# On the Impact of Additive Manufacturing Processes Complexity on Modelling
## 2 Framework  
The principles of model-based system engineering (MBSE) [69] were taken into account for the development of the framework. According to MBSE, a system model comprises increasing detailed models that are all part of a holistic system model, aiming to \"construct a model of a system that we can transform into the real thing\" [69]. Complexity is not a straightforward concept [70; 71], due to the fact that it is problem-oriented and its definition varies [72]. For sure, the factors that are implicated change with the type of the system. For instance, depending on the (deterministic herein) model used, the following factors affect the complexity:  
* Algebraic systems \(\mathrm{f}(\mathrm{x}_{\mathrm{n}})=0\): number of variables \(n\), form of equations \(f\), solvability. * Dynamic systems \(\mathrm{f}(\mathrm{x}_{\mathrm{n}}\)',\(\mathrm{x}_{\mathrm{n}})=0\): number of variables, order of derivatives, form of equations, solvability. * Distributed systems \(\mathrm{f}(\frac{\partial^{k}x_{\mathrm{n}}}{\partial q_{\mathrm{m}}}\mathrm{ x}_{\mathrm{n}})=0\): number of variables, number of independent variables, order of derivatives, form of equations, solvability  
Figure 3: The current meta modelling consideration. Control problems \(\mathrm{f}(\frac{\partial^{k}x_{q}}{\partial q^{k}}\mathrm{x}_{n},\mathrm{u}_{m})=0\): number of variables, number of independent variables (\(u\)), order of derivatives, number of inputs, form of equations, solvability, controllability  
Focusing on linear systems and considering the control problem in order to frame the quantities needed, the following quantities are indicative of the complexity, given that the equation is reduced to \(\sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{n=1}^{N}a_{n,k}\frac{\partial^{k}x_{q}}{ \partial q^{k}}+\sum_{k=1}^{Q}\sum_{m=1}^{M}\sum_{n=1}^{LU}a_{n,k}\frac{ \partial^{k}u_{n}}{\partial q^{k}}=0\) could be expressed through a heuristic metric like the following, without taking into account the solvability or the effect of the inputs on output. * \(\mathrm{I}=\mathrm{log}(\mathrm{KMNQU})\)  
However, this metric is not applicable to process modelling. In particular, the phase change in not represented in such equations without some extra manipulation. Also, it is not inclusive for multi-phenomena modelling. Finally, it regards only straightforward theoretical modelling, i.e., the behavior of the machine tool is not included. To this end, the following four-steps approach of Figure 4 is adopted. The goal is to have an estimation of the interactions complexity among the discretized variables. Hence, the black dots represent the discrete values that each variable may take. The key variables are a newly introduced term that attempts to describe variables deriving from modelling and can have either input, or output, or even neutral character; examples of such variables for the case of SLM are: B1 temporal derivative of temperature, B2 temperature, B3 gradient of temperature. Examples of input variables (or process parameters) can be the laser power and the head (or table) speed and output variables (or performance indicators) can be process time and quality. The graph formed is not indicative of the true relationships. The procedure followed to acquire a quantified version of this relationship's complexity is given below in Figure 4. There are four steps, three of which are analyzed below and the fourth one is explicitly applied in the case of AM in the next section. The first three steps are related to acquiring such a relationship and rendering its simplification and its quantification feasible. Hence, they correspond to standard modelling techniques, with simplification in mind as one of the final goals. Also, it has to be mentioned that there is a possibility that the input parameters are not always completely measurable, so they could be eventually replaced by other metrics. For instance, the path configuration itself is not a quantifiable aspect, thus in literature it can be substituted by discrete configurations and intermediate metrics [22; 73].