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

on the treatment of arbitrary boundary conditions using a fast direct mathcal h matrix solver in mom this communication adapts the formalism of hierarchical mathcal h matrices to the direct solution process lu decomposition of the method of moments mom in the frequency domain a novel clustering approach based on physical constraints is presented and extended to treat dielectric bodies as well as arbitrary boundary conditions comparisons with other numerical methods are provided the analyses show a considerable speedup of the computation times and a significant reduction of required memory compared with the traditional mom solution process especially for highly resonant structures where iterative solvers like the multilevel fast multipole algorithm show poor convergence

existence of solutions to some singular initial value problems on etudie l existence des solutions et leurs domaines de definition pour des problemes aux valeurs initiales pour des equations differentielles d ordre 1 et 2 qui sont singulieres soit dans les variables dependantes soit dans les variables independantes

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

well posedness for solid liquid phase transitions with a fourth order nonlinearity abstract a phase field system which describes the evolution of both the absolute temperature and the phase variable f during first order transitions in thermal insulators is considered a thermodynamic approach is developed by regarding the order parameter as a phase field and its evolution equation as a balance law by virtue of the special form of the internal energy a third order nonlinearity g 2 f appears in the energy balance in place of the customarily constant latent heat as a consequence the bounds 0 f 1 hold whenever is positive valued in addition a nonlinear fourier law with conductivity proportional to temperature is assumed well posedness for the resulting initial and boundary value problem are then established in a suitable setting

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

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

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 to equilibrium for a parabolic hyperbolic phase field system with dynamical boundary condition abstract this paper is concerned with the well posedness and the asymptotic behavior of solutions to the following parabolic hyperbolic phase field system 0 1 t 0 t t t 0 in 0 subject to the neumann boundary condition for 0 2 0 on 0 the dynamical boundary condition for 0 3 t 0 on 0 and the initial conditions 0 4 0 0 0 0 t 0 1 in where is a bounded domain in r 3 with smooth boundary is the outward normal direction to the boundary and is a real analytic function in this paper we first establish the existence and uniqueness of a global strong solution to 0 1 0 4 then we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate

asymptotic behaviour for a phase field system with hysteresis abstract the method of hysteresis operators in modelling phase transitions is applied here to the problem of asymptotic stabilization of solutions to a phase field system with hysteresis while it is known that a unique global strong solution exists for every initial data and that the evolution process described by this system is thermo dynamically consistent in the sense that the absolute temperature remains positive for all times and the clausius duhem inequality holds almost everywhere we study here the asymptotic behaviour of the solution as t

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

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

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

universal attractor for some singular phase transition systems a singular parabolic system describing the thermal diffusion in a substance possibly subject to a phase transition is introduced the physical process is described by the variables absolute temperature and order parameter the latter may have or not conserved total mass with respect to time in both cases after recalling and sometimes improving some known well posedness results the long time behavior of the system is studied it is shown that the process is dissipative and the compact universal attractor is constructed it turns out to attract the trajectories of the system in a rather strong metric which is strictly linked to the constraints imposed to both variables the techniques used in the proofs seem likely to be applied to other types of evolution systems containing maximal monotone nonlinearities

direct solve of electrically large integral equations for problem sizes to 1 m unknowns when unknowns are spatially grouped for electrically large bodies the blocked method of moments mom z matrix and its lu factors are comprised of blocks which can be well approximated by matrices of low rank expressed as the outer product of a column matrix times a row matrix this approximation is called matrix compression and can save significant memory and operations count and for monostatic scattering where there are many right hand sides rhs for electrically large bodies the blocked rhs v and current solution j are also well approximated by this low rank outer product form the blocked system equation zj luj v is lu factored and solved using this compressed block form where each outer product approximant is computed using the adaptive cross approximation aca this approach has been applied to a frequency domain efie rwg galerkin integral equation for 3 d pec surfaces preliminary results for this compressed block lu factor and solve technique are presented and compared to measured data bor code predictions and the same code without matrix compression complexity of this approach is studied numerically for an open pipe geometry with specific error tolerances using a pc workstation with unknowns n ranging from 2592 to 1 025 109 and the number of rhs varying from 310 to 6162 matrix fill time scales as n1 34 lu matrix memory storage scales as n1 5 lu factor time scales as n2 0 and the time per rhs solve scales as n1 67 complete solution wall time scales as n1 8

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

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

fast direct solution of method of moments linear system a novel algorithm the compressed block decomposition cbd is presented for highly accelerated direct noniterative method of moments mom solution of electromagnetic scattering and radiation problems the algorithm is based on a block wise subdivision of the mom impedance matrix impedance matrix subblocks corresponding to distant subregions of the problem geometry are not calculated directly but approximated in a compressed form subsequently the matrix is decomposed preserving the compression examples are presented of typical problems in the range of 5000 to 70000 unknowns the total execution time for the largest problem is about 1 h and 20 min for a single excitation vector the main strength of the method is for problems with multiple excitation vectors monostatic rcs computations due to the negligible extra cost for each new excitation for radiation and scattering problems in free space the numerical complexity of the algorithm is shown to be n 2 and the storage requirements scale with n 3 2

existence results for differential delay equations i abstract existence principles for systems of differential difference equations of delay type are established also applications are given to first order delay equations throughout our topological arguments are based on a nonlinear alternative

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

positive solutions to singular and non singular second order boundary value problems abstract the existence of positive solutions to second order singular boundary value problems is established our analysis is based on the topological transversality theorem of andrzej granas which relies on the notions of an essential map and a priori estimates on solutions

hierarchical matrix techniques based on matrix decomposition algorithm for the fast analysis of planar layered structures the matrix decomposition algorithm mda provides an efficient matrix vector product for the iterative solution of the integral equation ie by a blockwise compression of the impedance matrix the mda with a singular value decomposition svd recompression scheme i e so called mda svd method shows strong ability for the analysis of planar layered structures however iterative solution faces the problem of convergence rate an efficient hierarchical h lu decomposition algorithm based on the h matrix techniques is proposed to handle this problem exploiting the data sparse representation of the mda svd compressed impedance matrix h lu decomposition can be efficiently implemented by h matrix arithmetic h matrix techniques provide a flexible way to control the accuracy of the approximate h lu factors h lu decomposition with low accuracy can be used as an efficient preconditioner for the iterative solver due to its low computational cost while h lu decomposition with high accuracy can be used as a direct solver for dealing with multiple right hand side rhs vector problems particularly numerical examples demonstrate that the proposed method is very robust for the analysis of various planar layered structures

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

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

solvability of some nonlinear boundary value problems on etablit l existence des solutions positives de problemes a deux points limites singuliers pour des equations differentielles de la forme y f t y y o o t 1

a single level low rank ie qr algorithm for pec scattering problems using efie formulation this paper presents a single level matrix compression algorithm termed ie qr based on a low rank approximation to speed up the electric field integral equation efie formulation it is shown with the number of groups chosen to be proportional to n sup 1 2 where n is the number of unknowns the memory and cpu time for the resulting algorithm are both o n sup 1 5 the unique features of the algorithm are a the ie qr algorithm is based on the near rank deficiency property for well separated groups this near rank deficiency assumption holds true for many integral equation methods such as laplacian radiation and scattering problems in electromagnetics em the same algorithm can be adapted to other applications outside em with few or no modifications and b the rank estimation is achieved by a dual rank process which ranks the transmitting and receiving groups respectively thus the ie qr algorithm can achieve matrix compression without assembling the entire system matrix also a geometric neighboring preconditioner is presented in this paper this geometric neighboring preconditioner when used in conjunction with gmres is proven to be both efficient and effective for solving the compressed matrix equations

convergence to equilibrium for a parabolic hyperbolic phase field model with cattaneo heat flux law abstract in this paper we consider the well posedness and the asymptotic behavior of solutions to the following parabolic hyperbolic phase field system 0 1 t 3 0 t t div q 0 q t q 0 in 0 subject to the homogeneous neumann boundary condition for 0 2 n 0 on 0 and no heat flux boundary condition for q 0 3 q n 0 on 0 and the initial conditions 0 4 0 0 0 0 q 0 q 0 in where r 3 is a bounded domain with a smooth boundary and n is the outward normal direction to the boundary in this paper we first establish the existence and uniqueness of a global strong solution to 0 1 0 2 0 3 0 4 then we prove its convergence to an equilibrium as time goes to infinity

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

phase field systems with nonlinear coupling and dynamic boundary conditions we consider phase field systems of caginalp type on a three dimensional bounded domain the order parameter fulfills a dynamic boundary condition while the relative temperature is subject to a homogeneous boundary condition of dirichlet neumann or robin type moreover the two equations are nonlinearly coupled through a quadratic growth function here we extend several results which have been proven by some of the authors for the linear coupling more precisely we demonstrate the existence and uniqueness of global solutions then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors we also discuss the convergence of given solutions to a single equilibrium

continuability of solutions of perturbed differential equations on considere les systemes d equations differentielles suivants e x f t x p x f t x g t x per x f t x h t ou x f g h sont des n vecteurs f t x g t x sont continues sur 0 r n et h t est continue sur 0 on demontre des theoremes sur la perturbation de la continuite des solutions de e a p et per

a single level dual rank ie qr algorithm to model large microstrip antenna arrays a single level dual rank ie qr algorithm is introduced so that the resulting dense method of moments mom matrix is efficiently compressed for a system of n equations an amount of work of the order o n sup 2 has traditionally been required for both matrix assembly and matrix vector multiplication in an iterative matrix solver the algorithm of the present paper reduces the memory requirement and cpu time for both matrix assembly and matrix vector multiplication to o n sup 3 2 making it practical to solve for large antenna arrays with full wave approach in conjunction with a geometric neighboring preconditioner for matrix solution using gmres the current approach solves problems involving large antenna arrays using only a fraction of what are needed by conventional mom both in term of memory and total cpu time

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

long time behaviour for a model of phase field type in this paper we study a model of phase field type for the kinetics of phase transitions which was considered by halperin hohenberg and ma and which includes the phase field equations we study the well posedness of the corresponding initial boundary value problem in an open bounded subset in space dimension lower than or equal to 3 and prove that under suitable conditions the long time behaviour of the solutions to this problem is described by a maximal attractor

multiscale compressed block decomposition for fast direct solution of method of moments linear system the multiscale compressed block decomposition algorithm ms cbd is presented for highly accelerated direct non iterative solution of electromagnetic scattering and radiation problems with the method of moments mom the algorithm is demonstrated to exhibit n2 computational complexity and storage requirements scaling with n1 5 for electrically large objects several numerical examples illustrate the efficiency of the method in particular for problems with multiple excitation vectors the largest problem presented in this paper is the monostatic rcs of the nasa almond at 50 ghz for one thousand incidence angles discretized using 442 089 rwg basis functions being entirely algebraic ms cbd is independent of the greens function of the problem