Fractals and Additive Manufacturing
A. M. M. Sharif Ullah*1,†, D. M. D’Addona*2, Khalifa H. Harib*3, and Than Lin*4
*1Department of Mechanical Engineering, Kitami Institute of Technology
165 Koen-cho, Kitami, Hokkaido 090-8507, Japan
*2Department of Chemical, Material, Production, and Industrial Engineering, University of Naples Federico II
Piazzale Tecchio 80, 80125 Naples, Italy
*3Department of Mechanical Engineering, United Arab Emirates University
P.O. Box 15551-Al-Ain, UAE
*4School of Engineering and Technology, Asian Institute of Technology
P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand
Fractal geometry can create virtual models of complex shapes as CAD data, and from these additive manufacturing can directly create physical models. The virtual-model-building capacity of fractal geometry and the physical-model-building capacity of additive manufacturing can be integrated to deal with the design and manufacturing of complex shapes. This study deals with the manufacture of fractal shapes using commercially available additive manufacturing facilities and 3D CAD packages. Particular interest is paid to building physical models of an IFS-created fractal after remodeling it for manufacturing. This article introduces three remodeling methodologies based on binary-grid, convex/concave-hull, and line-model techniques. The measurements of the manufactured fractal shapes are also reported, and the degree of accuracy that can be achieved by the currently available technology is shown.
-  D. L. Bourell, D. W. Rosen, and M. C. Leu, “The Roadmap for Additive Manufacturing and Its Impact,” 3D Printing and Additive Manufacturing, Vol.1, No.1, pp. 6-9, 2014.
-  S. L. N. Ford, “Additive Manufacturing Technology: Potential Implications for U.S. Manufacturing Competitiveness,” J. of Int. Commerce and Economics, Vol.6, No.1, pp. 40-74, 2014.
-  A. M. M. S. Ullah, R. Omori, Y. Nagara, A. Kubo, and J. Tamaki, “Toward Error-free Manufacturing of Fractals,” Procedia CIRP, Vol.12, No.2013, pp. 43-48, 2013.
-  A. M. M. S. Ullah, Y. Sato, A. Kubo, and J. Tamaki, “Design for Manufacturing of IFS Fractals from the Perspective of Barnsley’s Fern-leaf,” Computer-Aided Design and Applications, Vol.12, No.3, pp. 241-255, 2015.
-  B. Mandelbrot, “The Fractal Geometry of Nature,” W. H. Freeman, 1982.
-  Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science, Vol.156, No.3775, pp. 636-638, 1967.
-  E. Stewart, “Towards numerically estimating Hausdorff dimensions,” The ANZIAM J., Vol.42, No.04, pp. 451-461, 2001.
-  G. A. Losa, “The living realm depicted by the fractal geometry,” Fractal Geometry and Nonlinear Analysis in Medicine and Biology, Vol.1, No.1, pp. 11-15, 2015.
-  J. E. Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics J., Vol.30, No.5, pp. 713-747, 1981.
-  M. Barnsley, “Fractals Everywhere,” Academic Press, 1993.
-  M. F. Barnsley and S. Demko, “Iterated Function Systems and the Global Contraction of fractals” Proc. of The Royal Society of London, Series A: Mathematical and Physical Sciences, Vol.399, No.1817, pp. 243-275, 1985.
-  S. Demko, L. Hodges, and B. Naylor, “Construction of Fractal objects with Iterated Function Systems,” Computer Graphics (ACM), Vol.19, No.3, pp. 271-278, 1985.
-  ISO Standard: Standard specification for additive manufacturing file format (AMF) Version 1.1, ISO/ASTM 52915:2013, 2013.
-  J. Vass, “On the Exact Convex Hull of IFS Fractals,” arXiv: 1502.03788v1, submitted 2015, http://arxiv.org/pdf/1502.03788. pdf [accessed September 30, 2015]
-  T. Martyn, “The attractor-wrapping approach to approximating convex hulls of 2D affine IFS attractors,” Computers & Graphics, Vol.33, No.1, pp. 104-112, 2009.
-  A. Mishkinis, C. Gentil, S. Lanquetin, and D. Sokolov, “Approximate convex hull of affine iterated function system attractors,” Chaos, Solitons & Fractals, Vol.45, No.11, pp. 1444-1451, 2012.
-  T. Martyn, “Realistic rendering 3D IFS fractals in real-time with graphics accelerators,” Computers & Graphics, Vol.34, No.2, pp. 167-175, 2010.
-  T. A. Campbell and J. A. Slotwinski, “Metrology for Additive Manufacturing: Opportunities in a Rapidly Emerging Technology,” Advances in Engineering Research, Vol.7, V. M. Petrova (Ed.), New York: Nova Science Publishers, 2013.
-  S. Buczkowski, S. Kyriacos, F. Nekka, and L. Cartilier, “The modified box-counting method: Analysis of some characteristic parameters,” Pattern Recognition, Vol.31, No.4, pp. 411-418, 1998.
-  H. Narahara, S. Takeshita, H. Fukumaru, H. Koresawa, and H. Suzuki, “Permeability performance on porous structure of injection mold fabricated by metal laser sintering combined with high speed milling,” Int. J. of Automation Technology, Vol.6, No.5, pp. 576-583, 2012.
-  H. Koresawa, H. Fukumaru, M. Kojima, J. Iwanaga, H. Narahara, and H. Suzuki, “Design method for inner structure of injection mold fabricated by metal laser sintering,” Int. J. of Automation Technology, Vol.6, No.5, pp. 584-590, 2012.
-  J. Potgieter, O. Diegel, F. Noble, and M. Pike, “Additive manufacturing in the context of hybrid flexible manufacturing systems,” Int. J. of Automation Technology, Vol.6, No.5, pp. 627-632, 2012.