Feature-based heterogeneous modelling

An important principle of heterogeneous volumetric modelling is fea- ture-based heterogeneous modelling where geometric features are defined within the volumes. Such features are often user-defined, and can be polylines, curves, points, surfaces or even solids. These features have prescribed material properties, and are called material features. The point set of all the points which do not belong to any material feature is called the material gap. Feature based methods are used to find values in these material gaps. Figure 2.9 shows the material gap (in brown) given two features (in blue and red).

Several authors introduced methods where colours or other attributes would fall off depending on the distance to a feature such as an edge or a line segment. Liu (2000) uses the distance to the boundary of the object to gradually change colours between the surface of the object and its interior. In Zhou et al.(2004), any surface, point or line can be used as a feature, and a function of the distance is associated with a feature. In Khoda et al. (2013), the medial axis of the shape is extracted, and

then a gradient porosity from the medial axis (dense) to the boundary of the object (sparse) is used to automatically control the following pro- cess to generate a structure with adaptive porosity. In Liu et al.(2004), two materials can be interpolated by letting the user define a feature (such as a solid), and the interpolation is performed within the distance to this feature. Kumar et al. (1998) also used distances to interpolate between two materials. Some geometric features represent a material, and the interpolation is based on a function of the ratio between the two distances. Bhashyam et al.(2000) provides a list of functions to interpo- late between various materials to optimize for some desired properties.

Park et al.(2001) introduced Volumetric Multi-Texturing (VMT) to de- fine gradient materials. Some features are defined using implicit surfaces such as algebraic surfaces, and then the values of the function are mapped onto material or density values.

In Siu and Tan (2002b), the features controlling the gradient materi- als are called grading sources. Siu and Tan (2002b) use any number of arbitrary features, of any type or dimensionality such as line segments, planes and points. Each feature provides a value for each modelled at- tribute, and the material distribution is expressed in function of the distance to those grading sources. The control of the graded materials is made easier through these simple control features, and Siu and Tan

(2002b) extends the usual operators (union, intersection and difference) with new capabilities for heterogeneous object modelling. The control of the gradual change is user-friendly. Moreover, this solution can easily be integrated into existing applications. However, it does not provide a global method, since the sources have a limited influence in space.

Wu et al. (2005) provided a similar structure, where the material com- position is successively built by providing material features with their material values and distances to control the length of the fall-off.

In Rvachev et al. (2001), transfinite interpolation was used to inter- polate material properties given any number of features. The prob- lem of transfinite interpolation is to construct a function which ”takes prescribed values and/or derivatives on some collection of point sets” (Rvachev et al. 2001). The point sets with prescribed values can be any

point set, such as points, lines, surfaces or spatial regions, each repre- sented by a real valued function. The interpolation is performed by using scalar fields generated from each material feature. The scalar fields can be distance fields (a minimal Euclidean distance from any point in space to the object boundary) as in Biswas et al. (2004), or scalar fields built with R-Functions (Rvachev 1982) as in Rvachev et al. (2001). The R- Functions are chosen to have C1 _{continuity, however, the R-Functions}

need to be carefully selected to preserve the distance properties. In fact, the authors used smooth R-functions to build ”smooth distance-like functions”. Both Rvachev et al. (2001) and Biswas et al. (2004) argue that distance approximation is necessary, and smoothness is desirable. As stated in Biswas et al. (2004), the distance provides predictability, and the smoothness provides smoothness to the attribute interpolation.

Hongmei et al. (2009) proposed a similar method which provides more control over the rate of change of the attributes.

There are several interpolation schemes, however, inverse distance weighting formulated in Rvachev et al.(2001) (eq. 2.4) is the most pop- ular (Kou and Tan 2007).

wi(p) = n Y j=1;j6=i dj(p) n X j=1 n Y k_=1;k6=j dk(p) (2.4) where • p a point in space

• di(p) is the unsigned distance of the point p to the i-th feature

• n is the number of features

If only two features are used, this simplifies to: wa(p) = db(p) da(p) + db(p) wb(p) = da(p) da(p) + db(p) (2.5)

The transfinite interpolation was applied in a similar fashion inFryazinov et al.

(2013) in order to interpolate between parameters of two given volumet- ric microstructures. This is a good example of the link between shape modelling and the modelling process related to an arbitrary volumet- ric attribute. Since distance properties are important to have intuitive interpolations,Fayolle et al. (2006) provides FRepC1 _{continuous opera-}

tions and primitives which approximate the distance. The continuity is necessary to avoid ”undesirable singularities in the material distribution, like stress or concentrations” (Fayolle et al. 2006).

2.2.6

Discussion

As Kou and Tan (2007) stated in their survey of the different repre- sentations, evaluated models are not exact nor compact which can be problematic. For instance, 3D printers are increasingly more and more precise. The Connex 3D printers by Stratasys can print at 16-micron for objects up to 25 centimetre wide. Evaluated models cannot simultane- ously be compact, accurate and computationally efficient. Unevaluated models are resolution independent, and scalar field based methods are capable of representing complex material distributions at various scales without loss of accuracy, and remains compact.

In the unevaluated representations, two scalar field approaches stand out: constructive methods and feature-based methods, both of which often rely on distances. As our survey shows, the use distance fields and feature based modelling for representing gradient material and other volumetric attributes is popular.

let the users control the material distribution everywhere. Additionally, feature based methods are easily integrated into existing CAD systems. However, these methods rely upon functions of the distance to the fea- tures, never taking into account the shape of the object being modelled. Additionally, using exact distance functions can cause abrupt changes, but building smooth functions is not always straightforward for arbitrary features.

The constructive methods can build a complex heterogeneous object where the gradient materials are ’following’ the shape. Additionally, several authors have shown that constructive methods can build com- plex heterogeneous objects alongside the object geometry. However, it requires the user to build the functions carefully and cannot be easily performed at a later stage of the modelling process. Additionally, the scalar fields produced by FRep trees do not generally approximate dis- tance fields, preventing the use of distance-based methods with FRep

trees.