# Towards an Automated Design Evaluation Method for Wire Arc Additive Manufacturing
## 3 Methods
### Application of WAAM Design Rules to 3D-MAT Geometry Indicators
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Feature** & **Indicator** & **Limits** & **Applied to Elements with...** \\ \hline wall thickness & \(s_{i}\) & \(\geq\)2 mm & \(0\leq\alpha_{i}\leq 65^{\circ}\) \\ clearance & \(s_{o}\) & \(\geq\)2 mm & \(0\leq\alpha_{o}\leq 65^{\circ}\) \\ edge radius & \(r_{i},r_{o}\) & \(\geq\)5 mm & \(30^{\circ}\leq\alpha_{i},\alpha_{o}\leq 175^{\circ}\) \\ wall angle & \(\alpha_{o}\) & \(\geq\)20\({}^{\circ}\) & \(s_{o}\leq 5\,\mathrm{mm}\) \\ thickness gradient & \(g_{r,i}\) & \(\leq\)0.23 & \(0\leq\alpha_{i}\leq 65^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of considered design rules. Figure 4: Different applications of Heuvers’ sphere. Figure 5: Comparison of various outer and inner angles and the secant used (red). Sections with an enclosed angle of \(0\leq\alpha\leq 65^{\circ}\) can be considered as tapered wall. raising [3]. Therefore, a lower limit of at least \(r_{\mathrm{i,o}}\geq 5\,\mathrm{mm}\) is defined for acceptable radii in lack of more precise guidance. Since the guidelines specify no differences for convex and concave corners, the same constraint is defined for inner and outer radii. For the classification of corners, the same considerations hold as for the detection of walls except that the inverse region is now of interest. Nevertheless, we chose an overlapping region of \(30^{\circ}\leq\alpha\leq 175^{\circ}\) to have a conservative approach that highlights a non-manufacturable element twice rather than neglecting it. Weld bead angles can be produced in arbitrary angles if path planning corrections are employed; however, \(\alpha\geq 20^{\circ}\) should be favored to avoid distortion [36], which is why we implement this recommendation on outer angles. The mentioned design rule relates to angle-induced distortion between distinct walls; therefore, the outer angle information is used. Since this distortion occurs only in weld bead's vertices, the restriction is limited to regions of small outer distance. We chose secant lengths of \(s_{\mathrm{o}}\leq 5\,\mathrm{mm}\) as a limit for these regions. Finally, the gradient indicator is used to spot regions of mass accumulation or sudden cross-sectional changes such as T-crossings which affect the heat transfer rate and in consequence warpage. At the current state of research, there are no quantitative recommendations for allowed gradients defined yet. Since the heat transfer problem in WAAM is similar to casting, the analogy of the Heuvers' spheres to the 3D-MAT is exploited to estimate maximal allowable gradients by the corresponding Heuvers' factors. Heuvers' factors limit the diameter ratio of touching inscribed circles. This factor \(k_{\mathrm{H}}=\frac{r_{2}}{r_{1}}\) can be transformed to a thickness gradient according to Equation (3):  
\[g_{\mathrm{r}}=\frac{r_{2}-r_{1}}{r_{1}+r_{2}}=\frac{k_{\mathrm{H}}-1}{k_{ \mathrm{H}}+1}. \tag{3}\]  
For a typical Heuvers' factor of \(k_{\mathrm{H}}=1.6\), the maximum limit of the thickness gradient is consequently \(g_{\mathrm{r}}=0.23\). This constraint is applied to wall elements as classified for the thickness measurement since corners are not regions of high mass accumulation despite high radius gradients.