Simulation Of Pattern Formation Research Articles

A numerical framework for the simulation of coupled electromechanical growth

Electro-mechanical response exists in growing materials such as biological tissues and hydrogels, influencing the growth process, pattern formation and geometry remodelling. To gain a better understanding of the mechanism of the coupled effects of growth and electric fields on the deformation behaviour, a finite element framework for coupled electro-elastic growth is established. Based on the extended volume growth theory, the governing equations of the growing electro-elastic solid are obtained. A coupled three-field mixed displacement-pressure-potential finite element formulation using inf–sup stable combinations is adapted. The finite element formulation is implemented in ABAQUS via a user element subroutine. The implementation is validated first by comparing the deformation and stress components of a growing tubular structure under axial strain and radial voltage. Using the example of a bi-layer beam actuator, it is illustrated that growth parameters and the external voltage can precisely control the bending angle. The framework is then applied to simulate pattern formation and transition behaviour, such as doubling and tripling of wrinkles, by specifying growth parameters and external voltage in a 3D stiff film/soft substrate structure. Furthermore, the suppression of wrinkles by applying external voltage is demonstrated. It is observed that the electric field plays a significant role in stress redistribution and guiding growth, resulting in the promotion or suppression of wrinkles, which is demonstrated by the numerical simulation of a long tubular structure. The proposed finite element scheme provides an accurate, efficient and stable tool for numerical simulation of electro-elastic solids incorporating growth effect, which can be used for understanding coupled growth phenomenon in biological soft matter and developing smart devices for medical treatment.

Pattern formation in reaction–diffusion systems on evolving surfaces

In this paper, we propose an explicit time-stepping scheme for the pattern formation in reaction–diffusion systems on evolving surfaces. The proposed numerical method is based on a simple discretization scheme of Laplace–Beltrami operator over triangulated surface. On the static and evolving domains, we perform various numerical experiments for effect of domain growth and pattern formations. The computational results demonstrate that our proposed method can simulate pattern formation in reaction–diffusion systems on evolving surfaces. The actual zebra skin pattern and computational results are compared. In the computational results, we can observe different pattern formations on the evolving surface with specific rotation speed.

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations.

We study traveling waves in a non-local cross-diffusion-type model, where organisms move along gradients in population densities. Such models are valuable for understanding waves of migration and invasion and how directed motion can impact such scenarios. In this paper, we demonstrate the emergence of traveling wave solutions, studying properties of both planar and radial wave fronts in one- and two-species variants of the model. We compute exact traveling wave solutions in the purely diffusive case and then perturb these solutions to analytically capture the influence directed motion has on these exact solutions. Using linear stability analysis, we find that the minimum wavespeeds correspond to the purely diffusive case, but numerical simulations suggest that advection can in general increase or decrease the observed wavespeed substantially, which allows a single species to more rapidly move into unoccupied resource-rich spatial regions or modify the speed of an invasion for two populations. We also find interesting effects from the non-local interactions in the model, suggesting that single species invasions can be enhanced with stronger non-locality, but that invasion of a competitive species may be slowed due to this non-local effect. Finally, we simulate pattern formation behind waves of invasion, showing that directed motion can have substantial impacts not only on wavespeed but also on the existence and structure of emergent patterns, as predicted in the first part of our study (Taylor et al. in Bull Math Biol, 2020).

Pattern Formation for a Local/Nonlocal Interaction Functional Arising in Colloidal Systems

In this paper we study pattern formation for a physical local/nonlocal interaction functional where the local attractive term is given by the $1$-perimeter and the nonlocal repulsive term is the Yukawa (or screened Coulomb) potential. This model is physically interesting as it is the $\Gamma$-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal systems \cite{BBCH,CCA,IR,GCLW}. Following a strategy introduced in~\cite{DR} we prove that in a suitable regime minimizers are periodic stripes, in any space dimension.

Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method

We present an efficient linear second-order method for a Swift–Hohenberg (SH) type of a partial differential equation having quadratic-cubic nonlinearity on surfaces to simulate pattern formation on surfaces numerically. The equation is symmetric under a change of sign of the density field if there is no quadratic nonlinearity. We introduce a narrow band neighborhood of a surface and extend the equation on the surface to the narrow band domain. By applying a pseudo-Neumann boundary condition through the closest point, the Laplace–Beltrami operator can be replaced by the standard Laplacian operator. The equation on the narrow band domain is split into one linear and two nonlinear subequations, where the nonlinear subequations are independent of spatial derivatives and thus are ordinary differential equations and have closed-form solutions. Therefore, we only solve the linear subequation on the narrow band domain using the Crank–Nicolson method. Numerical experiments on various surfaces are given verifying the accuracy and efficiency of the proposed method.

Coarse-grained modeling of sheared granular beds

In this work, a volume-filtered Eulerian–Lagrangian (VFEL) approach is employed to evaluate the model fidelity required to accurately predict the dynamics of sub-aqueous sedimentary flows. The VFEL approach allows for interphase exchange terms (e.g., volume fraction and drag) to be computed on a length scale independent of the grid resolution, enabling a comprehensive analysis of the errors associated with numerical discretization and physical models on bed height, particle flux, and bed wavelength. Simulations of initially flat granular beds under both laminar and turbulent shear flow are carried out in featureless and pattern forming regimes. Resulting errors are evaluated by comparing VFEL to the direct numerical simulation (DNS) data of [1,2]. Grid refinement demonstrates convergence of the bed height and particle flux in all of the cases considered. However, traditional drag laws based on local volume fraction and particle Reynolds number lead to systematic errors in the featureless laminar regime. A stochastic drag law that accounts for the variance in local particle configuration provides improvement of the steady state statistics. The use of a stochastic drag law was found to be less important when the rise in bed height is minimal. Finally, the VFEL approach with stochastic drag is used in a large-eddy simulation framework to simulate pattern formation in turbulent shear flow at relatively high Galileo numbers.

Islands growth control in adsorptive multilayer plasma-condensate systems

A model for adsorbate islands growth is proposed to simulate pattern formation in multilayer plasma-condensate system. It was shown that morphology of surface patterns and mean island size can be effectively controlled by adsorption rate (pressure of gaseous atmosphere), interaction strength of adsorbate and external electrical field, managing anisotropic diffusion of atoms between layers. Dynamics of patterns formation is analyzed by considering statistical properties of growing surface on each formed layer and mean island size having size from third up to two diffusion lengths of adatoms. It was shown that island growth relates to modified Ostawald ripening scenario with universal island size distribution. Obtained data relates well to experimental observation of adsorbate island formation at condensation of metals and semiconductors. Described model and results are general and can be used to describe real experimental observations of surface nanopatterning at condensation in plasma-condensate devices.

Study on the Compatibility of Multi-Bifurcations by Simulations of Pattern Formation

Study on the Compatibility of Multi-Bifurcations by Simulations of Pattern Formation

Phase field simulation of pattern formation on a curved surface and the effect of the curvature

Phase field simulation of pattern formation on a curved surface and the effect of the curvature

Capillary-mediated interface perturbations: Deterministic pattern formation

Leibniz–Reynolds analysis identifies a 4th-order capillary-mediated energy field that is responsible for shape changes observed during melting, and for interface speed perturbations during crystal growth. Field-theoretic principles also show that capillary-mediated energy distributions cancel over large length scales, but modulate the interface shape on smaller mesoscopic scales. Speed perturbations reverse direction at specific locations where they initiate inflection and branching on unstable interfaces, thereby enhancing pattern complexity. Simulations of pattern formation by several independent groups of investigators using a variety of numerical techniques confirm that shape changes during both melting and growth initiate at locations predicted from interface field theory. Finally, limit cycles occur as an interface and its capillary energy field co-evolve, leading to synchronized branching. Synchronous perturbations produce classical dendritic structures, whereas asynchronous perturbations observed in isotropic and weakly anisotropic systems lead to chaotic-looking patterns that remain nevertheless deterministic.

Statistical simulation of pattern formation by ion-beam sputtering

Purpose – The purpose of this paper is to investigate the influence of ion-flow parameters on surface topography and making numerical simulation at the times when the process of surface erosion becomes strongly nonlinear. Design/methodology/approach – The base of the mathematical model of target ion-sputtering is nonlinear evolutionary equation in which the erosion velocity dependence on ion flux is evaluated by means of a Monte Carlo method. The difference between this equation and the one of continuum theory is that the ion flux is not smooth function. Instead, it is a set of separate incident ions. Findings – Some simulations with using independent random points of arrival for the incident ions leads to results uncorrelated with the continuum model at early times. The ripples are not quite developed or observed. This phenomenon is explained by random fluctuations of the target sputtering depth. Sufficiently big values of the random fluctuations destroy the ripple structure on target surface. The simulation with using equally distributed sequence (Holton sequence) of points of arrival for the incident ions leads to results well correlated with the continuum model. Originality/value – The discrete model which goes into the equation of continuum theory within the appropriate asymptotic limit has been proposed. The discretization parameters influence on surface morphology formation has been studied. This paper may be interesting to researchers making the theoretical and numerical analysis of pattern formation on plane target surfaces undergoing ion-beam sputtering.

Pattern formation in the three-dimensional deformations of fibered sheets

We simulate pattern formation in the deformations of a pantographic lattice using a model of elastic surfaces that accounts for the geodesic bending of the constituent fibers. The theory predicts an unusual arrangement of coexistent phases observed in an actual lattice, manufactured by a 3D printing process, in which the fibers undergo part-wise uniform shears separated by internal transition layers controlled by geodesic bending stiffness.

Analysis of the drainage density of experimental and modelled tidal networks

Abstract. Based on controlled laboratory experiments, we numerically simulate the initiation and long-term evolution of back-barrier tidal networks in micro-tidal and meso-tidal conditions. The simulated pattern formation is comparable to the morphological growth observed in the laboratory, which is characterised by relatively rapid initiation and slower adjustment towards an equilibrium state. The simulated velocity field is in agreement with natural reference systems such as the micro-tidal Venice Lagoon and the meso-tidal Wadden Sea. Special attention is given to the concept of drainage density, which is measured on the basis of the exceedance probability distribution of the unchannelled flow lengths. Model results indicate that the exceedance probability distribution is characterised by an approximately exponential trend, similar to the results of laboratory experiments and observations in natural systems. The drainage density increases greatly during the initial phase of tidal network development, while it slows down when the system approaches equilibrium. Due to the larger tidal prism, the tidal basin has a larger drainage density for the meso-tidal condition (after the same amount of time) than the micro-tidal case. In both micro-tidal and meso-tidal simulations, it is found that there is an initial rapid increase of the tidal prism which soon reaches a relatively steady value (after approximately 40 yr), while the drainage density adjusts more slowly. In agreement with the laboratory experiments, the initial bottom perturbations play an important role in determining the morphological development and hence the exceedance probability distribution of the unchannelled flow lengths. Overall, our study indicates an agreement of the geometric characteristics between the numerical and experimental tidal networks.

Fast Simulation of Pattern Formation and Process Dependencies in Roller Nanoimprint Lithography

ABSTRACTWe contribute a fast numerical approach to simulating the roller-imprinting of complex patterns. The technique predicts the extent to which imprinted patterns are fully formed, as well as variation of the imprinted material’s residual layer thickness (RLT). The approach can be used for roll-to-roll and roll-to-plate configurations, and for rollers with or without elastomeric coatings. If patterns vary in pitch, shape or areal density across the roller, RLT and the completeness of pattern transfer can vary with position as well as with processing parameters, and our technique is able to model these effects. The technique has been successfully validated against published experimental data from two different roller-NIL processes: one involving an ultraviolet-curing resist film on a glass plate, and another involving a flexible thermoplastic web softened at its surface.

Stochastic simulations of pattern formation in excitable media.

We present a method for mesoscopic, dynamic Monte Carlo simulations of pattern formation in excitable reaction–diffusion systems. Using a two-level parallelization approach, our simulations cover the whole range of the parameter space, from the noise-dominated low-particle number regime to the quasi-deterministic high-particle number limit. Three qualitatively different case studies are performed that stand exemplary for the wide variety of excitable systems. We present mesoscopic stochastic simulations of the Gray-Scott model, of a simplified model for intracellular Ca oscillations and, for the first time, of the Oregonator model. We achieve simulations with up to particles. The software and the model files are freely available and researchers can use the models to reproduce our results or adapt and refine them for further exploration.

Techniques for analysing pattern formation in populations of stem cells and their progeny

BackgroundTo investigate how patterns of cell differentiation are related to underlying intra- and inter-cellular signalling pathways, we use a stochastic individual-based model to simulate pattern formation when stem cells and their progeny are cultured as a monolayer. We assume that the fate of an individual cell is regulated by the signals it receives from neighbouring cells via either diffusive or juxtacrine signalling. We analyse simulated patterns using two different spatial statistical measures that are suited to planar multicellular systems: pair correlation functions (PCFs) and quadrat histograms (QHs).ResultsWith a diffusive signalling mechanism, pattern size (revealed by PCFs) is determined by both morphogen decay rate and a sensitivity parameter that determines the degree to which morphogen biases differentiation; high sensitivity and slow decay give rise to large-scale patterns. In contrast, with juxtacrine signalling, high sensitivity produces well-defined patterns over shorter lengthscales. QHs are simpler to compute than PCFs and allow us to distinguish between random differentiation at low sensitivities and patterned states generated at higher sensitivities.ConclusionsPCFs and QHs together provide an effective means of characterising emergent patterns of differentiation in planar multicellular aggregates.

Phase-field simulation for crystallization of a highly supercooled forsterite-chondrule melt droplet

Chondrules are submillimeter-sized and spherical-shaped crystalline grains consisting mainly of silicate material observed in chondritic meteorites. We numerically simulated pattern formation of a forsterite (Mg2SiO4)-chondrule in the melt droplet using a phase-field method. Because of the large surface-to-volume ratio, the surface cooling term was introduced in the framework of this method. We reproduced an unique crystal growth pattern inside the droplet composed of two distinguishable parts; the rim that covers whole droplet surface, and dendrite inside the droplet. It was found that the rim was formed when there is a large temperature difference of ∼100 K between the center and surface of the droplet due to the large cooling flux at the surface. In order to obtain the temperature difference, we derived temperature distribution of the droplet analytically, and concluded that the rim was formed only when the droplet cools rapidly at a rate of Rcool∼103 K s−1. However, when the surface cooling was so large as the temperature at the droplet center still remains above the melting point, no dendrite was obtained, though the rim was formed. The double structure captures the distinctive features of barred-olivine textures observed in natural chondrules.

Pattern Formation in a Vibrated Granular Layer on an Inclined Base

We carry out the simulations of pattern formation in a two-dimensional vibrated granular layer on an inclined base by molecular dynamics. It is found that the maximum amplitude of the pattern is greater at the lower part than at the higher part of the base, and is proportional to the thickness of the layer. Meanwhile, the wavelength varies non-monotonically as the inclined angle of the base is increased.

CLUSTER SPLITTING TRANSITION IN A MARKOV CHAIN MODEL FOR LABOR DIVISION

A Markov chain model is constructed to simulate pattern formation arising from an evolutionary population of interactive homogeneous agents. In the structure of optimal evolution and stochastic properties, the model exhibits emergent properties with rich dynamical diagrams. We study the cluster splitting transition with the coarsening period-adding phenomena shown by the model, which gives an example of pattern formation in the evolution of complex system and reveals dynamical behavior of the Markov chain process.

Stabilization and destabilization effects of the electric field on stochastic precipitate pattern formation

Stabilization and destabilization effects of an applied electric field on the Liesegang pattern formation in low concentration gradient were studied with numerical model simulations. In the absence of an electric field pattern formation exhibits increasingly stochastic behaviour as the initial concentration difference between the outer and the inner electrolytes ( Δ ) approaches to zero. Our numerical simulation results have shown that, if the electric field promotes the transport of the reaction front of the outer electrolyte into the diffusion column, then the electric field stabilizes the stochastic pattern structure. This means that Liesegang patterns in the simulations will be decreasingly stochastic, in terms of reproducibility of band locations and of the band formation. Destabilization behaviour was observed in the reverse case, if the electric field retards the diffusion of the outer electrolyte. Simulation of pattern formation is presented using concentration perturbations in a deterministic model. The precipitation process is based on ion-product supersaturation theory (Ostwald's model).