**2. Reductionism**

The objective is to generate a mathematical representation of the modelled system, confining the description to a mathematical input/output representation. This behaviour description will be the result of a network of interacting basic entities. For these reasons, we apply reductionism to identify the smallest underlying relevant entities in the context of an intended application. Applying reductionism in model construction places the base entities' definition, the base building blocks, into the centre of development.

From the perspective of mathematics, describing the dynamic results in time-dependent differential equations for time-discrete systems uses time-difference equations, and for event-discrete systems, automata are used, and thus state-discrete difference equations. All the models must satisfy a fundamental condition: Independent of their nature, all equations must represent realisable systems, which, for physics, translates to *causal* or *nonanticipative systems*.

#### *2.1. Continuous Macroscopic Physical Systems*

Reductionism, when applied to continuous macroscopic systems, recursively splits the physical system into smaller volumes. The dividing process stops once one reaches a "suitable" granularity. The resulting base entities are three-dimensional dynamical systems that live in four-dimensional spacetime. Partial differential equations describe their behaviour and the applicable fundamental conservation principles. They describe the evolution of the state of the modelled smallest volume; the identified entity, namely, the state change is the consequence of the interaction with the environment. As a mechanism to establish the conservation and balance equations, one "walks" the surface of the volume and accounts for all interactions, and thus the conserved quantity crossing the boundary, which gives rise to the term "control volume". Overall, an accounting operation, that is based on the system and the conserved quantity. The assembly of extensive quantities then represents the fundamental state of the volume. The network of elementary control volumes, the base entities, and their interactions across common boundaries describe the behaviour of a modelled fundamental entity.

#### *2.2. Particulate Physical Systems*

In many cases, the modelling is not focused on a complexity described by a hierarchy of systems, but complexity arises from having many objects on the same level of granularity. Examples include molecular dynamics or large quantities of particles or models that approximate continuous systems with many particles, such as smoothed-particle hydrodynamics [4] or the like. One of the main problems with these systems is the formulation of boundary conditions and the forces acting between them. The fundamental entity in such systems is the particle.

Dynamic state equations describe the behaviour of the particles. With the particles' capacities, this results in ordinary differential equations, with population balances being a typical representation. As each particle has a state, the size of the equation set is an apparent computational problem.

#### *2.3. Control Systems*

Control is an enabling technology. It allows for steering and maintaining the state of the process. Processes must be driven; there must be driving forces acting on the process to keep it from its natural steady state. Thus, processes must be embedded in an environment that is not in equilibrium with the process. The parts that make up the environment are also not in equilibrium with each other, thereby enabling the process to "run" like a water wheel between an elevated water reservoir, ejecting it to the lower-level reservoir. This Carnot-kind of viewpoint is very generic.

The manipulation of the flows between the different constituent parts of the system makes it possible to move the process into any place in the attainable region defined by the environment (Refers to controllability.). Thus, the state of the environment determines the attainable region, and it is the main controls that act on the flows between the environment and the system that control the overall state of the process.

A control system implements the externally provided objectives in terms of target values for states or state-dependent quantities. Analysing the ideal controller, namely, requesting that the process instantaneously follows the given setpoint, shows that the controller would have to invert the plant. Since the plant exhibits capacity behaviour, the inversion is not causal, and therefore not feasible. Consequently, all controllers implement an approximate inverse plant, and their states mirror the plant's states.

The nature of the equations depends on the nature of the controlled system. In the case of a sampled system, time-discrete difference equations are used, while for event-observed systems, an automaton is used, and thus also a difference equation. Continuous control implies analogue controllers, which are physical systems and may also be modelled in the physical domain.

#### *2.4. Other Relevant Subjects*

The simulation of technical and natural systems is often augmented with analytical tools. For technical systems, these may be techno-economical or ecological analytical tools, such as Life-Cycle Analysis or financial return. Statistical tools also belong to this class of extension.

#### **3. Networks & Tokens**

Overall, reductionism describes the modelled physical system as a network of interacting capacities that partially project onto the control, and all other relevant subjects or disciplines.

If we attach the term "discipline" to physics and control, it is natural to subdivide each discipline into more specific "subdisciplines". For physics, this leads to a "tree" of disciplines, which captures continuous, macroscopic, particulate, small-scale, atomic, etc. For control, the control pyramid is used, with the event-discrete layers on top, planning, scheduling, optimisation and sampled systems below, as well as model-predictive control, optimising control, unit-level control and low-level control.

The subdivision into disciplines and subdisciplines for physics is somewhat involved. Carrying out subdivision on a global level leads to a very complex and large framework. The members of the European Material Modelling Council are working on such a global classification, called the European Materials Modelling Ontology (EMMO) (For information, see https://emmc.info/emmo-info/, accessed on 27 March 2021 and for the ontology https://github.com/emmo-repo/EMMO, accessed on 27 March 2021). ProMo enables the definition of subtrees of the EMMO discipline classification, aiming at the construction of smaller EMMO-related ontologies, which are more focused on specific application domains. Also, ProMo aims to handle large-scale models and is not limited to materials, although it has a substantial section that is associated with materials.

#### *3.1. Higher-Level Abstraction*

### 3.1.1. Tokens

The multidisciplinary nature of the problem requires higher-level abstraction. The network is the first element which we lift by adding the concept "tokens". The domainspecific "tokens" are the items living in the specific "networks". The "networks", have directed graphs as node capacities for the "tokens", whilst the edges are transporting "tokens". The "tokens" within a capacity, a graph's node, define the state of the node.

For physical systems, we use the conserved quantities as "tokens", primarily mass, energy and momenta, but also items that make up a population. In control, the token is a signal, and a monetary measure in economic systems. Thus, the abstraction of a (sub)-discipline is a network of capacities for tokens communicating tokens.

#### 3.1.2. Intra-Faces and Inter-Faces

The definition of networks and tokens provides the basis for the definition of the two types of interaction: "Intraface" is a communication path for tokens, while "interface" communicates state-dependent information.

The "intraface" thus couples similar networks, where *similar* implies that the two coupled "networks" contain the same "tokens". An "intraface" thus communicates "tokens". In contrast, if one couples two networks that are not similar, but have different "tokens", the link will transfer information, usually state-related information. To give an example, in the first case, one may transfer mass from a liquid to a gas phase. Mass is transferred through an "intraface". In contrast, for the second case, state information is passed to the controller through an "interface". The generated value for a manipulated variable in the physical system is the information given back to the physical system.

#### 3.1.3. Nodes & Arcs

"Networks" are directed graphs. The "nodes" or vertices represent the *capacities for the tokens*, and the "arcs" or edges represent the *flow of tokens*. The directed arcs define a reference coordinate for each flow of tokens. Tokens of the same type may be grouped and transported by one arc.

Figure 1 shows all elements that make up the graph of graphs in a minimal example. The physical part of the plant, labelled with "plant", shows two networks, a liquid phase and a gas phase. Those two communicate with each other mass through a common

"intraface". The liquid phase seeks material information from the material model, providing state information, typically species, pressure, temperature, and asking for a physical property, like density. Both channels operate through an "interface". The liquid phase also provides state information to a control network, again through "interfaces".

**Figure 1.** Network of networks representation of multi-disciplinary model graphs.

On the smallest scale, the "nodes" also represent basic entities, which result from applying reductionism.

*3.2. Formal Definitions—A Summary*

The discussion in this section yields the definition of a couple of items:


**Node**: *is a component of the "network", which exhibits the ability to store "tokens".*


**Intraface**: *communicates bidirectionaly tokens between nodes of two intra-related networks.*

**Interface**: *communicates unidirectionally state-related information from a node of one network to a node in the inter-related network.*

#### **4. Basic Entities**

"Entities" are discipline and application-specific, but also have some common properties, namely, the dependency on the free variables' time and the three spatial variables, and thus the four-dimensional spacetime.

#### *4.1. Time Aspects*

Any model of a dynamic system embraces three time scales: (i) "constant"—the parts that are not changing with time, (ii) "dynamic"—those parts that exhibit dynamic capacity behaviour, and (iii) "event-dynamic"—those parts that change instantaneously.

#### *4.2. Spatial Aspects*

For "physical entities", objects live in the four-dimensional spacetime. Continuous domains may then be categorised into classes.

**Lumped systems**: *are finite-dimensional volumes where the relevant intensive properties are not a function of the spatial coordinates, thus a spatial domain in which the relevant intensive properties are uniform in terms of the spatial distribution.*

In contrast:

**Distributed systems**: *are finite-dimensional volumes where the relevant intensive properties are a function of the spatial coordinates, thus a spatial domain in which the relevant intensive properties are not uniform in terms of the spatial distribution.*

Both "lumped systems" and "distributed systems" are basic building blocks (entities) when modelling macroscopic systems.

#### *4.3. Deterministic, Stochastic & Ergodic*

"Entities" may exhibit deterministic or stochastic behaviour. Given a deterministic input and a fixed entity, then the output is also deterministic. If any either the input or the entity's properties are stochastic, then the output will also be stochastic.

"Egodicity" is when the time average is identical to the state average of identical processes at one point in time, or in other terms: "all accessible microstates are "equiprobable" over a long period of "time" (Source wikipedia).

#### *4.4. Continuous Physical Systems*

The nature of "basic entities" is described above. The objective is to generate equations for numerical computations. Therefore, we need to define the input/output behaviour as a set of mathematical equations.

We create the equations on the background of system theory [6], leaning towards Willhelm's behaviour theory [7]. A state-space view serves the purpose of analysing and controlling the structure. The "state" takes a centre position. The definition of the term "state" is hard to trace, but one can find an early version in Caratheodory [8]: "... ein "Zustand" des Systems S, und wir wollen für die Zahlen *xi* selbst den Namen "Zustandskoordinaten" einführen." We shall have a look at the two essential domains, namely, continuous systems and control.

#### 4.4.1. Thermodynamic Systems

The domain of macroscopic, thermodynamic systems builds on the concepts of conserved mass, energy and momentum, while Newton's laws govern mechanical systems. Focusing on thermodynamic systems, one would typically also allow for the conversion of chemical or biological species, which, building on atomic mass conservation, gives rise to species balances.

Balances and conservations follow the same principle: one defines them by walking the system's boundary, accounting for all the quantities crossing the surface. The change occurring inside the system is then equated with the transfer, and, in the case of species conversion, it is augmented with the species' transposition, yielding a "species balance". Expressing these concepts verbally: "accumulation = net flow across the surface + net consumption". Conservation principles do not include the last term; it only appears as a consequence of allowing for conversion and inducing the atomic mass conservation, all of which turn into a "balance".

#### 4.4.2. Mechanical Systems

Mechanical systems are handled in much the same way, except that one balances momenta, and the energy analysis focuses mainly on kinetic and potential contributions. However, fields other than gravitational are also considered depending on the application.

#### *4.5. Particulate Physical Systems*

In many cases, reductionism yields particles as the smallest entities required to capture the system behaviour. Molecular dynamics is one example, and fluid models building on particles is another. It is particles that characterise these systems, and momentum balances with forces, representing the momentum flow, form the core of the description.

#### *4.6. A First Classification of Variables*

The generic conservation/balancing operation enables us to define a first set of variable classes, namely, the classes "state", "transport", "conversion". The "accumlation" term is represented by the spacetime derivative of the state variables.

If one draws up the scheme defining the model, one quickly recognises that one part is missing. For these reasons, the missing piece is often called "closure", which, by their nature, are state variable transformations. Looking at a diagrammatic representation, Figure 2 shows the mathematical representation of a generic deterministic physical system, with **x** being the "state" of the system in the form of a block diagram also showing the connections for a control system. The scheme has four external connections: the initial conditions **<sup>x</sup>**(0), the boundary conditions **<sup>y</sup>***b*and the two connections to the control system, namely, the manipulated variables **u**, the measurement **y**.

**Figure 2.** A generic dynamic physical system.

The dark-yellow box on the left indicates the balances and conservations. If the state is limited to the conserved quantities, then the yellow box is the conservation, and otherwise there is a balance. The green box on the top represents the transport laws for the extensive quantities, and the lower green box the reactions or transposition. Both are a function of a class of variables, which we call "secondary states". Typical members are the quantities driving the transport of extensive quantities, namely temperature, pressure and chemical potential, while concentration defines the probability function in reaction kinetics. It is the red box that closes the gap between the state and the secondary state. The red box's main components are thermodynamic models relating the extensive state to the intensive properties and geometrical relations that link geometrical variables to volume. The main physical–mathematical underlying frameworks are Hamiltonian systems for mechanical and contact geometry for the thermodynamic parts. Both define a configuration space. While classical mechanics is well established, contact geometry does not enjoy similar popularity. The energy formulation reads

$$M\left(S, V, \underline{\mathbf{n}}, \frac{\partial}{\partial S}, \frac{\partial}{\partial V}, \frac{\partial}{\partial V}, \frac{\partial}{\partial \underline{n}}\right) \tag{1}$$

with the last three elements being the temperature *T*, the negative pressure *p* and the chemical potential *μ*. The thermodynamic configuration space is the assembly, including *U*, *S*, *V*, **n**, *∂ U∂ S* , *∂ U∂ V* , *∂ U∂* **n** . An early introduction can be found in [9], followed by [10–12], which provides an extensive exposition of the subject.

Processes are controlled. A single-layer control scheme may be of the form shown in Figure 3. Some of the variable classes are different from the ones in the physical system. There is still the state and the differential state. One now has to classify outputs and inputs, setpoints and control error. Figure 3 also shows the interfaces transferring state information

in one direction, with control settings in the other direction, with the latter also being a function of the controller state via the output function.

**Figure 3.** A generic control system.

The two examples demonstrate that the variable classes are domain-specific. While one can streamline things to some extent by using a state-space approach, one needs to adapt the variable classes to the disciplines to effectively support the process of defining the variable and equation system.

#### **5. ProMo Ontology**

The previous sections provide the main elements used to define the structure of the ProMo ontology. The top definition is the tree of disciplines. For each discipline, we must define elements associated with the structural elements enabling efficient handling of the building blocks and the elements that provide the framework for defining each building block's behavioural equations.

We do not aim at generating a global ontology for all processes. Even if this was a feasible undertaking, the argumen<sup>t</sup> here is the improved targeting of groups of modellers (see also the discussion in Section 3), at least not in the first instance. Another argumen<sup>t</sup> is flexibility. An ontology is a living structure and will change with time, continuously aligning with new requirements. It is for these reasons that ProMo has its own ontology structure and a corresponding editor.

The organisation of the disciplines follows a couple of simple rules:


The sample ontology Figure 4 shows the discipline tree of an ontology designed for modelling a large group of macroscopic processes. The green disciplines are in an **inter-relation**, while the blue ones are in an **intra-relation**. On the top level, the ontology includes a physical and a control branch. We have the generic macroscopic systems, the material descriptions, and an example of a mixing domain devoted to empirical mixing models on the physical level.

**Figure 4.** A sample discipline tree. It is designed to capture a very wide range of processing plant models. The green nodes are in an inter-relation, and thus transfer tokens, while the blue ones are in an intra-relation, and thus transfer state information.

Each node has two branches of definitions: the branch associated with defining the process models' structure and the branch with the definition of the variable classes. Figure 5 shows the definition of the root of the discipline tree.

The overall system is defined as dynamic with basic entities belonging to all three timescale domains. For the representation of the network structures, we declare the variable class "network" and "projection". Both are associated with the "graph", the network. For the system theoretical definitions of the model equations, one defines the variable classes "frame" for the free variables, "state" and "constant". The two additional classes are currently explicitly defined, namely, classes associated with the spaces generated by differentiating the states and frame variables. These definitions are required because of the design choice of uniquely identifying the dimensions of all mathematical objects. The issue of indexing has been introduced and discussed in [1].

Moving down the discipline tree, one can extend and augmen<sup>t</sup> the definitions. Figure 6 shows the example of the node labelled with "physical". Thid is the parent to the nodes "macroscopic", "materialDB" and "mixing". It augments variable classes with "secondary state" and "effort". The former appears in the generic representation of the dynamic physical system, Figure 2, while the term "effort" is a class that captures the effort variables [13], also called the conjugates to the thermodynamic potentials or generalised forces. In the context of contact geometry Equation (1), the temperature, the pressure and the chemical potential are three elements of the odd configuration space [9,10].

The structure includes the extension of the nodes, which applies to all nodes in the tree. It captures the time-scale property of capacitive elements and details it with the distribution effect.

**Figure 5.** The **root** node in the discipline tree. It has two branches: the structure branch is used to define the ontology items used for designing the items associated with capturing the model as a directed graph; the behaviour branch serves the purpose of defining the variable classes used to capture the mathematical description of the entity behaviour models.

**Figure 6.** The physical node in the discipline tree. Both branches are expanded with additional items. In the structure branch, the basic entities are populated with tokens and specified with the modelled distribution effects.

#### **6. Entity Behaviour**

*6.1. Equations*

The ontology intrinsically defines the basic entities in the discipline-specific frame. For physical systems, this is the frame spacetime, while for control, the frame is usually limited to time.

ProMo's equation editor can enter and edit the mathematical behaviour description. As mentioned above, we start with a set of "port" variables: the constants, the frame variables, and the variables defining the configuration spaces' base. We define a small, tailored language to enter the equations, namely, a variable defined by an expression. The description of the language and the mechanisms associated with implementing the details is beyond this paper and will have to be reported separately. Further information can be found in [3]. Thus, in very brief terms: The equation editor implements a parser and a template machine. The parser generates an abstract syntax tree, and the template machine uses the abstract syntax tree and generates different compiled versions of the equations, including LaTex rendering of online documentation. The main features of the parser are that it implements the index (For the discussion on the indexing, we refer to [1]) and rigorous unit handling.

The variable/expression bipartite multi-graph: We construct the equation system starting with the "port" variables: frame, constants and the state variables. For physical systems, the base state variables are the ones defining the configuration spaces. The state variables appear in the block diagrams Figures 2 and 3 after the integral. We then follow the paths in the respective block diagram. For physical systems, shown in Figure 2, the next step is to define the equations representing the red box, labelled with " state variable transformations", whereby each equation can only be a function of already defined variables. For the control system, these are the nonlinear dynamic function and the nonlinear static output function.

We allow for the definition of more than one equation for the same variable. This approach allows us to implement the basic thermodynamic variables, for example, temperature as a partial derivative of internal energy with respect to entropy. Variables with an implicit model equation must first be defined with an explicit model equation. Only then can the implicit model be added. This enables the proper handling of indexing and unit handling. Further details can be found in [3].
