explicit streamline diffusion finite element methods for the compressible euler equations in conservation variables the paper concerns the streamline diffusion finite element method applied to one and two dimensional gas flow described by the inviscid euler equations in conservation variables we point out that the streamline diffusion method is a natural finite element analogue to upstream type finite difference volume schemes and in fact constitutes a general framework for a large class of them we study explicit implementations of the method and derive different choices of stabilizing streamline diffusion matrices in particular we propose a consistent fully multidimensional version a brief review of the theoretical background to the method is presented and some numerical results in two dimensions are given

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

a new restarting method in the arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix we have developed a new restarting method called minimum restarting method the choice of the initial vector in this procedure is different from all the other methods in the arnoldi algorithm this algorithm is very simple and can be use with personal computer up to high accuracy

one dimensional front tracking based on high resolution wave propagation methods a simple approach to shock tracking is presented in conjunction with conservative high resolution shock capturing methods in one space dimension an underlying uniform grid is used with additional grid interfaces introduced at appropriate points for tracked shocks conservative high resolution methods based on the large time step wave propagation approach are used on the resulting nonuniform grid this method is stable even if some of the small cells created by the tracked interface are orders of magnitude smaller than the regular cells used to determine the time step a fractional step method is used to handle source terms several calculations are presented to demonstrate the effectiveness of the method including an unstable detonation wave calculation where mesh refinement in the reaction zone is required in addition to shock tracking stability and accuracy results of the method are also shown for some sample problems the basic ideas described here can be extended to two space dimensions as will be

the enriched space time finite element method est for simultaneous solution of fluid structure interaction the paper introduces a weighted residual based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures the presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple flow immersed solid objects the fluid flow is described by the incompressible navier stokes equations the current configuration of the thin structure of linear elastic material with non linear kinematics is mapped to the flow using the zero iso contour of an updated level set function the formulation of fluid structure and coupling conditions uniformly uses velocities as unknowns the integration of the weak form is performed on a space time finite element discretization of the domain interfacial constraints of the multi field problem are ensured by distributed lagrange multipliers the proposed formulation and discretization techniques lead to a monolithic algebraic system well suited for strongly coupled fluid structure systems embedding a thin structure into a flow results in non smooth fields for the fluid based on the concept of the extended finite element method the space time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions this leads to the present enriched space time est method numerical examples of fluid structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems the test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail copyright 2007 john wiley sons ltd

weighted restarting method in the weighted arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix abstract in this paper we present a new algorithm which is called weighted restarting algorithm for computing the eigenvalues of a nonsymmetric matrix it has been applied to the weighted arnoldi method developed by a essai weighted fom and gmres for solving nonsymmetric linear systems numer algorithm 18 1998 277 292 the implementation of the algorithm has been tested by numerical examples the results show that the algorithm converges fast and works with high accuracy

the subpower membership problem for bands abstract fix a finite semigroup s and let a 1 a k b be tuples in a direct power s n the subpower membership problem smp for s asks whether b can be generated by a 1 a k for bands idempotent semigroups we provide a dichotomy result if a band s belongs to a certain quasivariety then is in p otherwise it is np complete furthermore we determine the greatest variety of bands all of whose finite members induce a tractable finally we present the first example of two finite algebras that generate the same variety and have tractable and np complete smp s respectively

a treatment of discontinuities for finite difference methods abstract in this paper we develop a new version of the treatment of discontinuities developed earlier by this author it is a variant of the x t version in j comput phys 83 148 1989 the new version differs from the x t version in the sense that it tracks discontinuity positions instead of the local conservation errors one of the main considerations of this paper is to show the equivalence between the two versions by revealing the link between the discontinuity positions and the local conservation errors in addition a stacking treatment is developed to deal with discontinuities that locate in the same grid cell numerical experiments for applying the new version to a two step scheme are presented and the results are qualitatively the same as for applying the x t version to one step schemes

numerical simulation of premixed combustion using an enriched finite element method in this paper we present a novel discretization technique for the simulation of premixed combustion based on a locally enriched finite element method fem use is made of the g function approach to premixed combustion in which the domain is divided into two parts one part containing the burned and another containing the unburned gases a level set or g function is used to define the flame interface separating burned from unburned gases the extended finite element method x fem is employed which allows for velocity and pressure fields that are discontinuous across the flame interface lagrange multipliers are used to enforce the correct essential interface conditions in the form of jump conditions across the embedded flame interface a persisting problem with the use of lagrange multipliers in x fem has been the discretization of the lagrange multipliers in this paper the distributed lagrange multiplier technique is adopted we will provide results from a spatial convergence analysis showing good convergence however a small modification of the interface is required to ensure a unique solution finally results are presented from the application of the method to the problems of moving flame fronts the darrieus landau instability and a piloted bunsen burner flame

a treatment of discontinuities for finite difference methods in the two dimensional case abstract this paper extends the treatment of discontinuities introduced by the author in 13 14 and 15 to the two dimensional case the main idea relies on the fact that on each side of a discontinuity the computations draw information from the same side a numerical method for ordinary differential equations modeling the movement of the discontinuity curve is incorporated into the algorithm to compute discontinuity positions the conservation feature of the treatment is studied for the case of an isolated discontinuity finally we study two dimensional scalar systems and systems of conservation laws and display some numerical results when our treatment is applied

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment summary a methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial dierential equation loses hyperbolicity the approach is limited to rate independent materials so that the transition occurs on a set of measure zero the discrete discontinuity is treated by the extendednite element method xfem whereby arbitrary discontinuities can be incorporated in the model without remeshing loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model a new method was developed for the case when the discontinuity ends within an element it facilitates the modelling of crack tips that occur within an element in a dynamic setting the method is applied to several dynamic crack growth problems including the branching of cracks copyright 2003 john wiley sons ltd

computational methods for propagating phase boundaries this paper considers numerical methods for computing propagating phase boundaries in solids described by the physical model introduced by abeyaratne and knowles the model under consideration consists of a set of conservation laws supplemented with a kinetic relation and a nucleation criterion discontinuities between two different phases are undercompressive crossing waves in the general terminology of nonstrictly hyperbolic systems of conservation laws this paper studies numerical methods designed for the computation of such crossing waves we propose a godunov type method combining front tracking with a capturing method we also consider glimm s random choice scheme both methods share the property that the phase boundaries are sharply computed in the sense that there are no numerical interior points for the description of a phase boundary this property is well known for the glimm s scheme on the other hand our front tracking algorithm is designed so that it tracks phase boundaries but captures shock waves phase boundaries are sensitive to numerical dissipation effects so the above property is essential to ensure convergence toward the correct entropy weak solution convergence of the godnuov type method is demonstrated numerically extensive numerical experiments show the practical interest of both approaches for computations of undercompressive crossing waves

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

a further look at newmark houbolt etc time stepping formulae this paper looks at some well known methods for the numerical integration of the vibration equation from the point of view of various groupings of the independent equations involved new forms are given of the general three parameter method introduced by zienkiewicz 1 some new one step methods are introduced these are of great practical importance because of the ease with which the size of time step can be changed

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

a finite set of functions with an exptime complete composition problem we exhibit a finite family of functions over a finite set i e a finite algebra such that the problem whether a given function can be obtained as a composition of the members of this family i e is a member of the clone generated by the algebra is exptime complete

a new restarting method in the harmonic projection algorithm for computing the eigenvalues of a nonsymmetric matrix the harmonic projection method can be used to find interior eigenpairs of large matrices given a target point or shift to which the needed interior eigenvalues are close the desired interior eigenpairs are the eigenvalues nearest and the associated eigenvectors however it has been shown that the harmonic projection method may converge erratically and even may fail to do so in this paper we present a new restarting method in the harmonic projection algorithm for computing the eigenvalues of a nonsymmetric matrix the implementation of the algorithm has been tested by numerical examples the results show that the algorithm converges fast and works with high accuracy

a kriging based error reproducing and interpolating kernel method for improved mesh free approximations an error reproducing and interpolating kernel method erikm which is a novel and improved form of the error reproducing kernel method erkm with the nodal interpolation property is proposed the erkm is a non uniform rational b splines nurbs based mesh free approximation scheme recently proposed by shaw and roy comput mech 2007 40 1 127 148 the erkm is based on an initial approximation of the target function and its derivatives by nurbs basis functions the errors in the nurbs approximation and its derivatives are then reproduced via a family of non nurbs basis functions the non nurbs basis functions are constructed using a polynomial reproduction condition and added to the nurbs approximation obtained in the first step in the erkm the interpolating property at the boundary is achieved by repeating the knot open knot vector however for most problems of practical interest employing nurbs with open knots is not possible because of the complex geometry of the domain and consequently erkm shape functions turn out to be non interpolating in erikm the error functions are obtained through localized kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint interpolating error functions so obtained are then added to the nurbs approximant while enriching the erkm with the interpolation property the erikm naturally possesses all the desirable features of the erkm such as insensitivity to the support size and ability to reproduce sharp layers the proposed erikm is finally applied to obtain strong and weak solutions for a class of linear and non linear boundary value problems of engineering interest these illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent

entropy satisfaction of a conservative shock tracking method in this paper we discuss the entropy satisfaction of the conservative shock tracking technique developed in d k mao j comput phys 92 1991 pp 422 455 d k mao j comput phys 103 1992 pp 359 369 and d k mao siam j numer anal 32 1995 pp 1677 1703 when it is applied to the godunov scheme we consider the scalar case in one space dimension and assume that both the flux and entropy functions are strictly convex we prove that when the tracked shocks are strong enough in comparison with the variation of the numerical solution around them the numerical solution satisfies the entropy condition in a certain sense we also discuss the entropy situation when the convexity of the flux function is very weak and the tracked shock neighbors to strong simple waves

toward front tracking based on conservation in two space dimensions a two dimensional front tracking method based on conservation for scalar conservation law and for the euler system of gas dynamics is being developed in the method discontinuities are tracked by enforcing the conservation properties of the pdes unlike the traditional front tracking methods for which the conservation is only a property that may or may not be preserved in this front tracking method the conservation is the mechanism by which the tracking is performed the method is also extended to treat the reflection wall boundary in the euler system as an attempt to treat the interactions of discontinuities by enforcing the conservation properties without solving two dimensional riemann problems three special cases of treatment of discontinuity interactions in scalar conservation law and in the euler system of gas dynamics are studied finally numerical experiments are implemented in both scalar and system cases to show the efficiency of the method

tractability and learnability arising from algebras with few subpowers a constraint language gamma on a finite set a has been called polynomially expressive if the number of n ary relations expressible by exists wedge atomic formulas over gamma is bounded by exp o n k for some constant k it has recently been discovered that this property is characterized by the existence of a k 1 ary polymorphism satisfying certain identities such polymorphisms are called k edge operations and include mal cev and near unanimity operations as special cases we prove that if gamma is any constraint language which for some k 1 has a k edge operation as a polymorphism then the constraint satisfaction problem for langle gamma rangle the closure of gamma under exists wedge atomic expressibility is globally tractable we also show that the set of relations definable over gamma using quantified generalized formulas is polynomially exactly learnable using improper equivalence queries

the interaction of nonlinear hyperbolic waves nonlinearities in wave equations lead to focusing and defocusing of solutions focusing causes sharply defined wave fronts the interaction of such sharply defined wave fronts and more generally of nonlinear hyperbolic waves is of fundamental importance and includes such phenomena as mach triple point formation shock wave diffraction patterns and the study of riemann problems in one and higher dimensions recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures this mathematical theory should be useful in the design of improved computational algorithms and in part was motivated by such considerations it is also of considerable interest for its own sake as new mathematical phenomena as well as in terms of the direct insight it provides into physical phenomena within the subject matter and point of view adopted here we have attempted to present a broad and we hope a representative account of recent progress

weighted fom inverse vector iteration method for computing a few smallest largest eigenvalues of pair a b the generalized eigenvalue problem ax lbx has special properties when a b is a symmetric and positive definite pair we have recently developed a new method for computing a few smallest largest eigenvalues of a symmetric positive definite problem ax lbx h saberi najafi a refahi fom inverse vector iteration method for computing a few smallest largest eigenvalues of pair a b appl math in press the method used three parameters which are weighted arnoldi inner and outer iterations parameters in this article those parameters have been optimized to increase the speed of convergence the method has been tested by some numerical examples

on time integration in the xfem the extended finite element method xfem is often used in applications that involve moving interfaces examples are the propagation of cracks or the movement of interfaces in two phase problems this work focuses on time integration in the xfem the performance of the discontinuous galerkin method in time space time finite elements fes and time stepping schemes are analyzed by convergence studies for different model problems it is shown that space time fe achieve optimal convergence rates special care is required for time stepping in the xfem due to the time dependence of the enrichment functions in each time step the enrichment functions have to be evaluated at different time levels this has important consequences in the quadrature used for the integration of the weak form a time stepping scheme that leads to optimal or only slightly sub optimal convergence rates is systematically constructed in this work copyright 2009 john wiley sons ltd

on the construction of blending elements for local partition of unity enriched finite elements for computational efficiency partition of unity enrichments are preferably localized to the sub domains where they are needed it is shown that an appropriate construction of the elements in the blending area the regionwhere the enriched elements blend to unenriched elements is often crucial for good performance of local partition of unity enrichments an enhanced strain formulation is developed which leads to good performance the optimal rate of convergence is achieved for polynomial enrichments it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence the methods are illustrated by several examples the examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element but other enrichments are also considered results show that both methods provide optimal rates of convergence

the immersed fictitious element method for fluid structure interaction volumetric consistency compressibility and thin members a weak form and an implementation are given for fluid structure interaction by the immersed fictitious element method for compressible fluids the weak form is applicable to models where the fluid is described by eulerian coordinates while the solid is described by lagrangian coordinates which suits their intrinsic characteristics a unique feature of the method is the treatment of the fictitious fluid by a lagrangian description which simplifies the interface conditions methods for enforcing volumetric consistency between the fluid and solid and treating thin members are given although a compressible viscous fluid is considered here the new developments can be applied to incompressible fluids

a local space time discontinuous finite element method we present a locally enriched space time finite element method for solving hyperbolic problems with discontinuities the space time formulation is coupled to a standard semidiscrete finite element method the discontinuities are captured by a space time version of the extended finite element method which treats arbitrary moving discontinuities since the discontinuities are local the enriched space time method is only needed around the discontinuities which provides significant computational savings the coupling is implemented through a weak enforcement of the continuity of the flux between the space time and semidiscrete domains in a manner similar to discontinuous galerkin methods the method is illustrated through one dimensional problems it displays remarkable accuracy in capturing moving discontinuities both in their amplitude and speed

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

arbitrary discontinuities in space time finite elements by level sets and x fem an enriched finite element method with arbitrary discontinuities in space time is presented the discontinuities are treated by the extended finite element method x fem which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper surface which is described by level sets a space time weak form for conservation laws is developed where the rankine hugoniot jump conditions are natural conditions of the weak form the method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non linear burgers equations by capturing the discontinuity in time as well as space results are improved over capturing the discontinuity in space alone and the method is remarkably accurate implications to standard semi discretization x fem formulations are also discussed copyright 2004 john wiley sons ltd

an eulerian lagrangian method for fluid structure interaction based on level sets abstract we describe a method for fluid structure interaction with the interface and free surfaces defined by level sets the fluid is treated by an eulerian mesh whereas the solid or structure by a lagrangian mesh the level set method provides a compact framework for writing the weak form and for developing and computing the discretized equations lagrange multiplier methods and penalty methods are used to couple the fluid and structure results are presented for a few compressible fluid problems representative of high pressure waves

space time least squares finite element method for convection reaction system with transformed variables we present a method to solve a convection reaction system based on a least squares finite element method lsfem for steady state computations issues related to recirculation flow are stated and demonstrated with a simple example the method can compute concentration profiles in open flow even when the generation term is small this is the case for estimating hemolysis in blood time dependent flows are computed with the space time lsfem discretization we observe that the computed hemoglobin concentration can become negative in certain regions of the flow it is a physically unacceptable result to prevent this we propose a quadratic transformation of variables the transformed governing equation can be solved in a straightforward way by lsfem with no sign of unphysical behavior the effect of localized high shear on blood damage is shown in a circular couette flow with blade configuration and a physiological condition is tested in an arterial graft flow

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

a new restarting method in the lanczos algorithm for generalized eigenvalue problem abstract in this paper we present a new restarting method in the lanczos algorithm for computing a few eigenvalues of the symmetric positive definite eigenvalue problem ax bx where a and b are large and sparse matrices the implementation of the algorithm has been tested by numerical examples the results show that the algorithm converges fast and works with high accuracy