# Computation of Implicit Representation of Volumetric Shells with Predefined Thickness
## 6 Shells with Variable Thickness  
It can sometimes be of interest to allow for a variable thickness of the shell [16; 17]. Applying the method outlined in this paper to shells with variable thickness would imply substituting \(h\) by the function \(h(\vec{x})\) in all stages of \(\phi^{(n)}(\vec{x},h(\vec{x}))\) before taking the derivative. The implicit dependence of \(h\) on \(\vec{x}\) would then automatically be taken into account. However, this approach would not produce the desired result, which can be explained as follows: The proposed method approximates the thickness of the shell by moving a predefined distance along the normal of \(\phi(\vec{x},h)=0\) for increasing \(h\). This procedure is only accurate under the assumption that the normal of \(\phi(\vec{x},h)=0\) does not deviate substantially from the normal of \(f(\vec{x})=0\), such that the procedure follows a straight line. However, if \(h\) is not constant, the normal becomes, through the chain rule of differentiation:  
\[\vec{n}=\frac{\
abla\phi(\vec{x},h(\vec{x}))}{|\
abla\phi(\vec{x},h(\vec{x}))| }=\frac{\
abla\phi(\vec{x},h)+\frac{\partial\phi(\vec{x},h)}{\partial h}\
abla h (\vec{x})}{|\
abla\phi(\vec{x},h)+\frac{\partial\phi(\vec{x},h)}{\partial h }\
abla h(\vec{x})|}. \tag{32}\]  
The direction of the normal will therefore be governed by how \(h\) changes with \(\vec{x}\), which is another way of saying that if \(h\) is not constant, \(\phi(\vec{x},h(\vec{x}))=0\) is naturally not parallel to \(f(\vec{x})=0\).  
Another and more straightforward approach to obtain variable thickness is to assume \(h\) to be constant in the derivation of \(\phi(\vec{x},h)\) and only replace it when evaluating \(\phi(\vec{x},h)=0\). This also has limited accuracy because if \(h\) is a function of \(\vec{x}\), it changes by a value \(\sim h\partial_{n}h\) over the thickness \(h\). For smooth variations of the thickness, it is reasonable to assume that \(\partial_{n}h=\mathcal{O}(h)\) such that \(h\partial_{n}h=\mathcal{O}(h^{2})\). Hence, the accuracy of the proposed method for variable thickness is bound to second order. This is still sufficiently accurate for many applications.  
Figure 8 shows the convergence for shells with a linearly increasing thickness along the \(x\) axis for the different TPMS. It is observed that the first-order method remains to be first-order accurate, albeit it performs better than expected for the Gyroid at larger  
Figure 7: Neovius convergence study. In both panels, the error in the measured offset \(h_{m}\) is plotted against different predefined normalized offsets. In the left figure, the offsets were measured starting from points on \(\vec{x}_{Noo}(\vec{x})=0\) to points on each surface \(\phi^{(n)}(\vec{x},h)=0\), respectively. In the right figure, the measurement direction is reversed. The triangles represent the corresponding slope for each order \(n\). The nonasymptotic behavior shown at offsets of larger values (e.g., \(h/L=0.1\)) is attributed to the high curvature of the original surface.  
thicknesses. The third-order method typically shows better results than the second-order method but it too converges only with second order. Since the intrinsic error in the thickness is already of order \(\mathcal{O}(h^{2})\), further increasing the order of \(\phi^{(n)}(\vec{x},h)\) would not improve the asymptotic nature of the result. For a shell thickness of 0.01 L, the second-order method already obtains errors smaller than 0.03% of the target thickness in all three cases. Figure 9 shows the resulting bodies with variable thickness.