## 4 Constraint-Based Process Modeling and Reasoning
### Knowledge modeling  
A process chain is created by stringing together several individual processes. Each process has input and output products that represent the links within the process chain. In manufacturing, these products are the individual production stages. Thus, a process chain represents a graph. This basic idea is taken up at this point by modeling the geometric transformations of the manufacturing stages by a single process as edges in the form of constraints and descriptive shape parameters of the manufacturing stages as well as transformation-influencing process parameters as nodes by variables with a specific domain. At this point, an abstraction and thus a lower resolution of the manufacturing processes is deliberately accepted in order to achieve a common level of abstraction within the process chain. If an abstraction is not possible for certain processes, it is also possible to use external surrogate models (see Section 5). Here a modular structure is followed, which groups all variables, domains, and constraints, which are assigned to a single manufacturing process, within the process containers. These process containers thus represent an abstraction of the capabilities of the respective process, e.g., resource-related manufacturing restrictions, achievable process windows, and available tools. In addition to the processes, the manufacturing stages are also important artifacts for describing a process chain, since they are the interfaces between the process containers. To formalize these interfaces, the manufacturing stages are explicitly annotated with specific geometry domains based on the variables. This geometric description can later be translated into three-dimensional CAD models via a link to a design system. The manufacturing resources significantly constrain the process capability and thus the feasible transformation of the manufacturing stages. The resources can also be formalized via characteristics. Limiting process parameters, such as a maximum weld-through thickness or formable diameters, are annotated via variables with a specific domain and linked to the geometry parameters of the manufacturing stages via constraints. These domains are defined by the available machinery and tooling. This modeling concept thus allows for the integration of the three artifacts--product, process, and resource--within a CSP. To enable the subsequent reasoning process (see Section 6), the modeling of the constraint network within the process containers follows a fixed scheme. When modeling constraints, CSP distinguishes between hard and soft constraints. Hard constraints represent the geometric transformations and physical or process-related restrictions that must not be violated. Thus, hard constraints define fixed limits for the manufacturable solution space for a precast geometry as the set of all possible manufacturing stage configurations. The soft constraints represent target requirements with which this solution space can be additionally restricted, starting from the available manufacturing resources. To model a process, the shaping characteristics of two successive manufacturing stages are linked via explicit binary hard constraints or higher order hard constraints. In this process, the shape change of the manufacturing stages can be modeled. The CSP must be designed in such a way that the consideration of all hard constraints results in at least one valid solution. In order to investigate the manufacturing suitability of the manufacturing stages, additional geometry-restricting manufacturing constraints are formalized within the process containers as binary soft constraints related to the manufacturing stages to be restricted. The soft constraints consist of a product-related (left side of the constraint) and a resource-related (right side of the constraint) side. At this point, resource-related manufacturing capabilities are integrated with available manufacturing resources.