## 2 Engineering Design Meta-Model for Interface Design
### Engineering Design Meta-Model Network for Methodology Representation  
The meta-model is represented mathematically as a network structure composed of layers as well as nodes and edges (Figure 1). **Definition 1**. : _The meta-model is represented as a network structure composed of layers as well as nodes and edges. The engineering design multilayer network is defined as a quadruplet \(M=(V_{M},E_{M},V,L)\) where \(V\) is the set of nodes, \(L\) is the set of layers, \(V_{M}\subseteq V\) is the set subset of active-nodes, and \(E_{M}\subseteq V_{M}\times V_{M}\) is the set of active edges._  
The set of nodes \(V\) in the network of the engineering design meta-model is composed of four models: _Conceptual model_, _Mathematical model_, _Computational model_, and _Experimental model_. The nodes of the set \(V\) are denoted as follows: \(V=\{\textit{Conc},\textit{Math},\textit{Comp},\textit{Expe}\}\), representing the Conceptual model, Mathematical model, Computational model, and Experimental model, respectively. The engineering design meta-model network has one dimension named the _domain dimension_. The domain set of the engineering design meta-model consists of the following elements: _User domain_, _Function domain_, _Physical domain_, and _Process domain_. A layer is used for every element of this set. The nodes of the set \(L\) are denoted as follows: \(L=\{\textit{User},\textit{Func},\textit{Phys},\textit{Proc}\}\), representing the User domain, Function domain, Physical domain, and Process domain, respectively. The set \(V\times L=\{(v\in V,l\in L)\}\) encompasses all possible combinations of model-domain pairs. If a model is absent in a given design methodology, it means it is also absent in a given domain layer. This implies that for every node and layer pairing, it is necessary to specify whether the node is active within that layer. If a model is used in a given design methodology, then it is represented by an _active-node_. Definition 2: The subset of active nodes \(V_{M}\subseteq V\) comprises solely those combinations of nodes and layers where a node is present in the corresponding layer. In the meta-model multilayer network, edges represent pairwise connections between all possible combinations of nodes in the set of models {_Conceptual_, _Mathematical_, _Computational_, _Experimental_} and layers in the set of domains {_User_, _Function_, _Physical_, _Process_}, as illustrated in Figure 1.  
A model node \(v\in V\) in domain layer \(l\in L\) can be connected to any node \(v\prime\in V\), where \(v\prime\
eq v\), in any layer \(l\in L\). This means that intra-relationships within a domain, as well as inter-relationships between domains, exist. Assuming that intra-domain relationships and inter-domain relationships are reciprocal, all edges are considered undirected. Figure 1: The set \(V\times L=\{(v\in V,l\in L)\}\). For readability purposes, all possible combinations of nodes were not represented in the figure. **Definition 3**. : _An edge set \(E_{M}\subseteq V_{M}\times V_{M}\), referred to as active-edges, of the engineering design multilayer network is defined as pairs of node-layer combinations that exist within the set \(V_{M}\)._  
The intra-layer edges are represented using solid curves and inter-layer edges using dotted curves. Figure 2a shows an illustrative example, where \(M_{83}\) is the methodology employed in the 83 th article studied [22] in the literature review. \(M_{83}\) is composed of the related quadruplet \(V_{M_{83}},E_{M_{83}},V,L\) where \(V_{M_{83}}\) (red color) is the set of active-nodes that are part of the methodology in place in the articles, and \(E_{M_{83}}\) (red color) is the set of edges that orders the nodes in the methodology. In design methodology, the design phases may not operate bidirectionally. Assuming that intra-domain relationships and inter-domain relationships are not reciprocal, all edges are considered directed. Thus, a directed network is used to model such methodologies. The edges of the directed network are ordered pairs. **Definition 4**.