includes manufacturing inaccuracies, the assembly process, thermal expansion variations between different engine components, temperature gradients during the fired state, and mechanical load during fired operations [16; 17]. More than 85% of the accumulated deformation in the liner is referred to the operational mechanical and thermal loads [13]. The shape, the way, and the value of this deformation, in both longitudinal and radial directions, must have a great influence on any design improvement. Hence, the prediction of those deformation factors is important to enhance engine performance. The rate of radial liner deformation along the longitudinal direction is influenced by the uneven distribution of combustion heat to the cylinder. The cylinder wall temperature increases toward the top section resulting in increased liner diameter in the upper part of the liner. As the piston diameter increases as well, eventually, the lower part causes more friction [18]. This problem can be solved by applying a specific honing profile on the liner, offering a greater liner diameter in the lower part of the liner. This technology is already implemented by some manufacturers [12; 18; 19]. However, detailed work on the quantification of the parallelism and straightness of the liner in the fired state has not been published so far. The present article describes a methodology to analyze the uneven deformation of the liner between the cold and a hot state with numerical methods. The aim of this article was to find a defined methodology to reduce the engine friction and achieve better PRCL conformation through increasing the liner's straightness, roundness, and parallelism during the fired operation state. To reach that, a tapered bore liner in the cold state should be used. The main hypothesis in this work is that a suitable formed conical liner will be deformed to a more cylindrical shape in the fired state, such that the friction will decrease. Furthermore, mixing the conical shape with elliptical cross sections can be supposed to enhance the roundness, parallelism, and straightness of the liner.