# On the Development of a Surrogate Modelling Toolbox for Virtual Assembly
## 2 State of the Art
### Prior Work for Surrogate Modelling for Joining Processes  
According to [PERSON] [47] a model is defined by the criteria _mapping_, _reduction_ and _pragmatism_. Models represent a real-world system in a simplified mathematical description and are used in order to predict a system output for a known system input. The connection between inputs and outputs is mapped by a simplified black-box, grey-box or white-box model. _Mapping_ means the representation of a real system by an artificial input-output connection that is reduced to the aspects that are relevant for the scope of the researched application (_reduction_). A model replaces the real system only for a particular application and is valid only under certain temporal and causal restrictions as well as for certain subjects (_pragmatism_). _Validity_ is a fourth criterion introduced by [PERSON] [48], which states that a model should represent the real system with a certain quality. The distinction between model and SM however is vague. In common practice, an SM means a simplified model for which no connection between inputs and outputs is known a-priori. Thus, the two main problems _model construction_ (finding a sufficient SM) and _model appraisal_ (assessing the model error) have to be dealt with [49]. By surrogate modelling, it is aimed to decrease the complexity for model formation and computation while maintaining a sufficient prediction quality [36]. Since SMs are less accurate due to simplification, they are mainly efficient for real-world problems that admit less detailed considerations. In the scope of this work, SMs for joining processes are researched. The _generic joining process_ according to [PERSON] et al. [50] consists of the four steps _Locating and Placement_ of parts in a fixture, _Clamping_ of the parts, _Joining_ and _Relensing_ from the fixture [27], as shown in Figure 3.  
SMs for geometrical properties map physical effects of the real-world assembly to geometrical deviations of the assembly. Influences on the geometry of the assembly from all four mentioned steps of the joining process need to be considered in the SM. A systematic overview of factors effecting the geometrical quality is given by [PERSON] et al. in [34]. Here, the authors mention influences from _material properties_, _part  
Figure 3: Generic joining process according to [PERSON] et al. [50]. variation, part collision, gravitational forces, the fixture_ and the _joining process_. A detailed discussion on influence factors is given in Section 3. Prior work on surrogate modelling comprises the research of [PERSON] et al., where the plastic deformations in the contact zone for a hip prosthesis assembly is considered [51; 52]. The effective contact surface is computed by erosion of the surface morphology up to a state, where assembly and contact forces are in equilibrium. The approaches of [PERSON] and [PERSON] [53; 54] as well as [PERSON] et al. [55] regard elastic deformation by implementing Finite Elements calculations in the contact simulation, implemented in the Cassino Unified Tolerance Analysis, and ANATOLEFLEX software, respectively. [PERSON] and [PERSON] introduced the Method of Influence Coefficients, where the connection between initial part deviations, assembly forces and resulting assembly deviations is linearized by means of a sensitivity matrix [56; 57]. [PERSON] et al. explore the use of neural networks for the geometrical variation propagation in assemblies concerning geometrical features from GD&T tolerances such as tolerance of size, location and form [58]. The mentioned concepts are either complex to apply or application-specific and thus indicate the need for a simplified, generic approach.