3D Fractals From Periodic Surfaces

2010 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Porous Silicon ◽  
Iterated Function Systems ◽  
Surface Model ◽  
Porous Structures ◽  
Implicit Surface ◽  
Periodic Surface ◽  
Irregular Structures ◽  
Periodic Surfaces ◽  
Iterated Function ◽  
3D Fractals

Fractals are ubiquitous as in natural objects and have been applied in designing porous structures such as micro antenna and porous silicon. The seemingly complex and irregular structures can be generated based on simple principles. In this paper, we present three approaches to construct 3D fractal geometries using a recently proposed periodic surface model. By applying iterated function systems to the implicit surface model in the Euclidean or parameter space, 3D fractals can be constructed efficiently. Porosity is also proposed as a metric in fractal design.

Degree Operations on Periodic Surfaces

2007 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Surface Model ◽  
Implicit Surface ◽  
Periodic Surface ◽  
Degree Elevation ◽  
Periodic Surfaces ◽  
Computer Aided ◽  
Particle Aggregates

In previous work, a periodic surface model for computer-aided nano-design (CAND) was developed. This implicit surface model can construct Euclidean and hyperbolic nano geometries parametrically and represent morphologies of particle aggregates and polymers. In this paper, we study the characteristics of degree elevation and reduction based on a generalized periodic surface model. Methods of degree elevation and reduction operations are developed in order to support multi-resolution representation and model exchange.

Minkowski Sums of Periodic Surface Models

2009 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Crystal Packing ◽  
Surface Model ◽  
Implicit Surface ◽  
Periodic Surface ◽  
Construction Methods ◽  
Periodic Surfaces ◽  
Nano Structures ◽  
Surface Models ◽  
Minkowski Sums ◽  
Volume Translation

Recently we proposed a periodic surface model to assist geometric construction in computer-aided nano-design. This implicit surface model helps create super-porous nano structures parametrically and support crystal packing. In this paper, we study construction methods of Minkowski sums for periodic surfaces. A numerical approximation approach based on the Chebyshev polynomials is developed and can be applied in both surface normal direction matching and volume translation formulations.

Feature-Based Crystal Construction in Computer-Aided Nano-Design

2008 ◽  
Author(s):  
Cheng Qi ◽  
Yan Wang
Keyword(s):  
Material Design ◽  
Surface Model ◽  
Periodic Surface ◽  
Periodic Surfaces ◽  
Nano Structures ◽  
Rapid Construction ◽  
Computer Aided ◽  
Feature Based ◽  
Basic Features ◽  
Complex Crystal

Providing nano engineers and scientists efficient and easy-to-use tools to create geometry conformations that have minimum energies is highly desirable in material design. Recently we developed a periodic surface model to assist the construction of nano structures parametrically for computeraided nano-design. In this paper, we present a feature-based approach for crystal construction. The proposed approach starts to create models of basic features by the aide of periodic surfaces followed by operations between basic features. The goal is to introduce a rapid construction method for complex crystal structures.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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