Geometric Modeling of Fractal Forms for CAD. Christian Gentil

Library Cataloguing-in-Publication Data

      A CIP record for this book is available from the British Library

      ISBN 978-1-78630-040-9

      Preface

      This work introduces a model of geometric representation for describing and manipulating complex non-standard shapes such as rough surfaces or porous volumes. It is aimed at students in scientific education (mathematicians, computer scientists, physicists, etc.), engineers, researchers or anyone familiar with the mathematical concepts addressed at early stages of the graduate level. However, many parts are accessible to all, in particular, all introductory sections that present ideas with examples. People with no prior background, whether they are artists, designers or simply curious, will be able to understand the philosophy of our approach, and discover a new universe of unsuspected and exciting forms.

      Geometric representation models are mathematical tools integrated into computer-aided geometric design (CAGD) software. They make the production of numerical representations of forms possible. By means of graphical interfaces or programming tools, users can draw and/or manipulate these shapes. They can also test or evaluate their physical properties (mechanical, electro-magnetic, acoustic, etc.) by communicating geometric descriptions to further specific numerical simulation software.

      The geometric representation model we present here is based on the fractal geometry paradigm. The principle behind this consists of studying the properties (signal, geometry, phenomena, etc.) at different scales and identifying the invariants from there. The objects are described as self-referential between two scales: each of the object features (namely, the lower scale level) is described as a reference to the object itself (namely, the higher scale). This approach is not conventional and often confusing at first. We come to perceive its richness and power very quickly, however. The universe of forms that can possibly be created is infinite and has still only partially been explored.

In this book, we present the mathematical foundations, so that the reader can access all the information to understand, test and make use of this model. Properties, theorems and construction methods are supplemented with algorithms and numerous examples and illustrations. Concerning the formalization, we have chosen to use precise and rigorous mathematical notations to remove any ambiguity and make understanding easier.

      Readers unwilling to be concerned with mathematical formalisms can get to grips with the philosophy of our approach by focusing on the sections found at the beginnings of the chapters, in which ideas and principles are intuitively presented, based on examples.

      This book is the result of 25 years of research carried out mainly in the LIRIS laboratories of the University of Lyon I and LIB of the University of Burgundy Franche-Comté. This research was initiated by Eric Tosan, who was instrumental at the origin of this formalism and to whom we dedicate this work.

      Christian GENTIL

      Gilles GOUATY

      Dmitry SOKOLOV

      February 2021

      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