Periodic Geometry Research Articles

Turbulent flow pattern and flow resistance characteristics of cross flow over square-arranged tube bundles with different pitch to diameter ratios

Helical tube bundles are used in steam generators and intermediate heat exchangers of high temperature gas-cooled reactors due to their compactness and good thermal expansion adaptation performance. However, the characteristics of cross flow over inline tube bundles are sensitive to pitch to diameter ratios (S/D) and Reynolds numbers (Re), and the influence of S/D and Re variation on the flow patterns (especially wake patterns) and pressure drop coefficients (ξ) remains unclear. In the current investigation, the flow characteristics of tube bundles with S/D = 1.58 and 1.88 with various Re are investigated with the aid of wind tunnel experiments and large eddy simulations (LES). For the tube bundle with S/D = 1.88, the results of LES with periodical geometry align well with wind tunnel experiments, successfully predicting the decreasing recirculation region size as Re increases. In contrast, for the tube bundle with S/D = 1.58, the recirculation region size remains almost constant with varying Re, filling the space between two adjacent tubes even when Re = 36 000. However, the LES with periodical geometry only successfully predicts flow phenomena up to Re = 25 285. The wakes become oblique when Re = 36 000, which differs from experiments. Based on the force balance on the recirculation region, the pressure drop coefficient (ξ) of the cross flow over tube bundles is related to the size of the recirculation region. For the tube bundle with S/D = 1.88, the recirculation region size decreases as Re increases when Re ≥ 18 000, resulting in a larger flow cross section area in the main flow region. The increasing main flow cross section area leads to an increased pressure coefficient along the separation streamlines. Since the Reynolds normal and shear stresses cancel each other out, the increasing pressure coefficients along the recirculation region boundary lead to the increasing surface pressure coefficients in the rear side of the tube and ultimately lead to the ξ decrease from 0.26 to 0.24 when Re increases from 18 000 to 44 571. Conversely, when Re ≤ 10 285, the recirculation area fills the space between two adjacent tubes, so the ξ barely change with Re. For tube bundle with S/D = 1.58, the recirculation area fills the space between tubes even when Re ≥ 18 000, leading to almost the unchanged ξ (≈0.27) with Re. The higher ξ at Re = 10 285 is related to the bounding wall effects on the near wall tubes.

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

Abstract In this paper, we study a nonlinear fluid-structure interaction problem between a ‘square-root’ viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions in both directions, while both the beam and the fluid are under the effect of time-periodic forces. By using a decoupling approach, at least one time-periodic weak solution to this problem is constructed which has a bounded energy and a fixed prescribed mass. The lack of a priori energy bounds is overcome by a series of estimates based on a careful choice of parameters. The most challenging one is the pressure estimate, which is obtained by utilizing the specific periodic geometry and the Bogovskiǐ operator on a fixed domain that has a uniform constant. With uniform estimates and improved regularity of the beam as in (Muha and Schwarzacher 2023 Ann. Inst. Henri Poin. Anal. Non Lineaire 39 1369–412), the time-periodic solution is constructed by a series of limit procedures, following the finite-dimensional time-space construction from (Feireisl et al 2012 Arch. Rational Mech. Anal. 204 74586).

Macrotransport of active particles in periodic channels and fields: Rectification and dispersion.

Transport and dispersion of active particles in structured environments, such as corrugated channels and porous media, are important for the understanding of both natural and engineered active systems. Owing to their continuous self-propulsion, active particles exhibit rectified transport under spatially asymmetric confinement. While progress has been made in experiments and particle-based simulations, a theoretical understanding of the effective long-time transport dynamics in spatially periodic geometries remains less developed. In this paper, we apply generalized Taylor dispersion theory to analyze the long-time effective transport dynamics of active Brownian particles (ABPs) in periodic channels and fields. We show that the long-time transport behavior is governed by an effective advection-diffusion equation. The derived macrotransport equations allow us to characterize the average drift and effective dispersion coefficient. For the case of ABPs subject to a no-flux boundary condition at the channel wall, we show that regardless of activity, the average drift is given by the net diffusive flux along the channel. For ABPs, their activity is the driving mechanism that sustains a density gradient, which ultimately leads to rectified motion along the channel. Our continuum theory is validated against direct Brownian dynamics simulations of the Langevin equations governing the motion of each ABP.

Size-dependency and lattice-discreetness effect on fracture toughness in 2D crystals under antiplanar loading

Fracture toughness is the material property characterizing resistance to failure. Predicting its value from the solid structure at the atomistic scale remains elusive, even in the simplest situations of brittle fracture. We report here numerical simulations of crack propagation in two-dimensional fuse networks of different periodic geometries, which are electrical analogs of bidimensional brittle crystals under antiplanar loading. Fracture energy is determined from Griffith’s analysis of energy balance during crack propagation, and fracture toughness is determined from fits of the displacement fields with Williams’ asymptotic solutions. Significant size dependencies are evidenced in small lattices, with fracture energy and fracture toughness both converging algebraically with system size toward well-defined material-constant values in the limit of infinite system size. The convergence speed depends on the loading conditions and is faster when the symmetry of the considered lattice increases. The material constants at infinity obey Irwin’s relation and properly define the material resistance to failure. Their values are approached up to ∼15% using the recent analytical method proposed in Nguyen and Bonamy (Phys Rev Lett 123:205503, 2019). Nevertheless, the deviation remains finite and does not vanish when the system size goes to infinity. We finally show that this deviation is a consequence of the lattice discreetness and decreases when the super-singular terms of Williams’ solutions (absent in a continuum medium but present here due to lattice discreetness) are taken into account.

Planar structured materials with extreme elastic anisotropy

Designing anisotropic structured materials by reducing symmetry results in unique behaviors, such as shearing under uniaxial compression. While rank-deficient materials such as hierarchical laminates have been shown to exhibit extreme elastic anisotropy, there is limited knowledge about the fully anisotropic elasticity tensors achievable with single-scale fabrication techniques. No established upper and lower bounds on anisotropic moduli exist. In this paper, we estimate the range of anisotropic stiffness tensors achieved by single-scale two-dimensional structured materials. We first develop a database of periodic anisotropic single-scale unit cell geometries using linear combinations of periodic cosine functions. The database covers a wide range of anisotropic elasticity tensors, which are then compared with the elasticity tensors of hierarchical laminates. We identify the regions in the property space where hierarchical design is necessary to achieve extremal properties. We demonstrate a method to construct various 2D functionally graded structures using this cosine function representation. These graded structures seamlessly interpolate between unit cells with distinct patterns, allowing for independent control of several functional gradients, such as porosity, anisotropic moduli, and symmetry. When designed with unit cells positioned at extreme parts of the property space, these graded structures exhibit unique mechanical behaviors such as selective strain energy localization, compressive strains under tension, and localized rotations.

Prediction of the failure behavior of pseudo-ductile composites using a multi-scale finite element model

The hybridization technique has recently been used to produce a new generation of composites called pseudo-ductile composites, which have shown higher failure strain compared to conventional composites, minimizing the risks of the occurrence of a catastrophic failure. The pseudo-ductility behavior in these composites is obtained by hybridization of fibers with high and low failure strains. In this study, a multi-scale finite element (FE) model incorporating micro and macro-scales is proposed to predict the failure behavior of pseudo-ductile composites. A micro-scale representative volume element (RVE), consisting of randomly distributed fibers, was generated using a Python code. Periodic boundary conditions (PBCs) were applied to the RVE generated with a periodic geometry. To account for fiber failure and ply fragmentation, the tensile strength of fibers was distributed based on the Weibull distribution function and a user-defined UMAT subroutine was developed. Tensile loading was then applied to the RVE to simulate the composite’s mechanical behavior. For validation, an RVE was developed based on experimental data from recent research on thinply and conventional thickness composites. Numerical results were compared to experimental data, demonstrating acceptable agreement. In the final step, following a sequential multi-scale modeling approach, a macro-scale model was constructed based on the outputs of the micro-scale model subjected to tensile and shear loads. The results were compared with experimental data, revealing good agreement. The proposed model allows for the optimization of pseudo-ductile composite structures to achieve a desired set of mechanical properties without the need for conducting extensive experimental material tests.

A comprehensive numerical study on heat transfer and friction characteristics of offset-strip fins

Offset-strip fins are among the most used geometries in compact heat exchangers. The geometric and flow parameters of the fins affect their heat transfer effectiveness and head losses. Hence, accurate predictive models are needed to guide the design process. However, most of the correlations available in the literature are valid only for a limited set of geometric configurations and flow regimes. This work discusses the derivation of multivariate response surfaces for the equivalent Darcy and Colburn factors in offset-strip fins. These surfaces feature clear applicability ranges and extend over wide Reynolds and Prandtl number ranges (50≤Re≤12000, 0.71≤Pr≤190). In addition, a novel empirical model for the Prandtl number scaling exponent is proposed. The analysis is carried out through a Design of Experiment approach, performed with computational techniques. Each numerical experiment is carried out by CFD analysis of a periodic fin geometry. The obtained response surfaces approximate CFD results with a mean deviation of ±8.4%. Moreover, application of the correlations to the analysis of complete heat exchangers issued mean and maximum deviations of ±7.8% and ±20%, respectively, thus highlighting the usefulness of the proposed models for the accurate modelling of offset-strip fins in heat transfer applications.

Hydrodynamic Porosity: A New Perspective on Flow through Porous Media, Part II

In this work, we build upon our previous finding that hydrodynamic porosity is an exponential function of pore-scale flow velocity (or interstitial Reynolds number). We previously discovered this relationship for media with a square cavity geometry—a highly idealized case of the dead-ended pore spaces in a porous medium. Thus, we demonstrate the applicability of this relationship to media with other cavity geometries. We do so by applying our previous analysis to rectangular and non-rectangular cavity geometries (i.e., circular, and triangular). We also study periodic flow geometries to determine the effect of upstream cavities on those downstream. We show that not only does our exponential relationship hold for media with a variety of cavity geometries, but it does so almost perfectly with a coefficient of determination (R2) of approximately one for each new set of simulation data. Given this high fit quality, it is evident that the exponential relationship we previously discovered is applicable to most, if not all, unwashed media.

Computational Homogenization for Inverse Design of Surface-based Inflatables

Surface-based inflatables are composed of two thin layers of nearly inextensible sheet material joined together along carefully selected fusing curves. During inflation, pressure forces separate the two sheets to maximize the enclosed volume. The fusing curves restrict this expansion, leading to a spatially varying in-plane contraction and hence metric frustration. The inflated structure settles into a 3D equilibrium that balances pressure forces with the internal elastic forces of the sheets. We present a computational framework for analyzing and designing surface-based inflatable structures with arbitrary fusing patterns. Our approach employs numerical homogenization to characterize the behavior of parametric families of periodic inflatable patch geometries, which can then be combined to tessellate the sheet with smoothly varying patterns. We propose a novel parametrization of the underlying deformation space that allows accurate, efficient, and systematical analysis of the stretching and bending behavior of inflated patches with potentially open boundaries. We apply our homogenization algorithm to create a database of geometrically diverse fusing patterns spanning a wide range of material properties and deformation characteristics. This database is employed in an inverse design algorithm that solves for fusing curves to best approximate a given input target surface. Local patches are selected and blended to form a global network of curves based on a geometric flattening algorithm. These fusing curves are then further optimized to minimize the distance of the deployed structure to target surface. We show that this approach offers greater flexibility to approximate given target geometries compared to previous work while significantly improving structural performance.

Numerical analysis of the axisymmetric lattice Boltzmann method for steady and oscillatory flows in periodic geometries

Compared to more typical computational fluid dynamics techniques, the lattice Boltzmann method (LBM) is relatively new and unexplored. In recent years, axisymmetric LBM formulations, which can simulate flow in rotationally symmetric 3D geometries, have been published. Here we verify a novel axisymmetric LBM implementation using numerical criteria. Hagen–Poiseuille and Womersley flow are considered within a straight tube where analytic solutions are available. With this, we establish sufficient accuracy of the approximated flow and study the effects of changing simulation parameters (e.g. Reynolds number, Womersley number) and spatial/temporal parameters (e.g. relaxation time, mesh nodes, time steps). Furthermore, steady and oscillatory flows within a periodically-varying, longitudinally asymmetric geometry are considered. Analytic solutions are not available in these cases; however, the validity of the axisymmetric LBM for curved boundaries is ensured through convergence, mesh independence and qualitative observations. Guaranteeing reasonable flow field determination for the aformentioned geometry is relevant to a larger problem where particulate suspension is pumped back and forth through a membrane of axisymmetric micropores. In these circumstances, experiments have induced directed particle transport even though there is no net flow of the carrier fluid. Hence, our work aims to improve current numerical simulations of these flow problems to better understand the factors that facilitate particle transport.

Digital Light Processing to Afford High Resolution and Degradable CO2‐Derived Copolymer Elastomers

AbstractVat photopolymerization 3D printing has proven very successful for the rapid additive manufacturing (AM) of polymeric parts at high resolution. However, the range of materials that can be printed and their resulting properties remains narrow. Herein, we report the successful AM of a series of poly(carbonate‐b‐ester‐b‐carbonate) elastomers, derived from carbon dioxide and bio‐derived ϵ‐decalactone. By employing a highly active and selective Co(II)Mg(II) polymerization catalyst, an ABA triblock copolymer (Mn=6.3 kg mol−1, ÐM=1.26) was synthesized, formulated into resins which were 3D printed using digital light processing (DLP) and a thiol‐ene‐based crosslinking system. A series of elastomeric and degradable thermosets were produced, with varying thiol cross‐linker length and poly(ethylene glycol) content, to produce complex triply periodic geometries at high resolution. Thermomechanical characterization of the materials reveals printing‐induced microphase separation and tunable hydrophilicity. These findings highlight how utilizing DLP can produce sustainable materials from low molar mass polyols quickly and at high resolution. The 3D printing of these functional materials may help to expedite the production of sustainable plastics and elastomers with potential to replace conventional petrochemical‐based options.

Digital Light Processing to Afford High Resolution and Degradable CO2-Derived Copolymer Elastomers.

Vat photopolymerization 3D printing has proven very successful for the rapid additive manufacturing (AM) of polymeric parts at high resolution. However, the range of materials that can be printed and their resulting properties remains narrow. Herein, we report the successful AM of a series of poly(carbonate-b-ester-b-carbonate) elastomers, derived from carbon dioxide and bio-derived ϵ-decalactone. By employing a highly active and selective Co(II)Mg(II) polymerization catalyst, an ABA triblock copolymer (Mn=6.3 kg mol-1, ÐM=1.26) was synthesized, formulated into resins which were 3D printed using digital light processing (DLP) and a thiol-ene-based crosslinking system. A series of elastomeric and degradable thermosets were produced, with varying thiol cross-linker length and poly(ethylene glycol) content, to produce complex triply periodic geometries at high resolution. Thermomechanical characterization of the materials reveals printing-induced microphase separation and tunable hydrophilicity. These findings highlight how utilizing DLP can produce sustainable materials from low molar mass polyols quickly and at high resolution. The 3D printing of these functional materials may help to expedite the production of sustainable plastics and elastomers with potential to replace conventional petrochemical-based options.

Coexistence of Defect Morphologies in Three-Dimensional Active Nematics.

We establish how active stress globally affects the morphology of disclination lines of a three-dimensional active nematic liquid crystal under chaotic flow. Thanks to a defect detection algorithm based on the local nematic orientation, we show that activity selects a crossover length scale in between the size of small defect loops and that of long and tangled defect lines of fractal dimension 2. This length scale crossover is consistent with the scaling of the average separation between defects as a function of activity. Moreover, on the basis of numerical simulation in a 3D periodic geometry, we show the presence of a network of regular defect loops, contractible onto the 3-torus, always coexisting with wrapping defect lines. While the length of regular defects scales linearly with the emerging active length scale, it verifies an inverse quadratic dependence for wrapping defects. The shorter the active length scale, the more the defect lines wrap around the periodic boundaries, resulting in extremely long and buckled structures.

Mapping Hydrogen Positions along the Proton Transfer Pathway in an Organic Crystal by Computational X-ray Spectra.

Understanding proton transfer (PT) dynamics in condensed phases is crucial in chemistry. We computed a 2D map of N 1s X-ray photoelectron/absorption spectroscopy (XPS/XAS) for an organic donor-acceptor salt crystal against two varying N-H distances to track proton motions. Our results provide a continuous spectroscopic mapping of O-H···N↔O-··· H+-N processes via hydrogen bonds at both nitrogens, demonstrating the sensitivity of N 1s transient XPS/XAS to hydrogen positions and PT. By reducing the O-H length at N1 by only 0.2 Å, we achieved excellent theory-experiment agreement in both XPS and XAS. Our study highlights the challenge in refining proton positions in experimental crystal structures by periodic geometry optimizations and proposes an alternative scaled snapshot protocol as a more effective approach. This work provides valuable insights into X-ray spectra for correlated PT dynamics in complex crystals, benefiting future experimental studies.

Topological wave equation eigenmodes in continuous 2D periodic geometries

In this paper, we address the topological characterization of the wave equation solutions in continuous two-dimensional (2D) periodic geometries with Neumann or Dirichlet boundary conditions. This characterization is relevant in the context of 2D vibrating membranes and our approach allows one to understand the topological behavior recently observed in acoustic three-dimensional artificial lattices. In particular, the dependence of the topological behavior on the experimental positioning of the coupling channels is explained using simple arguments and a simple method of construction of an equivalent effective tight-binding Hamiltonian is presented.

Numerical Investigation of Broad Mid-Frequency Flexural Bandgap in Composite Sandwich Structures with Periodic Hollow-Shaped Core Geometry

Numerical Investigation of Broad Mid-Frequency Flexural Bandgap in Composite Sandwich Structures with Periodic Hollow-Shaped Core Geometry

Topology of boron substitutional defects in single-walled carbon nanotubes: A first-principles study

This is a theoretical study of boron-doped single-walled carbon nanotubes. The same topology of primitive nanodomains, located at different positions on single-walled carbon nanotubes, leads to different electronic band structures. We propose a ϕ term. Density functional theory was corrected for van der Waals interactions and used to carry out the periodic boundary condition geometry optimization, where boron formed the topologies of primitive nanodomains. The calculated bulk structure and local structure spectroscopic parameters can be used for comparison with experimental results to confirm the theoretical models.

M-Voronoi and other random open and closed-cell elasto-plastic cellular materials: Geometry generation and numerical study at small and large strains

The present study deals with a numerical design strategy of a novel class of three-dimensional random Voronoi-type geometries, called M-Voronoi. These materials comprise random, non-quadratic convex void shapes and non-uniform intervoid ligament thicknesses, and can span high-to-low relative densities. The starting point for their generation is a random adsorption algorithm (RSA) construction with spherical voids embedded in an incompressible, nonlinear elastic matrix phase. The initial RSA geometry is subjected to large elastic volume changes by prescribing Dirichlet boundary conditions. Due to the incompressibility of the matrix phase, the externally imposed volume changes lead to significant void growth. The numerical growth process may be stopped at any desired porosity. The proposed M-Voronoi process is general and allows the formation of isotropic (or anisotropic) designs. As a byproduct of the developed approach, we also present a novel remeshing technique allowing to read arbitrary geometries of one or multiple phases. The elasto-plastic properties of the M-Voronoi porous materials are numerically investigated at small strains as well as large compressive and shear loads. Their response is assessed by comparison with other well-known random and periodic porous geometries such as polydisperse porous materials with spherical voids (RSA), classical TPMS Gyroid geometries and random Spinodoid topologies. The results show that M-Voronoi and RSA (with spherical voids) geometries exhibit the stiffest elastic and highest flow stress response compared to the other two geometries. This study shows unambiguously that randomness may or may not lead to enhanced mechanical response such as higher stiffness or flow stress.

Enhanced transmission capacity through a specialty multi-channel topological optical fiber

We propose a multi-channel specialty topological fiber based on 1D periodic geometry which supports three coexisting and non-interacting topologically-protected robust light states in the mid-infrared wavelength range. Each channel comprises two topologically-distinct 1D periodic lattices that are characterized by the topological invariant, Zak phase. The interface between the trivial and the nontrivial lattices is made unconventional to provide a strong transverse confinement of the light state. Moreover, the observed multiple interface states exhibit robust simultaneous transmission of light through all the channels keeping in contact with each other with an inter-channel cross-talk of ∼−27 dB. Such highly stable, scatter-free, and robust interface states have the potential to increase the transmission capacity of the existing optical communication channels, and can eventually be utilized as an alternative platform via the space-division multiplexing schemes.

Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains

The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation , where , both for the problem on the circle and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for and all real values of an advection parameter a when the data is small. We also derive new analytical solutions in both geometries when a = 0, and on the real line when , for various values of σ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet for a = 0 and σ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values of σ that are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory.