a moving mesh method for the solution of the one dimensional phase field equations a moving mesh method is developed for the numerical solution of one dimensional phase change problems modelled by the phase field equations the computational mesh is obtained by equidistribution of a monitor function tailored for the functional variation of the phase field in the interfacial region existence and uniqueness of the discretised equations using a moving mesh are also established numerical results are given for classical and modified stefan test problems the numerical algorithm is relatively simple and is shown to be far more efficient than fixed grid methods

numerical tests of a phase field model with second order accuracy numerical computations are performed for a recently derived phase field model for the interface between two phases the rigorous results indicate that solutions to this new phase field model should converge more rapidly than traditional ones to solutions of the corresponding sharp interface free boundary formulation for sufficiently small values of the approximation parameter e representing the thickness of the interfacial region in particular the distance between the sharp interface of the limiting model and the zero level set of the phase function in the phase field model is of order e 2 rather than e numerical computations within a three dimensional spherically symmetric setting compare the computed solutions of this new model with the known exact solutions for the limiting free boundary problem and confirm the second order accuracy predictions of the theory for sufficiently small e the sets of parameters include those of succinonitrile used in dendritic experiments

analysis of a variable time step discretization for the penrose fife phase relaxation problem in this paper we study a fully implicit time discrete scheme with variable time steps for a system of field equations of penrose fife type governing the dynamics of phase transitions with a nonconserved order parameter in particular fourier law for the heat flux and zero interfacial energy are assumed uniform positivity existence and uniqueness of the discrete solution is proved several a priori estimates are estabilished then passage to the limit is reached via compactness and monotonicity arguments optimal order estimates for the discretization error are shown

pinning control of complex dynamical networks with general topology abstract recently the researches on pinning control of complex dynamical networks have mainly focused on such networks with very specific coupling schemes e g symmetric coupling uniform coupling and linear coupling however most real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper pinning control of a continuous time complex dynamical network with general coupling topologies is studied some generic stability criteria based on master stability function msf are derived for such a general controlled network which guarantee that the whole network can be pinned to its equilibrium by placing feedback control only on a small fraction of nodes then these results are extended to discrete time case previous results about symmetric uniform or linear coupled networks in this area are included as special cases of the present work numerical simulations of directed networks with weighted coupling pinned by specifically selective pinning scheme are given for illustration and verification

nonlinear equations invariant under the poincare similitude and conformal groups in two dimensional space time all realizations of the lie algebras p 1 1 sim 1 1 and conf 1 1 are classified under the action of the group of local diffeomorphisms of r3 the result is used to obtain all second order scalar differential equations invariant under the corresponding poincare similitude and conformal groups the invariant equations are in general nonlinear and the requirement of linearity turns out to be very restrictive group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation

asymptotic behaviour of a phase field model with three coupled equations without uniqueness abstract we prove the existence of weak solutions for a phase field model with three coupled equations with unknown uniqueness and state several dynamical systems depending on the regularity of the initial data then the existence of families of global attractors level set depending for the corresponding multi valued semiflows is established applying an energy method finally using the regularizing effect of the problem we prove that these attractors are in fact the same

difference schemes for degenerate parabolic equations construction de schemas aux differences pour les equations paraboliques degenerees engendrant des semigroupes demonstration de la convergence des algorithmes completement discrets experiences numeriques

lmi based criteria for synchronization of complex dynamical networks this paper considers the problem of controlling a complex dynamical network by means of pinning we point out that the synchronization criterion given in the form of matrix eigenvalues can be equivalent to a linear matrix inequality criterion we further investigate network synchronization via pinning and prove several linear matrix inequality theorems in particular we theoretically provide two typical pinning strategies based on whether the graph which is made up of unpinned nodes and edges between them is irreducible or not numerical simulations including k regular networks star shaped networks and ba scale free networks are shown for illustration and verification

phase field computations of single needle crystals crystal growth and motion by mean curvature the phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase or order parameter and temperature respectively the parameters in the equations are related directly to the physical observables including the interfacial width epsilon which we can regard as a free parameter in computation the phase field equations can be used to compute a wide range of sharp interface problems including the classical stefan model its modification to incorporate surface tension and or surface kinetic terms the cahn allen motion by mean curvature the hele shaw model etc also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists by adjusting the parameters the computations can be varied continuously from single needle dendritic to faceted crystals the computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field e

error estimates for a finite element discretization of a phase field model for mixtures we derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region the main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system a suitable regularity of the exact solutions and a spectrum estimate for the operator of the allen cahn equation the spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model this is done by a technique introduced in x feng and a prohl math comp 73 2004 pp 541 567 for phase transitions of pure materials with linear constitutive relations

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

an interface tracking algorithm for the porous medium equation we study the convergence of a finite difference scheme for the cauchy problem for the porous medium equation u m av m 1 the scheme exhibits the following two features the first is that it employs a discretization of the known interface condition for the propagation of the support of the solution we thus generate approximate interfaces as well as an approximate solution the second feature is that it contains a vanishing viscosity term this term permits an estimate of the form i 1 a viii r c t we prove that both the approximate solution and the approximate interfaces converge to the correct ones finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters 1 introduction in this paper we derive and analyze a finite difference scheme for computing both the solution and the interfaces for the porous medium equation in one space dimension we demonstrate that the approximate solutions and the approximate interface curves converge to the correct ones and we obtain l00 bounds for the error in terms of the mesh parameter consider the laminar flow of a polytropic fluid of density x t u x t in a porous medium which is assumed to occupy the whole space and suppose that at time t 0 the fluid is contained in the slab 0 x fr 0 the phenomenon can be modeled by

synchronization in stochastic coupled systems theoretical results the stability of synchronized states including equilibrium point periodic orbit or chaotic attractor in stochastic coupled dynamical systems ordinary differential equations is considered a general approach is presented based on the master stability function gershg orin disc theory and the extreme value theory in statistics to yield constraints on the distribution of coupling to ensure the stability of synchronized dynamics three types of different behaviour global stable exponential stable and power stable are found depending on the nature of the distribution of the interactions between units systems with specific coupling schemes are used as examples to illustrate our general method

well posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations we obtain an existence and uniqueness result to fremond s phase transition model which take into account microscopic movements and accelerations moreover the irreversible evolution of the phase variable is considered next we perform an asymptotic analysis on the solution to the above problem as the power of the microscopic acceleration forces goes to zero

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

comparison of asymptotic solutions of a phase field model to a sharp interface model a one dimensional directional solidification problem is considered for the purpose of analyzing the relationship between the solution resulting from a phase field model to that from a sharp interface model the solidification problem is posed within a finite domain rather than an infinite extent as in classical stefan problems an asymptotic analysis based upon a small stefan number is performed on the sharp interface model in the phase field case the small stefan number expansion is coupled with a small interface thickness boundary layer expansion this approach enables us to develop analytical solutions to the phase field model the results show agreement at leading order between the two models for the location of the solidification front and the temperature profiles in the solid and liquid phases however due to the nonzero interface thickness in the phase field model corrections to the sharp interface location and temperature profiles develop these corrections result from the conduction of late

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

on a navier stokes allen cahn model with inertial effects abstract a mathematical model describing the flow of two phase fluids in a bounded container is considered under the assumption that the phase transition is influenced by inertial effects the model couples a variant of the navier stokes system for the velocity u with an allen cahn type equation for the order parameter relaxed in time in order to introduce inertia at the microscopic level the resulting model is characterized by second order material derivatives which constitute the main difficulty in the mathematical analysis in order to obtain a tractable problem a viscous relaxation term is then included in the phase equation the mathematical results consist in existence of weak solutions in 3d and under additional assumptions existence and uniqueness of strong solutions in 2d a partial characterization of the long time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed

convergence of the solution of the stochastic 3d globally modified cahn hilliard navier stokes equations abstract we study in this article the stochastic 3d globally modified cahn hilliard navier stokes model in a 3d dimensional bounded domain we prove the existence and uniqueness of strong solutions furthermore we discuss the relation of the stochastic 3d globally modified cahn hilliard navier stokes equations to the stochastic 3d cahn hilliard navier stokes equations by proving a convergence theorem that as the parameter n tends to infinity a subsequence of solutions of the stochastic 3d globally modified cahn hilliard navier stokes equations converges to a weak martingale solution of the stochastic 3d cahn hilliard navier stokes equations

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

a numerical procedure for the porous media equation introduction in this paper we contmue a study mltlated m 11 of numencal procedures for the porous media problem u u2l 0 x u x u x 2 0 t 0 x e r p it 1s well known 2 6 that p has a umque solution u which for each t is supported m a fimte interval c f t t inside the support u 1s a classical solution of p but across tf u 1s generally dlscontmuous in fact one has

weak solution of a stochastic 3d cahn hilliard navier stokes model driven by jump noise abstract we investigate a stochastic 2d and 3d cahn hilliard navier stokes system with a multiplicative noise of levy type the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard system for the order phase parameter we prove that the system the existence of weak martingale solution for both 2d and 3d cases the proof of the existence is based on a classical galerkin approximation as well as some compactness methods in the 2d case we prove the pathwise uniqueness of the weak solution

the phase field method in the sharp interface limit the phase field pf method for solidification phenomena is an open formulation based on a free energy functional two common choices for the pf potential here referred to briefly as the caginalp and kobayashi models are compared with respect to their numeric results within the classical sharp interface limit both qualitative and quantitative behavior are addressed and an assessment of the computational effort required to approximate a sharp interface problem is made it is shown that the specific form of the free energy potential does have a strong influence on the convergence of the pf results to their sharp interface limit compliance of the pf solutions with the linear kinetic model for the interface temperature is also investigated a simple one dimensional solidification problem in the presence of kinetic undercooling is solved by the pf model and also by a deforming grid method our results support the view that if care is exercised in formulating the phase temperature coupling there is a high degree of confidence in using the pf method for the numerical modeling of general solidification phenomena

instability and controllability of linearly coupled oscillators eigenvalue analysis the stability and controllability of synchronous chaos of linearly coupled oscillators are investigated in detail based on eigenvalue analysis the complicated coupled problem is reduced to two independent problems clarifying the unstable regions of a modified single site system and specifying the eigenvalue distribution of the linear coupling and control matrix both problems can be easily solved local injections pinnings are suggested to synchronize chaos the dependence of the controllability on pinning density and the diffusive and gradient couplings is studied it is shown that high control efficiency can be achieved in the strong gradient coupling case for certain classes of unstable region distributions s1063 651x 98 14410 0

the quasistationary phase field equations with neumann boundary conditions abstract we prove that the quasistationary phase field equations t u u f 2e 1 e w u where w t t 2 1 2 is a double well potential admit a solution when the space dimension n 3 and that the solutions converge for e 0 to solutions of the stefan problem with gibbs thomson law

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

existence of global solutions to the caginalp phase field system with dynamic boundary conditions and singular potentials our aim in this article is to prove the global in time existence of solutions to a caginalp phase field system with dynamic boundary conditions and a singular potential the main difficulty is to prove that the solutions are strictly separated from the singular values of the potential this is achieved by studying an auxiliary elliptic problem

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

the gibbs thompson relation within the gradient theory of phase transitions this paper discusses the asymptotic behavior as 0 of the chemical potentials associated with solutions of variational problems within the van der waals cahn hilliard theory of phase transitions in a fluid with free energy per unit volume given by 2 2 w where is the density the main result is that is asymptotically equal to e d o with e the interfacial energy per unit surface area of the interface between phases the constant sum of principal curvatures of the interface and d the density jump across the interface this result is in agreement with a formula conjectured by m gurtin and corresponds to the gibbs thompson relation for surface tension proved by g caginalp within the context of the phase field model of free boundaries arising from phase transitions

reconstruction of two time independent coefficients in an inverse problem for a phase field system abstract in this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain r n for the dimension n 3 with a single observation on a subdomain and the sobolev norm of certain partial derivatives of the solutions at a fixed positive time 0 t over the whole spatial domain the proof of these results relies on an appropriate carleman estimate for the phase field system

brief paper oscillation analysis of linearly coupled piecewise affine systems application to spatio temporal neuron dynamics this paper discusses oscillation analysis of a large number of linearly coupled piecewise affine pwa systems motivated by various kinds of reaction diffusion systems including cell signaling dynamics and neural dynamics we derive a sufficient condition under which the system shows an oscillatory behavior called y oscillation it is known that the analysis of pwa systems is difficult due to their switching nature an important feature of the result obtained is that under the assumption that every subsystem has a specific property in common the criteria can be rewritten in terms of coupling topology in an easily checkable way so it is applicable to large scale systems the results obtained are applied to theoretical investigation of the cardiac action potential generation propagation represented by spatio temporal fitzhugh nagumo equations

quasilinear systems with memory kernels we consider a quasilinear integrodifferential system in non normal form such a system is a generalization of a phase field model with memory and includes as a particular case the system describing the combustion of a material with memory in this paper we study both the direct and the inverse problems our fundamental tools are the theory of analytic semigroups optimal regularity results and fixed point arguments

exponential attractors for a conserved phase field system with memory we consider a conserved phase field system of caginalp type characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient respectively the model consists of a parabolic integrodifferential equation coupled with a fourth order evolution equation for the phase field this system supplemented with suitable boundary conditions can be interpreted as a dissipative dynamical system in a proper phase space in a previous joint work the last two authors have proved the existence of a global attractor of finite fractal dimension here we show the existence of an exponential attractor by means of the techniques we developed for nonconserved phase field systems with memory

transition from intermittency to periodicity in lag synchronization in coupled rossler oscillators the dynamical and statistical behavior of lag synchronization in two coupled self sustained chaotic rossler oscillators is reexamined the lack of uniqueness in the conventional characterization of lag synchronization based on the similarity function has caused much skepticism about the existence of lag synchronization we provide an evidence that the emergence of lag synchronization is associated with the transition from on off intermittency to a periodic structure in the laminar phase distribution

numerical methods for flows through porous media i the degenerate parabolic equation aut v oulvvu v 1 has been used to model the flow of gas through a porous medium error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved and several new regularity results are given 1 a porous medium equation introduction we shall study the porous medium equation 1 1 au at v i u i vu on 2 x 0 t 1 2 au an 0 on aa x o t 1 3 u x 0 uo x on 2 where v 1 is a parameter and q is a bounded domain in rn n 3 with a smooth boundary the initial function u0 is assumed to be nonnegative and four times continuously differentiable on u notice that the compatibility condition auo an 0 holds on asa our main result is the derivation of error estimates for numerical approximations to the problem 1 1 1 3 which we shall refer to as the porous medium equation or ipme the pme does not in general admit classical solutions existence and uniqueness of weak solutions was proved in one space dimension by oleinik kalashnikov and czou 15 16 and in several space dimensions by lions 12 these proofs concern the pme with different boundary conditions but the arguments carry over to the pme 1 1 1 3 the maximum principle implies that since u0 is nonnegative on 92 u x t is nonnegative for all x t e q x 0 t see 15 16 if u0 is nonzero the neumann boundary condition implies that u will eventually become strictly positive and 1 1 will become nondegenerate for all time t to to sufficiently large we can rewrite 1 1 in the form 1 4 au at ak u on qx 0 t received may 11 1979 revised july 24 1981 1980 mathematics subject classification primary 35k65 65n30 65n15 current address reservoir engineering branch petroleum engineering division bp exploration dos britannic house moore lane london ec2y 9bu united kingdom d1983 american mathematical society 0025 5718 82 0000 1 063 08 25 435 this content downloaded from 157 55 39 55 on tue 23 aug 2016 04 12 54 utc all use subject to http about jstor org terms

controlling synchronous patterns in complex networks although the set of permutation symmetries of a complex network could be very large few of them give rise to stable synchronous patterns here we present a general framework and develop techniques for controlling synchronization patterns in complex network of coupled chaotic oscillators specifically according to the network permutation symmetry we design a small size and weighted network namely the control network and use it to control the large size complex network by means of pinning coupling we argue mathematically that for any of the network symmetries there always exists a critical pinning strength beyond which the unstable synchronous pattern associated to this symmetry can be stabilized the feasibility of the control method is verified by numerical simulations of both artificial and real world networks and demonstrated experimentally in systems of coupled chaotic circuits our studies show the controllability of synchronous patterns in complex networks of coupled chaotic oscillators

the theory of wavelet transform method on chaotic synchronization of coupled map lattices the wavelet transform method originated by wei et al phys rev lett 89 284103 4 2002 was proved juang and li j math phys 47 072704 16 2006 juang et al j math phys 47 122702 11 2006 shieh et al j math phys 47 082701 10 2006 to be an effective tool to reduce the order of coupling strength for coupled chaotic systems to acquire the synchrony regardless the size of oscillators in juang et al ieee trans circuits syst i regul pap 56 840 2009 such method was applied to coupled map lattices cmls it was demonstrated that by adjusting the wavelet constant of the method can greatly increase the applicable range of coupling strengths the parameters range of the individual oscillator and the number of nodes for local synchronization of cmls no analytical proof is given there in this paper the optimal or near optimal wavelet constant can be explicitly identified as a result the above described scenario can be rigorously verified

finite element approximations of singular parabolic problems an efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented the stefan problem the hele shaw problem and the porous medium equation are included backward differences or linearization techniques are used for the time discretization of the problem the performances of these schemes are discussed with several numerical tests

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

stability of synchronized network of chaotic electromechanical devices with nearest and all to all couplings abstract a shift invariant set of n mutually nearest neighbour or all to all coupled moving coil electromechanical devices is analytically and numerically investigated the study of the stability of synchronization process is undertaken using the master stability function msf approach the emanating properties of this method make it possible to have a general study of the network dynamics and to explain de synchronization phenomena appearing in the synchronization stability parametric areas a detailed attention is paid to the effects of the dissipative component of the dispersive dissipative coupling tested here

abstract approach to evolution equations of phase field type and applications a general system of abstract nonlinear parabolic equations deriving from phase field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved then applications are presented to two physical situations first a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting second a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case moreover through convex analysis techniques it is performed a mathematical study of the subdifferential type relations linking the phase fields and the temperatures to the internal energies of the fluids email schimper dragon ian pv cnr it keywords phase field models subdifferential operators abstract evolution equations

realizations of galilei algebras all inequivalent realizations of the galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by shirokov the varieties of the deformed galilei algebras are discussed and families of one parametric deformations are presented in explicit form it is also shown that a number of well known and physically interesting equations and systems are invariant with respect to the considered galilei algebras or their deformations

velocity selection in 3d dendrites phase field computations and microgravity experiments abstract the growth of a single needle of succinonitrile scn is studied in three dimensional 3d space by using a phase field model for realistic physical parameters namely the large differences in the length scales i e the capillarity length 10 8 10 6 cm the radius of the curvature at the tip of the interface 10 3 10 2 cm and the diffusion length 10 3 10 1 cm resolution of the large differences in length scale necessitates a 500 3 grid on the supercomputer the parameters initial and boundary conditions used are identical to those of the microgravity experiments of glicksman et al for scn the numerical results for the tip velocity are i largely consistent with the space shuttle experiments ii compatible with the experimental conclusion that tip velocity does not increase with increased anisotropy iii different for 2d versus 3d by a factor of approximately 1 9 iv essentially identical for fully versus rotationally symmetric 3d

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

global existence and asymptotic behavior for the radially symmetric case of a two phase stefan problem with interfacial energy abstract this paper is concerned with global existence in time and asymptotic behavior for the radially symmetric case of a stefan problem with surface tension effects on the interface according to the static gibbs thomson law these problems arise in phase change theory

chaos synchronization of general complex dynamical networks recently it has been demonstrated that many large scale complex dynamical networks display a collective synchronization motion here we introduce a time varying complex dynamical network model and further investigate its synchronization phenomenon based on this new complex network model two network chaos synchronization theorems are proved we show that the chaos synchronization of a time varying complex network is determined by means of the inner coupled link matrix the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network especially we do not assume that the coupled configuration matrix is symmetric and its off diagonal elements are nonnegative which in a way generalizes the related results existing in the literature

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

a simple moving mesh method for one and two dimensional phase field equations a simple moving mesh method is proposed for solving phase field equations the numerical strategy is based on the approach proposed in li et al j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution the phase field equations are discretized by a finite volume method and the mesh moving part is realized by solving the conventional euler lagrange equations with the standard gradient based monitors numerical results demonstrate the accuracy and effectiveness of the proposed algorithm

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

optimization of synchronization in gradient clustered networks we consider complex clustered networks with a gradient structure where the sizes of the clusters are distributed unevenly such networks describe actual networks in biophysical systems and in technological applications more closely than the previous models theoretical analysis predicts that the network synchronizability can be optimized by the strength of the gradient field but only when the gradient field points from large to small clusters a remarkable finding is that if the gradient field is sufficiently strong synchronizability of the network is mainly determined by the properties of the subnetworks in the two largest clusters these results are verified by numerical eigenvalue analysis and by direct simulation of synchronization dynamics on coupled oscillator networks

phase transitions with semi diffuse interfaces in this paper we examine new phase field models with semi diffuse interfaces these models have the property that the 1 1 planar phase transitions take place over a finite interval the models also support multiple interface solutions with interfaces centered at arbitrary points l sub 1 l sub 2 l sub n these solutions correspond to local minima of an entropy functional rather than saddle points and are dynamically stable the classical models have no such exact solutions but they do support solutions with n equally spaced transition points where the order parameter transitions between valves p sub min n and p sub max n satisfying 1 p sub min n 0 p sub max n 1 these solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

analyzing and controlling the network synchronization regions abstract in this paper the commonly concerned issue of synchronization regions of complex dynamical networks is investigated for the case when the synchronous state is an equilibrium point some simple sufficient conditions for a network to have or have no unbounded synchronization regions of the form 1 are established where 1 is a constant in addition a sufficient condition for the existence of a bounded synchronization region of the form 2 3 is derived where 2 and 3 are constants by using the parameter dependent lyapunov function method furthermore some effective controller design methods are presented that can change the synchronization regions thereby managing the synchronizability of the network finally some numerical examples are given to show that a dynamical network may have disconnected synchronization regions particularly it may have the coexistence of unbounded and bounded synchronization regions in the form of 1 2 3

phase field models for hypercooled solidification abstract properties of the solidification front in a hypercooled liquid so called because the temperature of the resulting solid is below the melting temperature are derived using a phase field diffuse interface model certain known properties for hypercooled fronts in specific materials are reflected within our theories such as the presence of thin thermal layers and the trend towards smoother fronts with less pronounced dendrites when the undercooling is increased within the hypercooled regime both an asymptotic analysis to derive the relevant free boundary problems and a rigorous determination of the inner profile of the diffusive interface are given of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators these features necessitate a much richer mathematical analysis than previous theories anisotropic free boundary problems are derived from our models the simplest of which involves determining the evolution of a set a solid particle whose boundary moves with velocity depending on its normal vector considerable attention is given to the identification of surface tension to comparison with previous theories and to questions of stability

hyperbolic phase field dynamics with memory abstract we consider a non conserved phase field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations this model is derived from two basic assumptions the heat flux law is of gurtin pipkin type and the response of the order parameter to the variation of the free energy functional is delayed these hypotheses might be a reasonable attempt to describe for instance the melt of he 4 crystals a suitable initial and boundary value problem is then associated with the system and its well posedness is analyzed in detail

computation of sharp phase boundaries by spreading the planar and spherically symmetric cases abstract the sharp interface that arises from any of the major transition problems classical or modified stefan etc can be smoothed using the phase field approach as a numerical tool the basic idea is that the thickness of the interface can be regarded as a mathematical free parameter which can be stretched beyond its physical value for computational convenience the computations in one dimensional space and n dimensions with radial symmetry indicate that this is an efficient method for dealing with stiff equations and results in a very accurate interface determination without explicit tracking the question of optimizing the interfacial thickness with respect to grid size is also considered empirically the technique also provides a numerical verification of the concept of an unstable critical radius of solidification

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

on new galilei invariant equations in two dimensional spacetime we study realizations of galilei groups acting as transformation lie groups in the space of two independent variables and one dependent variable classification of realizations of the lie algebras and within the class of lie vector fields is carried out utilizing the classification results we have constructed the full sets of second order scalar differential equations in two dimensional spacetime invariant under the lie algebras and

lagrangian non oscillatory and fem schemes for the porous media equation abstract interface tracking schemes derived from a lagrangian formulation of the equation are considered existence and equivalence results valid even for the non smooth interface case are given and non oscillatory schemes needed in this case are discussed mixed euler lagrange fem discretization for mixed problems and numerical experiments are presented

a numerical procedure for solving 2d phase field model problems we present a general 2d phase field model but without anisotropy applied to freezing into a supercooled melt of pure nickel the complete numerical procedure and details of assigning the numerical parameters are provided convergence of the numerical method is demonstrated by conducting grid function convergence tests the physics of solidification problems such as conditions for nucleation and crystal growth rate are discussed theoretically and shown to display at least qualitative agreement numerically in particular comparison of the computed critical radius with the theoretical one and the consistency of the computational dendrite structure for different stefan numbers the relationship between the growth rate and the stefan number etc with the theoretical and experimental evidence indicate that phase field models are able to capture the physics of supercooled solidification

enhanced synchronizability by structural perturbations in this brief report we investigate the collective synchronization of a system of coupled oscillators on a barab asi albert scale free network we propose an approach of structural perturbations aiming at those nodes with maximal betweenness this method can markedly enhance the network synchronizability and is easy to realize the simulation results show that the eigenratio will sharply decrease by one half when only 0 6 of those hub nodes occur under three division processes when the network size n 2000 in addition the present study also provides numerical evidence that the maximal betweenness plays a major role in network synchronization

on a nonlocal phase field system abstract we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal phase field system on a bounded domain the system is a gradient flow for a free energy functional with nonlocal interaction also we study the asymptotic behavior of the solution and show the existence of an absorbing set in some metric space