## 2 Numerical Analysis
### Combustion Chamber Geometry and Computational Mesh  
The object of this study is a 2-stroke heavy-duty ME-GI DF marine engine. This is a DF engine that operates according to the diesel cycle. The engine can be operated smoothly in both diesel and DF modes. In the diesel mode, the engine works with diesel oil only. In the DF mode, diesel oil serves only as a pilot fuel to provide an ignition source, while NG serves as a primary fuel. In ME-GI engines, both the gaseous primary and liquid pilot fuels are directly injected into the engine cylinder by gas and pilot nozzles accordingly. In this study, the effects of SA and IP on the cylinder pressure, temperature, performance, and emission characteristics of the engine in both diesel and DF modes were analyzed. The CFD analysis is derived from the finite volume method (FVM). This method reduces the degrees of freedom (DoF) from infinite to finite with the help of discretization or meshing. In CFD analysis, a continuous fluid domain is divided into a discrete domain consisting of a finite number of elements called \"elements\", which are connected through \"nodes\". The governing equations of fluid dynamics (the continuity, momentum, and energy equations) in the computational domain will be calculated at these nodes. This means that the number of equations that need to be solved and thus the calculation time will be proportional to the number of nodes in the computational domain. Therefore, in most cases of CFD analysis, researchers will always try to reduce the number of nodes of the computational domain to be as low as possible while still ensuring the mesh quality criterion. For 3D computational domains that are symmetric (axis or plane), an effective way to reduce the number of nodes is to compute for only one symmetric part of the entire domain. The cutting surfaces of the computational domain will be assigned a \"cyclic\" or \"periodic\" boundary condition [33]. In ICE simulations, if the engine has an axis-symmetric combustion chamber, as typically found in CIE engines since they do not contain a spark plug geometry, it is highly recommended to use only one portion of the entire computational domain for simulation [34]. In this study, the engine uses 3 gas injectors for NG injection and 3 other pilot injectors for pilot fuel injection, spaced by 120 degrees evenly on the cylinder head. The piston surface is a U-type shape. Due to the axial symmetry of the engine, only a one-third section of the entire 3D model of the engine has been modeled in order to reduce the calculation time. The one-third section of the engine combustion chamber geometry is shown in Figure 1, while Table 1 shows the main specifications of the engine. Figure 2 shows the one-third portion of the entire 3D computational mesh of the combustion chamber at the top dead center (TDC) of the engine. The movement of the piston surface during the calculation was modeled by a dynamic mesh generated by the layering method. Due to the characteristics of the layering method, the computational mesh used in this study is a structured mesh with all elements in the computational domain as hexahedral cells. The movement of the piston surface was controlled by Equation (1). The mesh properties together with mesh quality are shown in Table 2.  
\[P_{l}=\sqrt{\left(r_{c}+L\right)^{2}-x_{offset}^{2}}-r_{c}\cos(\theta_{c})- \sqrt{L^{2}-\left(r_{c}\sin(\theta_{c})+x_{offset}\right)^{2}} \tag{1}\]  
where \(P_{l}\) is the piston location, \(r_{c}\) is the crank radius, \(L\) is the length of the connecting rod, \(x_{offset}\) is the piston pin offset, while \(\theta_{c}\) is the current crank angle degree (CAD). In this study, the computational mesh is re-meshed according to the movement of the piston based on the dynamic Layering method. During the piston movement, mesh layers will be added or collapsed depending on the movement direction of the piston (downward or upward movement).