a three point taylor algorithm for three point boundary value problems we consider second order linear differential equations x y f x y g x y h x in the interval 1 1 with dirichlet neumann or mixed dirichlet neumann boundary conditions given at three points of the interval the two extreme points x 1 and an interior point x s 1 1 we consider x f x g x and h x analytic in a cassini disk with foci at x 1 and x s containing the interval 1 1 the three point taylor expansion of the solution y x at the extreme points 1 and at x s is used to give a criterion for the existence and uniqueness of the solution of the boundary value problem this method is constructive and provides the three point taylor approximation of the solution when it exists we give several examples to illustrate the application of this technique

a three dimensional finite element method with arbitrary polyhedral elements the variable element topology finite element method vetfem is a finite element like galerkin approximation method in which the elements may take arbitrary polyhedral form a complete development of the vetfem is given here for both two and three dimensions a kinematic enhancement of the displacement based formulation is also given which effectively treats the case of near incompressibility convergence of the method is discussed and then illustrated by way of a 2d problem in elastostatics also the vetfem s performance is compared to that of the conventional fem with eight node hex elements in a 3d finite deformation elastic plastic problem the main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes making automatic mesh generation on complex domains a significantly simpler matter copyright 2006 john wiley sons ltd

a new spectral method for a class of linear boundary value problems in this paper we use bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear boundary value problems the proposed method has been applied to several bvps with dirichlet neumann and mixed types boundary conditions numerical examples provide favorable comparisons with other existing methods

taylor polynomial solutions of linear differential equations a matrix method which is called taylor matrix method for the approximate solution of linear differential equations with specified associated conditions in terms of taylor polynomials about any point the method is based on first taking truncated taylor series of the functions in equation and then substituting their matrix forms into the given equation thereby the equation reduces to a matrix equation which corresponds to a system of linear algebraic equations with unknown taylor coefficients to illustrate the method it is applied to certain linear differential equations and the generalized hermite laguerre legendre and chebyshev equations given by costa and levine int j math edu sci technol 20 1989 1

an extended pressure finite element space for two phase incompressible flows with surface tension we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set or vof method is applied then the interface which is implicitly given by the zero level of the level set function is in general not aligned with the triangulation that is used in the discretization of the flow problem this non alignment causes severe difficulties w r t the discretization of the localized surface tension force and the discretization of the flow variables in cases with large surface tension forces the pressure has a large jump across the interface in standard finite element spaces due to the non alignment the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure in many simulations these effects cause large oscillations of the velocity close to the interface so called spurious velocities in this paper for a simplified model problem we give an analysis that explains why known standard methods for discretization of the localized force term and for discretization of the pressure variable often yield large spurious velocities in the paper s grosz a reusken finite element discretization error analysis of a surface tension force in two phase incompressible flows preprint 262 igpm rwth aachen siam j numer anal accepted for publication we introduce a new and accurate method for approximation of the surface tension force in the present paper we use the extended finite element space xfem presented in n moes j dolbow t belytschko a finite element method for crack growth without remeshing int j numer meth eng 46 1999 131 150 t belytschko n moes s usui c parimi arbitrary discontinuities in finite elements int j numer meth eng 50 2001 993 1013 for the discretization of the pressure we show that the size of spurious velocities is reduced substantially provided we use both the new treatment of the surface tension force and the extended pressure finite element space

two point taylor expansions and one dimensional boundary value problems we consider second order linear differential equations x y f x y g x y h x in the interval 1 1 with dirichlet neumann or mixed dirichlet neumann boundary conditions we consider x f x g x and h x analytic in a cassini disk with foci at x 1 containing the interval 1 1 the two point taylor expansion of the solution y x at the extreme points 1 is used to give a criterion for the existence and uniqueness of solution of the boundary value problem this method is constructive and provides the two point taylor approximation of the solution s when it exists

a smoothed coupled ns nes fem for dynamic analysis of 2d fluid solid interaction problems abstract a node based smoothed finite element method ns fem t3 using triangular elements and a n sided polygonal edge based smoothed fem n es fem using polygonal elements is combined to give the smoothed coupled ns n es fem for dynamic analysis of two dimensional 2d fluid solid interaction problems based on the pressure displacement formulation in the present method the ns fem t3 is used for the fluid domain and the gradient of pressure is smoothed while the n es fem is used for the solid domain and the gradient of displacement is smoothed this gradient smoothing technique can provide proper softening effect which will effectively relieve the overly stiff behavior of the fem model and thus improve significantly the solution of coupled system some numerical examples have been presented to illustrate the effectiveness of the coupled ns n es fem compared with some existing methods for 2d fluid solid interaction problems

a stable generalized finite element method sgfem of degree two for interface problems abstract generalized or extended finite element methods gfem xfem of degree 1 for interface problems have been reported in the literature they i yield optimal order of convergence i e o h h is a discretization parameter ii are stable in a sense that conditioning is not worse than that of the standard fem and iii are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements however higher order gfem xfem with the properties i iii have not been successfully addressed yet various enrichment schemes for gfem xfem based on d or d p k d is a distance function or the absolute value of level set function and p k is the polynomial basis of degree k have been reported recently to obtain higher order convergence but they are not stable or robust in general in fact they even may not yield the optimal orders of convergence in this paper we propose a stable gfem xfem of degree 2 sgfem2 for the interface problems where we used the enrichment scheme based on d 1 x y instead of d or d 1 x y x 2 x y y 2 in the literature we prove that the sgfem2 yields the optimal order of convergence i e o h 2 for the interface problems with curved smooth interfaces a local principal component analysis technique has been proposed which ensures that the sgfem2 is stable and robust numerical experiments for straight and curved interfaces have been presented to illuminate these properties

two point taylor approximations of the solutions of two dimensional boundary value problems abstract we consider two dimensional elliptic boundary value problems of the form l u f in r 2 bounded and open with a dirichlet boundary condition u h where l is a second order linear differential operator whose coefficients as well as the functions f and h are differentiable up to a certain degree in a recent paper c kesan taylor polynomial solutions of second order linear partial differential equations appl math comput 152 2004 29 41 a matrix algorithm is introduced to compute the standard taylor polynomial of the solution u at a certain point in we propose an alternative formulation of the problem based on a redefinition of the unknown u and the use of the standard frobenius method that simplifies the computation of the taylor coefficients of u we also consider the use of a two point taylor representation of the solution instead of the classical one point taylor representation which gives a more uniform approximation of the solution

an extended residual based variational multiscale method for two phase flow including surface tension abstract in this study an extended residual based variational multiscale method is proposed for two phase flow including surface tension the extended residual based variational multiscale method combines a residual based form of the variational multiscale method and the extended finite element method xfem by extending the solution spaces it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface in particular we propose a quasi static enrichment to reproduce time dependent discontinuities kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here to capture the interface between the phases on a fixed grid a level set approach is used a residual based variational multiscale method is employed for computing both flow and interface motion the presented method is tested for various two phase flow examples exhibiting small and large density and viscosity ratios with and without surface tension a two phase couette flow a rayleigh taylor instability a sloshing tank and a three dimensional rising bubble to the best of our knowledge these are the first simulation results for representative time dependent three dimensional two phase flow problems using an extended finite element method stable and accurate results are obtained for all test examples

implicit multiple grid solution of the compressible navier stokes equations using k epsilon turbulence closure in the present work an algorithm for the numerical solution of the two dimensional mass weighted ensemble averaged compressible navier stokes equations is presented turbulence closure for the reynolds stresses is obtained using a low reynolds n umber two equation k s model which includes near wall effects the same algorithm is used for the integration of both the navier stokes and the turbulence transport equations the algorithm consists of a basic explicit finite volume scheme whose convergence is accelerated using local time step ping and multiple grid techniques convergence is further enhanced by the use of an implicit residual smoothing technique applied both in the fine and the coarser grids the application of the numerical scheme to the solution of the k s equations is examined in detail comparisons of computations with experimental data are presented for three shock wave turbulent boundary layer interaction flows algorithm convergence rate and cpu time require ments are discussed he numerical solution of the navier stokes equations for the simulation of turbulent flowfields is a subject of intense research and code development work 1 interest is focused both on code applicability to variegated config urations and on computational rapidity although direct simulation of turbulent flows2 or large eddy numerical sim ulations using subgrid turbulence modeling 3 4 have appeared in the literature current status of algorithm development and computer power indicate that for practical applications the ensemble averaged navier stokes equations should be used in conjunction with transport equations for turbulence closure 5 7 algebraic or mixing length models may provide satisfactory results for relatively simple flow geometries8 but require the determination of the shear layer thickness and of the distance of each grid point from the solid boundary thus they lack both geometric invariance and locality which are highly desirable properties from a computational viewpoint when addressing complex flowfields with many interacting solid boundaries although research on reynolds stre ss transport closure appears quite promising 9 10 its extension to three dimensions involves considerable increase of the number of transport equations to be solved also further research con cerning wall effects is required the more so since the use of wall functions is considered inadequate 11 on the other hand the use of two equation closures such as a 12 14 is quite widespread notwithstanding the fact that the use of the boussinesq hypothesis15 introduces a question able simplification of the flow physics 16 the major drawback of the k s model is probably the near wall formulation which fails to reproduce correctly the effects of the solid boundary on turbulence several variants of the original jones launder model12 have been developed in an attempt to improve upon near wall modeling some ten such models have been thor oughly examined in a critical review article by patel et al 17 who concluded that the models of launder and sharma 14

intersecting and trimming parametric meshes on finite element shells luiz cristovao g coelho marcelo gattass department of computer science puc rio rio de janeiro brazilluiz henrique de figueiredolncc laborato rio nacional de computac a o cient i ca p etro polis brazilabstractwe present an algorithm for intersecting nite element meshes de ned on parametric surface patches the inter section curves are modeled precisely and both meshes are adjusted to the newly formed borders without unwantedreparametrizations the algorithm is part of an interactive shell modeling program that has been used in the designof large offshore oil structures to achieve good interactive response we represent meshes with a topological datastructure that stores its entities in spatial indexing trees instead of linear lists these trees speed up the intersectioncomputations required to determine points of the trimming curves moreover when combined with the topologicalinformation they allow remeshing using only local queries

polytope finite elements in this paper finite elements based on arbitrary convex and non convex polytopes are introduced polytopes in combination with natural element coordinates necs permit a uniform element formulation of interpolation functions that are independent of the dimension of space localization and the number of vertices necs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes the differentiation and integration of these interpolation functions on the basis of necs is essential for finite element approximations the accuracy of the finite element interpolation or approximation can be controlled by either applying the h version or by utilizing the p version of the finite element method fem advantages in the handling of hanging nodes are discussed furthermore we present construction methods for lagrangian as well as for hierarchical interpolation functions based on necs numerical experiments on different convex and non convex decompositions will show the usability accuracy and convergence of the developed polytope fem copyright 2007 john wiley sons ltd

robustness in stable generalized finite element methods sgfem applied to poisson problems with crack singularities abstract in this paper we study the performance of the generalized finite element method gfem applied to the poisson problem with crack singularities recently a gfem with modified heaviside enrichments was proposed in gupta et al 2013 to approximate the solution of a 2d elasticity problem with a crack it was shown that the gfem is indeed a stable gfem sgfem i e it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method fem however the robustness of the conditioning of the gfem with respect to the position of the mesh relative to the crack was not addressed in gupta et al 2013 in this paper we observed that the conditioning of the gfem with the enrichments used in gupta et al 2013 is not robust when applied to approximate the solution of the poisson problem moreover the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces we proposed using additional singular enrichments at the nodes close to the crack near the crack tip we proved that the gfem with enrichments proposed in this paper yields optimal order of convergence irrespective of the position of the mesh moreover with a local orthogonalization procedure we have shown through numerical experiments that the conditioning of this gfem is not worse than that of the standard fem and the conditioning is robust with respect to the position of the mesh thus the gfem with the enrichments suggested in this paper is indeed an sgfem when applied to a poisson problem with the crack singularities

improved finite difference schemes for transonic potential flow calculations on utilise une methode de densite artificielle modifiee pour resoudre l equation potentielle sous forme conservatrice

implicit conservative schemes for the euler equations a new approach to a characteristic modeling scheme is presented that does not require the governing equations to be written in characteristi c variables or the flux terms to be split into positive and negative parts the method is based on the observation that for certain finite volume schemes the upwind influence can be accounted for through the conditions applied at the boundaries of individual cells instead of flux differencing the method is developed and applied to the one dimensional nozzle flow equations for subsonic transonic and supersonic flows and is then extended to two dimensions using an approximate factorization for the problem of a shock wave reflecting from a flat plate the present solution algorithm reduces the computational work per time step by 40 of that required by standard central difference implicit calculations i introduction n umerical methods that attempt to model the physics of the differential equations being approximated by properly accounting for the directions in which signals are propagated can be called characteristic modeling schemes the standard approach to characteristic modeling schemes is through the implementation of upwind schemes see refs 1 7 to devise an upwind scheme the primative fluxes are split into contributions whose jacobians have eigenvalues that are either 0 or 0 respectively at every point on the numerical mesh these fluxes are then differenced either upwind or downwind of that point

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

finite element analysis on implicitly defined domains an accurate representation based on arbitrary parametric surfaces in this paper we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis the novel contributions of this paper are threefold 1 describe and validate a method to represent arbitrary parametric surfaces implicitly 2 represent arbitrary solids implicitly including sharp features using level sets and boolean operations 3 impose arbitrary dirichlet and neumann boundary conditions on the resulting implicitly defined boundaries the methods proposed do not require local refinement of the finite element mesh in regions of high curvature ensure the independence of the domain s volume on the mesh do not rely on boundary regularization and are well suited to methods based on fixed grids such as the extended finite element method xfem numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries this approach is one step in the desired direction of tying numerical simulations to computer aided design cad similarly to the isogeometric analysis paradigm

quadrature schemes for arbitrary convex concave volumes and integration of weak form in enriched partition of unity methods abstract quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex concave volumes that arise among others in enriched partition of unity finite element methods epum the building block of the scheme involves the divergence theorem of multivariable calculus which is used to integrate the base functions an efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least squares problem the method is applied initially to integrate given polynomial functions over complex volumes and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with epum accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact numerical solutions and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method

strongly stable generalized finite element method ssgfem for a non smooth interface problem ii a simplified algorithm abstract the solution of interface problems with non smooth interfaces having corners connected by straight lines exhibit weak singularity along the interface and may have singularities at the corners in an earlier paper zhang et al 2019 a strongly stable generalized finite element method ssgfem was proposed to approximate solution of such problems the method required two separate enrichment functions at o h 1 number of nodes h is the mesh parameter and a local orthogonalization technique requiring additional operations was used to ensure its stability and robustness in this paper we present a simplified ssgfem that does not require two enrichment functions at any of the nodes as a result we obtain an optimally converging gfem that is stable i e its scaled condition number is of the same order that of a standard fem and robust i e the conditioning is independent of the position of meshes relative to the interface these properties are illuminated by well designed experiments the simplified ssgfem does not require lot and requires less degrees of freedom and is easier to implement in contrast to the ssgfem proposed in zhang et al 2019 we also proved the optimal convergence for the simplified ssgfem in this paper that required a more involved analysis based on a smart observation on approximation of weak singularities near the corner

nitsche xfem for the coupling of an incompressible fluid with immersed thin walled structures abstract in this paper we introduce a nitsche xfem method for fluid structure interaction problems involving a thin walled elastic structure lagrangian formalism immersed in an incompressible viscous fluid eulerian formalism the fluid domain is discretized with an unstructured mesh not fitted to the solid mid surface mesh weak and strong discontinuities across the interface are allowed for the velocity and pressure respectively the fluid solid coupling is enforced consistently using a variant of nitsche s method with cut elements robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization several coupling schemes with different degrees of fluid solid time splitting implicit semi implicit and explicit are investigated a series of numerical test in 2d involving static and moving interfaces illustrates the performance of the different methods proposed

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions in recent years a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes examples of such methods are the fictitious domain method the extended finite element method and nitsche s method in all these methods integrals must be computed over cut cells or subsimplices which is challenging to implement especially in three space dimensions in this note we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices as a canonical example of an overlapping mesh method we consider nitsche s method which we apply to poisson s equation and a linear elastic problem

steady shock tracking and newton s method applied to one dimensional duct flow abstract a new computational approach combining shock tracking and newton s method is applied to steady one dimensional flow through a variable area duct on a transformed computational grid where the shock is fixed the physical shock location appears explicitly as an unknown in a set of finite difference equations and is coupled to the other unknowns the space time characteristics for the associated time dependent problem are used in formulating the boundary conditions the resulting system is solved by newton s method the computed results agree very well with an exact solution and the newton iterates converge rapidly in comparison to some explicit shock capturing time asymptotic methods

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

an implicit lambda scheme this paper is concerned with the numerical simulation of transonic inviscid flows the unsteady euler equations are written in characteristi c type variables discretized in time and solved by means of an implicit technique or an alternating direction implicit technique for one and two dimensional flows respectively so that only block tridiagonal systems have to be solved at each time level the present method has been successfully applied to one dimension al transonic internal flows and two dimensional transonic external flows within the framework of thin airfoil theory and can be generalized to transonic two and three dimensional flows for arbitrary geometries in all computed cases the present technique has proved itself reliable and about one order of magnitude more efficient than its well known explicit counterpart

multi point taylor approximations in one dimensional linear boundary value problems abstract we consider second order linear differential equations in a real interval i with mixed dirichlet and neumann boundary data we consider a representation of its solution by a multi point taylor expansion the number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent x i we propose several algorithms to approximate the multi point taylor polynomials of the solution based on the power series method for initial value problems

the use of two point taylor expansions in singular one dimensional boundary value problems i abstract we consider the second order linear differential equation x 1 y f x y g x y h x in the interval 1 1 with initial conditions or boundary conditions dirichlet neumann or mixed dirichlet neumann the functions f x g x and h x are analytic in a cassini disk d r with foci at x 1 containing the interval 1 1 then the end point of the interval x 1 may be a regular singular point of the differential equation the two point taylor expansion of the solution y x at the end points 1 is used to study the space of analytic solutions in d r of the differential equation and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem this method is constructive and provides the two point taylor approximation of the analytic solutions when they exist

interface handling for three dimensional higher order xfem computations in fluid structure interaction three dimensional higher order extended finite element method xfem computations still pose challenging computational geometry problems especially for moving interfaces this paper provides a method for the localization of a higher order interface finite element fe mesh in an underlying three dimensional higher order fe mesh additionally it demonstrates how a subtetrahedralization of an intersected element can be obtained which preserves the possibly curved interface and allows therefore exact numerical integration the proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their extended axis aligned bounding boxes the intersection method is applied to a highly reduced number of intersection candidates the resulting linearized interface is used as input for an elementwise constrained delaunay tetrahedralization which computes an appropriate subdivision for each intersected element the curved interface is recovered from the linearized interface in the last step the output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element application of the interface algorithm currently concentrates on fluid structure interaction problems on low order and higher order fe meshes which may be composed of any arbitrary element types such as hexahedra tetrahedra wedges etc nevertheless other xfem problems with explicitly given interfaces or discontinuities may be tackled in addition multiple structures and interfaces per intersected element can be handled without any additional difficulties several parallelization strategies exist depending on the desired domain decomposition approach numerical test cases including various geometrical exceptions demonstrate the accuracy robustness and efficiency of the interface handling

calculation of shocked one dimensional flows on abruptly changing grids by mathematical programming abstract the steady state inviscid and nearly inviscid burgers equations and the euler equations for steady state shocked flow in a quasi 1 dimensional nozzle are discretized by cell centered finite differences on the cells of an arbitrarily spaced grid inviscid and viscous terms are approximated to second order by 2 point schemes and 4 point schemes respectively there results an overdetermined system of nonlinear algebraic equations this system is solved by a mathematical programming procedure that minimizes a weighted sum of the absolute values of the residuals the l 1 norm of the vector of residuals in this algorithm which is nonconservative no upwinding switches arbitrary constants or heuristic quantities are used the artificial viscosity used to solve the inviscid burgers equation is small 10 15 u no artificial viscosity of any kind is used for the euler equations the numerical solutions of both the viscous and the inviscid problems are accurate and nonoscillatory on grids with abrupt refinements in mesh length by factors as high as 10 4 shocks are invariably captured in one cell and this cell is rarely more than three cells away from the cell in which the physical shock occurs