Fractal-like Geometries Research Articles

Efficient and consistent adaptive mesh generation for geophysical models: A case study over the Gulf of Thailand

Geophysical domains typically exhibit intricate, irregular boundaries characterized by fractal-like geometries, while underlying physical processes operate across a broad spectrum of spatial scales. The challenge lies in generating spatial discretization of these domains that conform to their geographical constraints, utilizing anisotropic, fully adaptive meshes. This problem is compounded by the vast range of scales and a notably heterogeneous parameter space. Current methodologies often rely on ad hoc, model-specific, or application-dependent approaches, which lack comprehensive descriptions. Consequently, the development of new spatial domains is labor-intensive, prone to errors, challenging to replicate, and difficult to maintain consistency due to substantial human involvement. This predicament poses obstacles to the reproducibility of simulations and the establishment of provenance in data handling and model initialization, and it hinders rigorous model intercomparisons. Furthermore, the likelihood of discrepancies in model initialization and forcing parameters increases when employing flexible adaptive meshes. This paper introduces a systematic approach to the automated generation of adaptive meshes for geophysical models. This method is efficient in its generation process and readily reproducible, offering robust and consistent adherence to the source data. The proposed approach facilitates research in complex multi-scale geophysical domains, which would be challenging using existing methods. A simulation of monthly mean currents was carried out as a case study in the Gulf of Thailand. Results revealed that the simulated current circulations agreed with the observation. Examples of its application in various ongoing geophysical modeling endeavors illustrate its effectiveness.

3D topographies promote macrophage M2d-Subset differentiation

In vitro cellular models denote a crucial part of drug discovery programs as they aid in identifying successful drug candidates based on their initial efficacy and potency. While tremendous headway has been achieved in improving 2D and 3D culture techniques, there is still a need for physiologically relevant systems that can mimic or alter cellular responses without the addition of external biochemical stimuli. A way forward to alter cellular responses is using physical cues, like 3D topographical inorganic substrates, to differentiate macrophage-like cells. Herein, protein secretion and gene expression markers for various macrophage subsets cultivated on a 3D topographical substrate are investigated. The results show that macrophages differentiate into anti-inflammatory M2-type macrophages, secreting increased IL-10 levels compared to the controls. Remarkably, these macrophage cells are differentiated into the M2d subset, making up the main component of tumour-associated macrophages (TAMs), as measured by upregulated Il-10 and Vegf mRNA. M2d subset differentiation is attributed to the topographical substrates with 3D fractal-like geometries arrayed over the surface, else primarily achieved by tumour-associated factors in vivo. From a broad perspective, this work paves the way for implementing 3D topographical inorganic surfaces for drug discovery programs, harnessing the advantages of in vitro assays without external stimulation and allowing the rapid characterisation of therapeutic modalities in physiologically relevant environments.

Modeling of dendritic solidification and numerical analysis of the phase-field approach to model complex morphologies in alloys

Dendrites are one of the most widely observed patterns in nature, and occur across a wide spectrum of physical phenomena—from snow flakes to river basins; from bacterial colonies to lungs and vascular systems; and in solidification and growth patterns in metals and crystals. The ubiquitous occurrence of these “tree-like” structures can be attributed to their excellent space-filling properties, and at times, dendritic structures also spatially manifest fractal-like distributions. As is the case with many fractal-like geometries, the complex multi-level branching structures in dendrites pose a modeling challenge, and a full resolution of dendritic structures is computationally very demanding. In the literature, extensive theoretical models of dendritic formation and evolution, essentially as extensions of the classical moving boundary Stefan problem exist. Much of this understanding is from the analysis of dendrites occurring during the solidification of metallic alloys, as this is critical for understanding microstructure evolution during metal manufacturing processes that involve solidification of a liquid melt. Motivated by the problem of modeling microstructure evolution from liquid melts of pure metals and metallic alloys during metal additive manufacturing, we developed a comprehensive numerical framework for modeling a large variety of dendritic structures that are relevant to metal solidification. In this work, we present a numerical framework encompassing the modeling of Stefan problem formulations relevant to dendritic evolution using a phase-field approach and a finite element method implementation. Using this framework, we model numerous complex dendritic morphologies that are physically relevant to the solidification of pure melts and binary alloys. The distinguishing aspects of this work are—a unified treatment of both pure metals and alloys; novel numerical error estimates of dendritic tip velocity; and the study of error convergence of the primal fields of temperature and the order parameter with respect to numerical discretization. To the best of our knowledge, this is a first of its kind study of numerical convergence of the phase-field equations of dendritic growth in a finite element method setting. Further, using this numerical framework, various types of physically relevant dendritic solidification patterns like single equiaxed, multi-equiaxed, single columnar and multi-columnar dendrites are modeled in two-dimensional and three-dimensional computational domains.

Small-Angle Scattering from Fractional Brownian Surfaces

Recent developments in nanotechnology have allowed the fabrication of a new generation of advanced materials with various fractal-like geometries. Fractional Brownian surfaces (fBs) are often used as models to simulate and characterize these complex geometries, such as the surface of particles in dilute particulate systems (e.g., colloids) or the interfaces in non-particulate two-phase systems (e.g., semicrystalline polymers with crystalline and amorphous phases). However, for such systems, a realistic simulation involves parameters averaged over a macroscopic volume. Here, a method based on small-angle scattering technique is proposed to extract the main structural parameters of surfaces/interfaces from experimental data. It involves the analysis of scattering intensities and the corresponding pair distance distribution functions. This allows the extraction of information with respect to the overall size, fractal dimension, Hurst and spectral exponents. The method is applied to several classes of fBs, and it is shown that the obtained numerical values of the structural parameters are in very good agreement with theoretical ones.

Characterization of ultrafine particles emitted during laser-based additive manufacturing of metal parts

Particulate matter (PM) emitted during laser additive manufacturing with stainless steel powder materials has been studied in detail. Three different additive manufacturing techniques were studied: selective laser melting, direct metal deposition and laser cladding. Gas flow and temperature fields accompanying the processes were numerically modeled for understanding particle growth and oxidation. Transmission and scanning electron microscopy were used for primary particle and PM characterization. The PM collected in the atmosphere during manufacturing consisted of complex aggregates/agglomerates with fractal-like geometries. The overwhelming number of particles formed in the three processes had equivalent projected area diameters within the 4–16 nm size range, with median sizes of 8.0, 9.4 and 11.2 nm. The primary particles were spherical in shape and consisted of oxides of the main steel alloying elements. Larger primary particles (> 30 nm) were not fully oxidized, but where characterized by a metallic core and an oxidic surface shell.

Interfacial toughening effect of suture structures

Suture interfaces are one of the most common architectural designs in natural material-systems and are critical for ensuring multiple functionalities by providing flexibility while maintaining connectivity. Despite intensive studies on the mechanical role of suture structures, there is still a lack of understanding on the fracture mechanics of suture interfaces in terms of their interactions with impinging cracks. Here we reveal an interfacial toughening effect of suture structures by means of “excluding” cracks away from interfaces based on a dimensionless micro-mechanical model for single-leveled and hierarchical suture interfaces with triangular-shaped suture teeth. The effective stress-intensity driving forces for crack deflection along, versus penetration through, an interface at first impingement and on subsequent kinking are formulated and compared with the corresponding resistances. Quantitative criteria are established for discerning the cracking modes and fracture resistance of suture interfaces with their dependences on sutural tooth sharpness and interfacial toughness clarified. Additionally, the effects of structural hierarchy are elucidated through a consideration of hierarchical suture interfaces with fractal-like geometries. This study may offer guidance for designing bioinspired suture structures, especially for toughening materials where interfaces are a key weakness. Statement of significanceSuture interfaces are one of the most common architectural material designs in biological systems, and are found in a wide range of species including armadillo osteoderms, boxfish armor, pangolin scales and insect cuticles. They are designed to provide flexibility while maintaining connectivity. Despite many studies on the mechanical role of suture structures, there is still little understanding of their role in terms of interactions with impinging cracks. Here we reveal an interfacial toughening effect of suture structures by means of “excluding” cracks away from interfaces based on a dimensionless micro-mechanical model for single-leveled and hierarchical suture interfaces with triangular-shaped suture teeth. Quantitative criteria are established for discerning the cracking mode and fracture resistance of the interfaces with their dependences on sutural tooth sharpness and interfacial toughness clarified.

FIBONACCI OPTICAL LATTICE: A FRACTAL BIOMODULATION DEVICE

Natural scenes or structures share fractal-like geometries with scale-invariant statistical patterns that exhibit useful functional physiological properties in humans. This paper explores the self-assembling of alumina microelements on a nanostructured silica-embedded PET substrate and the application of Fibonacci fractal geometry as a photonic wave-guiding device. Two concentric Fibonacci circles impinged on the top of embedded silica clusters were investigated using fractal analysis. Calculating the fractal dimension of spatial scaling properties demonstrated the potential of a fractal photonic element. Simulating the propagation of spectral visible or low NIR incident light through the fractal trajectory shows that the fractal scaling properties generate novel nonlinear and double-twisted electromagnetic waves with biophilic interconnecting potential.

Resilient 3D hierarchical architected metamaterials

Hierarchically designed structures with architectural features that span across multiple length scales are found in numerous hard biomaterials, like bone, wood, and glass sponge skeletons, as well as manmade structures, like the Eiffel Tower. It has been hypothesized that their mechanical robustness and damage tolerance stem from sophisticated ordering within the constituents, but the specific role of hierarchy remains to be fully described and understood. We apply the principles of hierarchical design to create structural metamaterials from three material systems: (i) polymer, (ii) hollow ceramic, and (iii) ceramic-polymer composites that are patterned into self-similar unit cells in a fractal-like geometry. In situ nanomechanical experiments revealed (i) a nearly theoretical scaling of structural strength and stiffness with relative density, which outperforms existing nonhierarchical nanolattices; (ii) recoverability, with hollow alumina samples recovering up to 98% of their original height after compression to ≥ 50% strain; (iii) suppression of brittle failure and structural instabilities in hollow ceramic hierarchical nanolattices; and (iv) a range of deformation mechanisms that can be tuned by changing the slenderness ratios of the beams. Additional levels of hierarchy beyond a second order did not increase the strength or stiffness, which suggests the existence of an optimal degree of hierarchy to amplify resilience. We developed a computational model that captures local stress distributions within the nanolattices under compression and explains some of the underlying deformation mechanisms as well as validates the measured effective stiffness to be interpreted as a metamaterial property.

Natural Convection Fin Performance Using Fractal-Like Geometries

Fractal geometries are shown to increase surface area significantly with patterns such as the Sierpinski carpet providing the potential for mass reduction. Two fractal geometries were used in this study, the modified Koch snowflakeandtheSierpinskicarpet.Thisstudyexamines finperformanceforthebaselinecases(atriangular finand a square fin) relative to the first three fractal iterations for both geometries. Constant heat rate conditions were applied to the base of the fins, and the temperature distribution across the fins was observed using an infrared camera. Fin effectiveness and fi ne f ficiency were calculated for each fin geometry to quantify the effects of using fractal geometries to improve fin performance. Based on the observed results, fractal geometries can be used to improve fin performance, particularly when the decrease in fin mass is a performance criteria. As fins are used for passive thermal management in many industrial and electronic devices, the use of fractal-like geometries has wide reaching potential. Nomenclature A = area, m 2 C = constant d = constant h = heat transfer coefficient, W=m 2 -K k = thermal conductivity, W=m-K m = mass, kg Nu = Nusselt number, hw=k n = iteration index Q = heat rate, W T = temperature, K t = fin thickness, m w = fin base length, m � = width/thickness ratio � T = average fin to ambient temperature difference, K � = fin effectiveness = emissivity � = fi ne fficiency

FLOW OVER FRACTALS: DRAG FORCES AND NEAR WAKES

An experimental study of interactions between a high Reynolds number fluid flow and multi-scale, fractal, objects is performed. Studying such interactions is required to improve our current understanding of wind or ocean current effects on vegetation elements, which often display fractal-like branching geometries. The main objectives of the study are to investigate the effects of the range of scales (generation numbers) of the fractal object and of the incoming flow condition on the drag force and drag coefficient, and to observe flow features in the near wake region resulting from the interaction. In this study, Sierpinski carpets and triangles with the scale ratios of 1/3 and 1/2, respectively, are employed. The fractal dimensions of the Sierpinski carpet and triangle are D = 1.893 and 1.585, respectively. Each pre-fractal object is mounted on a load cell at the centerline in a wind tunnel. Two types of inflow conditions are considered: laminar flow and high-turbulence level, active-grid-generated, flow. As a first approximation, we find the drag coefficients are approximately constant of order unity, and do not depend upon generation number of the pre-fractal when defined using the actual frontal area that varies as function of generation number. Still, the drag coefficient of the Sierpinski carpet increases weakly with number of generations indicating that the drag force decreases less than the cross-sectional area. For the Sierpinski triangle a similar trend is observed at large scales. However, the drag coefficient displays a peak at the third generation and then shows a decreasing trend as smaller scales are included for higher generation cases. The drag coefficient for the turbulent flow is larger than that for the laminar flow for all the fractal generations observed. Flow features (mean velocity, mean vorticity, and turbulence root-mean-square distributions) are measured by using stereoscopic Particle Image Velocimetry to observe various scales of the motion in the near wake of the pre-fractal objects. Strong shear layers are formed behind the fractal objects depending on the hole locations of different generations, which results in the formation of various length scales of the dominant turbulence structures. The smaller scale wakes are found to merge behind the Sierpinski carpet, whereas they are merely damped behind the Sierpinski triangle.