Fractal Branch Research Articles

A permeability model for the fractal tree-like fracture network with self-affine surface roughness in shale gas reservoirs

The complex natural fracture network with self-affine rough surface and branching characteristics significantly impacts the gas transport in shale gas reservoirs. However, its effects on the permeability have not been studied so far. This study proposes an analytical permeability model for the fractal tree-like fracture network with self-affine surface roughness and branching characteristics. Firstly, the self-affine rough profiles of fracture surface are generated at different fractal dimensions by the Weierstrass–Mandelbrot function and a rough fractal tree-like fracture network is constructed with these surface profiles and branching characteristics. Then, an analytical permeability model is proposed to consider the effects of fracture surface roughness and tree-like branching characteristics on gas flow. This analytical model is verified by numerical simulations. Finally, the velocity distribution of the fracture network and the sensitivity of its structure parameters are analyzed. It is found that eddy flow is more easily formed on rougher fracture surfaces with larger fractal dimension when their fracture aperture is at millimeter scale. The eddy flow disappears when the fracture aperture is at micron scale. Bigger gas flow resistance and more energy loss are observed for smaller fracture aperture and rougher fracture surface. The gas velocity in rough fractures decreases by 60% at micron scale, but decreases by 50% at millimeter scale. Gas flow resistance also increases with the increase of branch angle, branch level and length ratio, but decreases with aperture ratio. As a result, permeability decreases with fractal dimension, branch angle, branch level and length ratio, but increases with aperture ratio.

A Novel Liquid Cooling Battery Thermal Management System With a Cooling Plate Based on Biomimetic Fractal Channels

Abstract An effective battery thermal management system (BTMS) is necessary to quickly release the heat generated by power batteries under a high discharge rate and ensure the safe operation of electric vehicles. Inspired by the biomimetic structure in nature, a novel liquid cooling BTMS with a cooling plate based on biomimetic fractal structure was proposed. By developing the physical model of the BTMS, numerical calculations were conducted to analyze the impacts of the structural parameters of the cooling plate and the inlet velocity of the coolant on the thermal performance of the batteries. The results showed that the cooling plate can meet the heat dissipation requirements of high-temperature uniformity for the batteries under high discharge rates, especially under the extremely uniform channel distribution mode for the adjacent fractal branch at the same level. Moreover, the increase in the group number of fractal branches can improve the cooling capacity of the cooling plate and reduce the pressure drop of the coolant. The increase in the level number of channels, the length ratio, and the inlet velocity of the coolant can enhance the cooling capacity. However, these methods of enhancing heat transfer require more pump power consumption. When the group number of fractal branches is 4, the level number of channels is 3, the length ratio is 1, and the inlet velocity of the coolant is 0.5 m/s, the BTMS can control the maximum temperature and maximum temperature difference of the batteries under 4C-rate discharge within 31.68 °C and 4.15 °C, respectively. Finally, orthogonal test was conducted on four factors: the group number of fractal branches, the level number of channels, the length ratio, and the inlet velocity of the coolant. The results showed that the level number of branches is the most important structural parameter.

A family of fractal non-contracting weakly branch groups

We construct a new example of an infinite family of groups acting on a d -adic tree, with d ≥ 2 that is non-contracting and weakly regular branch over the derived subgroup.

Electro-responsive surfaces with controllable wrinkling patterns for switchable light reflection–diffusion–grating devices

Dynamic microscale surface topographies are desired in smart optics, controlling surface wettability and preventing marine biofouling. Voltage-controlled reversible responses have demonstrated potential for reliable reproducibility and stability, fast response, and for actuating thin films fixed over large solid surfaces. To obtain reversible deformation with regular geometric patterns, however, electrical methods have had to be coupled with mechanical stretching/bending and other ways to induce anisotropy. There is a great need and potential for on-demand electrical generation of programmable complex responsive surface patterns. Here we demonstrate a responsive polymer coating over an underlying pattern of counter electrodes which can be activated selectively. We present a patternable electrode printing method to achieve localized and structured wrinkling deformation without mechanical pre-force deformation. We discover that below a minimal separation distance, electrodes below the polymer act as a single electrode. We establish parameters that govern the alignment of wrinkles and quantify the regularity and direction of the new patterns between electrodes separated by larger than this distance. We analyze and quantify the regularity of the formed wrinkling patterns by four disorder metrics: box-counting fractal dimension, tortuosity, angle distribution, and branch number. We demonstrate the application of such electrode/wrinkles-on-demand patterning with a working multi-state light reflection–diffusion–grating device.

Quantifying 3D-nanosized dispersion of SiO2 in elastomer nanocomposites by 3D-scanning transmission electron microscope (STEM)

Nanofiller/elastomer nanocomposites as strategically important materials have attracted intensive attentions due to high elasticity. However, due to the limitation of conventional two-dimensional characterization methods, the multi-scaled dispersion structure of nanofiller in elastomer nanocomposites hasn’t been comprehensively understood. Here, we established a highly-objective method to comprehensively quantify the three-dimensional (3D)-dispersion of nanoparticle in elastomer matrix. 3D-scanning transmission electron microscope (STEM) was applied to get the intrinsic 3D-dispersion structure of nano-silica (SiO2) in solution-polymerized styrene-butadiene rubber (SSBR). Equivalent sphere and fractal branch models were created to further quantify the poly-dispersity, inner connectivity and morphology of SiO2. A two-stage agglomeration evolution schematic was proposed to elucidate the development of nanosized dispersion structure of SiO2. With the increase of SiO2 volume fraction (Φsilica), the number, size and branching degree of SiO2 simultaneously increase (namely, self-agglomeration). Further increase Φsilica, adjacent SiO2 interconnect with each other, leading to sharp increases of connectivity and branching degree of SiO2 (namely, external agglomeration). This two-stage agglomeration mode interprets the well-known “Payne effect” well, which has not been quantified by 3D dispersion structure parameters before.

Compact fractal antenna for WiMAX 1.4 GHz and IEEE 802.11a using double branch line

This paper describes a novel configuration of a CPW-fed fractal antenna that exhibits dual-band performances of WiMAX 1.4 GHz and WLAN bands. The proposed fractal antenna consists of unit-cell based duble branch line attached to fractal patch as the main radiator, and a pair of mirror rectangular-shaped notch on the ground plane. Dual-band operations can be obtained by adjusting dimensions and length of fractal branch lines of patch, simultaneously. The fractal antenna is of a compact size with dimensions 17 × 17 mm2. The impedance bandwidth with S11 < -10 dB is about 2.3 GHz (1.1–2.2 GHz) for WiMAX 1.4GHz band and for WLAN-band 2.9 GHz (4.8–7 GHz). The measured peak gains are about 2.9dBi at WiMAX and 3dBi at WLAN-band. The fractal antenna has desirable S11 level, omni-directional radiation pattern and gain characteristics over the aforementioned bands. The proposed antenna is an excellent choice for multi-band wireless communication systems and mobile device.

Fractal dimensions and branching characteristics of placental chorionic surface arteries

We present a novel approach to characterize and compare the placental chorionic surface arteries (PCSA) of normal, preterm, diabetic and preeclamptic pregnancies using their fractal dimension (FD) and branch point number (BPN). Mean FD of PCSA of normal pregnancies was similar to those of diabetic and preeclamptic pregnancies, but significantly different from those of preterm pregnancies. In contrast, the BPN of PCSA of normal and preterm pregnancies was similar but significantly different from diabetic and preeclamptic pregnancies. The results suggest that branching properties of normal and pathological pregnancies are intrinsically different.

Analysis of Dendrite Images Formed by Electrochemical Migration in a Semiconductor Sensor

Dendrites were observed in the failure of semiconductor sensor devices. EDX analysis showed that the dendrites grown from bare sensor dice consisted of tin metal. The tin dendrites exhibited massive and dense branches. Dendrites grown from mechanically decapped parts consisted of silver. The silver dendrites exhibited delicate, lace-like structure. Binary and grey scale images of dendrites were analyzed for fractal dimension number and branch density. The tin dendrites had a higher, statistically significant branch density number than silver, due to tin’s more intricate branching pattern. Fractal numbers can be used to differentiate between tin and silver dendrites, even in the absence of EDX analysis equipment.

Thermal conductivity of Si/Ge nanocomposites with fractal tree-shaped networks by considering the phonon interface scattering

The fractal tree-shaped network is introduced to reconstruct the Si/Ge nanocomposites in this paper with a focus on reducing the thermal conductivity of thermo-electrical materials by enhancement of phonon scattering at the interfaces of nanostructures. The molecular dynamics simulation with the use of the empirical Tersoff potential is applied to study the thermal conductivity of Si/Ge nanocomposites with fractal tree-shaped networks. The effects of fractal dimensions and branch level on the thermal conductivity of nanocomposites are evaluated and analyzed. The results indicate that, compared with conventional rectangular core, the fractal tree-shaped network is more superior in the reduction of thermal conductivity of nanocomposites owing to stronger phonon interface scattering for larger interface density. In particular, the larger length and width fractal dimensions as well as more branch levels are preferred for the Si/Ge nanocomposites with fractal tree-shaped networks.

Optimized Ultrawideband and Uniplanar Minkowski Fractal Branch Line Coupler

The non-Euclidean Minkowski fractal geometry is used in design, optimization, and fabrication of an ultrawideband (UWB) branch line coupler. Self-similarities of the fractal geometries make them act like an infinite length in a finite area. This property creates a smaller design with broader bandwidth. The designed 3 dB microstrip coupler has a single layer and uniplanar platform with quite easy fabrication process. This optimized 180° coupler also shows a perfect isolation and insertion loss over the UWB frequency range of 3.1–10.6 GHz.

Self-Sensing Composites and Optimization of Composite Structures in Japan

I review research on self-sensing and structural optimizations of laminated carbon/epoxy composites in Japan. Self-sensing is one of the multiple functions of composites; i.e., carbon fiber is used as a sensor as well as reinforcement. I present a controversial issue in self-sensing and detail research results. Structural optimization of laminated CFRP composites is indispensable in reducing the weights of modern aerospace structural components. I present a modified efficient global search method using the multi-objective genetic algorithm and fractal branch and bound method. My group has focused its research on these subjects and our research results are presented here.

3309 フラクタル分枝限定法による複合材料構造の破壊強度最大化手法(J25-2 解析・設計の高度化・最適化II,ジョイントセッション,21世紀地球環境革命の機械工学:人・マイクロナノ・エネルギー・環境)

This paper deals with a new stacking sequence optimization method to maximize fracture strength. In authors' research group, a fractal branch and bound (FBB) method has been proposed for the stacking sequence optimization of laminated composites. The method successfully applied to maximize buckling loads and flutter limit speed of a composite wing. Although the method is quite effective on optimizations related to stiffness of the composite structures, the method leads large error for optimizations on fracture strength when stress-based failure criteria are adopted. In this study, first the mechanism of giving a large error will be revealed. Next, a modified FBB method for optimization to maximize fracture strength is proposed, by introduction of a new code into the FBB method. The method successfully optimizes the failure strength maximization problem.

Stacking sequence optimization to maximize the buckling load of blade-stiffened panels with strength constraints using the iterative fractal branch and bound method

Laminated carbon fiber reinforced polymer (CFRP) composites have widespread applications in aerospace structures, and thus optimization of the stacking sequences in these composites is indispensable. Here, a fractal branch and bound method (FBB) is proposed for optimizing the stacking sequences. This method requires only low computational costs, and an optimal result can be obtained rapidly by means of the deterministic process. For practical stacking sequence optimizations, more than two laminates have to be optimized, because a practical aerospace structural component usually comprises a panel and stiffeners made from composite laminates. Since the stacking sequences of the skin panel and stiffeners affect the buckling load of the stiffened panel, the optimization of both laminates must be performed simultaneously. In the present study, a new method to implement a strength constraint for the FBB method is proposed for the simultaneous optimization of more than two laminates (such as a panel and stiffeners). Moreover, a quadratic polynomial objective function, which includes lamination parameter variables of the two laminates: the stiffeners and the panel, is adopted. The strength constraint is implemented by means of a response surface. The new method is applied to the buckling load maximization of a blade-stiffened composite panel, in which the strength constraint is demonstrated as a feasibility study. The method successfully obtained optimal stacking sequences with the strength constraint at low computational cost.

Dimension and Stacking Sequence Optimization of Large Composite Structures Using Kriging and MOGA Method

Laminated composite structures are increasingly used for aerospace structures, such as interstage structures of launch vehicles. For an optimal design of the laminated composite structures, both dimensions and stacking sequences of the structures should be optimized simultaneously since they affect each other. The stacking sequence optimization of composite laminates is a combinatorial optimization problem with some constraints. That makes the optimization of dimensions and stacking sequences a complex mixed problem that contains both combinatorial and continuous-discrete variable optimizations. Therefore, the design of the laminated composite structures has generally been carried out using a genetic algorithm (GA). GA inherently requires many evaluations of optimization functions during the process. It must be time consuming especially for composite structural optimizations, which needs a computationally expensive analysis for the evaluation of the functions such as stress and buckling load. Authors have proposed an effective and global optimization method of the composite structures. The method utilizes a kriging method for approximating expensive functions of the optimization. Using the kriging models makes it possible not only to reduce the computational cost to evaluate the functions but also to search for the optimal design globally with using multi-objective genetic algorithm (MOGA). In the method, fractal branch and bound (FBB) method is also adopted, which is a practical and low-cost stacking sequence optimization method. In the present paper, the proposed method is applied to a complicated design problem of a large-sized composite structure. A design of a rocket structure under combined load is addressed here. The result shows that the method is effective in the large-sized optimization problems.

Two-Level Optimization of Dimensions and Stacking Sequences for Hat-Stiffened Composite Panel

This paper describes both the development of a new low-cost optimization method for the optimal design of laminated composite structures, and a result of applying this method to the minimum weight design of a hat-stiffened composite panel subject to a buckling constraint. The new method is based on two levels of optimization. In the low level optimization, the fractal branch and bound method is used for a stacking sequence optimization. In the high level optimization, the particle swarm optimization technique is used for optimizing configurations of the hat-stiffened panel. Moreover, a Kriging-based response surface is used to find the global optimum, and not just to approximate an objective function value. For this purpose, we use the ‘expected improvement’ (EI) criterion defined in the Kriging model. The validity of the proposed method is examined and the results show that the method provides valuable solutions with good precision and a low computational cost.

1826 複合材料の応力規準破壊則を用いた場合のフラクタル分枝限定法(OS18.計算力学と最適化(4),ポスターセッションP-5)

1826 複合材料の応力規準破壊則を用いた場合のフラクタル分枝限定法(OS18.計算力学と最適化(4),ポスターセッションP-5)

New iteration fractal branch and bound method for stacking sequence optimizations of multiple laminates

For laminated composite materials, the stacking sequence design is indispensable to make efficient use of the material properties. Laminated composites are usually fabricated from unidirectional plies of given thickness with limited fiber orientations to a small set of angles (e.g. 0°, +45°, 45°, and 90°). An improved version of fractal branch and bound method has been proposed to optimize multiple stacking sequences. The improved method, however, cannot be applied to complicated structures, since it is difficult to determine which is the most effective surface for designing such structures. In the present study, therefore, a new iteration fractal branch and bound method is proposed, which does not require empirical knowledge of the target structure. Here, this method is applied to a hat-stiffened panel to enable the stacking sequences of the two panel laminates and hat-type stiffener to be optimized simultaneously.

Extended fractal branch and bound method for optimization of multiple stacking sequences of stiffened composite panel

Stiffened composite panels are often used as structural components in aircraft in order to avoid buckling. It is well known that stacking sequence optimizations are indispensable for laminated composite structures. Stiffened composite panels usually have more than two stacking sequences because they consist of a panel skin laminate and stiffener laminates. This means that the stacking sequences need to be jointly optimized to achieve structural optimization of the stiffened composite panel. The authors have proposed a new stacking sequence optimization method, called the fractal branch and bound method, for optimizing a single laminate. In the present study, the fractal branch and bound method is extended to optimizing multiple stacking sequences. The extended method is applied for obtaining two optimal stacking sequences for the maximization of the buckling load of a hat-stiffened composite panel. The improved method successfully provides two optimal stacking sequences determined in a short period of time.

Stacking-Sequence Optimization of Composite Delta Wing to Improve Flutter Limit Using Fractal Branch and Bound Method

The authors developed a fractal branch and bound method for optimization of stacking sequences of laminated composites to maximize buckling load of composite structures. The method demands an approximation of a design space with a response surface comprising quadratic polynomials for pruning branches of stacking sequences. Approximation of the objective function with quadratic polynomials has been confirmed for buckling load maximizations and panel-flutter-speed-limit maximizations using lamination parameters as predictors. Flutter speed maximization of composite delta wing is employed in this study as an example of stacking-sequence optimization using the fractal branch and bound method. This paper describes the theoretical background of the fractal branch and bound method. Then approximations are performed using quadratic polynomials with lamination parameters as predictors. Subsequently, we investigated effectiveness of this method for supersonic flutter of a composite delta wing. Results indicate that the method was applied successfully; a practical optimal stacking-sequence was obtained using modified response surfaces.

修正応答曲面を用いたフラクタル分枝限定法による積層構成最適化

The fractal branch and bound method has been developed by the authors for optimization of stacking sequences to maximize buckling load of composite structures. The method demands approximation of a design space with a response surface comprising quadratic polynomials for pruning fractal branches of stacking sequences. The approximation of the design space with quadratic polynomials has been confirmed for the buckling load maximization and flutter speed limit maximization using lamination parameters as predictors. In the present study, flutter speed maximization for clamped panel flutter problem is employed as an example of stacking sequence optimization by means of the fractal branch and bound method for the flutter problem, which has complicated design space due to coupling of high-order vibration modes of flutter. The theoretical background of the Fractal branch and bound method, and the modified response surface method with newly proposed the adjusted coefficient of evaluation Eadj2 (E-square-adjusted) for complicated design-spaced problem is proposed; approximation using quadratic polynomials with lamination parameters as predictors. After that, the effectiveness of the method for the supersonic clamped panel flutters of composite laminates is investigated. As a result, the method is successfully applied, and the practical optimal stacking sequence is obtained using modified response surfaces.