Simulation of Crack Pattern Formation Due to Shrinkage in a Drying Material

Crack patterns observed in nature have attracted the interest of researchers in various fields, and the mechanism of the pattern formation has been investigated. However, the phenomenon is very complicated, and many factors affect the process. Therefore, we are motivated to construct a general simulation code with a simple algorithm. In this study, crack pattern formation due to shrinkage caused by the drying of a wet material was simulated. The process was simplified as follows: tensile force is generated in the model, and a crack is generated when the tension exceeds a critical value. The tensile forces in the x and y directions are independently evaluated. A crack propagates perpendicular to the tension until it reaches another crack or a boundary. Based on this modeling, simulations with a two-dimensional square domain were performed. Consequently, a cross-divided pattern was generated. Assuming zigzag crack propagation, more realistic patterns were obtained. The effects of the boundary and domain size were also considered, and various characteristic patterns were obtained. Furthermore, the orientation dependency was simulated, and 45&#730 declined patterns and rectangularly divided patterns were generated. The model presented in this study is very simplified and is expected to be applicable to various objects.

A perspective on regression and Bayesian approaches for system identification of pattern formation dynamics

We present two approaches to system identification, i.e. the identification of partial differentialequations (PDEs) from measurement data. The first is a regression-based variational systemidentification procedure that is advantageous in not requiring repeated forward model solves andhas good scalability to large number of differential operators. However it has strict data typerequirements needing the ability to directly represent the operators through the available data.The second is a Bayesian inference framework highly valuable for providing uncertaintyquantification, and flexible for accommodating sparse and noisy data that may also be indirectquantities of interest. However, it also requires repeated forward solutions of the PDE modelswhich is expensive and hinders scalability. We provide illustrations of results on a model problemfor pattern formation dynamics, and discuss merits of the presented methods.

Three-dimensional interfacial wave theory of dendritic growth: (Ⅱ). non-axi-symmetric global wave modes and selection criterion of pattern formation

This paper is ttie continuation of part （Ⅰ）, which completes the derivations of the 3D global wave modes solutions, yields the stability criterion and, on the basis of the results obtained, demonstrates the selection criterion of pattern formation.

Spatial pattern formation of a ratio-dependent predator-prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.

Laser-induced pattern formation from homogeneous polyisoprene solutions

This paper reports that the pattern formation in homogeneous solutions of polyisoprene in toluene saturated with C60 induced by a continuous-wave visible laser is observed experimentally. The transmitted beam patterns change with the increase of the laser irradiation time. In the initial phase, the patterns with concentric ring-shaped structure are formed. In the end, the patterns become speckle-shaped. The incubation time of the transmitted beam widening is inversely proportional to the laser power density and solution concentration. The pattern formation results from the optical-field-induced refractive index changes in the solutions, but the mechanism of optical-field-induced refractive index changes in the polymer solutions needs to be further studied.

Pattern formation in superdiffusion Oregonator model

Pattern formations in an Oregonator model with superdiffusion are studied in two-dimensional(2D) numerical simulations. Stability analyses are performed by applying Fourier and Laplace transforms to the space fractional reaction–diffusion systems. Antispiral, stable turing patterns, and travelling patterns are observed by changing the diffusion index of the activator. Analyses of Floquet multipliers show that the limit cycle solution loses stability at the wave number of the primitive vector of the travelling hexagonal pattern. We also observed a transition between antispiral and spiral by changing the diffusion index of the inhibitor.

Pattern Formation in a Dusty Plasma System

A rich variety of dust patterns have been observed in a capacitively coupled rf discharge dusty plasma system. Dust particles are synthesized through chemical reaction of the filled gas mixture during discharge. Different patterns are formed in different stages of particle growth. In the early stage of particle growth, dust cloud can be formed by a large number of small particles, and its behavior appears to be fluid-like. Such interesting nonlinear phenomena as dust void and complex dust cloud patterns are observed in this stage. As dust particles grow, the particle size and structure can be controlled to follow two different routes. In one of the routes, the particles grow up in a ball-like shape and can be formed into regular lattice and cluster patterns. In the other, the particles grow up in a fractal shape.

PatU3 plays a central role in coordinating cell division and differentiation in pattern formation of filamentous cyanobacterium Nostoc sp.PCC 7120

Spatial periodic signal for cell differentiation in some multicellular organisms is generated according to Turing's principle for pattern formation.How a dividing cell responds to the signal of differentiation is addressed with the filamentous cyanobacterium Nostoc sp.PCC 7120,which forms the patterned distribution of heterocysts.We show that differentiation of a dividing cell was delayed until its division was completed and only one daughter cell became heterocyst.A mutant of patU3,which encodes an inhibitor of heterocyst formation,showed no such delay and formed heterocyst pairs from the daughter cells of cell division or dumbbell-shaped heterocysts from the cells undergoing cytokinesis.The patA mutant,which forms heterocysts only at the filament ends,restored intercalary heterocysts by a single nucleotide mutation of patU3,and double mutants of patU3/patA and patU3/hetF had the phenotypes of the patU3 mutant.We provide evidence that HetF,which can degrade PatU3,is recruited to cell divisome through its C-terminal domain.A HetF mutant with its N-terminal peptidase domain but lacking the C-terminal domain could not prevent the formation of heterocyst pairs,suggesting that the divisome recruitment of HetF is needed to sequester HetF for the delay of differentiation in dividing cells.Our study demonstrates that PatU3 plays a key role in celldivision coupled control of differentiation.

Global Solvability,Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

In this paper,we deal with the following chemotaxis-haptotaxis system modeling cancer invasion with nonlinear diffusion,ut=Δum−χ∇·(u∇v)−ξ∇·(u∇ω)+μu(1−u−ω),inΩ×R^(+),vt−Δv+v=u,inΩ×R+,ωt=−vω,inΩ×R+,whereΩ⊂R^(N)is a bounded domain.We first supplement the results of global existence and uniform boundedness of solutions for the case m=2N N+2.Then for any m>0 and any spatial dimension,we consider the stability of equilibrium,and find that the chemotaxis has a destabilizing effect,that is for the ODEs,or the diffusion-ODE system without chemotaxis,the solutions tend to a linearly stable uniform steady state(1,1,0).When the chemotactic coefficientχis large,the equilibrium(1,1,0)become unstable.Then we study the existence of nontrivial stationary solutions via bifurcation techniques withχbeing the bifurcation parameter,and obtain nonhomogeneous patterns.At last,we also investigate the stability of these bifurcation solutions.

POSITIVITY-PRESERVING LOCAL DISCONTINUOUS GALERKIN METHOD FOR PATTERN FORMATION DYNAMICAL MODEL IN POLYMERIZING ACTIN FLOCKS

In this paper,we apply local discontinuous Galerkin(LDG)methods for pattern formation dynamical model in polymerizing actin focks.There are two main dificulties in designing effective numerical solvers.First of all,the density function is non-negative,and zero is an unstable equilibrium solution.Therefore,negative density values may yield blow-up solutions.To obtain positive numerical approximations,we apply the positivitypreserving(PP)techniques.Secondly,the model may contain stif source.The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method.However,for problems with stiff source,such time discretizations may require strictly limited time step sizes,leading to large computational cost.Moreover,the stiff source any trigger spurious filament polarization,leading to wrong numerical approximations on coarse meshes.In this paper,we combine the PP LDG methods with the semi-implicit Runge-Kutta methods.Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.

Noise transmission during the dynamic pattern formation in fly embryos

Background： Developmental patterning is highly reproducible and accurate at the single-cell level during fly embryogenesis despite the gene expression noise and external perturbations such as the variation of the embryo length, temperature and genes. To reveal the underlying mechanism, it is very important to characterize the noise transmission during the dynamic pattern formation. Two hypotheses have been proposed. The ＂channel＂ scenario requires a highly reproducible input and an accurate interpretation by downstream genes. In contrast, the ＂filter＂ scenario proposes a noisy input and a noise filter via the cross-regulation of the downstream network. It has been under great debates which scenario the fly embryogenesis follows. Results： The first 3-h developmental patterning of fly embryos is orchestrated by a hierarchical segmentation gene network, which rewires upon the maternal to zygotic transition. Starting from the highly reproducible maternal gradients, the positional information is refined to the single-cell precision through the highly dynamical evolved zygotic gene expression profiles. Thus the fly embryo development might strictly fit into neither the originally proposed ＂filter＂ nor ＂channel＂ scenario. The controversy that which scenario the fly embryogenesis follows could be further clarified by combining quantitative measurements and modeling. Conclusions： Fly embryos have become one of the perfect model systems for quantitative systems biology studies. The underlying mechanism discovered from fly embryogenesis will deepen our understanding of the noise control of the gene network, facilitate searching for more efficient and safer methods for cell programming and reprogramming, and have the great potential for tissue engineering and regenerative medicine.

A local discontinuous Galerkin method for the pattern formation dynamical model in polymerizing action flocks

In this paper,we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks.Optimal error estimates for the density and filament polarization in different norms are established.We use a semi-implicit spectral deferred correction time method for time discretization,which allows a relative large time step and avoids computation of a Jacobian matrix.Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.

Enslaved Phase-Separation Fronts and Liesegang Pattern Formation

We show that an enslaved phase-separation front moving with diffusive speeds U=C/√T can leave alternating domains of increasing size in their wake.We find the size and spacing of these domains is identical to Liesegang patterns.For equal composition of the components we are able to predict the exact form of the pattern analytically.To our knowledge this is the first fully analytical derivation of the Liesegang laws.We also show that there is a critical value for C below which only two domains are formed.Our analytical predictions are verified by numerical simulations using a lattice Boltzmann method.

Complex Pattern Formations by Spatial Varying Parameters

Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization.We numerically show the convergence of our algorithm.Pattern transitions in irregular domains are shown.We also provide various parameter settings on some irregular domains for different patterns appeared in nature.To further simulate patterns in reality,we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.

The roles of R2R3-MYBs in regulating complex pigmentation patterns in flowers

Pigmentation patterns are ubiquitous in nature.Visually striking pigmentation patterns are not only aesthetically appealing,but also crucial to pollinator interaction and plant fitness.The formation of complex floral pigmentation patterns mainly relies on the spatiotemporal expression of R2R3-MYB transcription factors and is often associated with certain floral development programs,such as floral organ identity,symmetry,which likely provide key information to initiate the patterning.For a complex pigmentation pattern to form,at least a pair of activator and inhibitor is required,despite their interaction might vary depending on the system being investigated.The regulation of pigmentation pattern involves multiple molecular mechanisms,such as transcriptional regulation,small RNA,transposon-mediated gene silencing,and methylation of gene body.Identifying these regulators can be facilitated by using single-cell and spatial transcriptomics as well as innovative plant transformation technologies.Moreover,plant organ development and pigmentation patterns are often interdependent,but current methods of describing patterns are static.Therefore,more precise and quantitative measurements are needed to elucidate the developmental mechanisms underlying complex pigmentation patterns in flowers.

Turing/Turing-like patterns:Products of random aggregation of spatial components

Turing patterns are typical spatiotemporal ordered structures in various systems driven far from thermodynamic equilibrium.Turing’s reaction-diffusion theory,containing a long-range inhibiting agent and a local catalytic agent,has provided an explanation for the formation of some patterns in nature.Numerical,experimental and theoretical studies about Turing/Turing-like patterns have been generally focused on systems driven far from thermodynamic equilibrium.The local dynamics of these systems are commonly very complex,which brings great difficulties to understanding of formation of patterns.Here,we investigate a type of Turing-like patterns in a near-equilibrium thermodynamic system experimentally and theoretically,and put forward a new formation mechanism and a quantitative method for Turing/Turing-like patterns.Specifically,we observe a type of Turing-like patterns in starch solutions,and study the effect of concentration on the structure of patterns.The experimental results show that,with the increase of concentration,patterns change from spots to inverse spots,and labyrinthine stripe patterns appear in the region of intermediate concentration.We analyze and model the formation mechanism of these patterns observed in experiments,and the simulation results agree with the experimental results.Our conclusion indicates that the random aggregation of spatial components leads to formation of these patterns,and the proportion of spatial components determines the structures.Our findings shed light on the formation mechanism for Turing/Turing-like patterns.

The Application of Pattern Recognition to the Formability of Quasicrystal Formation in Al-alloys

Four parameters of chemical bond havebeen used to span a feature space to classifyquasicrystal-forming Al-alloys from thatalloys without quasicrystal formationwith good result. Since the first quasicrystal-formingsystem, Al-Mn system, discovered by She-chtman in 1984[1], a series of quasicrystal-forming binary alloy systems have beenfound. Most of these systems are Al-contain-ing systems. Bancel has indicated thatthere are three factors affecting theformability of quasicrystals [2]: (1) ele-ctrochemical factor (this factor can be

Various concentric-ring patterns formed in a water-anode glow discharge operated at atmospheric pressure

Pattern formation is a very interesting phenomenon formed above a water anode in atmospheric pressure glow discharge.Up to now,concentric-ring patterns only less than four rings have been observed in experiments.In this work,atmospheric pressure glow discharge above a water anode is conducted to produce diversified concentric-ring patterns.Results indicate that as time elapses,the number of concentric rings increases continuously and up to five rings have been found in the concentric-ring patterns.Moreover,the ring number increases continuously with increasing discharge current.The electrical conductivity of the anode plays an important role in the transition of the concentric patterns due to its positive relation with ionic strength.Hence,the electrical conductivity of the water anode is investigated as a function of time and discharge current.From optical emission spectrum,gas temperature and intensity ratio related with density and temperature of electron have been calculated.The various concentric-ring patterns mentioned above have been simulated at last with an autocatalytic reaction model.

Controlling the transition between Turing and antispiral patterns by using time-delayed-feedback

The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model. We treat the time delay as a perturbation and analyse the effect of the time delay on the Turing and Hopf instabilities near the Turing Hopf codimension-two phase space. Numerical simulations show that the transition between the Turing patterns （hexagon, stripe, and honeycomb）, the dual-mode antispiral, and the antispiral by applying appropriate feedback parameters. The dual-mode antispiral pattern originates from the competition between the Turing and Hopf instabilities. Our results have shown the flexibility of the time delay on controlling the pattern formations near the Turing-Hopf codimension-two phase space.

Numerical Simulation of a Domain-Tessellation Pattern on a Spherical Surface Using a Phase Field Model

A numerical simulation scheme is proposed to analyze domain tessellation and pattern formation on a spherical surface using the phase-field method. A multi-phase-field model is adopted to represent domain growth, and the finite-difference method (FDM) is used for numerical integration. The lattice points for the FDM are distributed regularly on a spherical surface so that a mostly regular triangular domain division is realized. First, a conventional diffusion process is simulated using this lattice to confirm its validity. The multi-phase-field equation is then applied, and pattern formation processes under various initial conditions are simulated. Unlike pattern formation on a flat plane, where the regular hexagonal domains are always stable, certain different patterns are generated. Specifically, characteristic stable patterns are obtained when the number of domains, n, is 6, 8, or 12;for instance, a regular pentagonal domain division pattern is generated for n = 12, which corresponds to a regular dodecahedron.