## 3 Data Representation
### Multi-Material Objects  
The design methodology of multi-material objects is still unclear. This is not the objective of this study; however, insight into how the object was designed will be helpful for manufacturing. To address this gap, methods for defining multi-material objects have been developed.  
For representing multi-material objects, explicit hard boundaries corresponding to graded materials can be employed, for which the material properties should be estimates, or voxel-based representations can be employed.  
For explicit boundaries, the structure can be described using the boundary representation, BRep, with non-manifold internal boundaries. However, a few factors were considered. When the internal boundaries are not aligned with the build direction, which is unlikely, the squashing process defines overlapping areas where two materials are defined. The required alignment can be achieved using grading techniques or assigning material definition priorities such that one material will be chosen over another in the overlapping region. An interesting corollary to this is that holes can be defined as internal structures having an empty material. If the empty material has a higher priority than other materials, then this will force the hole to be open, although it may be larger than designed. In other cases, if the hole is oblique relative to the build direction, then the squashing process will tend to make the holes smaller. The problems of holes growing or diminishing in size are a side effect of the discrete nature of additive processes and must be considered by both designers and manufacturing planners.  
Another method of representing multi-material objects is to provide a functional definition using material properties at a given point that can be calculated. An example of this is shown in Figure 2 using two surfaces as a guide for calculation.  
Figure 1: The high-level structure of STEP-NC additive manufacturing.  
Voxel-based methods use a cellular structure with explicit material properties assigned to each cell. There are known disadvantages of using voxel-based cellular representations. One is that the precision of the representation depends on the cell size, higher precision demanding more memory, so the representation is scale dependent. Another disadvantage is that the material properties of each cell are absolute, leading to hard internal boundaries rather than smooth grading. Voxel-based representations are, to some extent, attractive because of their simplicity and clear intention. However, it is not practical to expect users to define cell material properties explicitly for each cell. Instead, some sort of functional approach to material definition is realistic, but then this method could be defined in either BRep or the functional gradient approach.  
For example, a BRep internal structure can use the ratio formulation [22] and requires a limited number of parameters to draw the material distribution.  
In addition to the polynomial or linear functions, it is possible to use NURBS geometry to create internal patterns. Figure 3 shows a formulated model between two boundary curves having the relationship of gradient distribution from \(l_{1}\) to \(l_{2}\). There is a correlation between each pair of vertices \(v_{1i}\) and \(v_{2i}\). A NURBS formulation is applied along the line \(R_{i}\). In order to define the distribution, the input parameters are knot vectors, degrees, and control points to configure the distribution in multiple dimensions. NURBS also allows for multi-dimensional space in one-dimensional two-dimensional space, as well as more accurate representation of a given feature. In the slice operation, it is necessary to determine the colours for each slice and each path. Figure 4 depicts the colour allocation of each slice (a) and of each path (b).  
Figure 2: Gradient surface of additive manufacturing.