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

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 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

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

the spectral connection matrix for classical orthogonal polynomials of a single parameter abstract in this paper we study the so called connection problem of given a polynomial expressed in the basis of one set of orthogonal polynomials computing the coefficients with respect to a different set of orthogonal polynomials we restrict our current study to the classical real orthogonal polynomials of the hermite laguerre and gegenbauer including legendre families the computational tool for this work is the class of quasiseparable matrices while the relationships between orthogonal polynomials and rank structured matrices such as quasiseparable matrices are very well known in this paper we investigate a more recently considered relationship we prove that while the matrix that implements the desired connection is not itself quasiseparable it is an eigenvector matrix of one that is quasiseparable we suggest to refer to this structured matrix as the spectral connection matrix finally we present a simple algorithm exploiting the computationally favorable properties of quasiseparable matrices to implement the desired change of basis by exploiting the quasiseparable structure this algorithm enjoys an order of magnitude reduction of complexity as compared to the simple method of inverting the connection matrix directly while not the focus of the paper some very preliminary numerical experimentation shows some positive indications that even with this reduction in complexity the accuracy of the resulting change of basis algorithm is comparable to that of inverting the connection matrix directly

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

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 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 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

implicit mesh discontinuous galerkin methods and interfacial gauge methods for high order accurate interface dynamics with applications to surface tension dynamics rigid body fluid structure interaction and free surface flow part i abstract in this two part paper a high order accurate implicit mesh discontinuous galerkin dg framework is developed for fluid interface dynamics facilitating precise computation of interfacial fluid flow in evolving geometries the framework uses implicitly defined meshes wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries and allows physically prescribed interfacial jump conditions to be imposed or captured with high order accuracy part one discusses the design of the framework including i high order quadrature for implicitly defined elements and faces ii high order accurate discretisation of scalar and vector valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient leading to optimal order accuracy in the maximum norm and discrete linear systems that are symmetric positive semi definite iii the design of incompressible fluid flow projection operators which except for the influence of small penalty parameters are discretely idempotent and iv the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method also discussed is a variety of aspects relating to moving interfaces including v dg discretisations of the level set method on implicitly defined meshes vi transferring state between evolving implicit meshes vii preserving mesh topology to accurately compute temporal derivatives viii high order accurate reinitialisation of level set functions and ix the integration of adaptive mesh refinement in part two several applications of the implicit mesh dg framework in two and three dimensions are presented including examples of single phase flow in nontrivial geometry surface tension driven two phase flow with phase dependent fluid density and viscosity rigid body fluid structure interaction and free surface flow a class of techniques known as interfacial gauge methods is adopted to solve the corresponding incompressible navier stokes equations which compared to archetypical projection methods have a weaker coupling between fluid velocity pressure and interface position and allow high order accurate numerical methods to be developed more easily convergence analyses conducted throughout the work demonstrate high order accuracy in the maximum norm for all of the applications considered for example fourth order spatial accuracy in fluid velocity pressure and interface location is demonstrated for surface tension driven two phase flow in 2d and 3d specific application examples include vortex shedding in nontrivial geometry capillary wave dynamics revealing fine scale flow features falling rigid bodies tumbling in unsteady flow and free surface flow over a submersed obstacle as well as high reynolds number soap bubble oscillation dynamics and vortex shedding induced by a type of plateau rayleigh instability in water ripple free surface flow these last two examples compare numerical results with experimental data and serve as an additional means of validation they also reveal physical phenomena not visible in the experiments highlight how small scale interfacial features develop and affect macroscopic dynamics and demonstrate the wide range of spatial scales often at play in interfacial fluid flow

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

multi region active contours with a single level set function segmenting an image into an arbitrary number of coherent regions is at the core of image understanding many formulations of the segmentation problem have been suggested over the past years these formulations include among others axiomatic functionals which are hard to implement and analyze and graph based alternatives which impose a non geometric metric on the problem we propose a novel method for segmenting an image into an arbitrary number of regions using an axiomatic variational approach the proposed method allows to incorporate various generic region appearance models while avoiding metrication errors in the suggested framework the segmentation is performed by level set evolution yet contrarily to most existing methods here multiple regions are represented by a single non negative level set function the level set function evolution is efficiently executed through the voronoi implicit interface method for multi phase interface evolution the proposed approach is shown to obtain accurate segmentation results for various natural 2d and 3d images comparable to state of the art image segmentation algorithms

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

continuity of type i intermittency from a measure theoretical point of view we study a simple dynamical system which displays a so called type i intermittency bifurcation we determine the bowen ruelle measure e and prove that the expectation e g of any continuous functiong and the kolmogoroff sinai entropyh e are continuous functions of the bifurcation parameter therefore the transition is continuous from a measure theoretical point of view those results could be generalized to any similar dynamical system

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

high order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles a high order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle by converting the implicitly defined geometry into the graph of an implicitly defined height function the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one dimensional root finding and one dimensional gaussian quadrature the computed quadrature scheme yields strictly positive quadrature weights and inherits the high order accuracy of gaussian quadrature a range of different convergence tests demonstrate orders of accuracy up to 20th order also presented is an application of the quadrature algorithm to a high order embedded boundary discontinuous galerkin method for solving partial differential equations on curved domains

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

threshold dynamics for networks with arbitrary surface tensions we present and study a new algorithm for simulating the n phase mean curvature motion for an arbitrary set of isotropic surface tensions the departure point is the threshold dynamics algorithm of merriman bence and osher for the two phase case a new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of n phases for arbitrary surface tensions and arbitrary isotropic mobilities for a large class of surface tensions the algorithm is shown to be consistent in the sense that at every time step it decreases an energy functional that is an approximation in the sense of gamma convergence of the interfacial energy a broad range of numerical tests shows good convergence properties an important application is the motion of grain boundaries in polycrystalline materials it is also established that the above mentioned large class of surface tensions contains the read shockley surface tensions both in the two dimensional and three dimensional settings 2015 wiley periodicals inc

an iterative algorithm for computing measures of generalized voronoi regions we present and analyze a fast algorithm for directly computing measures of generalized voronoi regions associated with generators of arbitrary codimension the algorithm consists of solving one eikonal equation to construct a kernel based operator whose iteration accumulates mass along the closest generator in particular the algorithm does not require the computation of the voronoi diagram or the gradient of the solution to the eikonal equation the algorithm is shown to be first order and converge very quickly by discretizing the distance to the generators on the grid and not the generators themselves very accurate geometric information is used even for coarse grids several examples are presented including the computation of the population influence associated with the los angeles county highway system the method is also one of the key ingredients for the fast computation of centroidal voronoi tessellations cvts of general rigid objects e g rigid curves and surfaces in higher dimensions w

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

a stable and optimally convergent latin cutfem algorithm for multiple unilateral contact problems in this paper we propose a novel unfitted finite element method for the simulation of multiple body contact the computational mesh is generated independently of the geometry of the interacting solids which can be arbitrarily complex the key novelty of the approach is the combination of elements of the cutfem technology namely the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty type regularisation of discrete operators and the latin hybrid mixed formulation of complex interface conditions furthermore the novel p1 p1 discretisation scheme that we propose for the unfitted latin solver is shown to be stable robust and optimally convergent with mesh refinement finally the paper introduces a high performance 3d level set cutfem framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries with more than two bodies interacting at a single point