high order numerical approximation of minimal surfaces we present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary the algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the laplace beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface for the steady state solution we measure the approximation error in a few cases where the exact solution is known in the framework of parametric interpolation the choice of interpolation points mesh nodes is directly affecting the approximation error and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth in our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases but the rate of convergence greatly differs with different choices of mesh update algorithms the present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces

finite element method for epitaxial growth with thermodynamic boundary conditions we develop an adaptive finite element method for island dynamics in epitaxial growth we study a step flow model which consists of an adatom adsorbed atom diffusion equation on terraces of different height thermodynamic boundary conditions on terrace boundaries including anisotropic line tension and the normal velocity law for the motion of such boundaries determined by a two sided flux together with the one dimensional anisotropic surface diffusion edge diffusion of edge adatoms along the step edges the problem is solved using independent meshes a two dimensional mesh for the adatom diffusion and one dimensional meshes for the boundary evolution a penalty method is used to incorporate the boundary conditions the evolution of the terrace boundaries includes both the weighted anisotropic mean curvature flow and the weighted anisotropic edge diffusion its governing equation is solved by a semi implicit front tracking method using parametric finite elements

convergence of dziuk s linearly implicit parametric finite element method for curve shortening flow convergence of dziuk s fully discrete linearly implicit parametric finite element method for curve shortening flow on the plane still remains open since it was proposed in 1991 though the correspo

geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

space time finite element methods for surface diffusion with applications to the theory of the stability of cylinders a family of space time finite element approximation schemes is presented for the nonlinear partial differential equations governing diffusion in the surface of a body of revolution the schemes share with the partial differential equations properties of conservation of volume and decrease of area numerical experiments are described showing that the result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory examples are given of cases in which longitudinal perturbations with high wave number spectra grow in amplitude after an initial rapid decay followed by a long incubation period and result in break up of the body into a necklace of beads the results of finite element calculations are compared with the predictions of a perturbation analysis

an improved razumikhin type theorem and its applications an improved razumikhin type theorem is established and the advantage of the established theorem for reducing conservatism is discussed for a class of time delay systems a simple asymptotic stability condition is deduced as an application of the established theorem two illustrative examples are given to show the superiority of the results to those in the literature

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

a coupled arbitrary lagrangian eulerian and lagrangian method for computation of free surface flows with insoluble surfactants a finite element scheme to compute the dynamics of insoluble surfactant on a deforming free surface is presented the free surface is tracked by the arbitrary lagrangian eulerian ale approach whereas the surfactant concentration transport equation is approximated in a lagrangian manner since boundary resolved moving meshes are used in the ale approach the surface tension which may be a linear or nonlinear function of surfactant concentration equation of state and the marangoni forces can be incorporated directly into the numerical scheme further the laplace beltrami operator technique which reduces one order of differentiation associated with the curvature is used to handle the curvature approximation a number of 3d axisymmetric computations are performed to validate the proposed numerical scheme an excellent surfactant mass conservation without any additional mass correction scheme is obtained the differences in using a linear and a nonlinear equation of state respectively on the flow dynamics of a freely oscillating droplet are demonstrated

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

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

finite element methods on very large dynamic tubular grid encoded implicit surfaces the simulation of physical processes on interfaces and a variety of applications in geometry processing and geometric modeling are based on the solution of partial differential equations on curved and evolving surfaces frequently an implicit level set type representation of these surfaces is the most effective and computationally advantageous approach this paper addresses the computational problem of how to solve partial differential equations on highly resolved level sets with an underlying very high resolution discrete grid these high resolution grids are represented in a very efficient dynamic tubular grid encoding format for a narrow band a reaction diffusion model on a fixed surface and surface evolution driven by a nonlinear geometric diffusion approach by isotropic or truly anisotropic curvature motion are investigated as characteristic model problems the proposed methods are based on semi implicit finite element discretizations directly on these narrow bands require only standard numerical quadrature and allow for large time steps to combine large time steps with a very thin and thus storage inexpensive narrow band suitable transparent boundary conditions on the boundary of the narrow band and a nested iteration scheme in each time step are investigated this nested iteration scheme enables the discrete interfaces to move in a single time step significantly beyond the domain of the narrow band of the previous time step furthermore algorithmic tools are provided to assemble finite element matrices and to apply matrix vector operators via fast cache coherent access to the dynamic tubular grid encoded data structure the consistency of the presented approach is evaluated and various numerical examples show its application potential

on delay independent stability of large scale systems with time delays a new criterion for delay independent stability of linear large scale time delay systems is deduced by employing an improved razumikhin type theorem and the m matrix properties an illustrative example is given to demonstrate the superiority of the obtained results to those in the literature

curve shortening flow coupled to lateral diffusion we present and analyze a semi discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve more precisely curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field the scheme is based on ideas of dziuk siam j numer anal 36 6 1808 1830 1999 for the curve shortening flow and dziuk and elliott ima j numer anal 27 2 262 292 2007 for the parabolic equation on the moving curve additional estimates are required in order to show convergence most notably with respect to the length element while in dziuk siam j numer anal 36 6 1808 1830 1999 an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation numerical simulation results support the theoretical findings

variational formulations for surface tension capillarity and wetting the interest in the simulation of flows with significant surface tension effects has grown significantly in recent years this has been driven by the substantial advances made in the measurement and manufacturing of microscopic systems since at small length scales surface phenomena are dominant in this article surface tension capillarity and wetting effects are discussed in terms of the virtual work principle and shape sensitivity starting from first principles and arriving at variational formulations that are adequate for numerical treatment by finite elements for example to make the exposition self contained some elements of differential geometry are recalled using a formulation that is fully in cartesian coordinates and may thus be more friendly to readers not familiar with covariant derivatives all necessary results are proved in this cartesian formulation several numerical examples computed with a finite element level set formulation are used to illustrate this challenging physical problem

a simple and efficient method for modeling constant mean curvature surfaces a constant mean curvature cmc surface is a critical point of surface area with respect to variations that preserve the volume bounded by the surface we present a simple and elegant method for constructing a triangle mesh approximation to a cmc surface by minimizing the discretized surface area subject to a fixed volume and other constraints such as fixed boundary curves the key idea is an optimization procedure in which the surface area functional is replaced by a least squares energy functional that is easier to minimize and whose critical points are uniformly parameterized surfaces or more generally surfaces parameterized in accord with an arbitrary density function a good initial solution estimate is not required in fact the input mesh quality may be almost arbitrarily poor the user specified volume is taken to be a shape control parameter and since the volume constraint need not be satisfied precisely it is simply treated as an additional equation in the least squares system the nonlinear

a note on discrete solutions of the plateau problem on demontre des theoremes pour la convergence de solutions discretes du probleme de plateau sous l hypothese que le contour de jordan soit rectifiable

a geometrical mechanical interpretation of gradient weighted moving finite elements the usual explanation of the gradient weighted moving finite element gwmfe method has been in terms of its variational interpretation this paper presents a more intuitive geometrical mechanical interpretation of gwmfe as a balance of forces on the nodes forces concentrated onto the nodes by the laws of leverage it also presents significant simplifications in the internodal viscosity terms for regularization of the nodal movements plus some simple linear internodal tensions for regularization of the long term nodal positioning these simplifications of the regularizations are especially important in two and three space dimensions one of the generalizations which follows from the geometrical mechanical interpretation is a promising but still untested second gwmfe formulation for systems of pdes the original mfe method is seen to be the small slope limit of gwmfe under vertical rescaling reporting on the design and extensive numerical trials of robust and versatile gwmfe systems codes in one and two dimensions is deferred to two forthcoming papers by carlson and the author design and application of a gradient weighted moving finite element code part i in 1 d siam j sci comput to appear and design and application of a gradient weighted moving finite element code part ii in 2 d siam j sci comput to appear here only a few illustrative examples are presented involving motion of surfaces by mean curvature i e by surface tension

the c1 and c2 quasi plateau problems abstract in this paper we study the c k quasi plateau problem to find the parametric surface of minimal area defined on a rectangular parametric domain among all the surfaces which are c k continuous on the border k 1 2 an approach is proposed based on the coons surface and the mra multi resolution analysis formed by the b spline of orders 3 and 4 the coons surface is constructed firstly to satisfy the prescribed border conditions then replacing the area functional with the dirichlet functional and using mra the problem reduces into solving a system of linear equations the linear equations have simple and sparse structure finally the method is concluded into six algorithms according to the different boundary conditions examples are provided to illustrate that the proposed method is effective and flexible

discrete gradient flows for shape optimization and applications we present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation its full discretization and the resulting linear algebraic systems our approach hinges on the following essential features shape differential calculus a semi implicit time discretization and a finite element method for space discretization we use shape differential calculus to express variations of bulk and surface energies with respect to domain changes the semi implicit time discretization allows us to track the domain boundary without an explicit parametrization and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity we propose a schur complement approach to solve the resulting linear systems efficiently we discuss applications of this framework to image segmentation optimal shape design for pde and surface diffusion along with the choice of suitable scalar products in each case we illustrate the method with several numerical experiments some developing pinch off and topological changes in finite time

variational discretization of axisymmetric curvature flows we present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail although numerical methods for geometric flows have been used frequently in axisymmetric settings numerical analysis results so far are rare in this paper we present stability equidistribution existence and uniqueness results for the introduced approximations numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one dimensional problem has to be solved the good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries

parametric fem for geometric biomembranes we consider geometric biomembranes governed by an l 2 gradient flow for bending energy subject to area and volume constraints helfrich model we give a concise derivation of a novel vector formulation based on shape differential calculus and corresponding discretization via parametric fem using quadratic isoparametric elements and a semi implicit euler method we document the performance of the new parametric fem with a number of simulations leading to dumbbell red blood cell and toroidal equilibrium shapes while exhibiting large deformations

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

discrete solution of the plateau problem and its convergence definition d une solution discrete du probleme de plateau et demonstration des theoremes de convergence de la solution discrete lorsque la frontiere satisfait une certaine condition exemple numerique

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

the discrete plateau problem convergence results we solve the problem of nding and justifying an optimal fully discrete nite element procedure for approximating minimal including unsta ble surfaces in a previous paper we introduced the general framework and some preliminary estimates developed the algorithm and give the numerical results in this paper we prove the convergence estimate

a convergent evolving finite element algorithm for mean curvature flow of closed surfaces a proof of convergence is given for semi and full discretizations of mean curvature flow of closed two dimensional surfaces the numerical method proposed and studied here combines evolving finite elements whose nodes determine the discrete surface like in dziuk s method and linearly implicit backward difference formulae for time integration the proposed method differs from dziuk s approach in that it discretizes huisken s evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space this numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five the error analysis combines stability estimates and consistency estimates to yield optimal order h 1 norm error bounds for the computed surface position velocity normal vector and mean curvature the stability analysis is based on the matrix vector formulation of the finite element method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments illustrate and complement the theoretical results

application of hybrid trefftz finite element method to non linear problems of minimal surface this investigation provides a hybrid trefftz finite element approach for analysing minimal surface problems the approach is based on combining trefftz finite element formulation with radial basis functions rbf and the analogue equation method aem in this method use of the analogue equation approach avoids the difficulty of treating the non linear terms appearing in the soap bubble equation making it possible to solve non linear problems with the trefftz method global rbf is used to approximate the inhomogeneous term induced from non linear functions and other loading terms finally some numerical experiments are implemented to verify the efficiency of this method copyright 2006 john wiley sons ltd

non parametric mean curvature evolution with boundary conditions soit un domaine borne de r n n 2 de frontiere lisse et soit a l operateur de surface minimale donne par a d i a i p a i p i 1 p 2 1 2 on considere des surfaces dont la vitesse en direction de leur normale unite est egale a la courbure moyenne on etudie l equation d evolution qui en resulte u 1 du 2 1 2 au 0 dans 0 t

a parametric finite element method for fourth order geometric evolution equations we present a finite element approximation of motion by minus the laplacian of curvature and related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple junctions on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches it turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points we state also an extension of our scheme to willmore flow of curves and discuss possible further generalizations

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

the hybrid trefftz finite element model and its application to plate bending this paper presents a new hybrid element approach and applies it to plate bending in contrast to more conventional models the formulation is based on displacement fields which fulfil a priori the non homogeneous lagrange equation trefftz method the interelement continuity is enforced by using a stationary principle together with an independent interelement displacement the final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs the formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve sided elements to be generated where relevant known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain for example particular singular corner elements elements presenting circular holes etc thus a high degree of accuracy may be achieved without a troublesome mesh refinement another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements triangles quadrilaterals etc presenting an increasing degree of accuracy the paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements the conclusions present some ideas concerning the adaptive version of the new elements extension to nonlinear problems and some other developments

discrete gradient flows for general curvature energies we consider the numerical approximation of the l 2 gradient flow of general curvature energies int g vec varkappa for a curve in mathbb r d d geq 2 here the curve can be either cl

geometric surface evolution with tangential contribution surface processing tools based on partial differential equations pdes are useful in a variety of applications in computer graphics digital animation computer aided modelling and computer vision in this work we deal with computational issues arising from the discretization of geometric pde models for the evolution of surfaces considering both normal and tangential velocities the evolution of the surface is formulated in a lagrangian framework we propose several strategies for tangential velocities yielding uniform redistribution of mesh points along the evolving family of surfaces preventing computational instabilities and increasing the mesh regularity numerical schemes based on finite co volume approximation in space will be considered finally we describe how this framework may be employed in applications such as mesh regularization morphing and features preserving surface smoothing

a level set method for anisotropic geometric diffusion in 3d image processing a new morphological multiscale method in three dimensional 3d image processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems its aim is to smooth the level sets of a 3d image while simultaneously preserving geometric features such as edges and corners on the level sets this is obtained by an anisotropic curvature evolution where time serves as the multiscale parameter thereby the diffusion tensor depends on a regularized shape operator of the evolving level sets as one suitable regularization local l2 projection onto quadratic polynomials is considered the method is compared to a related parametric surface approach and a geometric interpretation of the evolution and its invariance properties is given a spatial finite element discretization on hexahedral meshes and a semi implicit regularized backward euler discretization in time are the building blocks of the easy to code algorithm different a

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

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

finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement in this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces in particular we adopt a stabilizat

algorithms for computing motion by mean curvature we propose a finite element algorithm for computing the motion of a surface moving by mean curvature the algorithm uses the level set formulation so that changes in topology of the surface can be accommodated stability is deduced by showing that the discrete solutions satisfy both l infty and w 1 1 bounds existence of discrete solutions and connections with brakke flows are established some numerical examples and application to related problems such as the phase field equations are also presented

non rigid visible and infrared face registration via regularized gaussian fields criterion registration of multi sensor data particularly visible color sensors and infrared sensors is a prerequisite for multimodal image analysis such as image fusion typically the relationships between image pairs are modeled by rigid or affine transformations however this cannot produce accurate alignments when the scenes are not planar for example face images in this paper we propose a regularized gaussian fields criterion for non rigid registration of visible and infrared face images the key idea is to represent an image by its edge map and align the edge maps by a robust criterion with a non rigid model we model the transformation between images in a reproducing kernel hilbert space and a sparse approximation is applied to the transformation to avoid high computational complexity moreover a coarse to fine strategy by applying deterministic annealing is used to overcome local convergence problems the qualitative and quantitative comparisons on two publicly available databases demonstrate that our method significantly outperforms the state of the art method with an affine model as a result our method will be beneficial for fusion based face recognition highlightswe analyze the robustness of gaussian fields criterion both in theory and experiment the gaussian fields criterion is generalized from rigid to the non rigid case we propose a new non rigid registration method to deal with more real world problems a sparse approximation on the transformation is applied to speed up the method we customize and apply the proposed method to visible thermal ir face registration

a finite element method for surface diffusion the parametric case surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for parametric surfaces with or without boundaries the method is semi implicit requires no explicit parametrization and yields a linear system of elliptic pde to solve at each time step we next develop a finite element method propose a schur complement approach to solve the resulting linear systems and show several significant simulations some with pinch off in finite time we introduce a mesh regularization algorithm which helps prevent mesh distortion and discuss the use of time and space adaptivity to increase accuracy while reducing complexity

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

a hybrid level set front tracking approach for finite element simulations of two phase flows abstract in this paper we give details on the numerical realization of a new finite element method for the simulation of two phase flows which was recently introduced in basting and weismann 2013 the main ingredient is a hybrid representation of the interface between the fluid phases an implicit description of the interface is given by a level set function and an explicit representation is obtained from aligning edges of the computational mesh to the implicitly described interface this step is done by a black box optimization based mesh smoothing approach which does not change the topology of the mesh while guaranteeing optimal mesh quality furthermore we make use of quadratic isoparametric elements to increase the approximation quality of the discrete interface due to the alignment discontinuities of the solution variables pressure can be captured accurately while a variational treatment of the curvature allows for a precise approximation of surface tension we present our time discretization scheme for the coupled navier stokes level set equations and discuss our space discretization based on the so called subspace projection method spm to account for discontinuities across the interface we present two numerical examples for which reference solutions exist we consider the oscillation of a single droplet and provide our results for an established two phase flow benchmark problem

efficient kernel density estimation using the fast gauss transform with applications to color modeling and tracking many vision algorithms depend on the estimation of a probability density function from observations kernel density estimation techniques are quite general and powerful methods for this problem but have a significant disadvantage in that they are computationally intensive in this paper we explore the use of kernel density estimation with the fast gauss transform fgt for problems in vision the fgt allows the summation of a mixture of ill gaussians at n evaluation points in o m n time as opposed to o mn time for a naive evaluation and can be used to considerably speed up kernel density estimation we present applications of the technique to problems from image segmentation and tracking and show that the algorithm allows application of advanced statistical techniques to solve practical vision problems in real time with today s computers

on the parametric finite element approximation of evolving hypersurfaces in r 3 we present a variational formulation of motion by minus the laplacian of curvature and mean curvature flow as well as related second and fourth order flows of a closed hypersurface in r 3 on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the distribution of mesh points and if applicable volume conservation

on stable parametric finite element methods for the stefan problem and the mullins sekerka problem with applications to dendritic growth we introduce a parametric finite element approximation for the stefan problem with the gibbs thomson law and kinetic undercooling which mimics the underlying energy structure of the problem the proposed method is also applicable to certain quasi stationary variants such as the mullins sekerka problem in addition fully anisotropic energies are easily handled the approximation has good mesh properties leading to a well conditioned discretization even in three space dimensions several numerical computations including for dendritic growth and for snow crystal growth are presented

a hybrid variational front tracking level set mesh generator for problems exhibiting large deformations and topological changes we present a method for generating 2 d unstructured triangular meshes that undergo large deformations and topological changes in an automatic way we employ a method for detecting when topological changes are imminent via distance functions and shape skeletons when a change occurs we use a level set method to guide the change of topology of the domain mesh this is followed by an optimization procedure using a variational formulation of active contours that seeks to improve boundary mesh conformity to the zero level contour of the level set function our method is advantageous for arbitrary lagrangian eulerian ale type methods and directly allows for using a variational formulation of the physics being modeled and simulated including the ability to account for important geometric information in the model such as for surface tension driven flow furthermore the meshing procedure is not required at every time step and the level set update is only needed during a topological change hence our method does not significantly affect computational cost

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

on rotationally symmetric mean curvature flow the evolutionary motion of surfaces by their mean curvature has been studied by several authors from different points of view a measure theoretic approach is studied by k a brakke b the classical parametric problem is studied by g huisken hl non parametric mean curvature evolution with boundary conditions is treated in h2 for a detailed discussion see h3 we restrict our attention to the case in which the initial surface has rotational symmetry we shall show that under suitable initial conditions the solution degenerates in the sense that its curvature develops a singularity at exactly one point this confirms numerical results of the first author d for general surfaces in r3 see also the work of m a grayson gl p 286 g2 the proofs rely on the parabolic maximum principle the second author is gratefully indebted to s angencnt for bringing the problem to his attention and for pointing out the resemblance to so called quenching problems this research was financially via sfb 256 and sfb 123

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

convergence of finite elements on an evolving surface driven by diffusion on the surface for a parabolic surface partial differential equation coupled to surface evolution convergence of the spatial semidiscretization is studied in this paper the velocity of the evolving surface is not given explicitly but depends on the solution of the parabolic equation on the surface various velocity laws are considered elliptic regularization of a direct pointwise coupling a regularized mean curvature flow and a dynamic velocity law a novel stability and convergence analysis for evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed the stability analysis works with the matrix vector formulation of the method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments complement the theoretical results

stability of interconnected stochastic delay systems a decomposition of a large scale stochastic functional differential system into a family of isolated subsystems is made a set of criteria is obtained under which the uniform asymptotic stability with probability one of the trivial solution process of the large scale system is preserved these criteria are expressed in terms of the stability of isolated subsystems and bounds on interconnection components as a tool of our analysis razumikhin type stability conditions for stochastic functional differential systems are also obtained several examples are given to show the applicability of our results

gaussian fields a new criterion for 3d rigid registration abstract this paper introduces a new and simple criterion for rigid registration based on gaussian fields the criterion is always differentiable and convex in a large neighborhood of the alignment parameters allowing for the use of well proven optimization techniques using this method we can extend the size of the region of convergence so that no close initialization is needed thus overcoming local convergence problems of iterative closest point algorithms furthermore the gaussian energy function can be evaluated with linear complexity using the fast gauss transform which permits efficient implementation of the registration algorithm experimental analysis on real world data sets shows the usefulness and points the limits of the approach

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

on two methods for approximating minimal surfaces in parametric form etude de deux methodes d approximation des surfaces minimales sous forme parametrique minimisation de l aire et de l energie convergence transformation conforme approchee exemples

mixed finite element methods for nonlinear elliptic problems the h p version mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed existence and uniqueness of the approximate solution are demonstrated using a fixed point argument convergence and stability of the method are proved both with respect to mesh refinement and increase of the degree of the approximating polynomials the analysis is carried out in detail using raviart thomas nedelec spaces as an example numerical results for minimal surface problems are obtained using brezzi douglas marini spaces graphs of the approximate solutions are presented for two different problems

time integration for diffuse interface models for two phase flow we propose a variant of the q scheme for diffuse interface models for two phase flow together with three new linearization techniques for the surface tension these involve either additional stabilizing force terms or a fully implicit coupling of the navier stokes and cahn hilliard equation in the common case that the equations for interface and flow are coupled explicitly we find a time step restriction which is very different to other two phase flow models and in particular is independent of the grid size we also show that the proposed stabilization techniques can lift this time step restriction even more pronounced is the performance of the proposed fully implicit scheme which is stable for arbitrarily large time steps we demonstrate in a taylor flow application that this superior coupling between flow and interface equation can decrease the computation time by several orders of magnitude

variational convergence of discrete minimal surfaces this work is concerned with the convergence behavior of the solutions to parametric variational problems an emphasis is put on sequences of variational problems that arise as discretizations of either infinite dimensional optimization problems or infinite dimensional operator problems finally the results are applied to discretizations of the douglas plateau problem and of a boundary value problem in nonlinear elasticity

finite element approximations to surfaces of prescribed variable mean curvature we give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature which span a given boundary curve we work in the parametric setting and prove optimal estimates in the h 1 norm the estimates are verified computationally

parametric approximation of willmore flow and related geometric evolution equations we present various variational approximations of willmore flow in mathbb r d d 2 3 as well as the classic willmore flow we also consider variants that are a volume preserving and b volume and area preserving the latter evolution law is the so called helfrich flow in addition we consider motion by gaus curvature the presented fully discrete schemes are easy to solve as they are linear at each time level and they have good properties with respect to the distribution of mesh points finally we present numerous numerical experiments including simulations for energies appearing in the modeling of biological cell membranes

finite element method for epitaxial growth with attachment detachment kinetics an adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth such problems consist of an adatom adsorbed atom diffusion equation on terraces of different height boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment and the normal velocity law for the motion of such boundaries determined by a two sided flux together with the one dimensional surface diffusion the problem is solved using two independent meshes a two dimensional mesh for the adatom diffusion and a one dimensional mesh for the boundary evolution the diffusion equation is discretized by the first order implicit scheme in time and the linear finite element method in space a technique of extension is used to avoid the complexity in the spatial discretization near boundaries all the elements are marked and the marking is updated in each time step to trace the terrace height the evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion its governing equation is solved by a semi implicit front tracking method using parametric finite elements simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem numerical tests on pure geometrical motion mass balance and the stability of a growing circular island demonstrate that the method is stable efficient and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period

processing textured surfaces via anisotropic geometric diffusion a multiscale method in surface processing is presented which carries over image processing methodology based on nonlinear diffusion equations to the fairing of noisy textured parametric surfaces the aim is to smooth noisy triangulated surfaces and accompanying noisy textures as they are delivered by new scanning technology while enhancing geometric and texture features for an initial textured surface a fairing method is described which simultaneously processes the texture and the surface considering an appropriate coupling of the two smoothing processes one can take advantage of the frequently present strong correlation between edge features in the texture and on the surface edges the method is based on an anisotropic curvature evolution of the surface itself and an anisotropic diffusion on the processed surface applied to the texture here the involved diffusion tensors depends on a regularized shape operator of the evolving surface and on regularized texture gradients a spatial finite element discretization on arbitrary unstructured triangular grids and a semi implicit finite difference discretization in time are the building blocks of the corresponding numerical algorithm a normal projection is applied to the discrete propagation velocity to avoid tangential drifting in the surface evolution different applications underline the efficiency and flexibility of the presented surface processing tool

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

droplet footprint control controlling droplet shape via surface tension has numerous technological applications such as droplet lenses and lab on a chip this motivates a partial differential equation constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate we use shape differential calculus to derive an l 2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates we then develop a gradient based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint i e the liquid solid interface has a desired shape moreover we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint numerical results are also presented to showcase the method

removing mismatches for retinal image registration via multi attribute driven regularized mixture model in order to address the problem of retinal image registration this paper proposes and analyzes a novel and general matching algorithm called multi attribute driven regularized mixture model mad rmm mismatches removal can play a key role in image registration which refers to establish reliable matches between two point sets here the presented approach starts from multi feature attributes which are used to guide the feature matching to identify inliers correct matches from outliers incorrect matches and then estimates the spatial transformation in this paper motivated by the problem of feature matching that the initial correspondence is always contaminated by outliers thereby we formulate this issue as a probability deformable mixture model which consists of gaussian components for inliers and uniform components for outliers moreover the algorithm takes full advantage of using multiple attributes for better general matching performance here we are assuming all inliers are mapped into a high dimensional feature space namely reproducing kernel hilbert space rkhs and the closed form solution to the mapping function is given by the representation theorem with l2 norm regularization under the expectation maximization em algorithm finally we evaluate the performance of the algorithm by applying it to retinal image registration on several datasets where experimental results demonstrate that the mad rmm outperforms current state of the art methods and shows the robustness to outliers on real retinal images

constructing reference metrics on multicube representations of arbitrary manifolds reference metrics are used to define the differential structure on multicube representations of manifolds i e they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous this paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies the method is tested here by constructing reference metrics for compact orientable two dimensional manifolds with genera between zero and five these metrics are shown to satisfy the gauss bonnet identity numerically to the level of truncation error which converges toward zero as the numerical resolution is increased these reference metrics can be made smoother and more uniform by evolving them with ricci flow this smoothing procedure is tested on the two dimensional reference metrics constructed here these smoothing evolutions using volume normalized ricci flow with deturck gauge fixing are all shown to produce reference metrics with constant scalar curvatures at the level of numerical truncation error

stabilized finite element approximation of the mean curvature vector on closed surfaces the mean curvature vector of a surface is obtained by letting the laplace beltrami operator act on the embedding of the surface in bf r 3 in this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in l 2 the stabilization involves the jump in the tangent gradient in the direction of the outer co normals at each edge in the surface mesh we consider both standard meshed surfaces and so called cut surfaces that are level sets of piecewise linear distance functions we prove a priori error estimates and verify the theoretical results numerically

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

shape optimization of self avoiding curves this paper presents a softened notion of proximity or self avoidance for curves we then derive a sensitivity result based on shape differential calculus for the proximity this is combined with a gradient based optimization approach to compute three dimensional parameterized curves that minimize the sum of an elastic bending energy and a proximity energy that maintains self avoidance by a penalization technique minimizers are computed by a sequential quadratic programming sqp method where the bending energy and proximity energy are approximated by a finite element method we then apply this method to two problems first we simulate adsorbed polymer strands that are constrained to be bound to a surface and be locally inextensible this is a basic model of semi flexible polymers adsorbed onto a surface a current topic in material science several examples of minimizing curve shapes on a variety of surfaces are shown an advantage of the method is that it can be much faster than using molecular dynamics for simulating polymer strands on surfaces second we apply our proximity penalization to the computation of ideal knots we present a heuristic scheme utilizing the sqp method above for minimizing rope length and apply it in the case of the trefoil knot applications of this method could be for generating good initial guesses to a more accurate but expensive knot tightening algorithm a new softened version of proximity or self contact for 1 d curves is proposed sensitivity of the proximity with respect to shape differentiation is given a method for finding minimizing curve shapes that respect self contact is given the method is applied to simulating polymer strands adsorbed onto surfaces a heuristic method is given for approximating ideal knots

semi implicit complementary volume scheme for solving level set like equations in image processing and curve evolution we introduce linear semi implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems we study discretization of image selective smoothing equation of mean curvature flow type given by alvarez lions and morel 3 solution of the level set equation of osher and sethian 26 30 is also included in the study we prove l infty and w 1 1 estimates for the proposed scheme and give existence of its generalized solution in every discrete time scale step efficiency of the scheme is given by its linearity and stability preconditioned iterative solvers are used for computing arising linear systems we present computational results related to image processing and plane curve evolution

numerical simulation of dealloying by surface dissolution via the evolving surface finite element method in this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid electrolyte interface the evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a cahn hilliard equation modelling surface phase separation the method is based on a triangulated hypersurface whose evolution is calculated as well as the solution of the cahn hilliard equation by the evolving surface finite element method esfem the numerical experiments simulate complex morphology and concentration patterns providing evidence that the mathematical model may describe the formation of nanoporosity

eliminating spurious velocities with a stable approximation of viscous incompressible two phase stokes flow abstract we present a parametric finite element approximation of two phase flow this free boundary problem is given by the stokes equations in the two phases which are coupled via jump conditions across the interface using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface in addition the mesh quality of the parametric approximation of the interface does not deteriorate in general over time and an equidistribution property can be shown for a semidiscrete continuous in time variant of our scheme in two space dimensions moreover on using a simple xfem pressure space enrichment we obtain exact volume conservation for the two phase regions furthermore our fully discrete finite element approximation can be shown to be unconditionally stable we demonstrate the applicability of our method with some numerical results which in particular demonstrate that spurious velocities can be avoided in the classical test cases