adaptive unstructured volume remeshing ii application to two and three dimensional level set simulations of multiphase flow in part i adaptive unstructured volume remeshing i the method j comput phys in press we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles 2d and unstructured tetrahedra 3d with time and position in the computational domain in order to efficiently resolve the relevant physical scales here we illustrate the performance of an implementation of the algorithm in finite element level set simulations of deformable droplet and fluid fluid interface interactions breakup and coalescence in multiphase flows the wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp interface boundary integral numerical results the computational cost is found to be competitive even with respect to boundary integral methods for the first time using an interface capturing level set method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results

determination of point group harmonics for arbitrary j by a projection method i cubic group quantization along an axis of order 4 we give a method for systematically building cubic harmonics i e base vectors of irreducible representations of the group of invariance of a cube applications include solid state physics molecular chemistry spectroscopy etc the only necessary operators are cz4 and ry rotation of 2 around 0y projectors on representations are constructed then their rows are orthogonalized by a schmidt method and normalized simply by dividing by the square root of a diagonal element

a green s discrete transformation meshfree method for simulating transient diffusion problems this manuscript presents the formulation and application of the green s discrete transformation method gdtm for the meshfree simulation of transient diffusion problems including those with moving boundaries the gdtm implements a linear combination of time dependent green s basis functions defined on a set of source points to approximate the field in the form of a solution series a discrete transformation is implemented to evaluate unknown coefficients of this series which eliminates the need to use time integration schemes we will study the optimal number and location of the gdtm source points that yield the highest level of accuracy while maintaining a manageable condition number for the resulting linear system of equations the optimal values of these parameters which are inherently independent of the domain geometry are determined such that the basis functions have appropriate features for approximating the field a comprehensive convergence study is presented to show the precision and convergence rate of the gdtm for modeling various diffusion problems we also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies heat transfer in a turbine blade thermal response of a porous material and localized pitting corrosion in stainless steel copyright c 2015 john wiley sons ltd

a fast galerkin finite element method for a space time fractional allen cahn equation abstract in this paper a space time fractional allen cahn equation is investigated to account for the memory effect of certain materials or the anomalous diffusion processes in heterogeneous distributed media for the mixture of two immiscible phases due to the nonlocal features of fractional derivatives conventional solvers such as the gauss elimination method require o m 2 m n memory units and o m n 2 m 3 n operations to solve the space time fractional allen cahn equation where m is the number of spatial unknowns and n is the number of time steps by exploring the special structure of the stiffness matrix and utilizing the invariant energy quadratization approach we developed an efficient linearized divide and conquer galerkin finite element method without resorting to any lossy compression this new method is efficient as it significantly reduces the memory requirement to o m n and the operation count to o m n log 2 n log m we have developed a fast galerkin finite element method with local refinement that allows accurate capture of the coarsening dynamics at the interface furthermore we have investigated the modeling capacities and energy dissipation properties of the space time fractional allen cahn equation numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space time fractional allen cahn equation

isogeometric analysis of the cahn hilliard equation a convergence study herein we present a numerical convergence study of the cahn hilliard phase field model within an isogeometric finite element analysis framework using a manufactured solution a mixed formulation of the cahn hilliard equation and the direct discretisation of the weak form which requires a c 1 continuous approximation are compared in terms of convergence rates for approximations that are higher than second order in space the direct discretisation is found to be superior suboptimal convergence rates occur when splines of order p 2 are used this is validated with a priori error estimates for linear problems the convergence analysis is completed with an investigation of the temporal discretisation second order accuracy is found for the generalised method this ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the cahn hilliard equation the isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

determination of point group harmonics for arbitrary j by a projection method iii cubic group quantization along a ternary axis the method described in a first paper to obtain cubic harmonics quantized on an axis of order 4 is applied to the case of a ternary quantization axis projectors on irreducible representations are expressed with rotation matrices r 0 and r 0 arccos 1 3 acting on subspaces of su 2 invariant under d3 relations between the descriptions on the two axes of quantization are derived

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

the distribution of sampling rates for signals with equally wide equally spaced spectral bands the distribution of available sampling rates is analysed for a particular type of multiband signal which has bands of equal width centered at harmonics of a fundamental frequency

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

time fractional allen cahn and cahn hilliard phase field models and their numerical investigation abstract we study time fractional allen cahn and cahn hilliard phase field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials we develop an efficient finite difference scheme and a fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity which arise due to the nonlocal behavior of the time fractional models for time fractional cahn hilliard model we observe from the numerical results that the bigger the fractional order is the faster the energy decays however for time fractional allen cahn model we derived an opposite conclusion moreover we also study the coarsening dynamics for time fractional cahn hilliard model numerical results reveal that the scaling law for the energy decays as o t 3 which is consistent with the well known result o t 1 3 for integer order cahn hilliard model

guaranteed quality triangular mesh generation for domains with curved boundaries guaranteed quality unstructured meshing algorithms facilitate the development of automatic meshing tools however these algorithms require domains discretized using a set of linear segments leading to numerical errors in domains with curved boundaries we introduce an extension of ruppert s delaunay refinement algorithm to two dimensional domains with curved boundaries and prove that the same quality bounds apply with curved boundaries as with straight boundaries we provide implementation details for two dimensional boundary patches such as lines circular arcs cubic parametric curves and interpolated splines we present guaranteed quality triangular meshes generated with curved boundaries and propose solutions to some problems associated with the use of curved boundaries copyright 2002 john wiley sons ltd

a new approach for the numerical solution of diffusion equations with variable and degenerate mobility abstract we propose a novel approach for the numerical integration of diffusion type equations with variable and degenerate mobility or diffusion coefficient our focus is the cahn hilliard equation which plays a prominent role in phase field models of fluids and soft materials but the methodology has a more general applicability the central idea is a split method with a linearly implicit component and an analytic step to integrate out the variable mobility the proposed method is robust free of high order stability constraints and its cost is comparable to that of solving the linear heat equation with the backward euler method moreover by design the numerical solution is guaranteed to be strictly bounded by the stable constant states

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

an adaptive meshfree method for phase field models of biomembranes part i we present an adaptive meshfree method for phase field models of biomembranes max ent smooth approximants can deal with the high order phase field equations the method resolves adaptively the models at an affordable computational cost convergence to sharp interface limit is shown with small regularization parameters we present an adaptive meshfree method to approximate phase field models of biomembranes in such models the helfrich curvature elastic energy the surface area and the enclosed volume of a vesicle are written as functionals of a continuous phase field which describes the interface in a smeared manner such functionals involve up to second order spatial derivatives of the phase field leading to fourth order euler lagrange partial differential equations pde the solutions develop sharp internal layers in the vicinity of the putative interface and are nearly constant elsewhere thanks to the smoothness of the local maximum entropy max ent meshfree basis functions we approximate numerically this high order phase field model with a direct ritz galerkin method the flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions thus the proposed approach is more efficient than common tensor product methods e g finite differences or spectral methods and simpler than unstructured c 0 finite element methods applicable by reformulating the model as a system of second order pde the proposed method implemented here under the assumption of axisymmetry allows us to show numerical evidence of convergence of the phase field solutions to the sharp interface limit as the regularization parameter approaches zero in a companion paper we present a lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid

pressure boundary conditions for computing incompressible flows with sph in smoothed particle hydrodynamics sph methods for fluid flow incompressibility may be imposed by a projection method with an artificial homogeneous neumann boundary condition for the pressure poisson equation this is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries for this reason open boundary flows have rarely been computed using sph in this work we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries we resolve this problem by utilizing a rotational pressure correction scheme with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity we show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects and extends the scope of sph simulation beyond the usual periodic boundary conditions

sharp interface limits of the cahn hilliard equation with degenerate mobility in this work sharp interface limits for the degenerate cahn hilliard equation with a polynomial double well free energy and a mobility that vanishes at the minima of the double well are derived for the choice of a quadratic mobility the leading order sharp interface motion is not governed by pure surface diffusion as has been previously claimed in the literature but contains a contribution from nonlinear porous medium type bulk diffusion at the same order our analysis reveals that there are two subcases one where the solution for the order parameter is bounded between the minima proven to exist for the first mobility by elliott and garcke siam j math anal 27 1996 pp 404 423 and one where it converges to the classical stationary solution of the cahn hilliard equation consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers moreover the leading order sharp interface motion depends sensitively on the choice of mobility the asymptotic analysis shows that for example with a biquadratic mobility the leading order sharp interface motion is driven only by surface diffusion the sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model

an adaptive mesh algorithm for evolving surfaces simulation of drop breakup and coalescence abstract an algorithm is presented for the adaptive restructuring of meshes on evolving surfaces the resolution of the relevant local length scale is maintained everywhere with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations the resulting discretization depends on the instantaneous configuration of the surface but is insensitive to the deformation history application of the adaptive discretization algorithm is illustrated with three dimensional boundary integral simulations of deformable drops in stokes flow the results show that the algorithm can accurately resolve detailed features of deformed fluid interfaces including slender filaments associated with drop breakup and dimpled regions associated with drop coalescence our algorithm should be useful in a variety of fields including computational fluid dynamics image processing geographical information systems and biomedical engineering problems

a parallel domain decomposition based implicit method for the cahn hilliard cook phase field equation in 3d we present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional cahn hilliard cook chc equation which is a fourth order stochastic partial differential equation with a noise term the equation is discretized in space and time based on a fully implicit cell centered finite difference scheme with an adaptive time stepping strategy designed to accelerate the progress to equilibrium at each time step a parallel newton krylov schwarz algorithm is used to solve the nonlinear system we discuss various numerical and computational challenges associated with the method the numerical scheme is validated by a comparison with an explicit scheme of high accuracy and unreasonably high cost we present steady state solutions of the chc equation in two and three dimensions the effect of the thermal fluctuation on the spinodal decomposition process is studied we show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise we also show the evolution of the energies and statistical moments in terms of the parallel performance it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors

adaptive moving grid methods for two phase flow in porous media a b s t r a c t in this paper we present an application of the moving mesh method for approximating numerical solutions of the two phase flow model in porous media the numerical schemes combine a mixed finite element method and a finite volume method which can handle the nonlinearities of the governing equations in an efficient way the adaptive moving grid method is then used to distribute more grid points near the sharp interfaces which enables us to obtain accurate numerical solutions with fewer computational resources the numerical experiments indicate that the proposed moving mesh strategy could be an effective way to approximate two phase flows in porous media

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

computationally efficient approach for the minimization of volume constrained vector valued ginzburg landau energy functional the minimization of volume constrained vector valued ginzburg landau energy functional is considered in the present study it has many applications in computational science and engineering like the conservative phase separation in multiphase systems such as the spinodal decomposition phase coarsening in multiphase systems color image segmentation and optimal space partitioning a computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem the algorithm is based on the constrained nonmonotone l 2 gradient flow of ginzburg landau functional followed by a regularization step which is resulted from the tikhonov regularization term added to the objective functional that lifts the solution from the l 2 function space into h 1 space the regularization step not only improves the convergence rate of the presented algorithm but also increases its stability bound the step size selection based on the barzilai borwein approach is adapted to improve the convergence rate of the introduced algorithm the success and performance of the presented approach is demonstrated throughout several numerical experiments to make it possible to reproduce the results presented in this work the matlab implementation of the presented algorithm is provided as the supplementary material developing an efficient algorithm to solve conservative gradient flow of multiphase ginzburg landau functional performing the constrained minimization of energy functional on the h 1 space using non monotone optimization path by applying the barzilai borwein step size selection approach the matlab implementation of the presented algorithms is given as supplementary material

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

a nonconforming finite element method for the cahn hilliard equation this paper reports a fully discretized scheme for the cahn hilliard equation the method uses a convexity splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable and the scheme can preserve the mass conservation and energy dissipation properties of the original problem some typical phase transition phenomena are also observed through the numerical examples

coarsening unstructured meshes by edge contraction a new unstructured mesh coarsening algorithm has been developed for use in conjunction with multilevel methods the algorithm preserves geometrical and topological features of the domain and retains a maximal independent set of interior vertices to produce good coarse mesh quality in anisotropic meshes vertex selection is designed to retain the structure of the anisotropic mesh while reducing cell aspect ratio vertices are removed incrementally by contracting edges to zero length each vertex is removed by contracting the edge that maximizes the minimum sine of the dihedral angles of cells affected by the edge contraction rarely a vertex slated for removal from the mesh cannot be removed the success rate for vertex removal is typically 99 9 or more for two dimensional meshes both isotropic and anisotropic the new approach is an unqualified success removing all rejected vertices and producing output meshes of high quality mesh quality degrades only when most vertices lie on the boundary three dimensional isotropic meshes are also coarsened successfully provided that there is no difficulty distinguishing corners in the geometry from coarsely resolved curved surfaces sophisticated discrete computational geometry techniques appear necessary to make that distinction three dimensional anisotropic cases are still problematic because of tight constraints on legal mesh connectivity more work is required to either improve edge contraction choices or to develop an alternative strategy for mesh coarsening for three dimensional anisotropic meshes copyright 2003 john wiley sons ltd

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 nonstiff adaptive mesh refinement based method for the cahn hilliard equation we present a nonstiff fully adaptive mesh refinement based method for the cahn hilliard equation the method is based on a semi implicit splitting in which linear leading order terms are extracted and discretized implicitly combined with a robust adaptive spatial discretization the fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size we demonstrate the efficacy of the method with numerical examples both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy employing this approach we also identify several stationary solutions of that decomposition on the 2d torus

an adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

simulation of the phase field cahn hilliard and tumor growth models via a numerical scheme element free galerkin method abstract the main aim of this research work is to find the numerical solution based on a meshless technique for both the time dependent cahn hilliard and tumor growth partial differential equations the temporal variable is discretized using a second order method based on semi implicit backward differential formula and the stabilized term is added to the considered time discretization also an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization besides to approximate the spatial variables the element free galerkin method is considered for both mathematical models the result of full discrete schemes in studied models is solved via biconjugate gradient stabilized algorithm some numerical simulations are reported to show the capability of the numerical scheme presented here

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

an immersed discontinuous finite element method for the stokes problem with a moving interface abstract we present a discontinuous immersed finite element ife method for incompressible interfacial flows that are governed by the stokes equations the method is based on a cartesian mesh with elements cut by the moving interface on this fixed unfitted mesh we employ an immersed q 1 q 0 finite element space constructed according to the location of the interface and pertinent interface jump conditions as such the smearing of solution across the interface is greatly reduced the interface represented by a sequence of marker points is advected on the fixed background mesh by the local fluid velocity the mesh is locally refined near the interface to further improve accuracy compared with the phase field method on adaptive meshes our method can achieve the same level of accuracy with much less degrees of freedoms we present some numerical examples to validate and demonstrate the capability of the proposed method

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

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

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

robust patterns in recurrent sampling of multiband signals periodic nonuniform sampling can be used to achieve sub nyquist sampling of bandlimited multiband signals in this paper we examine the question of selecting the sampling pattern in such a scheme so that the reconstruction robustness measured by the condition number of the modulation matrix is maximized contrary to previous work where the sampling pattern was chosen from a discrete set we let the sampling patterns vary continuously but impose a structural constraint using this approach we derive necessary and sufficient conditions on the spectral support of the signal for which perfect conditioning exist namely for which a sampling pattern can be found so that the resulting modulation matrix has a condition number equal to 1 a simple test to check for these conditions is developed and the desired sampling patterns are found an algorithm for choosing the sampling pattern when the aforementioned conditions are not satisfied is also introduced finally we present some simulation results

determination of point group harmonics for arbitrary j by a projection method ii icosahedral group quantization along an axis of order 5 the method described in the previous paper to obtain cubic harmonics is applied to the similar problem for the icosahedral group the projection operators are expressed in terms of rotation matrix elements of r 0 and r 0 arctan 2 acting on subspaces invariant under d5

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

motion of interfaces governed by the cahn hilliard equation with highly disparate diffusion mobility we consider a two phase system governed by a cahn hilliard type equation with a highly disparate diffusion mobility it has been observed from recent numerical simulations that the microstructure evolution described by such a system displays a coarsening rate different from that associated with the cahn hilliard equation having either a constant diffusion mobility or a mobility that degenerates in both phases using the asymptotic matching method we derive sharp interface models of the system under consideration to theoretically analyze the interfacial motion with respect to different scales of time t in a very short time regime the transition layer stabilizes into the well known hyperbolic tangent single layer profile on an intermediate t o 1 time regime due to the small mobility in one of the phases the sharp interface limit is a one sided stefan problem determined by data in the phase with constant nonzero mobility on a slower t o varepsilon 1 time scale the leading order dynami

a topological geometric method for the obtention of symmetry adapted functions for point groups iv the dihedral groups abstract symmetry adapted icosahedral harmonics have been obtained by using a new method based on the geometrical and topological properties of icosaedral spatial arrangements of charged particles this study has allowed us to show that especially for groups of high symmetry our method proved to be more simple and advantageous than others existing in the literature as for instance the projection operator classical method

computationally efficient solution to the cahn hilliard equation adaptive implicit time schemes mesh sensitivity analysis and the 3d isoperimetric problem we present an efficient numerical framework for analyzing spinodal decomposition described by the cahn hilliard equation we focus on the analysis of various implicit time schemes for two and three dimensional problems we demonstrate that significant computational gains can be obtained by applying embedded higher order runge kutta methods in a time adaptive setting this allows accessing time scales that vary by five orders of magnitude in addition we also formulate a set of test problems that isolate each of the sub processes involved in spinodal decomposition interface creation and bulky phase coarsening we analyze the error fluctuations using these test problems on the split form of the cahn hilliard equation solved using the finite element method with basis functions of different orders any scheme that ensures at least four elements per interface satisfactorily captures both sub processes our findings show that linear basis functions have superior error to cost properties this strategy coupled with a domain decomposition based parallel implementation let us notably augment the efficiency of a numerical cahn hillard solver and open new venues for its practical applications especially when three dimensional problems are considered we use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions the framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3d sphere cylinder lamella doubly periodic surface with genus two lawson surface and triply periodic minimal surface p schwarz surface

approximation of liquid crystal flows the numerical solution of the flow of a liquid crystal governed by a particular instance of the ericksen leslie equations is considered convergence of finite element approximations is established under appropriate regularity hypotheses and numerical experiments exhibiting the interaction of singularities and the coupling of the director and momentum equations are presented

an arbitrary lagrangian eulerian method for three phase flows with triple junction points we present a moving mesh method suitable for solving two dimensional and axisymmetric three liquid flows with triple junction points this method employs a body fitted unstructured mesh where the interfaces between liquids are lines of the mesh system and the triple junction points if exist are mesh nodes to enhance the accuracy and the efficiency of the method the mesh is constantly adapted to the evolution of the interfaces by refining and coarsening the mesh locally dynamic boundary conditions on interfaces in particular the triple points are therefore incorporated naturally and accurately in a finite element formulation in order to allow pressure discontinuity across interfaces double values of pressure are necessary for interface nodes and triple values of pressure on triple junction points the resulting non linear system of mass and momentum conservation is then solved by an uzawa method with the zero resultant condition on triple points reinforced at each time step the method is used to investigate the rising of a liquid drop with an attached bubble in a lighter liquid

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

isogeometric analysis for phase field models of geometric pdes and high order pdes on stationary and evolving surfaces abstract in this paper we present an isogeometric analysis iga for phase field models of three different yet closely related classes of partial differential equations pdes geometric pdes high order pdes on stationary surfaces and high order pdes on evolving surfaces the latter can be a coupling of the former two classes in the context of geometric pdes we consider mean curvature flow and willmore flow problems and their corresponding phase field approximations which yield second order and fourth order nonlinear parabolic pdes through some numerical examples we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions moreover we study numerically the convergence of these phase field approximations to the sharp interface solutions as for the high order pdes on stationary surfaces we consider a model problem which is the cahn hilliard equation on a unit sphere where the surface is modeled using a diffuse interface approach finally as a model problem for high order pdes on evolving surfaces we consider a phase field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface the model consists of two fourth order nonlinear pdes which their direct finite element formulation in a galerkin framework necessitates smooth basis functions with at least global c 1 continuity a condition that can be easily satisfied using spline bases in iga we solve the coupled equations both in two dimensions where the vesicle is a curve and in three dimensions where the vesicle is a surface the simulation results agree with the numerical and experimental results from the literature

a second order in time uniquely solvable unconditionally stable numerical scheme for cahn hilliard navier stokes equation we propose a novel second order in time numerical scheme for cahn hilliard navier stokes phase field model with matched density the scheme is based on second order convex splitting for the cahn hilliard equation and pressure projection for the navier stokes equation we show that the scheme is mass conservative satisfies a modified energy law and is therefore unconditionally stable moreover we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme thanks to the simple coupling of the scheme we design an efficient picard iteration procedure to further decouple the computation of cahn hilliard equation and navier stokes equation we implement the scheme by the mixed finite element method ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

on a labeling for point group harmonics i cubic group expressions for the 4 6 and 8 representations are derived in terms of the 1 representations for neighboring angular momenta expressions for 5 7 and again for 8 are derived in terms of the 2 representations a third set of expressions for 8 is derived in terms of the 3 representations with an arbitrary choice of orthonormal sets of the 1 2 and 3 representations orthonormal sets of all other kinds of representations are thus well defined and can be entirely labeled using the parent representations all clebsch gordan coefficients are expressed in terms of those between parent representations and a few other ones the 4 and 5 representations defined here are not those conventionally used but they provide a simpler expression for the fictitious spin coefficients tables of the parent representations 1 2 and 3 are given for quaternary and ternary axes of quantization using the usual clebsch gordan coefficients of su 2 these tables allow us to obtain any representation for int

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

a sharp interface isogeometric solution to the stefan problem abstract in the present paper the stefan problem is solved by enriching an underlying nurbs based isogeometric approximation with an explicitly defined sharp interface on which a hybrid function derivative condition is isoparametrically described since the geometry of the enrichment is explicitly defined normals and curvatures are explicitly computed at any point on the interface thus the enriched approximation naturally captures the interfacial discontinuity in temperature gradient and naturally enables the imposition of gibbs thomson condition the blending of the enrichment with the underlying approximation requires an estimate of distance to the enriching geometry from a quadrature point and the parametric value of the footpoint on the enriching geometry these quantities are computed efficiently in the present paper using an algebraic estimate of distance coupled with an algebraic point projection method these algebraic schemes rely on implicitization of the parametric curve and are shown to be more efficient and robust than newton raphson iterations procedures for adaptive time stepping refinement and coarsening of geometry are developed to increase the stability and efficiency of the developed methodology several numerical examples of classical and dendritic stefan problem are presented to demonstrate the methodology

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

two phase flows on interface refined grids modeled with vof staggered finite volumes and spline interpolants abstract a two phase 2d model that combines the volume of fluid vof method with implicit staggered finite volumes discretization of the navier stokes equation is presented staggered finite volumes are developed on the basis of nonconforming crouzeix raviart finite elements where all components of the velocity lie in the middle of the element edges and the pressure degrees of freedom are found in the centers of mass of the elements staggered finite volumes extend marker and cell mac regular staggered grids to unstructured mesh a linear saddle point problem resulting from either the discretization or the newton method is solved for all unknown pressures and velocities interface is represented with spline interpolants which follow the vof distribution adaptive mesh refinement is used to obtain a high level of uniform refining at the domain of dependence of the interface the aligned grid is obtained by irregular refining of the cells which are intersected by a curve the boundaries of its elements coincide with the slope segments going through the intersections of the curve with the underlying regular elements boundary the deformable computational grids are used only to discretize the navier stokes equation the advection of volume fractions is done on the advection mesh which corresponds to highest regular refining on the computational grid approximation of the surface tension on spline interpolants offers a straightforward way to describe correctly the pressure jumps on interface fitted staggered grids this allows deletion of the anomalous currents around a statical bubble and their effective reduction in real simulations on the aligned grid the continuity of the viscous stress is modeled exactly due to the finite volume approach using the proposed numerical techniques single bubble rise is analyzed

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

exact solutions of the nonlinear boltzmann equation a review is given of research activities since 1976 on the nonlinear boltzmann equation and related equations of boltzmann type in which several rediscoveries have been made and several conjectures have been disproved subjects are i the bkw solution of the boltzmann equation for maxwell molecules first discovered by krupp in 1967 and the krook wu conjecture concerning the universal significance of the bkw solution for the large v t behavior of the velocity distribution functionf v t ii moment equations and polynomial expansions off v t iii model boltzmann equation for a spatially uniform system of very hard particles that can be solved in closed form for general initial conditions iv for maxwell and non maxwell type molecules there exist solutions of the nonlinear boltzmann equation with algebraic decrease at connections with nonuniqueness and violation of conservation laws v conjectured super h theorem and the bkw solution vi exactly soluble one dimensional boltzmann equation with spatial dependence

three dimensional fully adaptive simulations of phase field fluid models we present an efficient numerical methodology for the 3d computation of incompressible multi phase flows described by conservative phase field models we focus here on the case of density matched fluids with different viscosity model h the numerical method employs adaptive mesh refinements amr in concert with an efficient semi implicit time discretization strategy and a linear multi level multigrid to relax high order stability constraints and to capture the flow s disparate scales at optimal cost only five linear solvers are needed per time step moreover all the adaptive methodology is constructed from scratch to allow a systematic investigation of the key aspects of amr in a conservative phase field setting we validate the method and demonstrate its capabilities and efficacy with important examples of drop deformation kelvin helmholtz instability and flow induced drop coalescence

an extended finite element method for a diffuse interface model this work presents an extended finite element method xfem for a diffuse interface model which describes interfacial phenomena of multi phase flow the diffuse interface has a non zero thickness over which the phase field variable changes continuously the diffuse interface thickness is typically very small compared to the observed domain size resulting in a high gradient solution in this work the finite element approximation of the phase field variable is locally enriched with a tangent hyperbolic function which characterizes a high gradient solution of the diffuse interface model we study a one dimensional advection and two diffusion problems and demonstrate the remarkable improvement of the solution by the local enrichment

3d phase field simulations of interfacial dynamics in newtonian and viscoelastic fluids this work presents a three dimensional finite element algorithm based on the phase field model for computing interfacial flows of newtonian and complex fluids a 3d adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk it is key to accurate resolution of the interface at manageable computational costs the coupled navier stokes and cahn hilliard equations plus the constitutive equation for non newtonian fluids are solved using second order implicit time stepping within each time step newton iteration is used to handle the nonlinearity and the linear algebraic system is solved by preconditioned krylov methods the phase field model with a physically diffuse interface affords the method several advantages in computing interfacial dynamics one is the ease in simulating topological changes such as interfacial rupture and coalescence another is the capability of computing contact line motion without invoking ad hoc slip conditions as validation of the 3d numerical scheme we have computed drop deformation in an elongational flow relaxation of a deformed drop to the spherical shape and drop spreading on a partially wetting substrate the results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate excellent agreement is achieved provided that the 3d interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid since our model involves several coupled partial differential equations and we use a fully implicit scheme the matrix inversion requires a large memory this puts a limit on the scale of problems that can be simulated in 3d especially for viscoelastic fluids

a semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

inpainting of binary images using the cahn hilliard equation image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas we outline here the use of a model for binary inpainting based on the cahn hilliard equation which allows for fast efficient inpainting of degraded text as well as super resolution of high contrast images

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

c1 continuous h adaptive least squares spectral element method for phase field models abstract to describe the interfacial dynamics between two phases using the phase field method the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics due to the high gradients of the solution within the interfacial region and consequent high computational cost the use of the phase field method has been limited to the small scale problems whose characteristic length is similar to the interfacial thickness by using finer mesh at the interface and coarser mesh in the rest of computational domain the phase field methods can handle larger scale of problems with realistic interface thicknesses in this work a c 1 continuous h adaptive mesh refinement technique with the least squares spectral element method is presented it is applied to the navier stokes cahn hilliard nsch system and the isothermal navier stokes korteweg nsk system hermite polynomials are used to give global differentiability in the approximated solution and a space time coupled formulation and the element by element technique are implemented two refinement strategies based on the solution gradient and the local error estimators are suggested and they are compared in two numerical examples

a level set method for computing solutions to viscoelastic two phase flow abstract a finite element code based on the level set method is developed for simulating the motion of viscoelastic two phase flow problems this method is a generalization of the finite difference approach described in 1 4 for computing solutions to two phase problems of inviscid and viscous fluids the marchuk yanenko operator splitting technique is used to decouple the difficulties associated with the nonlinear convection term the incompressibility constraint the viscoelastic term and the interface motion problem the nonlinear convection problem is solved using a least squares conjugate gradient algorithm and the stokes like problem is solved using a conjugate gradient algorithm the constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor while the convection term in the constitutive equation is discretized using a third order upwinding scheme the code is verified by performing a convergence study to show that the results are independent of the mesh and time step sizes using our code we have studied the deformation of drops in simple shear and pressure driven flows and of bubbles in gravity driven flows over a wide range of dimensionless capillary ca and deborah numbers de for a newtonian bubble rising in a quiescent viscoelastic liquid we find that there are limiting values of the parameters de and ca above which the bubble assumes a characteristic shape with a cusp like trailing edge the front of the bubble however remains round as the local viscoelastic and viscous stresses act to round the bubble in a pressure driven flow the drop is stretched so that its front which is closer to the channel center remains round and the trailing edge which is closer to the channel wall becomes sharp these numerical results are in agreement with the experimental observations

on a labeling for point group harmonics ii icosahedral group the expressions for the 2 5 6 8 and 9 representations are given in terms of those of 1 representations for neighboring angular momenta the coefficients of the 4 representations are expressed in terms of those of 7 representations therefore with an arbitrary choice of orthonormal sets of 1 3 and 7 representations orthonormal sets of other kinds of representations are well defined and can be labeled with the labels of parent representations all clebsch gordan coefficients are expressed in terms of those between parent representations and a few others tables of all nondegenerate 1 3 and 7 representations are given for the axes of quantization of order 5 with some degenerate 3 and 7 representations which are also given any representation of integer or half integer angular momentum up to 27 can be obtained using some usual clebsch gordan coefficients of su 2

adaptive mesh refinement computation of solidification microstructures using dynamic data structures we study the evolution of solidification microstructures using a phase field model computed on an adaptive finite element grid we discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase field model at low undercooling in particular we show that the computational complexity of solving any phase boundary problem scales with the interface arclength when using an adapting mesh moreover the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions which would otherwise require lattices greater than 217 217elements we examine the convergence properties of our algorithm we also present two dimensional time dependent calculations of dendritic evolution with and without surface tension anisotropy we benchmark our results for dendritic growth with microscopic solvability theory finding them to be in good agreement with theory for high undercoolings at low undercooling however we obtain higher values of velocity than solvability theory at low undercooling where transients dominate in accord with a heuristic criterion which we derive

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

phase field model for three dimensional dendritic growth with fluid flow we study the effect of fluid flow on three dimensional 3d dendrite growth using a phase field model on an adaptive finite element grid in order to simulate 3d fluid flow we use an averaging method for the flow problem coupled to the phase field method and the semi implicit approximated projection method siapm we describe a parallel implementation for the algorithm using the charm fem framework and demonstrate its efficiency we introduce an improved method for extracting dendrite tip position and tip radius facilitating accurate comparison to theory we benchmark our results for 2d dendrite growth with solvability theory and previous results finding them to be in good agreement the physics of dendritic growth with fluid flow in three dimensions is very different from that in two dimensions and we discuss the origin of this behavior

adaptive time stepping and computational stability we investigate the effects of adaptive time stepping and other algorithmic strategies on the computational stability of ode codes we show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability a series of computational experiments with the standard implementation of dassl and a modified version including stepsize control based on digital filters is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense the inherent performance and stability of dassl are therefore much greater than the standard implementation seems to suggest

an explicit hybrid finite difference scheme for the allen cahn equation abstract in this paper we propose an explicit hybrid numerical method for solving the allen cahn equation which models antiphase domain coarsening process in a binary mixture the proposed method is based on an operator splitting method first we solve the linear diffusion part using the explicit euler method second we solve the nonlinear term using the closed form analytical solution we show the stability condition of the proposed numerical scheme we also show the pointwise boundedness of the numerical solution for the allen cahn equation under a solvability condition numerical experiments such as linear stability analysis traveling wave motion by mean curvature image segmentation and crystal growth are presented to demonstrate the performance of the proposed method

a rescaling scheme with application to the long time simulation of viscous fingering in a hele shaw cell in this paper we present a time and space rescaling scheme for the computation of moving interface problems the idea is to map time space such that the interfaces can evolve exponentially fast in the new time scale while the area volume enclosed by the interface remains unchanged the rescaling scheme significantly reduces the computation time especially for slow growth and enables one to accurately simulate the very long time dynamics of moving interfaces we then implement this scheme in a hele shaw problem examine the dynamics for a number of different injection fluxes and present the largest and most pronounced viscous fingering simulations to date

the numerical solutions for the energy dissipative and mass conservative allen cahn equation abstract there are three criteria when we model with phase field equations the first one is the evolution of phase separation the second one is energy dissipation law the last one is whether we have the law of conservation of mass or not if we minimize the ginzburg landau free energy functional the kinetics of phase separation inherently follows however the other properties such as energy dissipation and mass conservation are more difficult to model hitherto there have been a few proposals to conserve mass using the allen cahn ac equation but we do not have the energy dissipation law in these models when considered in light of local effects of mass correction for the reasons certain researchers still prefer to use the cahn hilliard equation e g biharmonic equation instead of using the allen cahn equation when we consider a mass conserving case this paper is concerned with numerical solutions of the ac equation in the mass conserving space where the solution is corrected at each iteration for mass conservation we introduce two different methods for solving the conservative ac equation with periodic boundary conditions in this work we have the ac equation in common but use two different mass correction operators note that both of mass correction operators satisfy the discrete energy dissipation property first we project the solution of the ac equation to the mass conserving space and then recover the linear convex splitting scheme for the conventional mass conserving equation the linear convex splitting scheme is well known but the convergence of numerical solutions is elaborately studied in terms of gradient projection next we propose the other energy minimizing method on the mass conserving space it translates the solution of the ac equation to the mass conserving space via an energy dissipation operator the dissipation of discrete energy functional for the proposed method is also established as far as the author knows no report has been presented for the numerical scheme having discrete energy dissipation property of the mass conserving ac equation without using the nonlocal and constant multiplier to verify the common properties of mass projection and proposed methods numerically we simulate the evolutions for phase separation dissipation of energy and mass conservation we moreover consider the short and long time stages of evolutions to see the difference between two methods these several tests are provided for its applications of the proposed method for modeling natural phenomena

energy law preserving c0 finite element schemes for phase field models in two phase flow computations we use the idea in 33 to develop the energy law preserving method and compute the diffusive interface phase field models of allen cahn and cahn hilliard type respectively governing the motion of two phase incompressible flows we discretize these two models using a c 0 finite element in space and a modified midpoint scheme in time to increase the stability in the pressure variable we treat the divergence free condition by a penalty formulation under which the discrete energy law can still be derived for these diffusive interface models through an example we demonstrate that the energy law preserving method is beneficial for computing these multi phase flow models we also demonstrate that when applying the energy law preserving method to the model of cahn hilliard type un physical interfacial oscillations may occur we examine the source of such oscillations and a remedy is presented to eliminate the oscillations a few two phase incompressible flow examples are computed to show the good performance of our method

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems