# Computation of Implicit Representation of Volumetric Shells with Predefined Thickness
## 7 Conclusions  
In this paper, we proposed a systematic methodology for deriving analytic implicit body shells of predefined thickness from implicit surfaces. The convergence order and thereby the accuracy of the thickness of the shell can be freely adjusted trough the extended Richardson extrapolation. The motivation for our method comes from engineering applications of additive manufacturing, where such infill volumes are required for the actual 3D-printing process on the one hand and for numerical simulations of such structures  
Figure 8: Convergence study for varied offset in the \(x\) direction. In each figure, the error in the measured offset \(h_{m}\) is plotted against different predefined normalized offsets governed by a linear function \(h(x)\). The offsets at \(x=0\) are set to zero, while the offsets at \(x=L\) vary from zero to a maximal value of 0.1 L. The offsets were measured starting from points on different offset surfaces \(\phi^{(n)}(\vec{x},h)=0\) obtained with approximations up to order three. No further improvement is observed when using an approximation method with an order higher than two. Figure 9: Infill volume enclosed between two offset contours from the original interface \(\widehat{g}(\vec{x})=0\) of the Gyroid, Schwarz P, and Neovius, respectively. For a unit cell, the offset distance linearly increases from 0 up to 0.1 in the \(x\) direction. In the above figures, the method used to obtain the implicit representation for each of the offset interfaces is the second-order method. on the other hand. For constant thicknesses, our method reaches a precision of smaller than \(10^{-6}\) for wall thicknesses measuring 1% of the wave length of the periodic structure. This precision by far exceeds the accuracy of contemporary 3D-printing technology. By opting for offset functions of lower-order computational time can be reduced. The memory requirements for storing the offset function of a TPMS is negligible compared to a discretized surface mesh. Being analytic, the offset function does not lose any quality when scaled. At the time of this writing, it is not straightforward to use the offset function directly as input for commercial 3D printers as they typically expect a tessellated mesh representation of the surface, which can, of course, easily be obtained from discretizing the zero contour of the offset function. Our method can be used to identify, for any point in the domain, whether it lies in the shell or outside the shell. As the shell can be expressed as a solid body in this way, the question of water tightness of the surface never arises. However, the slicing of the object for G-code generation and 3D printing was not addressed in this theoretical work. In the presented semianalytic form, our method can only be applied to surfaces provided as differentiable implicit functions. Such surfaces are always closed as they provide an inner and an outer side. In 3D printing, it is often of interest to include open surfaces [18]. In our approach, this is possible by cutting the resultant shell with an additional solid in a postprocessing step. Besides being useful for the definition of infill volumes for the printing process itself, the proposed method has other applications, for example, in the analysis and simulation of 3D-printed objects. In general, methods of adaptive manufacturing come with a finite smallest line width. In many applications, the line width of the printer is designed to coincide with the thickness of the shell structure under construction. This is, for example, common for 3D printing of slender concrete structures in applications of civil engineering and architecture [19; 20]. But the same is also true for wire arc additive manufacturing [21] and the various types of powder bed methods like selective laser melting [22] and selective binding processes [23; 24]. For structures built with these technologies, the thickness of the shell is often a simple result of the line width of the printer and is hence not specified as an infill volume.