analysis and applications of the voronoi implicit interface method we analyze a new mathematical and numerical framework the voronoi implicit interface method viim first introduced in saye and sethian 2011 r i saye j a sethian the voronoi implicit interface method for computing multiphase physics pnas 108 49 2011 19498 19503 for tracking multiple interacting and evolving regions phases whose motion is determined by complex physics fluids mechanics elasticity etc from a mathematical point of view the method provides a theoretical framework for moving interface problems that involve multiple junctions defining the motion as the formal limit of a sequence of related problems discretizing this theoretical framework provides a numerical methodolology which automatically handles multiple junctions triple points and quadruple points in two dimensions as well as triple lines etc in higher dimensions topological changes in the system occur naturally with no surgery required in this paper we present the method in detail and demonstrate several new extensions of the method to different physical phenomena including curvature flow with surface energy densities defined on a per phase basis as well as multiphase fluid flow in which density viscosity and surface tension can be defined on a per phase basis we test this method in a variety of ways we perform rigorous analysis and demonstrate convergence in both two and three dimensions for a variety of evolving interface problems including verification of von neumann mullins law in two dimensions and its analog in three dimensions as well as normal driven flow and curvature flow with and without constraints demonstrating topological change and the effects of different boundary conditions we couple the method to a second order projection method solver for incompressible fluid flow and study the effects of membrane permeability and impermeability large shearing torsional forces and the effects of varying density viscosity and surface tension on a per phase basis finally we demonstrate convergence in both space and time of a topological change in a multiphase foam

adaptive finite elements with high aspect ratio for the computation of coalescence using a phase field model a multiphase field model for the description of coalescence in a binary alloy is solved numerically using adaptive finite elements with high aspect ratio the unknown of the multiphase field model are the three phase fields solid phase 1 solid phase 2 and liquid phase a lagrange multiplier and the concentration field an euler implicit scheme is used for time discretization together with continuous piecewise linear finite elements at each time step a linear system corresponding to the three phases plus the lagrange multiplier has to be solved then the linear system pertaining to concentration is solved an adaptive finite element algorithm is proposed in order to reduce the number of mesh vertices the generated meshes contain elements with high aspect ratio the refinement and coarsening criteria are based on an error indicator which has already been justified theoretically for simpler problems numerical results on two test cases show the efficiency of the method

progress and investigation on lattice boltzmann modeling of multiple immiscible fluids or components with variable density and viscosity ratios lattice boltzmann models for simulating multiphase flows are relatively new and much work remains to be done to demonstrate their ability to solve fundamental test cases before they are considered for engineering problems from this perspective a hydrodynamic lattice boltzmann model for simulating immiscible multiphase flows with high density and high viscosity ratios up to o 1000 o 1000 and o 100 o 100 respectively is presented and validated against analytical solutions the method is based on a two phase flow model with operators extended to handle n immiscible fluids the current approach is o n o n in computational complexity for the number of different gradient approximations this is a major improvement considering the o n2 o n2 complexity found in most works a sequence of systematic and essential tests have been conducted to establish milestones that need to be met by the proposed approach as well as by other methods first the method is validated qualitatively by demonstrating its ability to address the spinodal decomposition of immiscible fluids second the model is quantitatively verified for the case of multilayered planar interfaces third the multiphase laplace law is studied for the case of three fluids fourth a quality index is developed for the three phase laplace young s law which concerns the position of the interfaces between the fluids resulting from the different surface tensions the current model is compatible with the analytical solution and is shown to be first order accurate in terms of this quality index finally the multilayered couette s flow is studied in this study numerical results can recover the analytical solutions for all the selected test cases as long as unit density ratios are considered for high density and high viscosity ratios the analytical solution is recovered for all tests except that of the multilayered couette s flow numerical results and a discussion are presented for this unsuccessful test case it is believed that other lb models may have the same problem in addressing the simulation of multiphase flows with variable density ratios

on the variational approximation of combined second and fourth order geometric evolution equations we present a variational formulation of combined motion by minus the laplacian of curvature and mean curvature flow as well as related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the equidistribution of mesh points and if applicable area conservation

large scale simulation of normal grain growth via diffusion generated motion diffusion generated motion is used to perform a very large scale simulation of normal grain growth in three dimensions with high accuracy the method is based on the diffusion of signed distance functions and shares similarities with level set methods the herring angle condition at junctions and topological transitions are naturally captured with this formulation this approach offers significant advantages over existing numerical methods and allows for accurate computations on scales not previously possible a fully resolved simulation of normal grain growth initially containing over 130 000 grains in three dimensions is presented and analysed it is shown that the average grain radius grows as the square root of time and the grain size distribution is self similar good agreement with other theoretical predictions experimental results and simulation results via other techniques is also demonstrated

a modified cumulant matrix for doa estimation presents a modification of the fourth order cumulant matrix c sub g proposed by one source the modification consists of taking certain submatrices of c sub g and using them to apply the esprit principle with general arrays which may not be composed of matched sensor doublets or to perform spatial smoothing to deal with the coherent sources in the cumulant domain computer simulations are carried out to illustrate the effectiveness of the modified cumulant matrix

a dynamic mesh algorithm for curvature dependent evolving interfaces a new finite element method is discussed for approximating evolving interfaces in2nwhose normal velocity equals mean curvature plus a forcing function the method is insensitive to singularity formation and retains the local structure of the limit problem and thus exhibits a computational complexity typical of3n 1without having the drawbacks of front tracking strategies a graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh time stepping is explicit but stability constraints force small time steps only when singularities develop whereas relatively large time steps are allowed before or past singularities when the evolution is smooth the explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step thereby making mesh updating simple and thus practical performance and potentials are fully documented via a number of numerical simulations in 2d 3d 4d and 8d with axial symmetries they include tori and cones for the mean curvature flow minimal and prescribed mean curvature surfaces with given boundary fattening for smooth driving force and volume constraint

multiphase flow modeling of spinodal decomposition based on the cascaded lattice boltzmann method abstract a new multiphase lattice boltzmann model based on the cascaded collision operator is developed to study the spinodal decomposition of critical quenches in the inertial hydrodynamic regime the proposed lattice boltzmann model is able to investigate simulations of multiphase spinodal decomposition with a very high reynolds number the law governing the growth of the average domain size i e l t is studied numerically in the late time regime when multiple immiscible fluids are considered in the spinodal decomposition it is found numerically that the growth exponent is inversely proportional to the number n of immiscible fluids in the system in fact 6 n 7 is a simple law that matches the numerical results very well even up to n 20 as the number of immiscible fluids increases the corresponding drop in the connectivity of the various fluid domains is believed to be the main factor that drives and slows down the growth rate various videos that accurately demonstrate spinodal decomposition with different transport mechanisms are provided see appendix a the remarks and statement made in this research are based on the analysis of 5120 numerical simulations and the postprocessing of about 3 5 tb of data

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

diffusion generated motion using signed distance functions we describe a new class of algorithms for generating a variety of geometric interfacial motions by alternating two steps construction of the signed distance function i e redistancing to the interface and convolution with a suitable kernel these algorithms can be seen as variants of merriman bence and osher s threshold dynamics 25 the new algorithms proposed here preserve the computational efficiency of the original threshold dynamics algorithm however unlike threshold dynamics the new algorithms also allow attaining high accuracy on uniform grids without adaptive refinement

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

diffusion generated motion for grain growth in two and three dimensions an efficient algorithm for accurately simulating curvature flow for large networks of curves in two dimensions and surfaces in three dimensions on uniform grids is proposed this motion arises in the technologically important problem of simulating grain boundary motion in polycrystalline materials in this formulation grain boundaries are zero level sets of signed distance functions curvature motion is achieved by first diffusing locally maintained signed distance functions followed by a reinitialization step a technique is devised to allow a single signed distance function to represent a large subset of spatially separated grains hundreds of thousands of grains can be simulated using a small number of signed distance functions in this work 32 in two dimensions and 64 in three dimensions are more than sufficient using modest computational hardware

a variational method for multiphase volume preserving interface motions we develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases whose volumes are preserved throughout time the foundation of the method is a thresholding algorithm of the bence merriman osher type the original algorithm is reformulated in a vector setting which allows for a natural inclusion of constraints even in the multiphase case moreover a new method for overcoming the inaccuracy of thresholding methods on non adaptive grids is designed since this inaccuracy becomes especially prominent in volume preserving motions formal analysis of the method and numerical tests are presented

a multiphase field concept numerical simulations of moving phase boundaries and multiple junctions we present numerical simulations which support the formal asymptotic analysis relating a multiorder parameter allen cahn system to a multiphase interface problem with curvature dependent evolution of the interfaces and angle conditions at triple junctions within the gradient energy of the allen cahn system the normal to an interface between phases i and j is modeled by the irreducible representations u i nabla u j u j nabla u i u i nabla u j u j nabla u i where ui and uj are the ith and jth components of the vectorial order parameter bu in rz n in the vectorial case the dependence of the limiting surface tensions and mobilities on the bulk potentials of the allen cahn system is not given explicitly but in terms of all the n components of the planar stationary wave solutions one of the issues of this paper is to find bulk potentials which allow a rather easy access to the resulting surface tensions and mobilities we compare numerical computations for planar and circular phase bound

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains in this paper we present a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains the method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high order time step stability constraints the continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains we describe the implementation of the proposed numerical scheme in detail the resulting system of discrete equations is solved by a nonlinear multigrid method we demonstrate the accuracy and robustness of the proposed dirichlet boundary formulation using various numerical experiments we numerically show the total energy decrease and the unconditionally gradient stability in particular the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

computer simulations of phase field drops on super hydrophobic surfaces abstract we present a novel quasi newton continuation procedure that efficiently solves the system of nonlinear equations arising from the discretization of a phase field model for wetting phenomena we perform a comparative numerical analysis that shows the improved speed of convergence gained with respect to other numerical schemes moreover we discuss the conditions that on a theoretical level guarantee the convergence of this method at each iterative step a suitable continuation procedure develops and passes to the nonlinear solver an accurate initial guess discretization performs through cell centered finite differences the resulting system of equations is solved on a composite grid that uses dynamic mesh refinement and multi grid techniques the final code achieves three dimensional realistic computer experiments comparable to those produced in laboratory settings this code offers not only new insights into the phenomenology of super hydrophobicity but also serves as a reliable predictive tool for the study of hydrophobic surfaces

preconditioning for allen cahn variational inequalities with non local constraints the solution of allen cahn variational inequalities with mass constraints is of interest in many applications this problem can be solved both in its scalar and vector valued form as a pde constrained optimization problem by means of a primal dual active set method at the heart of this method lies the solution of linear systems in saddle point form in this paper we propose the use of krylov subspace solvers and suitable preconditioners for the saddle point systems numerical results illustrate the competitiveness of this approach

efficient numerical solution of discrete multi component cahn hilliard systems in this work we develop preconditioners for the iterative solution of the large scale algebraic systems arising in finite element discretizations of microstructures with an arbitrary number of components described by the diffusive interface model the suggested numerical techniques are applied to the study of ternary fluid flow processes

a quasi dense approach to surface reconstruction from uncalibrated images this paper proposes a quasi dense approach to 3d surface model acquisition from uncalibrated images first correspondence information and geometry are computed based on new quasi dense point features that are resampled subpixel points from a disparity map the quasi dense approach gives more robust and accurate geometry estimations than the standard sparse approach the robustness is measured as the success rate of full automatic geometry estimation with all involved parameters fixed the accuracy is measured by a fast gauge free uncertainty estimation algorithm the quasi dense approach also works for more largely separated images than the sparse approach therefore it requires fewer images for modeling more importantly the quasi dense approach delivers a high density of reconstructed 3d points on which a surface representation can be reconstructed this fills the gap of insufficiency of the sparse approach for surface reconstruction essential for modeling and visualization applications second surface reconstruction methods from the given quasi dense geometry are also developed the algorithm optimizes new unified functionals integrating both 3d quasi dense points and 2d image information including silhouettes combining both 3d data and 2d images is more robust than the existing methods using only 2d information or only 3d data an efficient bounded regularization method is proposed to implement the surface evolution by level set methods its properties are discussed and proven for some cases as a whole a complete automatic and practical system of 3d modeling from raw images captured by hand held cameras to surface representation is proposed extensive experiments demonstrate the superior performance of the quasi dense approach with respect to the standard sparse approach in robustness accuracy and applicability

on the notion of a vector and a stress tensor for a general class of anisotropic diffuse interface models we show that under equilibrium conditions an anisotropic phase field model of a diffuse interface leads to the notion of a divergence free tensor whose components are negligible away from the interface near the interface it plays the role of a stress tensor and involves the phase field generalization of the cahn hoffman vector we show that this tensor may be used to derive the equilibrium conditions for edges in interfaces we then extend these ideas from the phase field model to a general class of multiple order parameter models in this broader context we demonstrate that it is also possible to construct a vector and we use it to derive the three dimensional form of the gibbs thomson equation in the sharp interface limit we derive the associated stress tensor for the multiple order parameter case and show that it also leads to the appropriate equilibrium conditions for edges in interfaces and multiple junctions in the sharp interface limit this approach shows that both multiple order parameter models and less physically based phase field models share a common framework the former providing a generalization of the notion of the vector for phase field models as well as providing a connection to the classical vector theory for a sharp interface for the special case of a phase field model with non convex surface energy we show that edges can be represented by weak shocks in which the spatial derivatives of the phase field are not continuous and we derive the associated jump conditions in all the situations considered the notion of the vector and the stress tensor play a central role

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

preconditioning for vector valued cahn hilliard equations the solution of vector valued cahn hilliard systems is of interest in many applications we discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities concerning the nonsmooth systems the necessary bound constraints are incorporated via the moreau yosida regularization technique we develop effective preconditioners for the efficient solution of the linear systems in saddle point form numerical results illustrate the efficiency of our approach in particular we numerically show mesh and phase independence of the developed preconditioner in the smooth case the results in the nonsmooth case are also satisfying and the preconditioned version always outperforms the unpreconditioned one

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

a numerical method for tracking curve networks moving with curvature motion abstract a finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions the prototypical example is that of phase interfaces that meet at prescribed angles although eutectic junctions and interactions through nonlocal effects are also considered the method is based on a direct discretization of the underlying parabolic problem and boundary conditions a linear stability analysis is presented for our scheme as well as computational studies that confirm the second order convergence to smooth solutions after a singularity in the curve network where the solution is no longer smooth we demonstrate almost second order convergence a numerical study of singularity types is done for the case of networks that meet at prescribed angles at triple junctions finally different discretizations and methods for implicit time stepping are presented and compared

multimaterial topology optimization by volume constrained allen cahn system and regularized projected steepest descent method abstract a new computational algorithm is introduced in the present study to solve multimaterial topology optimization problems it is based on the penalization of the objective functional by the multiphase volume constrained ginzburg landau energy functional the update procedure is based on the gradient flow of the objective functional by a fractional step projected steepest descent method in the first step the new design is found based on the projected steepest descent method to ensure the reduction in the objective functional simultaneously satisfying the control constraints in the second step regularization step an h 1 regularity of the solution is ensured while keeping the feasibility of solution with respect to the set of control constraints the presented algorithm could be accounted as a constrained h 1 optimization algorithm which according to our knowledge has not been reported to solve such kind of problems yet the success and efficiency of the presented method are shown through several test problems numerical results show that the presented algorithm ends with a near 0 1 topology and its computational cost scales sub linearly by the number of phases for the sake of reader convenience and the ease of further extension the matlab implementation of the presented algorithm is included in the appendix

anisotropic multi phase field model interfaces and junctions n this paper we bring together and extend two recent developments in phase field models namely a phase field model of a multiphase system i steinbach et al physica d 94 135 1996 and the extension of the cahn hoffman vector theory of anisotropic sharp interfaces to phase field models a a wheeler and g b mcfadden eur j appl math 7 369 1996 proc r soc london ser a 453 1611 1997 we develop the phase field model of a multiphase system proposed by steinbach et al to include both surface energy and interfacial kinetic anisotropy we show that this model may be compactly expressed in terms of generalized cahn hoffman vectors this generalized cahn hoffman vector formalism is subsequently developed to include the notion of a stress tensor which is used to succinctly derive the leading order conditions at both moving interfaces and stationary multijunctions in the sharp interface limit

on three phase boundary motion and the singular limit of a vector valued ginzburg landau equation we present a formal asymptotic analysis which suggests a model for three phase boundary motion as a singular limit of a vector valued ginzburg landau equation we prove short time existence and uniqueness of solutions for this model that is for a system of three phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point such models pertain to grain boundary motion in alloys the method we use based on linearization about the initial conditions applies to a wide class of parabolic systems we illustrate this further by its application to an eutectic solidification problem

a diffuse interface model for alloys with multiple components and phases a nonisothermal phase field model for alloys with multiple phases and components is derived the model allows for arbitrary phase diagrams we relate the model to classical sharp interface models by formally matched asymptotic expansions in addition we discuss several examples and relate our model to the ones already existing