Synthesis of fractal geometry and CAGD models for multi-scale topography modelling of functional surfaces

In order to support the functional design and simulation and the final fabrication processes for functional surfaces,it is necessary to obtain a multi-scale modelling approach representing both macro geometry and micro details of the surface in one unified model.Based on the fractal geometry theory,a synthesized model is proposed by mathematically combining Weierstrass-Mandelbrot fractal function in micro space and freeform CAGD model in macro space.Key issues of the synthesis,such as algorithms for fractal interpolation of freeform profiles,and visualization optimization for fractal details,are addressed.A prototype of the integration solution is developed based on the platform of AutoCAD's Object ARX,and a few multi-scale modelling examples are used as case studies.With the consistent mathematic model,multi-scale surface geometries can be represented precisely.Moreover,the visualization result of the functional surfaces shows that the visualization optimization strategies developed are efficient.

SOME PROBLEMS ON FRACTAL GEOMETRY AND TOPOLOGICAL DYNAMICAL SYSTEMS

This paper is a continuation of [14], some new problems on fractal geometry and topological dynamical systems have been posed.

Prediction of heat transfer of nanofluid on critical heat flux based on fractal geometry

Analytical expressions for nucleate pool boiling heat transfer of nanofluid in the critical heat flux （CHF） region are derived taking into account the effect of nanoparticles moving in liquid based on the fractal geometry theory. The proposed fractal model for the CHF of nanofluid is explicitly related to the average diameter of the nanoparticles, the volumetric nanoparticle concentration, the thermal conductivity of nanoparticles, the fractal dimension of nanoparticles, the fractal dimension of active cavities on the heated surfaces, the temperature, and the properties of the fluid. It is found that the CHF of nanofluid decreases with the increase of the average diameter of nanoparticles. Each parameter of the proposed formulas on CHF has a clear physical meaning. The model predictions are compared with the existing experimental data, and a good agreement between the model predictions and experimental data is found. The validity of the present model is thus verified. The proposed fractal model can reveal the mechanism of heat transfer in nanofluid.

FRACTAL GEOMETRY STUDY OF CORRELATION BETWEEN IMPACT TOUGHNESS OF STEEL AND PARAMETERS OF FREE-CUTTING PHASE

Studies were made of the calculation of fractal dimension of transverse impact fracture sur- face,and of the correlation between impact toughness of steel and parameters of free-cutting phase by means of the developed fractal geometry model of crack propagation.It is believed that the area fraction,f,of free-cutting phase is negligibly influential to the longitudinal im- pact toughness,as f1 .While the aspect ratio,saying ratio of length to width,of free-cut. ting phase is inversely influential to the transverse impact toughness.This may .be the reason why the transverse impact toughness of free-cutting steel containing more rare earth contrast to sulphur is even higher than the low sulphur containing steel.

Visual Evaluation of the Morphologic Structure of Nonwovens Using Image Analysis and Fractal Geometry

Nonwovens are fiber materials which are based on nonwoven technologies. For the complexity and randomness of nonwovens morphologic structures, it is difficult to express them effectively using classical method. Fractal geometry gives us a new idea and a powerful tool to study on irregularity of geometric objects. Therefore, we studied on the pore size, pore shape, pore size distribution and fiber orientation distribution of real nonwovens using fractal geometry combined with computer image analysis to evaluate nonwovens’ morphologic structures.

Fractal geometry and topology abstracted from hair fibers

Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2）-circle fractal sets as subsets. As far as the （9＋2） topological patterns are concerned, the following propositions are proved： The （9＋2） topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent （or fundamental）. Besides, we demonstrate that the （3, 9＋2）-circle and （9＋2, 3）-circle fractal sets are golden ones with symmetry breaking.

PRIMARY RESEARCH ON FRACTAL GEOMETRY OF MERIDIAN THEORY

In meridian theory of traditional Chinese medicine (TCM), the geometrical descriptions can be traced back to the remote ancient times in China, mainly in The Yellow Emperor’s Internal Classic (The Internal Classic in short). Euclid’s geometry, topology and other classic mathematics are all at their wit’s end to explain the high complexity and non clinear phenomenon of the meridian. In recent over 2000 years, the meridian phenomenon has been being the challenge to fundamental mathematics. Fractral geometry, founded by Mandelbrot (1975), is a branch of learning for investigating irregular geometrical curves. It has successfully solved some qualitative and quantitative problems about the topographical structure of molecular Brown’s movement curve and other irregular complicated curves and geometrical characters. The characteristics of geometrical topographical structure of meridian and its phenomenon belong to the research category of Fractal Geometry. The author of this paper believes that Fractal Geometry may provide a useful mathematical tool and a possible way for revealing the enigma of acup moxibustion meridian theory. The human body is of basic characters of Fractal Geometry in structure, while meridian is the expression form of Fractal structure of the human body. The basic Fractal geometrical characters of meridian are: self similarity, self affinity, symmetry, minute structure and self avoidance, which has been applied for thousands of years in clinic, such as “taking the acupoints on the right side of the body in cases of disorders appearing on the left side and vice versa". The basic characters of meridians are 1) symmetry of the 12 regular meridians on the bilateral sides of the body (symmetry); 2) similarity in characters and actions of acupoints of the same one meridian (self similarity); 3) taking acupoints on the lower part of the body when disorders occurring on the upper part of the body; and taking acupoints on the upper part of the body if disorders appearing on the lower part (self affinity); 4) micro acupuncture system including hand acupuncture, foot acupuncture, scalp acupuncture, auricular acupuncture and eye acupuncture (minute structure); and 5) systematical running of needling sensation (self avoidance).

Experimental realization of fractal fretwork metasurface for sound anomalous modulation

Natural creatures and ancient cultures are full of potential sources to provide inspiration for applied sciences.Inspired by the fractal geometry in nature and the fretwork frame in ancient culture,here we design the acoustic metasurface to realize sound anomalous modulation,which manifests itself as an incident-dependent propagation behavior:sound wave propagating in the forward direction is allowed to transmit with high efficiency while in the backward direction is obviously suppressed.We quantitatively investigate the dependences of asymmetric transmission on the propagation direction,incident angle and operating frequency by calculating sound transmittance and energy contrast.This compact fractal fretwork metasurface for acoustic anomalous modulation would promote the development of integrated acoustic devices and expand versatile applications in acoustic communication and information encryption.

The lift-off effect analysis of flexible differential pick-up koch fractal eddy current probe

This study utilized finite element simulation and experimental methods to investigate the evolution of crack detection performanceof a flexible differential fractal Koch eddy current probe at different excitation frequencies as the lift-off distance increases.As the lift-off distanceincreased,the distribution shape of induced eddy currents changed,leading to reduced similarity in the shape of the excitation coil and an expandeddistribution range of induced eddy currents,ultimately resulting in weakened output signal strength.The experimental results showed that forexcitation frequencies of 10 kHz,20 kHz,50 kHz,100 kHz,200 kHz,500 kHz,and1000 kHz,the maximum lift distances of the real partof the output signal when cracks were detected were 5.0 mm,7.0 mm,8.0 mm,8.0 mm,8.0 mm,6.5 mm,and 4.0 mm,respectively.Theimaginary parts were 6.5 mm,6.5 mm,7.5 mm,5.5 mm,8.0 mm,6.5 mm,and 6.5 mm,respectively.

Quasi-static and dynamic compressive behaviour of additively manufactured Menger fractal cube structures

This paper presents the first-ever investigation of Menger fractal cubes'quasi-static compression and impact behaviour.Menger cubes with different void ratios were 3D printed using polylactic acid(PLA)with dimensions of 40 mm×40 mm×40 mm.Three different orders of Menger cubes with different void ratios were considered,namely M1 with a void ratio of 0.26,M2 with a void ratio of 0.45,and M3with a void ratio of 0.60.Quasi-static Compression tests were conducted using a universal testing machine,while the drop hammer was used to observe the behaviour under impact loading.The fracture mechanism,energy efficiency and force-time histories were studied.With the structured nature of the void formation and predictability of the failure modes,the Menger geometry showed some promise compared to other alternatives,such as foams and honeycombs.With the increasing void ratio,the Menger geometries show force-displacement behaviour similar to hyper-elastic materials such as rubber and polymers.The third-order Menger cubes showed the highest energy absorption efficiency compared to the other two geometries in this study.The findings of the present work reveal the possibility of using additively manufactured Menger geometries as an energy-efficient system capable of reducing the transmitting force in applications such as crash barriers.

FRACTAL KINEMATICS OF CRACK PROPAGATION IN GEOMATERIALS

Experimental results indicate that propagation paths of cracks in geomaterials are often irregular. producing rough fracture surfaces which are fractal. A formula is derived for the fractal kinematics of crack propagation in geomaterials. The formula correlates the dynamic and static fracture toughnesses with crack velocity, crack length and a microstructural parameter, and allows the fractal dimension to be obtained. From the equations for estimating crack velocity and fractal dimension it can be shown that the measured crack ve1ocity, Vo, should be much smaller than the fractal crack velocity, V. It can also be shown that the fractal dimension of the crack propagation path can be calculated directly from Vo and from the fracture toughness.

Performance-based fractal fracture model for complex fracture network simulation

The paper presents a novel hydraulic fracturing model for the characterization and simulation of the complex fracture network in shale gas reservoirs. We go beyond the existing method that uses planar or orthogonal conjugate fractures for representing the ＇＇complexity＇＇ of the network. Bifurcation of fractures is performed utilizing the Lindenmayer system based on fractal geometry to describe the fracture propagation pattern, density and network connectivity. Four controlling parameters are proposed to describe the details of complex fractures and stimulated reservoir volume（SRV）. The results show that due to the multilevel feature of fractal fractures, the model could provide a simple method for contributing reservoir volume calibration. The primary-and second-stage fracture networks across the overall SRV are the main contributions to the production, while the induced fracture network just contributes another 20% in the late producing period. We also conduct simulation with respect to different refracturing cases and find that increasing the complexity of the fracture network provides better performance than only enhancing the fracture conductivity.

Dual-band and polarization-insensitive terahertz absorber based on fractal Koch curves

We report the design, fabrication, and characterization of a dual-band and polarization-insensitive metamaterial ab-sorber （MA）, which consists of periodically arranged fractal Koch curves acting as the top resonator array and a metallic ground plane separated by a dielectric spacer. Compared with conventional MAs, a more compact size and multi-frequency operation are achieved by using fractal geometry as the unit cell of the MA. Both the effective medium theory and the multi- reflection interference theory are employed to investigate the underlying physical mechanism of the proposed terahertz MA, and results indicate that the latter theory is not suitable for explaining the absorption mechanism in our investigated struc-ture. Two absorption peaks are observed at 0.226 THz and 0.622 THz with absorptivities of 91.3% and 95.6% respectively and good agreements between the full-wave simulation and experimental results are achieved.

Fractal identification of wear debris group in sliding wear process

Fractal geometry was used to describe the distribution characteristics of wear debris group collected from pin-on-disc wear tester under dry friction conditions, and experimental study and theoretical analysis were made for the distribution features of wear debris group. It was found that the wear debris size distribution conforms to the fractal distribution law. Two numerical parameters, fractal dimension D and scale coefficient C, were defined with their geometric and tribological meanings and calculating methods given. It was discovered that these two parameters can be used to describe the variation law of wear status, which provide the basis for diagnosis and prognosis of tribological systems.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

Using texture synthesis in fractal pattern design

Traditional fractal pattern design has some disadvantages such as inability to effectively reflect the characteristics of real scenery and texture. We propose a novel pattern design technique combining fractal geometry and image texture synthesis to solve these problems. We have improved Wei and Levoy (2000)’s texture synthesis algorithm by first using two-dimensional autocorrelation function to analyze the structure and distribution of textures, and then determining the size of L neighborhood. Several special fractal sets were adopted and HSL (Hue, Saturation, and Light) color space was chosen. The fractal structure was used to manipulate the texture synthesis in HSL color space where the pattern’s color can be adjusted conveniently. Experiments showed that patterns with different styles and different color characteristics can be more efficiently generated using the new technique.

FRACTAL PROPERTIES OF ROCK FRACTURE SURFACES

To give a better understanding of the morphological features of rock fracture surfaces within the framework of fractal geometry, the fractal characters of the rough surfaces in rock are analyzed according to the variogram method. The study elaborates the significance of the geometric parameters-fractal dimension D and the intercept A on a log-log plot to the surface structure. Investigation extends to the anisotropy and heterogeneity of rock fracture surfaces, and the scale effect on the fractal estimation. The present study indicates that fractal dimension alone may not be sufficient to characterize the surface roughness of rock Joints. A reliable estimation should take into account the combination of D and A.

Study on Permeability of 2.5D Woven Reinforcement Materials with Fractal Theory

Permeability is one of the key issues in the design of molds and in the molding process for composite manufacture. As a disordered fibrous assembly, 2.5- dimension （2.5 D） woven reinforcement materials have complex structure. It poses a challenge to the study of pore structure and the establishment of the theoretical permeability model. Toward addressing this problem, a powerful tool called fractal theory emerged. According to the analysis of 2.5 D woven reinforcement material stmcture using fractal theory, it is found that the structure has an obvious fractal character. Therefore, a permeability fractal model of 2.5D woven reinforcement material was established by cormbining the Hagen-Poiseulle equation with Darcy law according to the capillary vessel fractal model in this paper. The permeability was expressed as a function of the fractal dimension and microstructure parameter of the porous media in this model. The theoretical model is verified by experimental tests and the measurement data are in good agreement with the results obtained from the fractal medel .

Application of fractal theory to size effect of disordered materials

For disordered materials it is impossible to measure constant material properties using the Euclidian geometrical dimension of the test specimens. Based on the theory of fractal geometry, the fractal dimension of the damaged microstructure is applied to measure the strength and fracture toughness of imitation marbles, which turn out to be scale invariant material constants. In this paper, the experimental data are treated and interpreted by the theory of fractal geometry. Reasonable results are obtained and the size effects on strength and fracture energy are observed.

Fractal Description of the Shearing-Surface of Tools

In this paper, the basic methods are introduced to calculate the fractaldimensions of the shearing surface of some tools. The fractal dimension of the shearing surface ofexperimental sampling is obtained and the fractal characteristics are also discussed. We can applythe fractal method to identify types of tools used by burglars and to do the job of individualrecognition. New theories and methods are provided to measure and process the shearing surfaceprofile of tools.