Читать книгу Geometric Modeling of Fractal Forms for CAD - Christian Gentil - Страница 9

Introduction

I.1. Fractals for industry: what for?

This book shows our first steps toward the fundamental and applied aspects of geometric modeling. This area of research addresses the acquisition, analysis and optimization of the numerical representation of 3D objects.

Figure I.1. 3D tree built by iterative modeling

(source: project MODITERE no. ANR-09-COSI-014)

Figure I.1 shows an example of a structure that admits high vertical loads, while minimizing the transfer of heat between the top and bottom of the part. Additive manufacturing (3D printing) allows, for the first time, the creation of such complex objects, even in metal (here with a high-end laser printer EOS M270). This type of technology will have a high societal and economic impact, enabling better systems to be created (engines, cars, airplanes, etc.), designed and adapted numerically for optimal functionality, thus consuming less raw material, for their manufacturing, and energy, when used.

Current computer-aided design is, however, not well suited to the generation of such types of objects. For centuries, for millennia, humanity has produced goods with axes, files (or other sharp or planing tools), by removing bits from a piece of wood or plastic. Tools subsequently evolved into complex numerical milling machines. However, at no point during these manufacturing processes did we need sudden stops or permanent changes in the direction of the cutting tool. The patterns were always “regular”, hence the development of mathematics specific to these problems and our excellent knowledge of the modeling of smooth objects. This is why it was necessary to wait until the 20th century to have the mathematical knowledge needed to model rough surfaces or porous structures: we were just not able to produce them earlier.

Thus, since the development of computers in the 1950s, computer-aided geometric design (CAGD or CAD) software has been developed to represent geometric shapes intended to be manufactured by standard manufacturing processes. These processes are as follows:

 – subtractive manufacturing, using machine tools such as lathes or milling machines;

 – molding, where molds themselves are made using machine tools;

 – deformation-based manufacturing: stamping or swaging (but again, dies are usually manufactured using machine tools), folding, etc.;

 – cutting, etc.

Each of these processes imposes constraints, for example, concerning collision issues in milling machines (even a five-axis mill cannot produce any geometry). These manufacturing processes inevitably influenced the way we design the geometries of objects, in order to actually manufacture them. For example, CAD software has integrated these design methodologies by developing appropriate numerical models or tools. Currently, most CAD software programs are based on the representation of shapes by means of surfaces defining their edges. These surfaces are usually described using a parametric representation called non-uniform rational B-spline (NURBS). These surface models are very powerful and very practical. It is possible to represent any volume enclosed by a quadric (cylinders, cones, spheres, etc.) and complex shapes, such as car bodies or airplane wings. They were originally designed for this.

However, the emergence of additive manufacturing techniques has caused an upheaval in this area, opening up the possibility of potentially “manufacturable” forms. By removing the footprint constraint of the tool, it then becomes possible to produce very complex shapes with gaps or porosity. These new techniques have called into question the way objects are designed. New types of objects, such as porous objects or rough surfaces, can have many advantages, due to their specific physical properties. Fractal structures are used in numerous fields such as architecture (Rian and Sassone 2014), jewelry (Soo et al. 2006), heat and mass transport (Pence 2010), antennas (Puente et al. 1996; Cohen 1997) and acoustic absorption (Sapoval et al. 1997).

I.2. Fractals for industry: how?

The emergence of techniques such as 3D printers allows for new possibilities that are not yet used or are even unexplored. Different mathematical models and algorithms have been developed to generate fractals. We can categorize them into three families, as follows:

 – the first groups algorithms for calculating the attraction basins of a given function. Julia and Mandelbrot (Peitgen and Richter 1986) or the Mandelbulb (Aron 2009) sets are just a few examples;

 – the second is based on the simulation of phenomena such as percolation or diffusion (Falconer 1990);

 – the last corresponds to deterministic or probabilistic algorithms or models based on the self-similarity property associated with fractals such as the terrain generator (Zhou et al. 2007), the iterated function system (Barnsley et al. 2008) or the L-system (Prusinkiewicz and Lindenmayer 1990).

In the latter family of methods, shapes are generated from rewriting rules, making it possible to control the geometry. Nevertheless, most of these models have been developed for image synthesis, with no concerns for “manufacturability”, or have been developed for very specific applications, such as wood modeling (Terraz et al. 2009). Some studies approach this aspect for applications specific to 3D printers (Soo et al. 2006). In (Barnsley and Vince 2013b), Barnsley defines fractal homeomorphisms of [0, 1]² onto the modeling space [0, 1]². The same approach is used in 3D to build 3D fractals. A standard 3D object is integrated into [0, 1]³ and then transformed into a 3D fractal object. This approach preserves the topology of the original object, which is an important point for “manufacturability”.

The main difficulty associated with traditional methods for generating fractals lies in controlling the forms. For example, it is difficult to obtain the desired shape using the fractal homeomorphism system proposed by Barnsley. Here, we develop a modeling system of a new type based on the principles of existing CAD software, while expanding their capabilities and areas of application. This new modeling system offers designers (engineers in industry) and creators (visual artists, designers, architects, etc.) new opportunities to quickly design and produce a model, prototype or unique object. Our approach consists of expanding the possibilities of a standard CAD system by including fractal shapes, while preserving ease of use for end users.

We propose a formalism based on standard iterated function systems (IFS) enhanced by the concept of boundary representation (B-rep). This makes it possible to separate the topology of the final forms from the geometric texture, which greatly simplifies the design process. This approach is powerful, and it generalizes subdivision curves and standard surfaces (linear, stationary), allowing for additional control. For example, we have been able to propose a method for connecting a primal subdivision scheme surface with a dual subdivision scheme surface (Podkorytov et al. 2014), which is a difficult subject for the standard subdivision approach.

The first chapter recalls the notion of self-similarity, intimately linked to that of fractality. We present the IFS, formalizing this property of self-similarity. We then introduce enhancements into this model: controlled iterated function systems (C-IFS) and boundary controlled iterated function systems (BC-IFS). The second chapter is devoted to examples. It provides an overview of the possibilities of description and modeling of BC-IFS, but also allows better understanding the principle of the model through examples. The third chapter presents the link between BC-IFS, the NURBS surface model and subdivision surfaces. The results presented in this chapter are important because they show that these surface models are specific cases of BC-IFS. This allows them to be manipulated with the same formalism and to make them interact by building, for example, junctions between two surfaces of any kind. In the fourth chapter, we outline design tools that facilitate the description process, as well as examples of the applications, of the design of porous volumes and rough surfaces.