# Computation of Implicit Representation of Volumetric Shells with Predefined Thickness
## 1 Introduction  
Implicit surfaces are of considerable interest in engineering as they provide an efficient means for the modeling of complex shapes. However, if transferred to the physical world by any means of production, any surface must have a finite thickness and hence, becomes a volume. This can be done by tools for computer aided design by finding surface points and extruding them in the normal direction. For example, [PERSON] and [PERSON] [1] proposed an algorithm to compute a distance field for a discretized surface such that the offset surface could be extracted as an isosurface of the distance field. [PERSON] applied the marching cube algorithm to generate volume models from triply periodic minimal surfaces [2]. [PERSON] et al. [3] used generalized Voronoi diagrams to compute offset functions of variable thickness. [PERSON] presented an overview on offsetting methods [4], which was later extended by [PERSON] [5]. However, these reviews do not address the analytic offsetting of implicit surfaces.  
[PERSON] et al. [6] proposed a method for offsetting implicit functions by computing level sets of the distance function. The distance function is obtained by normalizing the original implicit surface by Rvachev's normalization [7]. [PERSON]'s method is based on numerical procedures that require sampling of the initial implicit surface. They apply the Euler forward method to propagate the offset into the direction of the normal to the surface.  
[PERSON] [8] provides a good overview over fast marching methods used for offsetting implicit surfaces. He also discusses how the accuracy of the the fast marching method can be improved by discretizing the required normal derivatives with finite differences of higher order. However, this by itself does not guarantee an improvement of the order of convergence of the method.  
In the mentioned methods, the resulting new surface is either a discretized surface with an explicit surface description or a discretized form of the implicit surface. In both cases, many of the appealing properties of the original implicit surface are lost.  
Explicit surface representations usually require much more data than implicit representations, especially if the resolution requirements are high. Applications for implicit surfaces are in 3D printing [9; 10; 11] and simulation software. In both domains, tessellatedsurfaces are currently the state of the art. However, simulation software can suffer from artifacts when curved surfaces are approximated by triangles. For example, the accurate simulation of flow around a sphere at a high Reynolds number can be qualitatively wrong if the sphere is approximated by triangles due to spurious flow separation at the edges of triangles, while accurate results can be obtained with the sphere defined as geometric primitive [12], which also requires less memory.  
The purpose of the current paper is to propose a method to obtain a new implicit surface at a predefined distance to a given implicit surface.  
The reminder of the paper is organized as follows. In Section 2, we introduce a low-order approximation of the offset function to the implicit surface. In Section 3, it is shown how the approximation of the low-order scheme can be improved by Richardson extrapolation in order to obtain a second-order convergent method. In Section 4, it is shown how this methodology is extended to obtain offset functions at any order of convergence. Section 5 demonstrates empirically that the predicted convergence orders are in fact obtained by computing the error norm for offset functions for three different triply periodic minimal surfaces. In Section 6, we discuss the case of a variable thickness of the shell and demonstrate second-order of convergence. Conclusions are given in Section 7.