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

three dimensional non planar crack growth by a coupled extended finite element and fast marching method a numerical technique for non planar three dimensional linear elastic crack growth simulations is proposed this technique couples the extended finite element method x fem and the fast marching method fmm in crack modeling using x fem the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two dimensional asymptotic crack tip displacement fields the initial crack geometry is represented by two level set functions and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation crack modeling is performed without the need to mesh the crack and crack propagation is simulated without remeshing crack growth is conducted using fmm unlike a level set formulation for interface capturing no iterations nor any time step restrictions are imposed in the fmm planar and non planar quasi static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique copyright 2008 john wiley sons ltd

an adaptive level set segmentation on a triangulated mesh level set methods offer highly robust and accurate methods for detecting interfaces of complex structures efficient techniques are required to transform an interface to a globally defined level set function in this paper a novel level set method based on an adaptive triangular mesh is proposed for segmentation of medical images special attention is paid to an adaptive mesh refinement and redistancing technique for level set propagation in order to achieve higher resolution at the interface with minimum expense first a narrow band around the interface is built in an upwind fashion an active square technique is used to determine the shortest distance correspondence sdc for each grid vertex simultaneously we also give an efficient approach for signing the distance field then an adaptive improvement algorithm is proposed which essentially combines two basic techniques a long edge based vertex insertion strategy and a local improvement these guarantee that the refined triangulation is related to features along the front and has elements with appropriate size and shape which fit the front well we propose a short edge elimination scheme to coarsen the refined triangular mesh in order to reduce the extra storage finally we reformulate the general evolution equation by updating 1 the velocities and 2 the gradient of level sets on the triangulated mesh we give an approach for tracing contours from the level set on the triangulated mesh given a two dimensional image with n grids along a side the proposed algorithms run in o kn time at each iteration quantitative analysis shows that our algorithm is of first order accuracy and when the interface fitted property is involved in the mesh refinement both the convergence speed and numerical accuracy are greatly improved we also analyze the effect of redistancing frequency upon convergence speed and accuracy numerical examples include the extraction of inner and outer surfaces of the cerebral cortex from magnetic resonance imaging brain images

a numerical method for computing minimal surfaces in arbitrary dimension in this paper we propose a numerical method for computing minimal surfaces with fixed boundaries the level set method is used to evolve a codimension 1 surface with fixed codimension 2 boundary in r n under mean curvature flow for n 3 the problem has been approached in d l chopp 1993 and l t cheng d l chopp computing minimal surfaces via level set curvature flow j comput phys 106 1 1993 77 91 and l t cheng the level set method applied to geometrically based motion materials science and image processing ucla cam report 00 20 using the level set method but with a more complicated boundary conditions the method we present can be generalized straightforward to arbitrary dimension and the framework in which it is presented is dimension independent examples are shown for n 2 3 4

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 level set method for simulating island coarsening modeling of microstructural evolution during thin film deposition requires a knowledge of several key activation energies surface diffusion island edge atom diffusion adatom migration over descending step edges etc these and other parameters must be known as a function of crystal orientation in order to generate values for these parameters we have developed a numerical simulation in tandem with physical experiments by tuning the simulation to the results from experiments we can extract and verify approximate values for these parameters the numerical method we use is based upon the level set method our model is a continuum model in directions parallel to the crystal facet and resolves each discrete atomic layer in the normal direction the model includes surface diffusion step edge dynamics and attachment detachment rates all of which may depend upon the local geometry of the step edge the velocity field for advancing the island edges in the level set framework is generated by computing the equilibrium adatom density on the flat terraces resulting in laplace s equation with mixed boundary conditions at the step edges we have turned to the finite element method for solving this equation which results in very good agreement with analytically known solutions and with experiment

a high order elliptic pde based level set reinitialisation method using a discontinuous galerkin discretisation abstract in this paper an efficient high order accurate level set reinitialisation method is proposed based on the elliptic reinitialisation method basting and kuzmin 2013 1 which is discretised spatially using the discontinuous galerkin dg symmetric interior penalty method sipg in order to achieve this a number of improvements have been made to the elliptic reinitialisation method including reformulation of the underlying minimisation problem driving the solution adoption of a lagrange multiplier approach for enforcing a dirichlet boundary condition on the implicit level set interface and adoption of a narrow band approach numerical examples confirm the high order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the sipg method that is h p 1 and h p in the l 2 and dg norms respectively furthermore the degree to which the level set function satisfies the eikonal equation improves proportionally to h p and the often ignored homogeneous dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders

level set based topology optimization for electromagnetic dipole antenna design this paper presents a level set framework for a typical electromagnetic design problem of dipole antenna in this study the geometrical configuration of an antenna is represented by the zero level contour of a higher dimensional level set function the governing equation for the induced current flow on a metal surface is the electric field integral equation efie which takes into account the electric component of the incident wave the design objective is formulated in terms of the surface current and incident electric field the normal velocity of the level set model which reflects the sensitivity of the objective function is derived from the adjoint variable method and shape derivative by optimizing the objective function the area with the highest current density to which the voltage feeding should be applied can be reshaped the advantages of adopting the level set technique for electromagnetic design lie in its capacity for capturing sophisticated topological changes and facilitation in mathematical representation of the design configuration the demonstrative examples of dipole antenna design show that the level set method results in a fairly smooth optimization process where the vacuum metal interface gradually attains its optimal configuration a series of design cases with self adjoint and non self adjoint sensitivity analyses are studied and compared to the benchmarking problems in dipole antenna

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

a lagrangian particle level set method we present a novel particle level set method for capturing interfaces the level set equation is solved in a lagrangian frame using particles that carry the level set information a key aspect of the method involves a consistent remeshing procedure for the regularization of the particle locations when the particle map gets distorted by the advection field the lagrangian description of the level set method is inherently adaptive and exact in the case of solid body motions the efficiency and accuracy of the method is demonstrated in several benchmark problems in two and three dimensions involving pure advection and curvature induced motion of the interface the simplicity of the particle description is shown to be well suited for real time simulations of surfaces involving cutting and reconnection as in virtual surgery environments

minimal surface computation using a finite element method on an embedded surface we suggest a finite element method for finding minimal surfaces based on computing a discrete laplace beltrami operator operating on the coordinates of the surface the surface is a discrete repres

minimal surfaces based object segmentation a geometric approach for 3d object segmentation and representation is presented the segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3d image the model is based on curvature motion and the computation of surfaces with minimal areas better known as minimal surfaces the space where the surfaces are computed is induced from the 3d image volumetric data in which the objects are to be detected the model links between classical deformable surfaces obtained via energy minimization and intrinsic ones derived from curvature based flows the new approach is stable robust and automatically handles changes in the surface topology during the deformation

simulations of solitons using quadratic spline finite elements abstract a numerical solution to the korteweg de vries equation is obtained using the bubnov galerkin method with quadratic spline finite elements the element matrices are set up algebraically using reduce a linear stability analysis of the scheme shows the method to be unconditionally stable classical problems concerning the development motion and interaction of solitons are used to validate the method the numerical scheme is compared with published algorithms and shown to be both accurate and efficient

a structural optimization method based on the level set method using a new geometry based re initialization scheme structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function and updated by solving the so called hamilton jacobi equation based on a eulerian coordinate system these new methods can allow topological alterations such as the number of holes during the optimization process whereas the boundaries of the target structure are clearly defined however the re initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures in this paper we propose a new structural optimization method based on the level set method using a new geometry based re initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the finite element method the stiffness maximization eigenfrequency maximization and eigenfrequency matching problems are considered as optimization problems several design examples are presented to confirm the usefulness of the proposed method copyright 2010 john wiley sons ltd

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

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

coarse to fine boundary location with a som like method a coarse to fine boundary location with a self organizing map som like method is proposed in this paper inspired from the conventional som and universal gravitation given a small quantity of supervision seeds from the desired boundaries neurons are used to evolve to the desired boundaries in a coarse to fine framework the major components of this framework are the designs of union action and evolving rate in the course of neuron evolution the union actions acting on these neurons will offer them the evolving directions also controlled by the corresponding referenced gradients the neurons evolving rates are adaptively adjusted at different positions with the union actions and evolving rates the neurons will evolve with appropriate manners to expand the set of feature points on the desired boundaries the newly expanded feature points will cause the generation updates for feature points and neurons and offer new information to guide the new generation of neurons to the boundaries what is more the proposed multiround evolution is as well a coarse to fine way for boundary location experiments and comparisons show that the proposed method performs well in complex long concavities inhomogeneous and weak boundary location with good initialization flexibility

shape from shading level set propagation and viscosity solutions we present a new implementation of an algorithm aimed at recovering a 3d shape from its 2d gray level picture in order to reconstruct the shape of the object an almost arbitrarily initialized 3d function is propagated on a rectangular grid so that a level set of this function tracks the height contours of the shape the method imports techniques from differential geometry fluid dynamics and numerical analysis and provides an accurate shape from shading algorithm the method solves some topological problems and gracefully handles cases of non smooth surfaces that give rise to shocks in the propagating contours real and synthetic images of 3d profiles were submitted to the algorithm and the reconstructed surfaces are presented demonstrating the effectiveness of the proposed method

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

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

level set reinitialization at a contact line when a level set signed distance function is reinitialized in the vicinity of a contact line there is a blind spot that prevents an accurate reconstruction of a signed distance function the numerical method can create parasitic velocity currents near this region if additional contact line physics are included the parasitic velocity currents would pollute the solution and alter the physical behavior in this study a modified reinitialization routine is proposed that combines the standard hamilton jacobi equation with a relaxation equation for those grid cells along a wall in the blind spot two test cases an angled fluid wedge zero curvature and a circular fluid arc constant curvature are used to evaluate the numerical error induced by different methods the proposed method has less numerically induced interface distortion than other techniques examined furthermore this routine can be easily extended to three dimensions drops sliding on a wall are simulated in both two and three dimensions to demonstrate the advantages of this method a spreading fluid interface further shows that this method allows contact lines to merge naturally

fast geodesic active contours we use an unconditionally stable numerical scheme to implement a fast version of the geodesic active contour model the proposed scheme is useful for object segmentation in images like tracking moving objects in a sequence of images the method is based on the weickert romeney viergever additive operator splitting aos scheme it is applied at small regions motivated by the adalsteinsson sethian level set narrow band approach and uses sethian s 1996 fast marching method for re initialization experimental results demonstrate the power of the new method for tracking in color movies

the vector distance functions we present a novel method for representing and evolving objects of arbitrary dimension the method called the vector distance function vdf method uses the vector that connects any point in space to its closest point on the object it can deal with smooth manifolds with and without boundaries and with shapes of different dimensions it can be used to evolve such objects according to a variety of motions including mean curvature if discontinuous velocity fields are allowed the dimension of the objects can change the evolution method that we propose guarantees that we stay in the class of vdf s and therefore that the intrinsic properties of the underlying shapes such as their dimension curvatures can be read off easily from the vdf and its spatial derivatives at each time instant the main disadvantage of the method is its redundancy the size of the representation is always that of the ambient space even though the object we are representing may be of a much lower dimension this disadvantage is also one of its strengths since it buys us flexibility

gas bubble with a moving contact line rising in an inclined channel at finite reynolds number abstract the dynamics of a gas bubble with a moving contact line rising in an inclined channel is investigated the solution of this free boundary problem is determined numerically by using a level set method coupled with a finite difference solution of the navier stokes equations results are presented as a function of reynolds number re bond number and the angle of inclination steadily rising bubbles are found for small values of both the reynolds and bond number with increasing reynolds number the steady solution first bifurcates to a time periodic oscillation and then the numerical results imply period doubling large values of reynolds or bond number cause the bubble to either detach from the wall or to rupture

a discontinuous galerkin method for strain gradient dependent damage study of interpolations and convergence abstract a discontinuous galerkin method has been developed for strain gradient dependent damage the strength of this method lies in the fact that it allows the use of c 0 interpolation functions for continuum theories involving higher order derivatives while in a conventional framework at least c 1 interpolations are required the discontinuous galerkin formulation thereby offers significant potential for engineering computations with strain gradient dependent models when using basis functions with a low degree of continuity jump conditions arise at element edges which are incorporated in the weak form in addition to the formulation itself a detailed study of the convergence properties of the method for various element types is presented and an error analysis is undertaken numerical results of some one and two dimensional problems are presented and discussed

parallel re initialization of level set functions on distributed unstructured tetrahedral grids level set functions are employed to track interfaces in various application areas including simulation of two phase flows and image segmentation often a re initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function in this note we present a parallel algorithm for re initializing level set functions on unstructured three dimensional tetrahedral grids the main idea behind this new domain decomposition approach is to combine a parallel brute force re initializing algorithm with an efficient way to compute distances between the interface and grid points time complexity and error analysis of the algorithm are investigated detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes

geodesic active contours and level sets for the detection and tracking of moving objects this paper presents a new variational framework for detecting and tracking multiple moving objects in image sequences motion detection is performed using a statistical framework for which the observed interframe difference density function is approximated using a mixture model this model is composed of two components namely the static background and the mobile moving objects one both components are zero mean and obey laplacian or gaussian law this statistical framework is used to provide the motion detection boundaries additionally the original frame is used to provide the moving object boundaries then the detection and the tracking problem are addressed in a common framework that employs a geodesic active contour objective function this function is minimized using a gradient descent method a new approach named hermes is proposed which exploits aspects from the well known front propagation algorithms and compares favorably to them very promising experimental results are provided using real video sequences

a nonparametric statistical method for image segmentation using information theory and curve evolution in this paper we present a new information theoretic approach to image segmentation we cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities subject to a constraint on the total length of the region boundaries we assume that the probability densities associated with the image pixel intensities within each region are completely unknown a priori and we formulate the problem based on nonparametric density estimates due to the nonparametric structure our method does not require the image regions to have a particular type of probability distribution and does not require the extraction and use of a particular statistic we solve the information theoretic optimization problem by deriving the associated gradient flows and applying curve evolution techniques we use level set methods to implement the resulting evolution the experimental results based on both synthetic and real images demonstrate that the proposed technique can solve a variety of challenging image segmentation problems furthermore our method which does not require any training performs as good as methods based on training

weighted minimal hypersurface reconstruction many problems in computer vision can be formulated as a minimization problem for an energy functional if this functional is given as an integral of a scalar valued weight function over an unknown hypersurface then the sought after minimal surface can be determined as a solution of the functional s euler lagrange equation this paper deals with a general class of weight functions that may depend on surface point coordinates as well as surface orientation we derive the euler lagrange equation in arbitrary dimensional space without the need for any surface parameterization generalizing existing proofs our work opens up the possibility of solving problems involving minimal hypersurfaces in a dimension higher than three which were previously impossible to solve in practice we also introduce two applications of our new framework we show how to reconstruct temporally coherent geometry from multiple video streams and we use the same framework for the volumetric reconstruction of refractive and transparent natural phenomena bodies of flowing water

regularized laplacian zero crossings as optimal edge integrators we view the fundamental edge integration problem for object segmentation in a geometric variational framework first we show that the classical zero crossings of the image laplacian edge detector as suggested by marr and hildreth inherently provides optimal edge integration with regard to a very natural geometric functional this functional accumulates the inner product between the normal to the edge and the gray level image gradient along the edge we use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models we also incorporate a 2d geometric variational explanation to the haralick edge detector into the geometric active contour framework

curvature flow and entropy conditions applied to grid generation we describe a numerical technique to generate logically rectangular body fitted interior and exterior grids the technique is based on solving a hamilton jacobi type equation for a propagating level set function using techniques borrowed from hyperbolic conservation laws coordinate grid lines are kept smooth through curvature terms which regularize the equation of motion and upwind difference schemes which satisfy the correct entropy conditions of front propagation the resulting algorithm can be used to generate two and three dimensional interior and exterior grids around reasonably complex bodies which may contain sharp corners and significant variations in curvature the technique may also be easily extended to problems of boundary fitted moving grids

new methods for the computer assisted 3 d reconstruction of neurons from confocal image stacks exact geometrical reconstructions of neuronal architecture are indispensable for the investigation of neuronal function neuronal shape is important for the wiring of networks and dendritic architecture strongly affects neuronal integration and firing properties as demonstrated by modeling approaches confocal microscopy allows to scan neurons with submicron resolution however it is still a tedious task to reconstruct complex dendritic trees with fine structures just above voxel resolution we present a framework assisting the reconstruction user time investment is strongly reduced by automatic methods which fit a skeleton and a surface to the data while the user can interact and thus keeps full control to ensure a high quality reconstruction the reconstruction process composes a successive gain of metric parameters first a structural description of the neuron is built including the topology and the exact dendritic lengths and diameters we use generalized cylinders with circular cross sections the user provides a rough initialization by marking the branching points the axes and radii are fitted to the data by minimizing an energy functional which is regularized by a smoothness constraint the investigation of proximity to other structures throughout dendritic trees requires a precise surface reconstruction in order to achieve accuracy of 0 1 microm and below we additionally implemented a segmentation algorithm based on geodesic active contours that allow for arbitrary cross sections and uses locally adapted thresholds in summary this new reconstruction tool saves time and increases quality as compared to other methods which have previously been applied to real neurons

a variational level set method for the topology optimization of steady state navier stokes flow the smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density based approaches though well established in structural designs inadequate this paper presents a level set method for topology optimization of steady state navier stokes flow subject to a specific fluid volume constraint the solid fluid interface is implicitly characterized by a zero level contour of a higher order scalar level set function and can be naturally transformed to other configurations as its host moves a variational form of the cost function is constructed based upon the adjoint variable and lagrangian multiplier techniques to satisfy the volume constraint effectively the lagrangian multiplier derived from the first order approximation of the cost function is amended by the bisection algorithm the procedure allows evolving initial design to an optimal shape and or topology by solving the hamilton jacobi equation two classes of benchmarking examples are presented in this paper 1 periodic microstructural material design for the maximum permeability and 2 topology optimization of flow channels for minimizing energy dissipation a number of 2d and 3d examples well demonstrated the feasibility and advantage of the level set method in solving fluid solid shape and topology optimization problems

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

motion of curves constrained on surfaces using a level set approach the level set method has been successfully applied to a variety of problems that deal with curves in r2 or surfaces in r3 we present here a combination of these two cases creating a level set representation for curves constrained to lie on surfaces we study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves i e merging and breaking to occur applications include finding geodesic curves and shortest paths and curve shortening on surfaces further applications can be arrived at by extending those for curves moving in r2 to surfaces the problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces the method can also be easily extended to higher dimensions

on the reinitialization procedure in a narrow band locally refined level set method for interfacial flows a local level set algorithm for simulating interfacial flows described by the two dimensional incompressible navier stokes equations is presented the governing equations are solved using a finite difference discretization on a cartesian grid and a second order approximate projection method the level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid which is reconstructed every time step and refined using a simple uniform cell splitting operation within the band instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band the influence of different pde based reinitialization strategies on the accuracy of the results is investigated the ability of the proposed method to accurately compute interfacial flows is discussed using different tests namely the advection of a circle of fluid in two different time reversed vortex flows the advection of zalesak s rotating disk the propagation of small amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water and a classical test of rayleigh taylor instability with and without surface tension effects the interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method copyright 2005 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

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 topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method

an assumed gradient finite element method for the level set equation the level set equation is a non linear advection equation and standard finite element and finite difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution we recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front interface being captured as with the original level set equation the use of an extensional velocity helps maintain this signed distance function for some interface evolution problems this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve further we find that sufficient accuracy is available through a standard galerkin formulation without any stabilization or discontinuity capturing terms several numerical experiments are conducted to assess the ability of the proposed assumed gradient level set method to capture the correct solution particularly in the presence of discontinuities in the extensional velocity or level set gradient we examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a hamilton jacobi equation with convex non convex hamiltonian importantly discretizations based on structured and unstructured finite element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well copyright 2005 john wiley sons ltd

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

a second order shape optimization approach for image segmentation the problem of segmentation of a given image using the active contour technique is considered an abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented the speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or newton type directions for the underlying optimization problem the newton type speed function is found by solving an elliptic problem on the current active contour in every time step numerical experiments comparing the classical gradient method with newton s method are presented

an oriented particle level set method based on surface coordinates in this work we present a new method for tracking evolving interfaces like the adaptive oriented particle level set method aopls ianniello and mascio 2010 it uses a combination of unconnected particles containing geometric information and a level set function to track the interface in the present method the geometric information solved for in each particle is a set of tangent vectors and the corresponding curvature tensor of the interface compared to the aopls method the amount of geometric data stored in each particle is roughly halved while still retaining a description of the interface that includes both orientation and curvature a mechanism for ensuring a sufficiently dense distribution of particles on the interface is also included from the information stored in each particle a continuous representation of the interface is obtained by transferring the information to a level set function stored on an eulerian grid unlike the aopls method the present method takes the curvature into account when the zero level set is constructed this results in significant improvements in regions of high curvature two different formulations of the new method have been explored in the first the tangent vectors deform purely according to the velocity field in the second the tangent vectors are forced to coincide with the principal directions of the surface both formulations are shown to be applicable