redistancing by flow of time dependent eikonal equation construction of signed distance to a given interface is a topic of special interest to level set methods there are currently however few algorithms that can efficiently produce highly accurate solutions we introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations we provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction which for example are critical in local level set methods we further apply the same ideas to extension of values off interfaces together our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems

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

high order time marching reinitialization for regional level set functions abstract in this work the time marching reinitialization method is extended to compute the unsigned distance function in multi region systems involving arbitrary number of regions high order and interface preservation are achieved by applying a simple mapping that transforms the regional level set function to the level set function and a high order two step reinitialization method which is a combination of the closest point finding procedure and the hj weno scheme the convergence failure of the closest point finding procedure in three dimensions is addressed by employing a proposed multiple junction treatment and a directional optimization algorithm simple test cases show that our method exhibits 4th order accuracy for reinitializing the regional level set functions and strictly satisfies the interface preserving property the reinitialization results for more complex cases with randomly generated diagrams show the capability our method for arbitrary number of regions n with a computational effort independent of n the proposed method has been applied to dynamic interfaces with different types of flows and the results demonstrate high accuracy and robustness

a remark on computing distance functions we propose a new method for the reconstruction of the signed distance function in the context of level set methods the new method is a modification of the algorithm which makes use of the pde equation for the distance function introduced by m sussman p smereka and s osher 1994 j comput phys 119 146 it is based mainly on the use of a truly upwind discretization near the interface comparison with the previous algorithm shows a definite improvement when used with a first order upwind scheme the method provides first order accuracy for the signed distance function in the whole computational domain and second order accuracy in the location of the interface a second order version of the method is also presented

a finite element level set redistancing method based on gradient recovery we introduce a new redistancing method for level set functions this method applies in a finite element setting and uses a gradient recovery technique based on the recovered gradient a quasi normal field on the zero level of the finite element level set function is defined and from this an approximate signed distance function is determined for this redistancing method rigorous error bounds are derived for example the distance between the original zero level and the zero level after redistancing can be shown to be bounded by ch k 1 if finite elements of degree k are used in the discretization

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

algorithms for area preserving flows we propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane such as area preserving motion by curvature these schemes are based on a new class of diffusion generated motion algorithms using signed distance functions in particular they alternate two very simple and fast operations namely convolution with the gaussian kernel and construction of the distance function to generate the desired geometric flow in an unconditionally stable manner we present applications of these area preserving flows to large scale simulations of coarsening

random walks on ultrametric spaces low temperature patterns the authors present the numerically determined distribution of the number of sites visited by thermally activated walkers on ultrametric spaces which mimic the energetic disorder found in amorphous materials at high temperatures the results resemble those found for sierpinski type fractals at low temperatures the distributions are highly structured showing discontinuities which increase with the number of steps in contrast to an intuitive picture that would lead to smoothing they apply the results to the energy transfer in disordered systems and study the decay due to trapping

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 simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

revisiting the redistancing problem using the hopf lax formula this article presents a fast new numerical method for redistancing objective functions based on the hopf lax formula 1 the algorithm suggested here is a special case of the previous work in 2 and an extension that applies the hopf lax formula for computing the signed distance to the front we propose the split bregman approach to solve the minimization problem as a solution of the eikonal equation obtained from hopf lax formula our redistancing procedure is expected to be generalized and widely applied to many fields such as computational fluid dynamics the minimal surface problem and elsewhere

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

tree methods for moving interfaces fast adaptive numerical methods for solving moving interface problems are presented the methods combine a level set approach with frequent redistancing and semi lagrangian time stepping schemes which are explicit yet unconditionally stable a quadtree mesh is used to concentrate computational effort on the interface so the methods move an interface withndegrees of freedom ino nlogn work per time step efficiency is increased by taking large time steps even for parabolic curvature flows the methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging anisotropy faceting and curvature

multivariate spline approximation of the signed distance function the signed distance function can effectively support many geometry processing tasks such as decimates smoothing and shape reconstruction since it provides efficient access to distance estimates in this paper we present an adaptive method to approximate the signed distance function of a smooth curve by using polynomial splines over type 2 triangulation the trimmed offsets are also studied

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

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 self adaptive oriented particles level set method for tracking interfaces a new method for tracking evolving interfaces by lagrangian particles in conjunction with a level set approach is introduced this numerical technique is based on the use of time evolution equations for fundamental vector and tensor quantities defined on the front and represents a new and convenient way to couple the advantages of the eulerian description given by a level set function f to the use of lagrangian massless particles the term oriented points out that the information advected by the particles not only concern the spatial location but also the local outward normal vector n to the interface c and the second fundamental tensor the shape operator n the particles are exactly located upon c and provide all the requested information for tracking the interface on their own in addition a self adaptive mechanism suitably modifies at each time step the markers distribution in the numerical domain each particle behaves both as a potential seeder of new markers on c so as to guarantee an accurate reconstruction of the interface and a de seeder to avoid any useless gathering of markers and to limit the computational effort the algorithm is conceived to avoid any transport equation for f and to confine the level set function to the role of a mere post processing tool thus all the numerical diffusion problems usually affecting the level set methodology are removed the method has been tested both on 2d and 3d configurations it carries out a fast reconstruction of the interface and its accuracy is only limited by the spatial resolution of the mesh

an improved implicit re initialization method for the level set function applied to shape and topology optimization of fluid this paper presents an accurate implicit re initialization approach in the framework of a level set method the improved method includes two schemes for keeping the zero level set unperturbed the first scheme is to derive and use a new formula for the smoothing parameter in the conventional re initialization equation based on the principle of the interface not moving the second scheme is to reduce the local time step to avoid the interface moving across grid points when the sign of the level set function near the interface is changed the example presented suggests that the new algorithm has a better approximation to the signed distance function and obtains a more accurate interface than the algorithm presented by sussman et al 1994 for shape and topology optimization of fluid the effectiveness of the developed method in this paper was demonstrated by the internal and external flow examples

short note on reinitializing level set functions in this paper we consider reinitializing level functions through equation f t sgn f 0 f 1 0 16 the method of russo and smereka 11 is taken in the spatial discretization of the equation the spatial discretization is simply speaking the second order eno finite difference with subcell resolution near the interface our main interest is on the temporal discretization of the equation we compare the three temporal discretizations the second order runge kutta method the forward euler method and a gauss seidel iteration of the forward euler method the fact that the time in the equation is fictitious makes a hypothesis that all the temporal discretizations result in the same result in their stationary states the fact that the absolute stability region of the forward euler method is not wide enough to include all the eigenvalues of the linearized semi discrete system of the second order eno spatial discretization makes another hypothesis that the forward euler temporal discretization should invoke numerical instability our results in this paper contradict both the hypotheses the runge kutta and gauss seidel methods obtain the second order accuracy and the forward euler method converges with order between one and two examining all their properties we conclude that the gauss seidel method is the best among the three compared to the runge kutta it is twice faster and requires memory two times less with the same accuracy

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

fast and accurate redistancing by directional optimization a fast and accurate algorithm for the reinitialization of the signed distance function in two and three spatial dimensions is presented the algorithm has computational complexity o n log n for the reinitialization of n grid points the order of accuracy of the reinitialization is demonstrated to depend primarily on the interpolation algorithm used bicubic interpolation is demonstrated to result in fourth order accuracy for smooth interfaces simple numerical examples demonstrating the convergence and computational complexity of the reinitialization algorithm in two and three dimensions are presented as verification of the algorithm

a topology preserving level set method for shape optimization the classical level set method which represents the boundary of the unknown geometry as the zero level set of a function has been shown to be very effective in solving shape optimization problems the present work addresses the issue of using a level set representation when there are simple geometrical and topological constraints we propose a logarithmic barrier penalty which acts to enforce the constraints leading to an approximate solution to shape design problems

reinitialization free level set evolution via reaction diffusion this paper presents a novel reaction diffusion rd method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution lse a diffusion term is introduced into lse resulting in an rd lse equation from which a piecewise constant solution can be derived in order to obtain a stable numerical solution from the rd based lse we propose a two step splitting method to iteratively solve the rd lse equation where we first iterate the lse equation then solve the diffusion equation the second step regularizes the level set function obtained in the first step to ensure stability and thus the complex and costly reinitialization procedure is completely eliminated from lse by successfully applying diffusion to lse the rd lse model is stable by means of the simple finite difference method which is very easy to implement the proposed rd method can be generalized to solve the lse for both variational level set method and partial differential equation based level set method the rd lse method shows very good performance on boundary antileakage the extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed rd lse approach

the accurate inversion of vandermonde matrices two modifications are suggested in the commonly used algorithms such as the o n 2 parker algorithm for the explicit inversion of vandermonde matrices resulting in an algorithm whose accuracy is no worse than those of the existing algorithms but which is significantly more accurate in many pathological situations the first modification circumvents to some extent the subtraction of two big like signed numbers which in turn reduces round off errors while the second modification exploits the structure of the inverse and uses two recursive formulae instead of one to bring about an increase in accuracy numerical results are presented to demonstrate the increase in accuracy that results from these two modifications although the modified algorithm is always at least as accurate as the parker algorithm it does unfortunately involve an increase in complexity from o n 2 to o n 3 so that use of this algorithm to increase the relative accuracy is recommended only in situations where the standard algorithms fail to yield accurate results

gradient augmented level set method for phase change simulations abstract a numerical method for the simulation of two phase flow with phase change based on the gradient augmented level set gals strategy is presented sharp capturing of the vaporization process is enabled by i identification of the vapor liquid interface t at the subgrid level ii discontinuous treatment of thermal physical properties except for and iii enforcement of mass momentum and energy jump conditions where the gradients of the dependent variables are obtained at t and are consistent with their analytical expression i e no local averaging is applied treatment of the jump in velocity and pressure at t is achieved using the ghost fluid method the solution of the energy equation employs the sub grid knowledge of t to discretize the temperature laplacian using second order one sided differences i e the numerical stencil completely resides within each respective phase to carefully evaluate the benefits or disadvantages of the gals approach the standard level set method is implemented and compared against the gals predictions the results show the expected trend that interface identification and transport are predicted noticeably better with gals over the standard level set this benefit carries over to the prediction of the laplacian and temperature gradients in the neighborhood of the interface which are directly linked to the calculation of the vaporization rate however when combining the calculation of interface transport and reinitialization with two phase momentum and energy the benefits of gals are to some extent neutralized and the causes for this behavior are identified and analyzed overall the additional computational costs associated with gals are almost the same as those using the standard level set technique

a positivity preserving and conservative variational scheme for phase field modeling of two phase flows abstract we present a positivity preserving variational scheme for the phase field modeling of incompressible two phase flows with high density ratio the variational finite element technique relies on the allen cahn phase field equation for capturing the phase interface on a fixed eulerian mesh with mass conservative and energy stable discretization the mass conservation is achieved by enforcing a lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase field equation to make the scheme energy stable in a variational sense we discretize the spatial part of the lagrange multiplier in the phase field equation by the mid point approximation the proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second order accuracy of both spatial and temporal discretizations we integrate the allen cahn phase field equation with the incompressible navier stokes equations for modeling a broad range of two phase flow and fluid fluid interface problems the coupling of the implicit discretizations corresponding to the phase field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase indicator field we perform a standalone test to verify the accuracy and stability of the allen cahn two phase solver we examine the convergence and accuracy properties of the coupled phase field solver through the standard benchmarks of the laplace young law and a sloshing tank problem two and three dimensional dam break problems are simulated to assess the capability of the phase field solver for complex air water interfaces involving topological changes on unstructured meshes finally we demonstrate the phase field solver for a practical offshore engineering application of wave structure interaction

a high order and interface preserving discontinuous galerkin method for level set reinitialization abstract a high order numerical method for interface preserving level set reinitialization is presented in this paper in the interface cells the gradient of the level set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved in the non interface cells we compute the gradient of the level set function by solving a hamilton jacobi equation as a conservation law system using the discontinuous galerkin method following the work by hu and shu siam j sci comput 21 1999 660 690 the missing constant is then recovered by the continuity of the level set function while taking into account the characteristics to handle highly distorted initial conditions we develop a hybrid numerical flux that combines the lax friedrichs flux and the penalty flux our method is stable for non trivial test cases and handles singularities away from the interface very well when derivative singularities are present on the interface a second derivative limiter is designed to suppress the oscillations at least n 1 th order accuracy in the interface cells and nth order in the whole domain are observed for smooth solutions when nth degree polynomials are used two dimensional test cases are presented to demonstrate superior properties such as accuracy long term stability interface preserving capability and easy treatment of contact lines we also show some preliminary results on the pinch off process of a pendant drop where topological changes of the fluid interface are involved our method is readily extendable to three dimensions and adaptive meshes

geometric integration over irregular domains with application to level set methods we present a geometric approach for calculating integrals over irregular domains described by a level set function this procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms this approach produces results that are second order accurate and robust to the perturbation of the interface location on the grid moreover since we use a cell wise approach this procedure can be easily extended to quadtree and octree grids we demonstrate the second order accuracy and the robustness of the method in two and three spatial dimensions

a robust incompressible navier stokes solver for high density ratio multiphase flows abstract this paper presents a robust adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined staggered cartesian grids that adapt to the evolving interfaces and track regions of high vorticity the algorithm combines the interface capturing level set method with a variable coefficient incompressible navier stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude the discretization approach ensures second order pointwise accuracy for both velocity and pressure with several physical boundary treatments including velocity and traction boundary conditions the paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver to ensure the stability of the numerical scheme in the presence of high density and viscosity ratios we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations this consistency is achieved by solving an additional mass balance equation which we approximate via a strong stability preserving runga kutta time integrator and by employing the same mass flux obtained from the mass equation in the discrete momentum equation the scheme uses higher order total variation diminishing tvd and convection boundedness criterion cbc satisfying limiter to avoid numerical fluctuations in the transported density field the high order bounded convective transport is done on a dimension by dimension basis which makes the scheme simple to implement we also demonstrate through several test cases that the lack of consistent mass and momentum transport in non conservative formulations which are commonly used in practice or the use of non cbc satisfying limiters can yield very large numerical error and very poor accuracy for convection dominant high density ratio flows our numerical scheme also uses well balanced surface tension and gravity force discretizations in the hydrostatic limit we show that the well balanced formulation mitigates spurious flow currents and achieves discrete force balance between the pressure gradient and surface tension or gravity

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions

an efficient mass preserving interface correction level set ghost fluid method for droplet suspensions under depletion forces abstract aiming for the simulation of colloidal droplets in microfluidic devices we present here a numerical method for two fluid systems subject to surface tension and depletion forces among the suspended droplets the algorithm is based on an efficient solver for the incompressible two phase navier stokes equations and uses a mass conserving level set method to capture the fluid interface the four novel ingredients proposed here are firstly an interface correction level set icls method global mass conservation is achieved by performing an additional advection near the interface with a correction velocity obtained by locally solving an algebraic equation which is easy to implement in both 2d and 3d secondly we report a second order accurate geometric estimation of the curvature at the interface and thirdly the combination of the ghost fluid method with the fast pressure correction approach enabling an accurate and fast computation even for large density contrasts finally we derive a hydrodynamic model for the interaction forces induced by depletion of surfactant micelles and combine it with a multiple level set approach to study short range interactions among droplets in the presence of attracting forces

electrohydrodynamics of three dimensional vesicles a numerical approach a three dimensional numerical model of vesicle electrohydrodynamics in the presence of dc electric fields is presented the vesicle membrane is modeled as a thin capacitive interface through the use of a semi implicit gradient augmented level set jet scheme the enclosed volume and surface area are conserved both locally and globally by a new navier stokes projection method the electric field calculations explicitly take into account the capacitive interface by an implicit immersed interface method formulation which calculates the electric potential field and the transmembrane potential simultaneously the results match well with previously published experimental analytic and two dimensional computational works

a novel active contour model for image segmentation using distance regularization term abstract in this paper a novel active contour model r drlse model based on level set method is proposed for image segmentation the r drlse model is a variational level set approach that utilizes the region information to find image contours by minimizing the presented energy functional to avoid the time consuming re initialization step the distance regularization term is used to penalize the deviation of the level set function from a signed distance function the numerical implementation scheme of the model can significantly reduce the iteration number and computation time the results of experiments performed on some synthetic and real images show that the r drlse model is effective and efficient in particular our method has been applied to mr kidney image segmentation with desirable results

fast tree based redistancing for level set computations level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface such as the signed distance to the interface this paper presents efficient algorithms for this redistancing problem the algorithms use quadtrees and triangulation to compute global approximate signed distance functions a quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants given a polygonal interface with n elements our algorithms run in o n space and o n log n time two dimensional numerical results show they are highly efficient in practice

computing minimal surfaces via level set curvature flow given a fixed closed curve or curves in r sp3 there is an associated problem of finding a surface with mean curvature zero which has that curve or curves as its boundary in this paper a new approach to numerically solving the minimal surface problem is introduced the surface is represented as a level set of a global function phi r sp3 to r the family of surfaces evolve in time according to mean curvature flow a solution is found when a steady state condition is achieved a new system of interpolatory boundary conditions are used to maintain the connection between the surface and the fixed boundary contour several computer surfaces are shown including examples of topological changes

a hybrid level set volume constraint method for incompressible two phase flow a second order hybrid level set volume constraint method hlsvc for numerically simulating deforming boundaries is presented we combine the hlsvc interface advection algorithm with a two phase flow solver in order to numerically capture deforming bubbles and drops whose actual volume s fluctuate about fixed target volume s three novel developments are described 1 a new method for enforcing a volume constraint in which the number of bubbles and drops can change due to merging or splitting 2 a new second order semi lagrangian narrow band level set reinitialization algorithm and 3 validation of a two phase flow numerical method by comparison with linear stability analysis results for a co flowing liquid jet in gas the new interface capturing method is tested on benchmark problems in which the velocity is prescribed passive advection of interfaces and in which the velocity is determined by the incompressible navier stokes equations for two phase flow the error in interface position when using the hybrid level set volume constraint method is reported for many benchmark problems in polar coordinates cylindrical coordinates and on an adaptive grid in which one criteria of adaptivity is the magnitude of the interface curvature

an alternative numerical implementation of the burnett family of acoustic infinite elements in a series of pioneering papers burnett and holford presented the formulation and numerical implementation of three dimensional acoustic infinite elements for modeling acoustic fields in exterior unbounded domains surrounding a structure they are all based on a multipole expansion outside a closed coordinate surface that circumscribes the structure the multipole expansion has a radial variable r and two angular variables that describe the surfaces r const in the finite element representation the pressure field is discretized using a product of radial and angular shape functions the angular variables are discretized directly using nodal angular values however in some cases and especially in problems involving symmetry nodes are located in certain positions where one of the angular variables is undefined thus leading to errors in the value of the angular variable in the interior of the elements connected to these nodes and hence in the computed acoustic field to overcome this problem we propose an alternative numerical formulation in this work which is based on the interpolation of the position vector expressed in the global cartesian coordinate system as in conventional finite elements instead of the interpolation of angular variables we present numerical solutions for a few benchmark problems using the proposed method

the formulae and algorithms for lagrange power basis transformation and lagrange newton transformation abstract the interpolation polynomials based on lagrange newton and power basis play important roles in applied mathematics computing method and many other emerging applications in this paper we present some coordinate transformation formulae and algorithms as demonstrated below firstly we put forward the formulae of the lagrange power basis transformation and its inverse transformation and as a byproduct we provide a new method to arrive at the inversion of the vandermonde matrix secondly we give the formulae of lagrange newton transformation and its inverse transformation moreover we construct related algorithms of lagrange power basis transformation lagrange newton transformation and their inverse transformations