an unconditionally energy stable scheme based on an implicit auxiliary energy variable for incompressible two phase flows with different densities involving only precomputable coefficient matrices abstract we present an energy stable scheme for numerically approximating the governing equations for incompressible two phase flows with different densities and dynamic viscosities for the two fluids the proposed scheme employs a scalar valued auxiliary energy variable in its formulation and it satisfies a discrete energy stability property more importantly the scheme is computationally efficient within each time step it computes two copies of the flow variables velocity pressure phase field function by solving individually a linear algebraic system involving a constant and time independent coefficient matrix for each of these field variables the coefficient matrices involved in these linear systems only need to be computed once and can be pre computed additionally within each time step the scheme requires the solution of a nonlinear algebraic equation about a scalar valued number using the newton s method the cost for this nonlinear solver is very low accounting for only a few percent of the total computation time per time step because this nonlinear equation is about a scalar number not a field function extensive numerical experiments have been presented for several two phase flow problems involving large density ratios and large viscosity ratios comparisons with theory show that the proposed method produces physically accurate results simulations with large time step sizes demonstrate the stability of computations and verify the robustness of the proposed method an implication of this work is that energy stable schemes for two phase problems can also become computationally efficient and competitive eliminating the need for expensive re computations of coefficient matrices even at large density ratios and viscosity ratios

a level set approach for simulations of flows with multiple moving contact lines with hysteresis the level set method of sussman et al j comput phys 148 1999 81 is extended such that flows with multiple moving contact lines can be simulated accounting for inertia a relation between contact line speed and contact angle slip and contact line hysteresis the convergence properties of the method are investigated with particular attention on the resolution of the contact line stress singularity results are compared with a lubrication theory for spreading droplets

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

heterogeneous multiscale method for the modeling of complex fluids and micro fluidics the framework of the heterogeneous multiscale method hmm is used to develop numerical methods for the study of macroscale dynamics of fluids in situations where either the constitutive relation or the boundary conditions are not explicitly available and have to be inferred from microscopic models such as molecular dynamics continuum hydrodynamics is used as the macroscopic model while molecular dynamics serves as the microscopic model and is used to supply the necessary data e g the stress or the boundary condition for the macroscopic model scale separation is exploited so that the macroscopic variables can be evolved in macroscopic spatial temporal scales using data that are estimated from molecular dynamics simulation on microscale spatial temporal domains this naturally decouples the micro and macrospatial and temporal scales whenever possible applications are presented for models of complex fluids contact line dynamics and a simple model of non trivial fluid solid interactions

an l bound for solutions of the cahn hilliard equation in this paper we consider the equation partial t u varepsilon delta u varepsilon sum limits i j 1 n partial ij f ij u varepsilon x varepsilon 2 delta 2 u varepsilon with fij constant when u e is large and we obtain l bounds independent of 0

an efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity abstract in this paper we develop an efficient numerical method for the two phase moving contact line problem with variable density viscosity and slip length the physical model is based on a phase field approach which consists of a coupled system of the cahn hilliard and navier stokes equations with the generalized navier boundary condition 1 2 5 to overcome the difficulties due to large density and viscosity ratio the navier stokes equations are solved by a splitting method based on a pressure poisson equation 11 while the cahn hilliard equation is solved by a convex splitting method we show that the method is stable under certain conditions the linearized schemes are easy to implement and introduce only mild cfl time constraint numerical tests are carried out to verify the accuracy stability and efficiency of the schemes the method allows us to simulate the interface problems with extremely small interface thickness three dimensional simulations are included to validate the efficiency of the method

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

linear and unconditionally energy stable schemes for the binary fluid surfactant phase field model abstract in this paper we consider the numerical solution of a binary fluid surfactant phase field model in which the free energy contains a nonlinear coupling entropy a ginzburg landau double well potential and a logarithmic flory huggins potential the resulting system consists of two nonlinearly coupled cahn hilliard type equations we develop a first and a second order time stepping schemes for this system using the invariant energy quadratization approach in particular the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi explicitly both schemes are linear and lead to symmetric positive definite systems in space at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable in the discrete sense various 2d and 3d numerical experiments are performed to validate the accuracy and energy stability 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

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

a variable order infinite acoustic wave envelope element abstract a new infinite wave envelope element for analyzing acoustic radiation and scattering problems in an unbounded domain is presented the element is based on a finite to infinite geometry mapping and a wave representation within the shape function it allows for the specification of an arbitrary number of acoustic degrees of freedom in the radial infinite direction yielding a 1 r expansion for the proper modelling of the amplitude decay both two dimensional and axisymmetric problems are presented to show the use and accuracy of this n th order infinite wave envelope element numerical results are compared with analytical and boundary element solutions

numerical simulation of moving contact line problems using a volume of fluid method abstract moving contact lines are implemented in a volume of fluid scheme with piecewise linear interface construction interfacial tension is treated as a continuous body force computed from numerical derivatives of a smoothed volume of fluid function two methods for implementing the contact angle condition are investigated the first extrapolates the volume of fluid function beyond the flow domain on the basis of the condition that its gradient is perpendicular to the interface and that the normal to the interface at the wall is determined by the contact angle the second method treats the problem as a three phase situation and mimics the classical argument of young it is found that the latter approach introduces an artificial localized flow and the extrapolation method is preferable slip is a crucial factor in the spreading of contact lines the numerical method introduces slip at the discrete level effectively introducing a slip length on the order of the mesh size

energy stable predictor corrector schemes for the cahn hilliard equation abstract in this paper we construct a new class of predictor corrector time stepping schemes for the cahn hilliard equation which are linear second order accurate in time unconditionally energy stable and uniquely solvable then we present the stability and error estimates of the semi discrete numerical schemes for solving the cahn hilliard equation with general nonlinear bulk potentials the semi discrete scheme is further discretized using the compact central finite difference method several numerical examples are shown to verify the theoretical results in particular the numerical simulations show that the predictor corrector schemes reach the second order convergence rate at relatively larger time step sizes than the classical linear schemes the numerical strategies and theoretical tools developed in this article could be readily applied to study other phase field models or models that can be cast as gradient flow problems

a time stepping scheme involving constant coefficient matrices for phase field simulations of two phase incompressible flows with large density ratios we present an efficient time stepping scheme for simulations of the coupled navier stokes cahn hilliard equations for the phase field approach the scheme has several attractive characteristics i it is suitable for large density ratios and numerical experiments with density ratios up to 1000 have been presented ii it involves only constant time independent coefficient matrices for all flow variables which can be pre computed during pre processing so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity iii it completely de couples the computations of the velocity pressure and the phase field function strategy for spectral element type spatial discretizations to overcome the difficulty associated with the large spatial order of the cahn hilliard equation is also discussed ample numerical simulations demonstrate that the current algorithm together with the navier stokes cahn hilliard phase field approach is an efficient and effective method for studying two phase flows involving large density ratios moving contact lines and interfacial topology changes

a gradient stable scheme for a phase field model for the moving contact line problem in this paper an efficient numerical scheme is designed for a phase field model for the moving contact line problem which consists of a coupled system of the cahn hilliard and navier stokes equations with the generalized navier boundary condition 1 2 4 the nonlinear version of the scheme is semi implicit in time and is based on a convex splitting of the cahn hilliard free energy including the boundary energy together with a projection method for the navier stokes equations we show under certain conditions the scheme has the total energy decaying property and is unconditionally stable the linearized scheme is easy to implement and introduces only mild cfl time constraint numerical tests are carried out to verify the accuracy and stability of the scheme the behavior of the solution near the contact line is examined it is verified that when the interface intersects with the boundary the consistent splitting scheme 21 22 for the navier stokes equations has the better accuracy for pressure

propagation of gibbsianness for infinite dimensional gradient brownian diffusions we study the strong gibbsian character on mathbb r mathbb z d of the law at time t of an infinite imensional gradient brownian diffusion when the initial distribution is gibbsian

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

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

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

spectral approximation of pattern forming nonlinear evolution equations with double well potentials of quadratic growth this paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as l 2 gradient flow for the modica mortola regularization of the functional v bv t d 1 1 e v 2 td v 1 2 k d k v k 2 here is the interfacial energy per unit length or unit area t d is the flat torus in d and is a nonnegative fourier multiplier that is continuous on d symmetric in the sense that for all d and that decays to zero at infinity such functionals feature in mathematical models of pattern formation in micromagnetics and models of diblock copolymers the resulting evolution equation is discretized by a fourier spectral method with respect to the spatial variables and a modified crank nicolson scheme in time optimal order a priori bounds are derived on the global error in the l 0 t l 2 t d norm

on imposing dynamic contact angle boundary conditions for wall bounded liquid gas flows we present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall bounded flows of two immiscible incompressible fluids with large density ratios first we develop an algorithm for imposing the dynamic contact angle boundary conditions to the cahn hilliard equation our algorithm consists of two components i we ignore the boundary conditions and transform the cahn hilliard equation into two nominally de coupled helmholtz type equations ii we treat the dynamic contact angle boundary conditions in such a manner that the two helmholtz type equations are truly de coupled then we combine this algorithm together with a scheme for variable density navier stokes equations we developed recently to form an efficient method for the coupled system of navier stokes and cahn hilliard equations for contact line problems involving large density ratios the overall method can deal with moving contact lines under dynamic and also static contact angle boundary conditions it is endowed with several attractive features that make the method very efficient in particular computations for all flow variables are completely decoupled the resultant linear algebraic systems after discretization for all flow variables involve only constant and time independent coefficient matrices which can be pre computed during pre processing even though the coupled navier stokes cahn hilliard system involves variable density and variable viscosity ample numerical simulations of wall bounded air water two phase flows have been presented to demonstrate the capability of the method for dealing with contact line problems under dynamic and static contact angle conditions involving large density ratios

acoustic infinite elements for non separable geometries we present new infinite element formulations for solving acoustic scattering and radiation problems in the exterior of long slender bodies the new infinite elements are geometrically constructed from a prolate spheroid inscribed by the scatterer these elements need not begin on a level surface of the prolate spheroidal coordinate system instead they may be attached to any convex surface including that of the scatterer itself this scheme reduces or even completely eliminates finite element modeling of the exterior medium the formulations may easily be extended to the cases of an interior oblate spheroid or ellipsoid we present both conjugated and unconjugated formulations without any weighting factors although it would be simple to include them we describe a fast numerical scheme for computing the element integrals based on chebychev approximation we include numerical results for scattering from spheres and capped cylinders these results demonstrate the accuracy and the dramatic reduction in computational expense of our new formulations compared to other coupled finite element infinite element methods

the conjugated vs the unconjugated infinite element method for the helmholtz equation in exterior domains abstract this work is devoted to a convergence study of infinite element ie discretizations for the helmholtz equation in exterior domains the different behavior of the conjugated and the unconjugated ie formulation is analyzed in context of 1 1 a formulation following a mathematical existence theory by leis 2 2 a formulation not based on an available existence theory following burnett four variational formulations are presented and the differences in implementing them are discussed the effect of using or not using the complex conjugate in the weak formulation is carefully studied the numerical and theoretical results clearly indicate which of the four presented formulations is the most efficient that can still give reliable results

coarsening kinetics from a variable mobility cahn hilliard equation application of a semi implicit fourier spectral method an efficient semi implicit fourier spectral method is implemented to solve the cahn hilliard equation with a variable mobility the method is orders of magnitude more efficient than the conventional forward euler finite difference method thus allowing us to simulate large systems for longer times we studied the coarsening kinetics of interconnected two phase mixtures using a cahn hilliard equation with its mobility depending on local compositions in particular we compared the kinetics of bulk diffusion dominated and interface diffusion dominated coarsening in two phase systems results are compared with existing theories and previous computer simulations

numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method abstract the molecular beam epitaxial model is derived from the variation of a free energy that consists of either a fourth order ginzburg landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function one challenge in solving the mbe model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time discrete level in this paper we resolve this issue by developing a first and second order time stepping scheme based on the invariant energy quadratization ieq method the novelty is that all nonlinear terms are treated semi explicitly and the resulted semi discrete equations form a linear system at each time step moreover the linear operator is symmetric positive definite and thus can be solved efficiently we then prove that all proposed schemes are unconditionally energy stable the semi discrete schemes are further discretized in space using finite difference methods and implemented on gpus for high performance computing various 2d and 3d numerical examples are presented to demonstrate stability and accuracy of the proposed schemes

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

a least squares finite element method for the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line in this article we discuss the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall the method we employ combines a finite element space approximation with a time discretization by operator splitting to solve the cahn hilliard part of the problem we use a least squares conjugate gradient method we also show that the scheme has the total energy decaying in time property under certain conditions our numerical experiments indicate that the method discussed here is accurate stable and efficient

efficient and accurate numerical schemes for a hydro dynamically coupled phase field diblock copolymer model in this paper we consider numerical approximations of a hydro dynamically coupled phase field diblock copolymer model in which the free energy contains a kinetic potential a gradient entropy a ginzburglandau double well potential and a long range nonlocal type potential we develop a set of second order time marching schemes for this system using the invariant energy quadratization approach for the double well potential the projection method for the navierstokes equation and a subtle implicit explicit treatment for the stress and convective term the resulting schemes are linear and lead to symmetric positive definite systems at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

time integration for diffuse interface models for two phase flow we propose a variant of the q scheme for diffuse interface models for two phase flow together with three new linearization techniques for the surface tension these involve either additional stabilizing force terms or a fully implicit coupling of the navier stokes and cahn hilliard equation in the common case that the equations for interface and flow are coupled explicitly we find a time step restriction which is very different to other two phase flow models and in particular is independent of the grid size we also show that the proposed stabilization techniques can lift this time step restriction even more pronounced is the performance of the proposed fully implicit scheme which is stable for arbitrarily large time steps we demonstrate in a taylor flow application that this superior coupling between flow and interface equation can decrease the computation time by several orders of magnitude

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

mapped wave envelope elements for acoustical radiation and scattering wave envelope elements provide an alternative to boundary element methods for the solution of exterior wave problems applications to date have been to axisymmetric geometries and have involved the use of large but finite wave envelope meshes compatibly matched to denser near field meshes of conventional elements in the present paper the wave envelope idea is extended by the inclusion of an element mapping taken directly from infinite element methodology and by modifications to the residual formulation taken in conjunction these two measures permit the outer boundary of the wave envelope region to be moved to infinity while preserving the boundedness of the acoustic mass and stiffness integrals the present formulation is developed initially with reference to a simple one dimensional model and is then applied in two and three dimensions comparisons between computed results and those obtained from analytic solutions where applicable and from boundary element codes demonstrate the effectiveness of the method

decoupled energy stable schemes for phase field models of two phase complex fluids we consider in this paper numerical approximations of phase field models for two phase complex fluids we first reformulate the phase field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws then we construct two classes stabilized and convex splitting of decoupled time discretization schemes for the coupled nonlinear systems these schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step furthermore these elliptic equations are linear for the stabilized version stability analysis and ample numerical simulations are presented

numerical approximations for a phase field moving contact line model with variable densities and viscosities we consider the numerical approximations of a two phase hydrodynamics coupled phase field model that incorporates the variable densities viscosities and moving contact line boundary conditions the model is a nonlinear coupled system that consists of incompressible navierstokes equations with the generalized navier boundary condition and the cahnhilliard equations with moving contact line boundary conditions by some subtle explicitimplicit treatments to nonlinear terms we develop two efficient unconditionally energy stable numerical schemes in particular a linear decoupled energy stable scheme for the system with static contact line condition and a nonlinear energy stable scheme for the system with dynamic contact line condition an efficient spectral galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes various numerical results show that the proposed schemes are efficient and accurate

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

fully decoupled linear and unconditionally energy stable time discretization scheme for solving the magneto hydrodynamic equations abstract in this paper we consider numerical approximations for solving the magneto hydrodynamic equations which couples the navier stokes equations and maxwell equations together a challenging issue to solve this model numerically is the time discretization 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 a linear fully decoupled first order time marching scheme by combining the projection method for navier stokes equations and some subtle implicit explicit treatments for nonlinear coupling terms we further prove that the scheme is unconditional energy stable and derive the optimal error estimates of the semi discretization rigorously various numerical simulations are implemented to demonstrate the stability and the accuracy

an ellipsoidal acoustic infinite element abstract in previous papers the authors presented new 3 d time harmonic prolate and oblate spheroidal acoustic infinite elements based on new prolate and oblate spheroidal multipole expansions for modeling acoustic fields in unbounded domains here we present the development of an ellipsoidal infinite element which is the logical generalization of those elements this development is also based on a new multipole expansion as well as a new system of ellipsoidal coordinates the element stiffness radiation damping and mass matrices are developed and presented in sufficient detail to enable their software implementation both the new coordinate system and the new element include the previous spheroidal coordinate systems and spheroidal elements as limiting cases therefore all the previously reported performance data for the spheroidal elements apply to this ellipsoidal element when used in spheroidal form since the three axes of an ellipsoid can be chosen independently an ellipsoid can circumscribe any structural shape at least as closely and generally more closely than a prolate or oblate spheroid the resulting reduction in size of the finite computational domain will result in even greater computational speeds than those already reported for the spheroidal elements the element may be used to model problems in free space 4 steradians half space 2 quarter space or eighth space 2 since this infinite element provides maximum computational efficiency for structures of all shapes software element libraries would need only this one element for all problems in unbounded domains

application of the finite element method to acoustic scattering problems for aircraft scattering of engine and rotor noise from the fuselage and wings can considerably influence community and interior noise levels here the finite element method fem is considered as a modeling technique because of its ability to solve for a broad range of governing equations and its potential efficiency advantages over the more commonly used boundary element method with fem however the data storage requirements become very demanding for three dimensional problems involving realistic geometries and frequency ranges to enable large fem problems to be tackled using relatively modest computing facilities the computational cost is addressed at several levels the extent of the mesh is reduced using infinite elements the data are manipulated using a sparse storage scheme and an iterative complex solver in combination with a preconditioner is implemented validation of the code is illustrated by comparison with classical analytical solutions for plane wave scattering from a sphere to demonstrate the potential of the technique results are calculated for three dimensional acoustic scattering off a fuselage geometry at useful frequencies

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

analysis of a coupled finite infinite element method for exterior helmholtz problems this analysis of convergence of a coupled fem iem is based on our previous work on the fem and the iem for exterior helmholtz problems the key idea is to represent both the exact and the numerical solution by the dirichlet to neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle the investigation of convergence can then be related to a spectral analysis of these dtn operators we give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere our main goal is to explore the convergence mechanism in this context we show well posedness of both the continuous and the discrete models we further show that the discrete inf sup constants have a positive lower bound that does not depend on the number of dof of the iem the proofs are based on lemmas on the spectra of the continuous and the discrete dtn operators where the spectral characterization of the discrete dtn operator is given as a conjecture from numerical experiments in our convergence analysis we show algebraic in terms of n convergence of arbitrary order and generalize this result to exponential convergence

finite element formulations for acoustical radiation finite and infinite element techniques are applied to linear acoustical problems involving infinite anechoic boundaries theory is presented for a simple one dimensional model based on webster s horn equation results are then presented both for the one dimensional model and for two axisymmetric test cases comparisons with exact solutions indicate that both the infinite element and wave envelope schemes are effective in correctly predicting the near field the wave envelope scheme is also shown to be capable of resolving the far field radiation pattern