## 3 Functionality of Key Modules
### Completion of Model Based on Mesh Input  
The planar input mesh has to be in the \(xy\)-plane, and both linear and quadratic shape functions are supported, as shown in Figure 5a. The mesh type is assigned according to the chosen (plane stress or plane strain) formulation. In the sub-case approach (Method 2), the mesh is split by a circle and re-meshing is performed for the inner and outer volumes. This can affect the element quality, especially near the defect contour (Figure 5b). In the superposition approach (Method 3), the input mesh pattern is retained.  
Boundary conditions are defined only for the corner nodes, and multi-point equations (MPC) within the adjacent corners are set up. This involves the use of the same MPC for both methods in all load cases.  
Figure 5: Setup of finite element models for both methods based on the same input mesh (circular hole-in-plate; referenced input mesh ‘circle_in_R1 mm_v1’): (**a**) Planar input mesh-pattern; (**b**) Re-meshed sub-case model with boundary conditions; and (**c**) Model of superposition method with boundary conditions and connectors.  
The multi-point constraints are applied to all exterior edges and relate the displacements of the edge node <\(P\)> to the corners <\(A\)> and <\(B\)>. The displacement of the edge node <\(P\)> is based on the actual displacements of corner node <\(A\)> and corner node <\(B\)>, where the relative position \(l_{AP}/l_{AB}\) of the edge node on the corner vector is maintained (Equation (9)). The selected solver, Abaqus(r), supports a simplified multipoint constraint keyword '_SLIDER,P,A,B_', which is called in the generated input file after the \"MPC' section for each edge node.  
\[(\overline{u}_{P}-\overline{u}_{A})l_{AB}=(\overline{u}_{B}-\overline{u}_{A}) l_{AP} \tag{9}\]  
The boundary conditions for the tension load in the \(x\)- and \(y\)-axes are defined for the four corner nodes. The bottom-left corner is fixed, the bottom-right corner node displacement in the \(y\)-direction is set to zero, and the top-left corner node is fixed to zero-displacement in the \(x\)-direction. The other degrees of freedom at the boundary edges are defined according to the multi-point constraints. The tension loading is applied as a distributed load on the edges for the tension load cases. These boundary constraints are independent of the chosen method.  
As the derived model also requires transverse loading on the edges, the load is applied as distributed surface loading in the \(xy\)-shear load case. However, the constraints are further modified, according to our intention to set up only one model with three different load steps, instead of three models dealing only with a single load step each. In Abaqus(r), multi-point constraints cannot be activated or deactivated for certain load steps; therefore, an additional weld connector element--between the corner and a free node with an offset in the \(z\)-axis--is put into operation at the top-right corner node (Figure 5c). For the tension load cases, the displacement of the top and right edges is bound to this free node, which is linked itself as a rigid weld connector to the upper-right corner node. Thus, under tension loading, the edge orientations are unchanged. However, in the case of \(xy\)-shear loading, the connector element is de-activated by the \"_MODEL_CHANGE_' option, and a new boundary condition at the top-right corner ensures that this node is moved only on the diagonal axis of the bounding box. Thus, the shear loading is correctly applied.  
The displayed deformation scale in Figure 6 was purposely set to a value of 2000\(\times\) in order to visualize the effect of the applied boundary conditions more clearly for each load case. Unity load magnitude was applied for all load cases and methods.  
Figure 6: Displacement magnitude and principal stress results for different load angles \(\varphi\) (circular hole-in-plate; mesh ‘circle_lin_R1 mm_v1’): (**a**) Sub-case approach; (**b**) Basic load cases; and (**c**) Superposition of derived basic load cases.