# Digital Twin-Based Clamping Sequence Analysis and Optimization for Improved Geometric Quality
## 2 Method
### Proposed Clamping Sequence Model  
To model the assembly process in the geometry assurance digital twin, Section 1.3, the nominal parts are integrated with the corresponding nodal displacements from the 3D-scanned data. This way, each mesh node in the assembly is associated with an offset. The parts are then positioned in a fixture. There are two types of fixturing elements during the modeling process: the orientation elements, where the in-plane movements are locked, and the clamping elements, where the normal-to-surface movements are locked. The positioning of the non-rigid parts is based on the N-2-1 positioning systems, where over-constraining (for rigid bodies, the 3-2-1 positioning systems are used to lock the 6 degrees of freedom; here, N can have a value larger than 3) is allowed in the normal-to-surface direction to compensate for bending, e.g., due to gravitational force. The overview of the enhanced MIC approach is presented in Figure 1.  
Utilizing the stress-strain law of the elastic bodies, the deformations are captured,  
\[\mathbf{f}=\mathbf{K}\mathbf{u}, \tag{1}\]  
where \(\mathbf{f}\) is the force, \(\mathbf{K}\) is the global stiffness matrix, and \(\mathbf{u}\) is the vector of the initial gaps. Similarly, the clamping and joining forces are calculated. Because, during the assembly, virtual penetrations between the components need to be constrained, contact modeling is utilized to capture the contact forces. The contact forces are captured by defining the computational nodes, representing the areas where the parts are in contact nominally. During assembly, the penetration is checked in the following nodes, and the reaction forces are applied to the assembly. The contact reaction forces are derived through the quadratic optimization of the minimum forces required to bring out the surfaces from the penetrated state. \[ \begin{split}\underset{F}{\text{minimize}}&\frac{1} {2}\mathbf{f}^{T}\mathbf{S}\mathbf{f}+\mathbf{f}^{T}\mathbf{u}\\ \text{subject to}&-\mathbf{S}\mathbf{f}\leq\mathbf{u}\\ &\mathbf{f}\geq\mathbf{0}.\end{split}  \tag{2}\]  
Here, \(\mathbf{S}\) is the assembly sensitivity matrix derived from the fixtured assembly expression,  
\[\mathbf{K}\mathbf{u}=\mathbf{f}_{\mathbf{c}}+\mathbf{f}_{\mathbf{c}}+\mathbf{f}_{\mathbf{w}}, \tag{3}\]  
where \(\mathbf{f}_{\mathbf{c}}\) is the clamping force, \(\mathbf{f}_{\mathbf{c}}\) is the contact force, and \(\mathbf{f}_{\mathbf{w}}\) is the force required for welding the parts together. In this formulation, \(\mathbf{S}=\mathbf{K}^{-1}\) holds only with respect to the relevant rows in the stiffness matrix [5]. The solution to the optimization problem (2) is achieved by an interior point solver. This solver breaks down the quadratic program into a set of linear problems by introducing slack variables. This unconstrained problem is then solved iteratively by the Mehrotra predictor-corrector method [39]. The fixture is a stiff part (rigid part) fixed in the space to which the parts are positioned and clamped (in the CAT models, a fictive part without any geometry is often created as a fixture because only six positioning points in each part need to be positioned on the fixture; for this reason, the fixture can be without any geometry). The above steps consider the clamping points to be applied simultaneously, thus not allowing for a sequence analysis of the clamping conditions. To enable a clamp sequence analysis, we introduce a stiff fixture with the same geometry as the nominal parts for each included part in the assembly. Now, the  
Figure 1: Overview of the assembly digital twin model. ficturing force in Equation (3) is decomposed into two segments. First, the forces caused by the locating elements in the fixture are derived. Later, the forces caused by the clamping elements are stated. We denote these by \(f_{\mathbf{l}}\) and \(f_{\mathbf{c}l}\), respectively.