finite element solution of compressible viscous flows using conservative variables abstract a finite element method for solving the compressible navier stokes equations is presented these equations are solved in conservation form and using conservative variables appropriate finite element approximations are discussed when a galerkin variational formulation is used for high speed flows the formulation is stabilized using the stream line upwind petrov galerkin method for which we propose an analytical expression for the matrix a new discontinuity capturing operator is also proposed to accurately solve flow problems exhibiting sharp gradients the non linear systems of equations arising from the discretization are solved using an iterative strategy based on the generalized minimal residual gmres algorithm numerical results for transonic and supersonic flows are presented that demonstrate the effectiveness of the proposed finite element method

discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

time discontinuous stabilized space time finite elements for timoshenko beams stabilized space time e nite element methods for the transient computational analysis of the elastodynamics of timoshenko beams have been developed the underlying time discontinuous galerkin formulation is implicit unconditionally stable higher order accurate and robust the employed interpolations are continuous in space and time inside time slabs but discontinuous in time between adjacent time slabs to suppress spurious numerical oscillations near discontinuities or high gradients galerkin least squares stabilization has been applied further improvement of the numerical representation of the transient elastic wave propagation phenomena has been achieved by specially designed galerkin gradient least squares operators because of the reduced phase and amplitude errors of these e nite elements much coarser spatial meshes may be used without loss of accuracy the resulting reduction of the number of unknowns leads to a signie cant decrease of computer time furthermore a globaladaptivetime stepping strategy has beenemployed basedon thetemporal jump residual of thetimee niteelement method as error estimator numerical examples of transientelastic wave propagation in timoshenko beams involving a wide spectrum of wave numbers and frequencies demonstrate the good performance of the developed methods

discontinuity capturing operators for elastodynamics finite element methods are presented which include discontinuity capturing operators to accurately solve elastodynamics problems exhibiting sharp gradients in their solution e g wave propagation the formulation involves finite element discretization of the temporal domain as well as the usual finite element discretization of the spatial domain linear least squares terms are included that enhance the stability of the space time finite element method nonlinear discontinuity capturing operators are presented that result in more accurate capturing of wave fronts in transient solutions while maintaining the high order accuracy of the underlying linear algorithm in smooth regions stability of the discontinuity capturing formulation is proved and a convergence proof is given for a particular choice of the discontinuity capturing operator numerical results are presented that demonstrate the effectiveness of the proposed discontinuity capturing operators for elastodynamics

on the recovery and resolution of exponential relaxation rates from experimental data iii the effect of sampling and truncation of data on the laplace transform inversion the method of singular function expansions which in previous papers in this series was used for the inversion of the laplace transform for the cases respectively of continuous data and continuous solution and discrete data and discrete solution is extended to cover the case of discrete data and continuous solution two discrete data points distributions are considered uniform and geometric for both we prove that the singular values and the singular functions in the solution space of the problem with discrete data and continuous solution converge to the singular values and singular functions of the problem with continuous data and continuous solution when the number of points tends to infinity and the distance between adjacents points tends to zero furthermore we show by means of numerical computations that for geometrically sampled data it is possible to obtain even better approximations of the greatest singular values than in the discrete to discrete case using a similar number of data points excellent approximations of the continuous to continuous case singular functions are also obtained implementation of the inversion procedure gives continuous solutions with high computational efficiency

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

smoothed projection methods for the moment problem we discuss the problem of reconstructing a functionf from a finite set of moments problems of this kind typically arise as discretizations of integral equations of the first kind we propose an algorithm which is based on a pointwise optimization of the pointspread function which makes it particularly suitable for local reconstructions the method is compared with known methods as backus gilbert and projection methods convergence of the method is proved and the rate of convergence is determined the influence of noisy data is examined and numerical examples show the usefulness of the method

on the recovery and resolution of exponential relaxation rates from experimental data a singular value analysis of the laplace transform inversion in the presence of noise the problem of numerical inversion of the laplace transform is considered when the inverse function is of bounded strictly positive support the recent eigenvalue analysis of mcwhirter and pike for infinite support has been generalized by numerical calculations of singular values a priori knowledge of the support is shown to lead to increased resolution in the inversion and the number of exponentials that can be recovered in given levels of noise is calculated

structural dynamic analysis by a time discontinuous galerkin finite element method this paper studies a time discontinuous galerkin finite element method for structural dynamic problems by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels a new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations by using the jumps of the displacements and the velocities in the total energy norm as error indicators an adaptive time stepping procedure for selecting the proper time step size is described numerical examples including both single dof and multi dof problems are used to illustrate the performance of these algorithms comparisons with the exact results and or the results by the newmark integration scheme are given it is shown that the time discontinuous galerkin finite element method discussed in this study possesses good accuracy third order and stability properties its numerical implementation is not difficult and the higher computational cost needed in each time step is compensated by use of a larger time step size

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

design of galerkin generalized least squares methods for timoshenko beams abstract a class of finite element methods the galerkin generalized least squares methods are developed and applied to model the steady state response of timoshenko beams an optimal method is designed using a linear interpolation of the response such that there is zero finite element dispersion error the classical method of selective reduced integration and a modified version of selective reduced integration with mass lumping are shown to fall under the galerkin generalized least squares framework numerical experiments in wave propagation demonstrate the dramatic superiority of the new optimal method over the standard approaches the goal of the new methods is to decrease the computational burden required to achieve a desired accurcy level at a particular frequency thereby enabling larger scale higher frequency computations for a given platform

a space time finite element method for structural acoustics in infinite domains part 1 formulation stability and convergence abstract a space time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated in particular time discontinuous galerkin and galerkin least squares gls variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence the formulation employs a finite computational fluid domain surrounding the structure and incorporates time dependent non reflecting boundary conditions on the fluid truncation boundary energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid structure problem with absorbing boundaries the methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space time meshes are used to track waves propagating along space time characteristics an important feature of the space time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time for additional stability least squares operators based on local residuals of the structural acoustics equations including the non reflecting boundary conditions are incorporated the energy decay estimates and high order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

linear regularizing algorithms for positive solutions of linear inverse problems linear regularization methods provide a simple technique for determining stable approximate solutions of linear ill posed problems such as fredholm equations of the first kind cauchy problems for elliptic equations and backward solution of forward parabolic equations in most of these problems the solution must be positive to satisfy physical plausibility in this paper we consider ill posed first kind convolution equations and related problems such as numerical differentiation radon transform and laplace transform inversion we investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data for noisy data the solution is not necessarily positive because the appearance of negative values can then only be an effect of the noise the negative part of the solution should be negligible with a suitable choice of the regularization parameter a price to pay for ensuring positivity is always however a reduction in resolution

a fictitious domain finite element method for particulate flows the paper presents a finite element method for the direct numerical simulation of 3d incompressible fluid flows with suspended rigid particles it uses the fictitious domain approach extending the fluid domain into the domain occupied by the particles although the idea for the present formulation came from the fictitious domain formulation of glowinski et al j comput phys 169 2001 363 it finally yielded a new scheme which allows for a more efficient integration of the system of equations and which is easier to implement algorithmically it also avoids the need to grid the rigid particles and solves the entire problem on a single eulerian grid the spatial discretization is performed with p2 p1 finite elements but any other stable pair of approximations can be used the present scheme is compared to a second order scheme based on the approach in glowinski et al loc cit using the test case of a single sphere in a viscous liquid its accuracy costs performance is excellent

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

constraint networks in vision applications in machine vision of constraint networks based on an augmented lagrangian formulation are discussed only those applications that have a fundamental significance are addressed the first of these provides a generalization of the harris coupled depth slope analog model of visual reconstruction because of the generality of the approach one can derive many more alternative structures and the mathematical setting places this approach within the bounds of mixed finite element theory this offers many advantages in terms of the associated mathematical theory and implementation on digital machines the second use is in data fusion which is a crucial task for systems using multiple sensors or methods of analysis of data

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

a unified set of single step asymptotic annihilation algorithms for structural dynamics when solving the equations of structural dynamics using direct time integration meth6ds algorithmic damping is useful in controlling spurious high frequency oscillations ideally an algorithm should possess asymptotic annihilation of the highfrequency response i e spurious oscillations in the high frequencies are eliminated after one time step in this paper a new class of asymptotic annihilation algorithms for structural dynamics is presented that possesses little numerical dissipation in the low frequency regime the algorithms are based upon using finite elements in time the displacement and velocity fields are interpolated independently using time discontinuous functions the equations of motion displacement velocity compatibility and the time continuity of displacement and velocity are satisfied weakly asymptotic annihilation is achieved by choosing the displacement and velocity fields to be of equal order algorithms of any desired order of temporal accuracy can be obtained by appropriate choice of the finite element interpolations in time an analysis of the proposed class of algorithms is given proving the asymptotic annihilation property and the spectral equivalence of the algorithms to the upper diagonal of the pad6 approximation table results from finite difference analyses are presented showing the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low frequency regime

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability