## 2 Development of Beam Element-Based Simulation
### Model Description
#### 2.1.3 Boundary Conditions  
In the model, the contact between the bed and print can be modeled in two ways. The bottom layer of the print can be fixed to the bed, simulating a perfect bond between the first layer of the part and the bed. To fix vertical displacement, vertical springs are added extending from the centroid of the lowest bead to the height of the bed. The bottom ends of these elements are fixed. To lock horizontal displacements, constraints of the form of Equations (14) and (15) are added, where the first bead is the upper layer and the constraint constants and values for the lower bead are all zero.  
The second option for boundary conditions in the model simulates no adhesion of the part to the bed, which means the part should be able to slide horizontally and freely lift off from the print bed but not penetrate into it. The simulation of this condition has significant practical importance, as it is not unusual to sandwich a bond breaker between a part and the print bed. To ensure numerical stability, very soft horizontal springs are attached to the lowest bead. To allow only positive vertical displacement, bilinear beam on elastic foundation (BBOEF) elements were used for the lowest bead that have distributed vertical springs with a very large compressive stiffness and very small tensile stiffness relative to other elements in the model. The derivation of the contribution of the springs to the element stiffness matrix is as follows. From the discretized weak form given in Equation (5), the contribution of the springs to element stiffness is given in Equation (20),  
\[K_{se}=\int_{0}^{l}kN_{b}^{T}N_{b}dx. \tag{20}\]  
Because \(k\) is not constant over the length of the element, the integral must be split as shown in Equation (21),  
\[K_{se}=\int_{L_{c}}k_{c}N^{T}Ndx+\int_{L_{t}}k_{t}N^{T}Ndx, \tag{21}\]  
where \(L_{c}\) and \(L_{t}\) are the portions of the element where the springs are in compression and tension, respectively. The compressive stiffness \(k_{c}\) is large to prevent inter-penetration of the bed, and \(k_{t}\) is set to a very small value to allow free separation of the part from the bed if it is unbonded. Because \(L_{c}\) and \(L_{t}\) are not known a priori, the use of these elements introduces nonlinearity, requiring an iterative solution strategy as discussed in Section 2.1.4.  
The formulation of the BBOEF element was verified through comparison of the model prediction using 100 elements with the analytical solution of [PERSON] and [PERSON] [74] for the displacement of a free-free beam on a tensionless foundation with a point load at the center. The authors normalize all values by \(\beta\), given in Equation (22),  
\[\beta=\left(\frac{k_{c}}{4 EI}\right)^{\frac{1}{4}}, \tag{22}\]  
where \(E\) is the Young's modulus of the beam material and \(I\) is the second moment of area of the beam cross-section. Important values are normalized as shown in Equation (23),  
\[\begin{split}& l=\beta L\\ & F=\frac{P}{4\beta^{2}EI}\\ &\zeta=\beta x\\ & w=\beta v\end{split} \tag{23}\]  
where \(L\) is physical beam length, \(P\) is the magnitude of the applied point load and \(v\) is transverse displacement. [PERSON] and [PERSON] [74] chose to show the results of a beam with normalized length of four and two cases of normalized applied load of 0.1 and 0.2. For the BBOEF beam, \(EI\) was chosen to be 4300 kN-m\({}^{2}\) and \(k_{c}\) was 1.76 \(\times\) 10\({}^{6}\) kN/m\({}^{2}\). From this,\(\beta,\ L\) and \(P\) were calculated and the simulation was run. The two solutions are compared in Figure 5, which indicates excellent agreement.