a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

bundles and finite foliations by a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume we recall that if is a kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space otherwise is geometrically in nite if happens to be a surface group then we say it is quasi fuchsian if the limit set for the group action is a jordan curve c and preserves the components of s n c the starting point for this work is the following theorem which is a combination of theorems due to marden thurston and bonahon

a hyperelastic based large strain elasto plastic constitutive formulation with combined isotropic kinematic hardening using the logarithmic stress and strain measures summary this paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity the combined isotropic kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions i the formulation is hyperelastic based ii the stress strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the hencky strain tensor and its elastic work conjugate stress tensor and iii the multiplicative decomposition of the deformation gradient is adopted since no stress rates are present the formulation is of course numerically objective in the time integration it is shown that the model gives adequate physical behaviour and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor

double reduction of pdes from the association of symmetries with conservation laws with applications abstract we use the definition of the association of symmetries of partial differential equations with conservation laws a kara f mahomed the relationship between symmetries and conservation laws int j theor phys 39 1 2000 23 40 to find a double reduction of partial differential equations with two independent variables applications to the linear heat equation the bbm equation the sine gordon equation and a system of differential equations from one dimensional gas dynamics illustrate the new theory

increasing the number of fibered faces of arithmetic hyperbolic 3 manifolds we exhibit a closed hyperbolic 3 manifold which satisfies a very strong form of thurston s virtual fibration conjecture in particular this manifold has finite covers which fiber over the circle in arbitrarily many ways more precisely it has a tower of finite covers where the number of fibered faces of the thurston norm ball goes to infinity in fact faster than any power of the logarithm of the degree of the cover and we give a more precise quantitative lower bound the example manifold m is arithmetic and the proof uses detailed number theoretic information at the level of the hecke eigenvalues to drive a geometric argument based on fried s dynamical characterization of the fibered faces the origin of the basic fibration m s 1 is the modular elliptic curve e x 0 49 which admits multiplication by the ring of integers of q 7 we first base change the holomorphic differential on e to a cusp form on gl 2 over k q 3 and then transfer over to a quaternion algebra d k ramified only at the primes above 7 the fundamental group of m is a quotient of the principal congruence subgroup of o d of level 7 to analyze the topological properties of m we use a new practical method for computing the thurston norm which is of independent interest we also give a noncompact finite volume hyperbolic 3 manifold with the same properties by using a direct topological argument

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

an eulerian projection method for quasi static elastoplasticity a well established numerical approach to solve the navier stokes equations for incompressible fluids is chorin s projection method 1 whereby the fluid velocity is explicitly updated and then an elliptic problem for the pressure is solved which is used to orthogonally project the velocity field to maintain the incompressibility constraint in this paper we develop a mathematical correspondence between newtonian fluids in the incompressible limit and hypo elastoplastic solids in the slow quasi static limit using this correspondence we formulate a new fixed grid eulerian numerical method for simulating quasi static hypo elastoplastic solids whereby the stress is explicitly updated and then an elliptic problem for the velocity is solved which is used to orthogonally project the stress to maintain the quasi staticity constraint we develop a finite difference implementation of the method and apply it to an elasto viscoplastic model of a bulk metallic glass based on the shear transformation zone theory we show that in a two dimensional plane strain simple shear simulation the method is in quantitative agreement with an explicit method like the fluid projection method it is efficient and numerically robust making it practical for a wide variety of applications we also demonstrate that the method can be extended to simulate objects with evolving boundaries we highlight a number of correspondences between incompressible fluid mechanics and quasi static elastoplasticity creating possibilities for translating other numerical methods between the two classes of physical problems

operator split methods for the numerical solution of the elastoplastic dynamic problem the elastoplastic dynamic problem is first formulated in a form that facilitates the application of product formula techniques the additive decomposition of the dynamic equations into elastic and plastic parts is taken as a basis for the definition of product algorithms that exploit such decomposition in the context of a finite element discretization these product algorithms entail for every time step the solution of an elastic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive relations suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity the proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions with or without corners the stability properties of the product algorithm are identical to those of the elastic algorithm used whereas the computational expense is practically equal to that of an elastic problem

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

quasi self adjoint nonlinear wave equations in this paper we generalize the classification of self adjoint second order linear partial differential equation to a family of nonlinear wave equations with two independent variables we find a class of quasi self adjoint nonlinear equations which includes the self adjoint linear equations as a particular case the property of a differential equation to be quasi self adjoint is important e g for constructing conservation laws associated with symmetries of the differential equation

symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics abstract exact solutions of some classical pdes with two independent variables are achieved by exploiting a double reduction method which is applied after deducing conserved vectors associated with lie point symmetries these are generally obtained through the application of noether s theorem after introducing the notions of adjoint equation and extended lagrangian alternatively associated conserved vectors can be obtained by a direct method

on the cuspidal cohomology of arithmetic subgroups of mathrm sl 2n and the first betti number of arithmetic 3 manifolds