high accuracy solutions to energy gradient flows from material science models a computational framework is presented for materials science models that come from energy gradient flows the models of interest lead to the evolution of structure involving two or more phases the framework includes higher order derivative models and vector problems solutions are considered in periodic cells and standard fourier spectral discretization in space is used implicit time stepping is used with adaptivity based on local error estimates the implicit system at every time step is solved iteratively with newton s method the resulting linear systems are solved in inner iterations with the conjugate gradient method using a novel preconditioner that is a constant coefficient version of the system taking values for the coefficients at the pure phase states solutions with high spatial and temporal accuracy are obtained the dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model that represents the scaled width of transition layers between phases is investigated numerically and with formal asymptotics in a simple setting the asymptotic results require a conjecture on the rank of a modified square distance matrix results from a fast graphical processing unit implementation for a three dimensional model are shown a comparison to time stepping with operator splitting into convex and concave parts that guarantees energy decrease in the numerical scheme is done

an energy stable and convergent finite difference scheme for the modified phase field crystal equation we present an unconditionally energy stable finite difference scheme for the modified phase field crystal equation a generalized damped wave equation for which the usual phase field crystal equation is a special degenerate case the method is based on a convex splitting of a discrete pseudoenergy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present a local in time error estimate that ensures the pointwise convergence of the scheme

the scalar auxiliary variable sav approach for gradient flows abstract we propose a new approach which we term as scalar auxiliary variable sav approach to construct efficient and accurate time discretization schemes for a large class of gradient flows the sav approach is built upon the recently introduced ieq approach it enjoys all advantages of the ieq approach but overcomes most of its shortcomings in particular the sav approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step the scheme is not restricted to specific forms of the nonlinear part of the free energy so it applies to a large class of gradient flows numerical results are presented to show that the accuracy and effectiveness of the sav approach over the existing methods

efficient adaptive energy stable schemes for the incompressible cahn hilliard navier stokes phase field models in this paper we develop a fully adaptive energy stable scheme for cahn hilliard navier stokes system which is a phase field model for two phase incompressible flows consisting a cahn hilliard type diffusion equation and a navier stokes equation this scheme which is decoupled and unconditionally energy stable based on stabilization involves adaptive mesh adaptive time and a nonlinear multigrid finite difference method numerical experiments are carried out to validate the scheme for problems with matched density and non matched density and also demonstrate that cpu time can be significantly reduced with our adaptive approach

decoupled linear and energy stable finite element method for the cahn hilliard navier stokes darcy phase field model in this paper we consider the numerical approximation for a phase field model of the coupled two phase free flow and two phase porous media flow this model consists of cahn hilliard navier stokes equations in the free flow region and cahn hilliard darcy equations in the porous media region that are coupled by seven interface conditions the coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step a fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system in order to deal with the difficulties arising from the interface conditions the decoupled scheme needs to be constructed appropriately for the interface terms and a modified discrete energy is introduced with an interface component furthermore the scheme is linearized and energy stable hence at each time step one need only solve a linear elliptic system for each of the two decoupled equations stability o

efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model abstract in this paper we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles in which the free energy of the system consists of an elastic bending energy a surface area constraint and a volume constraint one major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi discrete level we develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic pde system based on the invariant energy quadratization approach in particular the resulted semi discretizations lead to linear systems in space with symmetric positive definite operators at each time step thus can be efficiently solved in addition the proposed schemes are rigorously proved to be unconditionally energy stable various numerical experiments in 2d and 3d are presented to demonstrate the stability and accuracy of the proposed schemes

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

fast provably unconditionally energy stable and second order accurate algorithms for the anisotropic cahn hilliard model abstract in this paper we consider numerical approximations for solving the anisotropic cahn hilliard model we combine the scalar auxiliary variable sav approach with the stabilization technique to arrive at a novel stabilized sav approach where three linear stabilization terms which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient are added to enhance the stability while keeping the required accuracy the schemes are very easy to implement and fast in the sense that all nonlinear terms are treated in a semi explicit way and one only needs to solve three decoupled linear equations with constant coefficients at each time step we further prove the unconditional energy stabilities rigorously and present numerous 2d and 3d numerical simulations to demonstrate the stability and accuracy

linear cone invariant control systems and their equivalence abstractin this paper we study invariant control systems that generalise positive systems a characterisation of linear control systems invariant on polyhedral cones corner regions in the state space called cone invariant linear control systems is established both for the inputs taking values in a polyhedral cone in the control space and for the inputs taking values in an affine polyhedral cone the problem of equivalence between control systems invariant on corner regions is introduced for cone invariant linear control systems we study invariance preserving state equivalence and invariance preserving feedback equivalence and present characterisations of both notions of equivalence

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

efficient non iterative and second order accurate numerical algorithms for the anisotropic allen cahn equation with precise nonlocal mass conservation abstract we propose two efficient non iterative and second order accurate algorithms to solving the anisotropic allen cahn equation with the nonlocal mass conservation the first scheme is the stabilized sav approach which combines the sav approach with the stabilization technique in which three linear stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps the scheme not only can conserve the mass precisely but also is very easy to implement and non iterative where one only needs to solve three decoupled fourth order biharmonic equations with constant coefficients at each time step we further prove the scheme is unconditionally energy stable rigorously the second scheme is based on the linear stabilization approach where all nonlinear terms are treated in an explicit way therefore the scheme is quite efficient and stable that allows for large time steps in computations for both schemes we present a number of 2d and 3d numerical simulations to show stability and accuracy

linear sources for mesh generation sources offer a convenient way to prescribe a size distribution in space for each newly created mesh point the mesh generator queries the local size distribution either to create a new point or element depending on the underlying mesh generation method to smooth the mesh or to get a local relevant length scale sources may have different shapes such as points edges triangles or boxes they provide the size distribution given some user defined parameters and the distance of a point location to the source traditionally the source strength is considered as constant in this work extensions to linear sources in space are proposed it is shown that in the case of curvature refined mesh generation substantial savings may occur due to the much better approximation of the curvature variation for a simple modification of traditional sources even though curvature refinement is the main application of this work improvements through linear sources are relevant to other contexts such as user defined sou

about nonnegative realizations we will state sufficient conditions such that a discrete time time invariant finite dimensional system with a nonnegative impulse response has a nonnegative finite dimensional realization the conditions are in terms of the minimal polynomial

wall bounded multiphase flows of n immiscible incompressible fluids we present an effective method for simulating wall bounded multiphase flows consisting of n n2 immiscible incompressible fluids with different densities viscosities and pairwise surface tensions the n phase physical formulation is based on a modified thermodynamically consistent phase field model that is more general than in a previous work and it is developed by considering the reduction consistency if some of the fluid components were absent from the system we propose an n phase contact angle boundary condition that is reduction consistent between n phases and m phases 2mn1 we also present a numerical algorithm for solving the n phase governing equations together with the contact angle boundary conditions developed herein extensive numerical experiments are presented for several flow problems involving multiple fluid components and solid wall boundaries to investigate the wettability effects with multiple types of contact angles in particular we compare simulation results with the de gennes theory for the contact angle effects on the liquid drop spreading on wall surfaces and demonstrate that our method produces physically accurate results

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

decoupled non iterative and unconditionally energy stable large time stepping method for the three phase cahn hilliard phase field model abstract in this paper we consider numerical approximations for a three phase phase field model where three fourth order cahn hilliard equations are nonlinearly coupled together through a lagrange multiplier term and a sixth order polynomial bulk potential by combining the recently developed sav approach with the linear stabilization technique we arrive at a novel stabilized sav scheme at each time step the scheme requires solving only four linear biharmonic equations with constant coefficients making it the first to the best of the author s knowledge totally decoupled second order accurate linear and unconditionally energy stable scheme for the model we further prove the unconditional energy stability rigorously and demonstrate the stability and the accuracy of the scheme numerically through the comparisons with the non stabilized sav scheme for simulating numerous benchmark numerical examples in 2d and 3d

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

an entropy satisfying boundary layer surface mesh generation a method is presented to generate surface boundary layers the novelty of the method consists in a uniform treatment of the concave and convex regions based on offsetting the initial front in particular concave situations are enforced to respect the entropy satisfying solution local solutions of the offset of edges in the planes and triangles in the three dimensional space are also given showing the relationship between normal computation distance extrusion and the generalized voronoi diagram the boundary layer mesh generation is cast as a weak solution of the eikonal equation computed locally layer by layer to mimic the hyperbolic character of the eikonal equation augmented by a possibly nonbijective mapping between layers this provides a theoretical framework to rely on for practical answers such as configurations where the adjacent topologies to a boundary layer may be curved or at least not aligned with the offset direction or if the boundary layer mesh must be generated as a mapping betw

positively invariant sets for constrained continuous time systems with cone properties this note deals with some properties of particular bounded sets w r t linear continuous time systems described by x spl dot t a 0 x t c t where c t spl isin spl omega spl sub r sup n spl omega a compact set and matrix e sup ta 0 has the property of leaving a proper cone k positively invariant that is e sup ta 0 k spl sub k the considered bounded sets spl dscr k a b are described as the intersection of shifted cones necessary and sufficient conditions are given they guarantee that such sets are positively invariant w r t the considered system the trajectories starting from x sub 0 spl isin r sup n spl dscr k a b respectively to x sub 0 spl isin r sup n are studied in terms of attractivity and contractivity of the set spl dscr k a b the results are applied to the study of the constrained state feedback regulator problem

a three dimensional parametric mesher with surface boundary layer capability abstract a novel parametric surface meshing technique is presented its distinctive feature relies on successive approximations of the cad geometry through a hierarchical process where geometric information is gathered incrementally a detailed review of zero and first order surface approximations and their impact on parametric surface meshing algorithms is performed the proposed approach emphasizes the use of three dimensional information in order to be as independent as possible of the parametrization to overcome limitations of meshing purely in the parametric plane the presented technique includes semi structured boundary layer surface mesh generation which is a critical capability for accurate solutions to flows around geometries that have leading edge features numerous examples illustrate the method s robustness and ability to high quality meshes for complex cad geometries

phase field computations for ternary fluid flows we present a new phase field model for three component immiscible liquid flows with surface tension in the phase field approach the classical sharp interface between the two immiscible fluids is replaced by a transition region across which the properties of fluids change continuously the proposed method incorporates a chemical potential which can eliminate the unphysical phase field profile and a continuous surface tension force formulation from which we can calculate the pressure field directly from the governing equations the capabilities of the method are demonstrated with several examples we compute the ternary phase separation via spinodal decomposition equilibrium phase field profiles pressure field distribution and a three interface contact angle resulting from a spreading liquid lens on an interface the numerical results show excellent agreement with analytical solutions

linear second order and unconditionally energy stable schemes for the viscous cahn hilliard equation with hyperbolic relaxation using the invariant energy quadratization method abstract in this paper we consider numerical approximations for the viscous cahn hilliard equation with hyperbolic relaxation this type of equations processes energy dissipative structure the main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms allowing large time marching step while preserving the energy stability i e the energy dissipative structure at the time discrete level we resolve this issue by developing two second order time marching schemes using the recently developed invariant energy quadratization approach where all nonlinear terms are discretized semi explicitly in each time step one only needs to solve a symmetric positive definite spd linear system all the proposed schemes are rigorously proven to be unconditionally energy stable and the second order convergence in time has been verified by time step refinement tests numerically various 2d and 3d numerical simulations are presented to demonstrate the stability accuracy and efficiency of the proposed schemes

remarks on strongly flow invariant sets abstract conditions ensuring the strong flow invariance property for a compact set m with respect to the differential system x t x are considered an answer to a question raised in gard applicable anal 10 1980 285 293 is given in the case of autonomous systems

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

convergence and error analysis for the scalar auxiliary variable sav schemes to gradient flows we carry out convergence and error analysis of the scalar auxiliary variable sav methods for l 2 and h 1 gradient flows with a typical form of free energy we first derive h 2 bounds under certain assumptions suitable for both the gradient flows and the sav schemes which allow us to establish the convergence of the sav schemes under mild conditions we then derive error estimates with further regularity assumptions we also discuss several other gradient flows which cannot be cast in the general framework used in this paper but for which convergence and error analysis can still be established using a similar procedure

constrained regulation of linear continuous time dynamical systems abstract in this paper the linear constrained regulation problem lcrp for continuous time dynamical systems is studied the first part of the paper deals with the existence of linear state feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space while respecting linear constraints on both state and control vectors then the lcrp is formulated as an optimization problem which can be solved by applying linear programming techniques

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

thermodynamically consistent time stepping algorithms for non linear thermomechanical systems we present the basic theory for developing novel monolithic and staggered time stepping algorithms for general non linear coupled thermomechanical problems the proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics for isolated systems they preserve the total energy and the entropy never decreases furthermore if the governing equations of the problem have symmetries the proposed integrators preserve them too the formulation of such methods is based on two ideas expressing the evolution equation in the so called general equations for non equilibrium reversible irreversible coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields the new methods can be considered as an extension of the energy momentum integration algorithms to coupled thermomechanical problems to which they reduce in the purely hamiltonian case in the article the new ideas are applied to a simple coupled problem a double thermoelastic pendulum with symmetry numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability which stems precisely from their ability to preserve the structure of the evolution equations they discretize copyright 2009 john wiley sons ltd