a convergence rate theorem for finite difference approximations to delta functions we prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in r n the type of manifold involved here is defined by the zero level set of a smooth mapping u r email protected r our approximations are based on the two finite difference methods for discretizing delta functions presented in 16 we included a convergence proof in that paper but only proved rates of convergence in some greatly simplified situations numerical experiments indicated that our two methods were at least first and second order accurate respectively in this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses

optimal reconstruction of material properties in complex multiphysics phenomena we develop an optimization based approach to the problem of reconstructing temperature dependent material properties in complex thermo fluid systems described by the equations for the conservation of mass momentum and energy our goal is to estimate the temperature dependence of the viscosity coefficient in the momentum equation based on some noisy temperature measurements where the temperature is governed by a separate energy equation we show that an elegant and computationally efficient solution of this inverse problem is obtained by formulating it as a pde constrained optimization problem which can be solved with a gradient based descent method a key element of the proposed approach the cost functional gradients are characterized by mathematical structure quite different than in typical problems of pde constrained optimization and are expressed in terms of integrals defined over the level sets of the temperature field advanced techniques of integration on manifolds are required to evaluate numerically such gradients and we systematically compare three different methods as a model system we consider a two dimensional unsteady flow in a lid driven cavity with heat transfer and present a number of computational tests to validate our approach and illustrate its performance

self consistent field theory simulations of polymers on arbitrary domains we introduce a framework for simulating the mesoscale self assembly of block copolymers in arbitrary confined geometries subject to neumann boundary conditions we employ a hybrid finite difference volume approach to discretize the mean field equations on an irregular domain represented implicitly by a level set function the numerical treatment of the neumann boundary conditions is sharp i e it avoids an artificial smearing in the irregular domain boundary this strategy enables the study of self assembly in confined domains and enables the computation of physically meaningful quantities at the domain interface in addition we employ adaptive grids encoded with quad oc trees in parallel to automatically refine the grid where the statistical fields vary rapidly as well as at the boundary of the confined domain this approach results in a significant reduction in the number of degrees of freedom and makes the simulations in arbitrary domains using effective boundary conditions computationally efficient in terms of both speed and memory requirement finally in the case of regular periodic domains where pseudo spectral approaches are superior to finite differences in terms of cpu time and accuracy we use the adaptive strategy to store chain propagators reducing the memory footprint without loss of accuracy in computed physical observables

a fixed mesh method for incompressible flow structure systems with finite solid deformations a fixed mesh algorithm is proposed for simulating flow structure interactions such as those occurring in biological systems in which both the fluid and solid are incompressible and the solid deformations are large several of the well known difficulties in simulating such flow structure interactions are avoided by formulating a single set of equations of motion on a fixed eulerian mesh the solid s deformation is tracked to compute elastic stresses by an overlapping lagrangian mesh in this way the flow structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system these forces which depend on the solid elastic stress distribution are computed on the lagrangian mesh by a standard finite element method and then transferred to the fixed eulerian mesh where the joint momentum and continuity equations are solved by a finite difference method the constitutive model for the solid can be quite general for the force transfer standard immersed boundary and immersed interface methods can be used and are demonstrated we have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum the interface is still effectively sharp for this approach the spatial convergence of the method is observed to be between first and second order as in most immersed boundary methods for membrane flows the algorithm is demonstrated by the simulations of an advected elastic disk a flexible leaflet in an oscillating flow and a model of a swimming jellyfish

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

order preserving approximations to successive derivatives of periodic functions by iterated splines a method is presented for the iterative use of periodic splines of order 2r to accurately approximate successive derivatives of a periodic function on a uniform mesh asymptotic expansions are derived for the error in this method of approximating the mth derivative at the meshpoints it is found that the leading order term in this error expansion is o h 2r independently of m a numerical example is given showing excellent agreement with theoretical predictions

a conservative discretization of the poisson nernst planck equations on adaptive cartesian grids in this paper we present a novel hybrid finite difference finite volume method for the numerical solution of the nonlinear poisson nernst planck pnp equations on irregular domains the method is described in two spatial dimensions but can be extended to three dimensional problems as well the boundary of the irregular domain is represented implicitly via the zero level set of a signed distance function and quadtree data structures are used to systematically generate adaptive grids needed to accurately capture the electric double layer near the boundary to handle the nonlinearity in the pnp equations efficiently a semi implicit time integration method is utilized an important feature of our method is that total number of ions in the system is conserved by carefully imposing the boundary conditions by utilizing a conservative discretization of the diffusive and more importantly the nonlinear migrative flux term several numerical experiments are conducted which illustrate that the presented method is first order accurate in time and second order accurate in space moreover these tests explicitly indicate that the algorithm is also conservative finally we illustrate the applicability of our method in the study of the charging dynamics of porous supercapacitors

a level set approach for diffusion and stefan type problems with robin boundary conditions on quadtree octree adaptive cartesian grids we present a numerical method for simulating diffusion dominated phenomena on irregular domains and free moving boundaries with robin boundary conditions on quadtree octree adaptive meshes in particular we use a hybrid finite difference and finite volume framework that combines the level set finite difference discretization of min and gibou 2007 13 with the treatment of robin boundary conditions of papac et al 2010 19 on uniform grids we present numerical results in two and three spatial dimensions on the diffusion equation and on a stefan type problem in addition we present an application of this method to the case of the simulation of the ehrlich schwoebel barrier in the context of epitaxial growth

finite difference methods for approximating heaviside functions we present a finite difference method for discretizing a heaviside function h u x where u is a level set function u r email protected r that is positive on a bounded region email protected r n there are two variants of our algorithm both of which are adapted from finite difference methods that we proposed for discretizing delta functions in j d towers two methods for discretizing a delta function supported on a level set j comput phys 220 2007 915 931 j d towers discretizing delta functions via finite differences and gradient normalization preprint at http www miracosta edu home jtowers j d towers a convergence rate theorem for finite difference approximations to delta functions j comput phys 227 2008 6591 6597 we consider our approximate heaviside functions as they are used to approximate integrals over w we prove that our first approximate heaviside function leads to second order accurate quadrature algorithms numerical experiments verify this second order accuracy for our second algorithm numerical experiments indicate at least third order accuracy if the integrand f and email protected are sufficiently smooth numerical experiments also indicate that our approximations are effective when used to discretize certain singular source terms in partial differential equations we mostly focus on smooth f and u by this we mean that f is smooth in a neighborhood of w u is smooth in a neighborhood of email protected and the level set u x 0 is a manifold of codimension one however our algorithms still give reasonable results if either f or u has jumps in its derivatives numerical experiments indicate approximately second order accuracy for both algorithms if the regularity of the data is reduced in this way assuming that the level set u x 0 is a manifold numerical experiments indicate that dependence on the placement of w with respect to the grid is quite small for our algorithms specifically a grid shift results in an o h p change in the computed solution where p is the observed rate of convergence

adaptively weighted numerical integration in the finite cell method abstract with adaptively weighted aw numerical integration for a given set of quadrature nodes order and domain of integration the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense the moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain and then are corrected for the deviation from the original domain using shape sensitivity analysis in this paper we demonstrate the application of aw integration scheme in the context of the finite cell method which must perform numerical integration over arbitrary domains without meshing the standard integration technique employed in fcm is the characteristic function method that converts the continuous integrand over a complex domain into a discontinuous integrand over a simple box domain then well known integrand adaptivity techniques are employed to integrate the resulting discontinuous integrand over the box domain although this method is simple to implement it becomes computationally very expensive for realistic complex 3d domains such as sculptures bones and engines in contrast in aw scheme the quadrature weights directly adapt to the complex geometric domain without the need to making the integrand discontinuous leading to superior computational properties in this paper we demonstrate the computational efficiency of aw over the characteristic function method as it requires fewer subdivisions and less time to achieve a given accuracy in both two and three dimensions in addition aw offers a number of advantages including flexibility in the choice of quadrature points and basis functions

a high order method for the approximation of integrals over implicitly defined hypersurfaces we introduce a novel method to compute approximations of integrals over implicitly defined hypersurfaces the new method is based on a weak formulation in l 2 0 1 that uses the coarea formula to circumvent an explicit integration over the hypersurfaces as such it is possible to use standard quadrature rules in the spirit of hp spectral finite element methods and the expensive computation of explicit hypersurface parametrizations is avoided we derive error estimates showing that high order convergence can be achieved provided the integrand and the hypersurface defining function are sufficiently smooth the theoretical results are supplemented by numerical experiments including an application for plasma modeling in nuclear fusion

the rectangle complexity of functions on two dimensional lattices let x be a non empty set let f z2x all vectors which occur have integer coefficients and for a1 a2 b1 b2 we write or 0 a block is a set of the form b z2 0 such that the complexity of f does not exceed b1b2 in this paper we prove the statement for b n 2 where n is any positive integer

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

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

a second order discretization of the nonlinear poisson boltzmann equation over irregular geometries using non graded adaptive cartesian grids in this paper we present a finite difference scheme for the discretization of the nonlinear poisson boltzmann pb equation over irregular domains that is second order accurate the interface is represented by a zero level set of a signed distance function using octree data structure allowing a natural and systematic approach to generate non graded adaptive grids such a method guaranties computational efficiency by ensuring that the finest level of grid is located near the interface the nonlinear pb equation is discretized using finite difference method and several numerical experiments are carried which indicate the second order accuracy of method finally the method is used to study the supercapacitor behaviour of porous electrodes

simplicial isosurfacing in arbitrary dimension and codimension the level set method has been successfully used for moving interface problems the final step of the method is to construct and visualize the isosurface of a discrete function 0 n n rm there have existed many practical isosurfacing algorithms when n 3 m 1 or n 2 m 1 recently we have begun to see the development of isosurfacing algorithms for higher dimensions and codimensions this paper introduces a unified theory and an efficient isosurfacing algorithm that works in arbitrary number of dimensions and codimensions the isosurface of a discrete function is defined as the isosurface of its simplicial interpolant 0 n n rm with this simplicial definition is geometrically a piecewise intersection of a simplex and m hyperplanes is constructed as the union of simplices the construction costs o nn with a uniform grid and o nn m log n with a dyadic grid in numerical space and time when n m 1 or m 2 is projected down into r3 and can be visualized for surface visualizations a simple formula is presented calculating the normal vector field of the projection of into r3 which gives light shadings

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

adaptively weighted numerical integration over arbitrary domains abstract in adaptively weighted numerical integration for a given set of quadrature nodes order and domain of integration the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense the moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain and then are corrected for the deviation from the original domain using shape sensitivity analysis using divergence theorem the moments reduce to integrals over the boundary of the simplified domain the proposed approach supports accurate and efficient computation of quadrature weights for integration of a priori unknown functions over arbitrary 2d and 3d solid domains experimental results 2d indicate that adaptively weighted integration compares favorably with more traditional approaches because the adaptively weighted integration avoids excessive domain subdivision it is useful in many applications and meshfree analysis in particular

boundary integral techniques for multi connected domains abstract several boundary integral techniques are available for the computation of the solution to laplace s equation in multi connected domains however for cases where the domain is changing such as incompressible inviscid fluid flow with free surfaces iterative methods are highly attractive the paper describes one such formulation and tests it on circular and elliptic annuli it is necessary to use interpolated quadrature points to maintain accuracy when regions of the annuli are thin

a second order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids application to shape optimization we present a numerical method for solving the equations of linear elasticity on irregular domains in two and three spatial dimensions we combine a finite volume and a finite difference approaches to derive discretizations that produce second order accurate solutions in the l norm our discretization is sharp in the sense that the physical boundary conditions mixed dirichlet neumann type are imposed at the interface and the solution is computed inside the irregular domain only without the need of smearing the solution across the interface the irregular domain is represented implicitly using a level set function so that this approach is applicable to free moving boundary problems we provide a simple example of shape optimization to illustrate this capability in addition we provide an extension of our method to the case of adaptive meshes in both two and three spatial dimensions we use non graded quadtree 2d and octree 3d data structures to represent the grid that is automatically refined near the irregular domain s boundary this extension to quadtree octree grids produces second order accurate solutions albeit non symmetric linear systems due to the node based sampling nature of the approach however the linear system can be solved with simple linear solvers in this work we use the bicgstab algorithm

geometric discretization of the multidimensional dirac delta distribution application to the poisson equation with singular source terms abstract we present a discretization method for the multidimensional dirac distribution we show its applicability in the context of integration problems and for discretizing dirac distributed source terms in poisson equations with constant or variable diffusion coefficients the discretization is cell based and can thus be applied in a straightforward fashion to quadtree octree grids the method produces second order accurate results for integration superlinear convergence is observed when it is used to model dirac distributed source terms in poisson equations the observed order of convergence is 2 or slightly smaller the method is consistent with the discretization of dirac delta distribution for codimension one surfaces presented in 1 2 we present quadtree octree construction procedures to preserve convergence and present various numerical examples including multi scale problems that are intractable with uniform grids

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

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

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

asymptotic expansions for interpolating periodic splines a detailed study is made of the structure of smooth periodic interpolating splines of odd degree over a uniform mesh for splines of degree 2r 1 all terms of the asymptotic expansion of the error at the mesh points of the first 2r 2 derivatives are given this result is used to derive a number of other high order spline approximations

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

a poisson boltzmann solver on irregular domains with neumann or robin boundary conditions on non graded adaptive grid we introduce a second order solver for the poisson boltzmann equation in arbitrary geometry in two and three spatial dimensions the method differs from existing methods solving the poisson boltzmann equation in the two following ways first non graded quadtree in two spatial dimensions and octree in three spatial dimensions grid structures are used second neumann or robin boundary conditions are enforced at the irregular domain s boundary the irregular domain is described implicitly and the grid needs not to conform to the domain s boundary which makes grid generation straightforward and robust the linear system is symmetric positive definite in the case where the grid is uniform nonsymmetric otherwise in this case the resulting matrix is an m matrix thus the linear system is invertible convergence examples are given in both two and three spatial dimensions and demonstrate that the solution is second order accurate and that quadtree octree grid structures save a significant amount of computational power at no sacrifice in accuracy

efficient symmetric discretization for the poisson heat and stefan type problems with robin boundary conditions we present a novel and efficient method for solving the poisson equation the heat equation and stefan type problems with robin boundary conditions over potentially moving arbitrarily shaped domains the method utilizes a level set framework thus it has all of the benefits of a sharp implicitly represented interface such as the ease of handling complex topological changes this method is straightforward to implement and leads to a linear system that is symmetric and positive definite which can be inverted efficiently with standard iterative methods this approach is second order accurate for both the poisson and heat equations and first order accurate for the stefan problem we demonstrate the accuracy in the l 1 and l norms

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

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

nivat s conjecture and pattern complexity in algebraic subshifts abstract we study nivat s conjecture on algebraic subshifts and prove that in some of them every low complexity configuration is periodic this is the case in the ledrappier subshift the 3 dot system and more generally in all two dimensional algebraic subshifts over f p defined by a polynomial without line polynomial factors in more than one direction we also find an algebraic subshift that is defined by a product of two line polynomials that has this property the 4 dot system and another one that does not

theory of cellular automata a survey this article surveys some theoretical aspects of cellular automata ca research in particular we discuss classical and new results on reversibility conservation laws limit sets decidability questions universality and topological dynamics of ca the selection of topics is by no means comprehensive and reflects the research interests of the author the main goal is to provide a tutorial of ca theory to researchers in other branches of natural computing to give a compact collection of known results with references to their proofs and to suggest some open problems

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

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

short note on the performance of a simple parallel implementation of the ilu pcg for the poisson equation on irregular domains we report on the performance of a parallel algorithm for solving the poisson equation on irregular domains we use the spatial discretization of gibou et al 2002 6 for the poisson equation with dirichlet boundary conditions while we use a finite volume discretization for imposing neumann boundary conditions ng et al 2009 purvis and burkhalter 1979 8 10 the parallelization algorithm is based on the cuthill mckee ordering its implementation is straightforward especially in the case of shared memory machines and produces significant speedup about three times on a standard quad core desktop computer and about seven times on a octa core shared memory cluster the implementation code is posted on the authors web pages for reference

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

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

highly accurate computation of volume fractions using differential geometry abstract this paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso contour of a sufficiently smooth level set function after spatial discretization local approximation of the hypersurface and application of the gaussian divergence theorem the volume integrals are transformed to surface integrals application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature we discuss the theoretical foundations and provide details of the numerical algorithm finally we present numerical results for convex and non convex hypersurfaces embedded in cuboidal domains showing both high accuracy and third to fourth order convergence in space

solving elliptic interface problems with jump conditions on cartesian grids abstract we present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient the source term the solution and its flux are discontinuous across an irregular interface the algorithm produces second order accurate solutions and first order accurate gradients in the l norm on cartesian grids the condition number is bounded regardless of the ratio of the diffusion constant and scales like that of the standard 5 point stencil approximation on a rectangular grid with no interface numerical examples are given in two and three spatial dimensions

an efficient fluid solid coupling algorithm for single phase flows we present a simple and efficient fluid solid coupling method in two and three spatial dimensions in particular we consider the numerical approximation of the navier stokes equations on irregular domains and propose a novel approach for solving the hodge projection step on arbitrary shaped domains this method is straightforward to implement and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity we demonstrate the accuracy of our method in the l 1 and l norms and present its removing the errors associated with the conventional rasterization type discretizations we apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces we also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate reynolds number to reproduce the known steady axisymmetric and non axisymmetric flow regimes we further apply this algorithm to the coupling of flows with moving rigid bodies

an accurate robust and easy to implement method for integration over arbitrary polyhedra application to embedded interface methods we present an accurate method for the numerical integration of polynomials over arbitrary polyhedra using the divergence theorem the method transforms the domain integral into integrals evaluated over the facets of the polyhedra the necessity of performing symbolic computation during such transformation is eliminated by using one dimensional gauss quadrature rule the facet integrals are computed with the help of quadratures available for triangles and quadrilaterals numerical examples in which the proposed method is used to integrate the weak form of the navier stokes equations in an embedded interface method eim are presented the results show that our method is as accurate and generalized as the most widely used volume decomposition based methods moreover since the method involves neither volume decomposition nor symbolic computations it is much easier for computer implementation also the present method is more efficient than other available integration methods based on the divergence theorem efficiency of the method is also compared with the volume decomposition based methods and moment fitting methods to our knowledge this is the first article that compares both accuracy and computational efficiency of methods relying on volume decomposition and those based on the divergence theorem

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

the complexity of functions on lattices abstract let f z 0 1 be a given function in 1938 morse and hedlund observed that if the number of distinct vectors f x 1 f x n x z called complexity is at most n for some positive integer n then f is periodic with period at most n this result is best possible functions with low complexity have been studied to a large extent and relations with or applications to many branches of mathematics computer science and physics are known in the present paper we discuss the above phenomenon in greater generality to begin with we observe that f is periodic if the number of distinct vectors f x a 1 f x a n x z is at most n for some n and given integers a 1 n but that the period cannot be bounded as a function of n only our main topic are multi dimensional functions f z k 0 1 with the property that for some n and distinct vectors a 1 a n z k the number of distinct vectors f x a 1 f x a n x z k is bounded by n we show that such a function with arbitrary k is periodic if n 3 for n 4 there are non periodic examples which we determine completely finally limitations to the general periodicity principle are discussed a conjecture for convex bodies a 1 a n in z 2 is made and we prove it for n 4

a stable projection method for the incompressible navier stokes equations on arbitrary geometries and adaptive quad octrees we present a numerical method for solving the incompressible navier stokes equations on non graded quadtree and octree meshes and arbitrary geometries the viscosity is treated implicitly through a finite volume approach based on voronoi partitions while the convective term is discretized with a semi lagrangian scheme thus relaxing the restrictions on the time step a novel stable implementation of the projection step is introduced making use of the marker and cell layout for the data the solver is validated numerically in two and three spatial dimensions

on a conjecture on bidimensional words we prove that given a double sequence w over the alphabet a i e a mapping from z2 to a if there exists a pair n0 m0 z2 such that pw n0 m0 1 100n0m0 then w has a periodicity vector where pw is the complexity function in rectangles of w