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

two methods for discretizing a delta function supported on a level set this paper presents two new methods for discretizing a dirac delta function which is concentrated on the zero level set of a smooth function u r n r the function u is only known at the discrete set of points belonging to a regular mesh covering r n these two methods are used to approximate integrals over the manifold defined by the level set both methods are conceptually simple and easy to implement we present the results of numerical experiments indicating that as the mesh size h goes to zero the rate of convergence is at least o h for the first method and o h 2 for the second method we perform a limited analysis of the proposed algorithms including a proof of convergence for both methods

extended discontinuous galerkin methods for two phase flows the spatial discretization summary this work discusses a discontinuous galerkin dg discretization for two phase flows the fluid interface is represented by a level set and the dg approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply this adaption of the original dg space which can be performed on the fly for arbitrary interface shapes is referred to as extended discontinuous galerkin by combining this ansatz with a special quadrature technique one can regain spectral convergence properties for low regularity solutions which is demonstrated by numerical examples this work focuses on the aspects of spatial discretization and special emphasis is devoted on how to overcome problems related to quadrature small cut cells and condition number of linear systems temporal discretization will be discussed in future works copyright 2016 john wiley sons ltd

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

highly accurate surface and volume integration on implicit domains by means of moment fitting summary we introduce a new method for the numerical integration over curved surfaces and volumes defined by a level set function the method is based on the solution of a small linear system based on a simplified variant of the moment fitting equations numerical experiments suggest that the accuracy of the resulting quadrature rules exceeds the accuracy of traditional methods by orders of magnitude using moments up to an order of p the measured experimental orders of convergence exceed hp consequently their construction is very efficient because only coarse computational grids are required the conceptual simplicity allows for the application on very general grid types which is demonstrated by numerical experiments on quadrilateral triangular and hexahedral grids copyright 2013 john wiley sons ltd

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

the diffuse nitsche method dirichlet constraints on phase field boundaries summary we explore diffuse formulations of nitsche s method for consistently imposing dirichlet boundary conditions on phase field approximations of sharp domains leveraging the properties of the phase field gradient we derive the variational formulation of the diffuse nitsche method by transferring all integrals associated with the dirichlet boundary from a geometrically sharp surface format in the standard nitsche method to a geometrically diffuse volumetric format we also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem from which the stabilization parameter can be estimated automatically in each element we advertise metastable phase field solutions of the allen cahn problem for transferring complex imaging data into diffuse geometric models in particular we discuss the use of mixed meshes that is an adaptively refined mesh for the phase field in the diffuse boundary region and a uniform mesh for the representation of the physics based solution fields we illustrate accuracy and convergence properties of the diffuse nitsche method and demonstrate its advantages over diffuse penalty type methods in the context of imaging based analysis we show that the diffuse nitsche method achieves the same accuracy as the standard nitsche method with sharp surfaces if the inherent length scales i e the interface width of the phase field the voxel spacing and the mesh size are properly related we demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body this article is protected by copyright all rights reserved

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

delta function approximations in level set methods by distance function extension in a k tornberg b engquist numerical approximations of singular source terms in differential equations j comput phys 200 2004 462 488 it was shown for simple examples that the then m

higher order xfem for curved strong and weak discontinuities the extended finite element method xfem enables the accurate approximation of solutions with jumps or kinks within elements optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces higher order convergence for arbitrary curved interfaces relies on two major issues i an accurate quadrature of the galerkin weak form for the cut elements and ii a careful formulation of the enrichment which should preclude any problems in the blending elements for i we employ a strategy of subdividing the elements into subcells with only one curved side reference elements that are higher order on only one side are then used to map the integration points to the real element for ii we find that enrichments for strong discontinuities are easily extended to higher order accuracy in contrast problems in blending elements may hinder optimal convergence for weak discontinuities different formulations are investigated including the corrected xfem numerical results for several test cases involving strong or weak curved discontinuities are presented quadratic and cubic approximations are investigated optimal convergence rates are achieved using the standard xfem for the case of a strong discontinuity close to optimal convergence rates for the case of a weak discontinuity are achieved using the corrected xfem copyright 2009 john wiley sons ltd

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

poisson equations in irregular domains with robin boundary conditions solver with second order accurate gradients abstract we present a conservative method for solving the poisson equation in irregular domains with robin boundary conditions second order accurate solutions and gradients in the l norm are obtained

high order numerical methods to two dimensional delta function integrals in level set methods in this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in x wen high order numerical methods to a type of delta function integrals j comput phys 226 2007 1952 1967 we establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods

algorithms for piecewise polynomials and splines with free knots methode de resolution de certains systemes non lineaires garantissant la convergence lors de l initialisation avec des points arbitraires

on the elimination of quadrature subcells for discontinuous functions in the extended finite element method the introduction of discontinuous non differentiable functions in the extended finite element method allows to model discontinuities independent of the mesh structure however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity line is commonly adopted in the paper it is shown how standard gauss quadrature can be used in the elements containing the discontinuity without splitting the elements into subcells or introducing any additional approximation the technique is illustrated and developed in one two and three dimensions for crack and material discontinuity problems

high order numerical quadratures to one dimensional delta function integrals we study high order numerical quadratures to one dimensional delta function integrals in this paper this is motivated by the fact that traditional numerical quadratures give only first order accuracy in general we provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures discrete delta functions and support size formulas satisfying these criteria are designed numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures

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

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

numerical approximation of internal discontinuity interface problems this work focuses on the finite element discretization of boundary value problems whose solution features either a discontinuity or a discontinuous conormal derivative across an interface inside the computational domain the interface is characterized via a level set function the discontinuities are accounted for by using suitable extension operators whose numerical implementation requires a very low computational effort after carrying out the error analysis numerical results to validate our approach are presented in one two and three dimensions

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

solving poisson type equations with robin boundary conditions on piecewise smooth interfaces abstract we present two finite volume schemes to solve a class of poisson type equations subject to robin boundary conditions in irregular domains with piecewise smooth boundaries the first scheme results in a symmetric linear system and produces second order accurate numerical solutions with first order accurate gradients in the l norm for solutions with two bounded derivatives the second scheme is nonsymmetric but produces second order accurate numerical solutions as well as second order accurate gradients in the l norm for solutions with three bounded derivatives numerical examples are given in two and three spatial dimensions

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

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

a stabilized nitsche cut element method for the wave equation abstract we give a weak formulation for solving the wave equation u 2 u f on a 2 dimensional immersed domain in the spatial finite element discretization boundaries do not conform to element boundaries dirichlet and neumann boundary conditions are enforced weakly by nitsche s method additional penalty terms act on the gradient jumps over the interior faces of the elements cut by the boundary these terms ensure a non stiff temporal system which makes it possible to perform explicit time stepping we give optimal a priori error estimates second order accuracy for u u h and u u h and first order accuracy for u u h in l 2 norm numerical results verify this

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

higher order noether symmetries and constants of the motion the authors identify a class of symmetries for lagrangian systems called higher order noether symmetries which yield a corresponding first integral without further integrations first a version of noether s theorem is recalled in which noether symmetries are considered to be symmetries of the two form d theta theta being the cartan form nth order noether symmetries are then defined by the requirement lynd theta 0 the picture is further generalised by exploring various ways in which the computation of repeated lie derivatives of d theta can lead to the identification of first integrals a number of illustrative examples are discussed

a high order discontinuous galerkin method for compressible flows with immersed boundaries summary we present a higher order discretization scheme for the compressible euler and navier stokes equations with immersed boundaries our approach makes use of a discontinuous galerkin dg discretization in a domain that is implicitly defined by means of a level set function the zero iso contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary fitted cells in order to retain the full order of convergence of the scheme it is crucial to perform volume and surface integrals in boundary cells with high accuracy this is achieved using a linear moment fitting strategy moreover we apply a non intrusive cell agglomeration technique that averts problems with very small and ill shaped cuts the robustness accuracy and convergence properties of the scheme are assessed in several two dimensional test cases for the steady compressible euler and navier stokes equations approximation orders range from zero to four even though the approach directly generalizes to even higher orders in all test cases with a sufficiently smooth solution the experimental order of convergence matches the expected rate for dg schemes this article is protected by copyright all rights reserved

a high order discontinuous galerkin nitsche method for elliptic problems with fictitious boundary we present a discontinuous galerkin method based on the classical method of nitsche for elliptic problems with an immersed boundary representation on a structured grid in such methods very small elements typically occur at the boundary leading to breakdown of the discrete coercivity as well as numerical instabilities in this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain this construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates furthermore we prove a bound of the condition number of the stiffness matrix all the results are valid for polynomials of arbitrary order we also discuss the implementation of the method and present numerical examples in three dimensions

numerical quadrature over smooth closed surfaces the numerical approximation of definite integrals or quadrature often involves the construction of an interpolant of the integrand and its subsequent integration in the case of one dimension it is natural to rely on polynomial interpolants however their extension to two or more dimensions can be costly and unstable an efficient method for computing surface integrals on the sphere is detailed in the literature reeger fornberg 2016 stud appl math 137 174 188 doi 10 1111 sapm 12106 the method uses local radial basis function interpolation to reduce computational complexity when generating quadrature weights for any given node set this article generalizes this method to arbitrary smooth closed surfaces

high order numerical methods to three dimensional delta function integrals in level set methods in this paper we propose a class of high order numerical methods to delta function integrals appearing in level set methods in the three dimensional case by extending the idea for designing high order methods to two dimensional delta function integrals in x wen j comput phys 228 2009 pp 4273 4290 the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the three dimensional delta function integral can be rewritten as a two dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral a key idea in designing high order methods in this paper is that the high order accuracy of the methods is not ensured in each mesh cell and the methods are designed in a consistent way to foster error cancelations in mesh cells where high order accuracy is not ensured in the individual cell this issue is essentially related to the construction of integral area for the two dimensional ordinary integral in each mesh cell which is achieved by considering the intersection points between the zero level set and the edges or sides of the mesh cell the mesh cell restrictions of the three dimensional delta function integral are then approximated by applying standard two dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals numerical examples are presented showing that our methods in this paper achieve or exceed the expected second to fourth order accuracy and demonstrating the stability of shifting mesh for our methods and the advantage of using our high order numerical methods