weierstrass formula and zero finding methods classical weierstrass formula 29 has been often the subject of investigation of many authors in this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions we are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros conditions for the safe convergence of weierstrass method depending only on initial approximations are given in particular we study polynomials with interval coefficients using an interval version of weierstrass method enclosures in the form of disks for the complex valued set containing all zeros of a polynomial with varying coefficients are obtained we also present weierstrass like algorithm for approximating simultaneously all zeros of a class of analytic functions in a given closed region to demonstrate the proposed algorithms three numerical examples are included

a coupled momentum method for modeling blood flow in three dimensional deformable arteries abstract blood velocity and pressure fields in large arteries are greatly influenced by the deformability of the vessel moreover wave propagation phenomena in the cardiovascular system can only be described considering wall deformability since blood is usually described as an incompressible fluid however computational methods for simulating blood flow in three dimensional models of arteries have either considered a rigid wall assumption for the vessel or significantly simplified or reduced geometries computing blood flow in deformable domains using standard techniques like the ale method remains a formidable problem for large realistic anatomic and physiologic models of the cardiovascular system we have developed a new method to simulate blood flow in three dimensional deformable models of arteries the method couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain we consider a strong coupling of the degrees of freedom of the fluid and the solid domains and a linear membrane model enhanced with transverse shear for the vessel wall the effect of the vessel wall boundary is therefore added in a monolithic way to the fluid equations resulting in a remarkably robust scheme we present here the mathematical formulation of the method and discuss issues related to the fluid solid coupling membrane formulation time integration method and boundary and initial conditions implementation is discussed and results with simple geometries as well as large subject specific models are presented

linear stability analysis in fluid structure interaction with transpiration part ii numerical analysis and applications this paper constitutes the numerical counterpart of the mathematical framework introduced in part i we address the problem of flutter analysis of a coupled fluid structure system involving an incompressible newtonian fluid and a reduced structure we use the linearization principle approach developed in part i particularly suited for fluid structure problems involving moving boundaries thus the stability analysis is reduced to the computation of the leftmost eigenvalues of a coupled eigenproblem of minimal complexity this eigenproblem involves the linearized incompressible navier stokes equations and those of a reduced linear structure the coupling is realized through specific transpiration interface conditions the eigenproblem is discretized using a finite element approximation and its smallest real part eigenvalues are computed by combining a generalized cayley transform and an implicit restarted arnoldi method finally we report three numerical experiments a structure immersed in a fluid at rest a cantilever pipe conveying a fluid flow and a rectangular bridge deck profile under wind effects the numerical results are compared to former approaches and experimental data the quality of these numerical results is very satisfactory and promising

state observers of a vascular fluid structure interaction model through measurements in the solid we analyze the performances of two types of luenberger observers namely the so called direct velocity feedback and schur displacement feedback procedures originally devised for elasto dynamics to estimate the state of a fluid structure interaction model for hemodynamics when the measurements are assumed to be restricted to displacements or velocities in the solid we first assess the observers using hemodynamics inspired test problems with the complete model including the navier stokes equations in arbitrary lagrangian eulerian formulation in particular then in order to obtain more detailed insight we consider several well chosen simplified models each of which allowing a thorough analysis emphasizing spectral considerations while illustrating a major phenomenon of interest for the observer performance namely the added mass effect for the structure the coupling with a lumped parameter boundary condition model for the fluid flow and the fluid dynamics effect per se whereas improvements can be sought when additional measurements are available in the fluid domain in order to more effectively deal with strong uncertainties in the fluid state in the present framework this establishes luenberger observer methods as very attractive strategies compared e g to classical variational techniques to perform state estimation and more generally for uncertainty estimation since other observer procedures can be conveniently combined to estimate uncertain parameters

dirichlet inversion and lattice inversion problem abstract another application of dirichlet multiplication is considered in this note we show that dirichlet inversion in number theory plays an important role in lattice inversion problem with the help of this concept lattice inversion problem becomes straightforward

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

dirichlet multiplication and easily generated function analysis abstract following sine cosine functions sawtooth wave square wave triangular wave and trapezoidal wave become new easily generated periodic functions can a signal be considered to be a superposition of easily generated functions with different frequencies in order to answer this question we generalize fourier analysis to easily generated function analysis including easily generated functions series easily generated function transformation and discrete transformation for easily generated functions the results in this paper make it possible to represent a signal by use of easily generated functions a lot of techniques based on sine cosine functions can be translated into techniques based on easily generated functions because dirichlet multiplication in number theory plays a basic role in easily generated function analysis we briefly introduce this concept and present a related formula the main contents of this paper include dirichlet multiplication and a related formula relations between basic waveforms in electronics easily generated function series easily generated function transformation discrete transformation for easily generated functions and techniques of easily generated function analysis

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

linearized euler predictions of unsteady aerodynamic loads in cascades a linearized euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and vibratory blade motion is presented using the linearized euler technique one decomposes the flow into a mean or steady flow plus an unsteady harmonically varying small disturbance flow linear variable coefficient equations describe the small disturbance behavior of the flow and are solved using a pseudotime time marching lax wendroff scheme for the blade motion problem a harmonically deforming computational grid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition

efficient methods for the inclusion of polynomial zeros abstract using a suitable zero relation and the inclusion isotonicity property new interval iterative methods for the simultaneous inclusion of simple complex zeros of a polynomial are derived these methods produce disks in the complex plane that contain the polynomial zeros in each iteration providing in this manner an information about upper error bounds of approximations starting from the basic method of the fourth order two accelerated methods with newton s and halley s corrections having the order of convergence five and six respectively are constructed this increase of the convergence rate is obtained without any additional operations which means that the methods with corrections are very efficient the convergence analysis of the basic method and the methods with corrections is performed under computationally verifiable initial conditions which is of practical importance two numerical examples are presented to demonstrate the convergence behavior of the proposed interval methods

some improved inclusion methods for polynomial roots with weierstrass corrections one decade ago the third order method without derivatives for the simultaneous inclusion of simple zeros of a polynomial was proposed in 1 following nourein s idea 2 some modifications of this method with the increased convergence are proposed the acceleration of convergence is attained by using weierstrass corrections without additional calculations which provides a high computational efficiency of the modified methods it is proved that their r orders of convergence are asymptotically greater than 3 5 the presented interval methods are realized in circular complex arithmetic

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

square wave analysis about two hundred years ago the french mathematician j b j fourier asserted that any function with period 2ir can be expressed as a trigonometric series this great idea has had an important influence upon many sciences including mathematics and physics in electronics fourier analysis has been playing an important role and a signal is often considered as a superposition of many sine and cosine waves with different frequencies however with the development of switching circuits and pulse technique it becomes very easy to generate and process square waves for example it s quite convenient to obtain a system of square waves with different frequencies from a common high frequency pulse by means of counters whereas it s complicated to do the same thing with sine and cosine functions again multipliers and summers for square waves are very simple

frequency analysis based on general periodic functions electrical waveforms may have various shapes and therefore it is necessary to study the frequency analysis based on general periodic functions in this chapter we introduce the frequency series and frequency transform based on general periodic functions considering the fact that the theory for fourier series in l2 and the theory for fourier transform in l2 are both simple and elegant we shall develop a theory for general frequency series in l2 and a theory for general frequency transform in l2 in the following we shall discuss when a frequency system is a complete system or an unconditional basis in l2 and when a frequency transform can be carried out in l2 we present a number of complete systems and unconditional bases formed by common waveforms almost every common waveform is considered as an example

improved methods for the simultaneous inclusion of multiple polynomial zeros using a new fixed point relation the interval methods for the simultaneous inclusion of complex multiple zeros in circular complex arithmetic are constructed using the concept of the r order of convergence of mutually dependent sequences we present the convergence analysis for the total step and the single step methods with schroder s and halley like corrections under computationally verifiable initial conditions the suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations two numerical examples are given to demonstrate convergence characteristics of the proposed method

implicitly restarted arnoldi with purification for the shift invert transformation the need to determine a few eigenvalues of a large sparse generalised eigenvalue problem az bx with positive semidefinite b arises in many physical situations for example in a stability analysis of the discretised navier stokes equation a common technique is to apply arnoldi s method to the shift invert transformation but this can suffer from numerical instabilities as is illustrated by a numerical example in this paper a new method that avoids instabilities is presented which is based on applying the implicitly restarted arnoldi method with the b semi inner product and a purification step the paper contains a rounding error analysis and ends with brief comments on some extensions

on the equivalence between a minimal codomain cardinality riesz basis construction a system of hadamard sylvester operators and a class of sparse binary optimization problems piecewise low order polynomial riesz basis families are constructed such that they share the same coefficient functionals of smoother orthonormal bases in a localized indexing subset it is shown that a minimal cardinality basis codomain can be realized by inducing sparsity via l1 regularization in the distributional derivatives of the basis functions and that the optimal construction can be found numerically by constrained binary optimization over a suitably large dictionary furthermore it is shown that a subset of these solutions are equivalent to a specific constrained analytical solution derived via sylvester type hadamard operators

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

a linearized method for the frequency analysis of three dimensional fluid structure interaction problems in all flow regimes abstract we present a computational fluid dynamics cfd based linearized method for the frequency analysis of three dimensional fluid structure interaction problems this method is valid in the subsonic transonic and supersonic flow regimes and is insensitive to the frequency or damping level of the sought after coupled eigenmodes it is based on the solution by an orthogonal iteration procedure of a complex eigenvalue problem derived from the linearization of a three field fluid structure moving mesh formulation the key computational features of the proposed method include the reuse of existing unsteady flow solvers a second order approximation of the flux jacobian matrix and a parallel domain decomposition based iterative solver for the solution of large scale systems of discretized fluid structure equations while the frequency analysis method proposed here is primarily targeted at the extraction of the eigenpairs of a wet structure we validate it with the flutter analysis of the agard wing 445 6 for which experimental data is available

coupled fluid structure interaction hemodynamics in a zero pressure state corrected arterial geometry abstract hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the distensible arterial wall a numerical procedure for solving the fluid structure interaction problem encountered in cardiovascular flows is presented we consider a patient specific carotid bifurcation geometry obtained from 3d reconstruction of in vivo acquired tomography images which yields a geometrical representation of the artery corresponding to its pressurized state to recover the geometry of the artery in its zero pressure state which is required for a fluid structure interaction simulation we utilize inverse finite elastostatics time dependent flow simulations with in vivo measured inflow volume flow rate in the 3d undeformed artery are performed through the finite element method the coupled momentum method for fluid structure interaction is adopted to incorporate the influence of wall compliance in the numerical computation of the time varying flow domain to demonstrate the importance in recovering the zero pressure state of the artery in hemodynamic simulations we compute the time varying flow field with compliant walls for the original and the zero pressure state corrected geometric configurations of the carotid bifurcation the most important resulting effects in the hemodynamic environment are evaluated our results show a significant change in the wall shear stress distribution and the spatiotemporal extent of the recirculation regions

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

linear stability of incompressible flow using a mixed finite element method we analyzed the linear stability for incompressible flow using the mixed finite element method first by means of the fractional step finite element method the two dimensional base flows were numerically computed over a range of reynolds numbers and then they were perturbed with three dimensional disturbances by using linear stability and normal mode analysis we obtained the partial differential equations governing the evolution of perturbation from linearized navier stokes equations with slight compressibility in terms of the mixed finite element discretization in which six node quadratic lagrange triangular elements with quadratic interpolation for velocities p2 and three node linear lagrange triangular elements for pressure p1 were used a nonsingular generalized eigenproblem was formulated from these equations whose solution gives the dispersion relationship between the complex growth rate and the wave number we examined the stability of lid driven cavity flow in this study the stability results obtained from thep2p1elements were compared with those fromp1p1elements the spatial distribution of the pressure disturbance can be smoothed by using ap2p1element such as that in the simulation of incompressible flows by the mixed finite element method however the choice of elemental type has little influence on the stability results this study presents stability curves to identify the critical reynolds number and the critical wavelength of the neutral mode and discusses the instability mechanism the taylor goertler like vortices in the cavity were obtained by means of the reconstruction of three dimensional flows the stability results and the reconstructed three dimensional flows were in agreement with the observed ones and the other numerical stability results respectively

stable and time dissipative finite element methods for the incompressible navier stokes equations in advection dominated flows we examine a new galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1 d scalar advection diffusion and that leads to non oscillatory solutions as the streamline upwinding algorithms for 2 d scalar advection diffusion and incompressible navier stokes this method retains the satisfaction of the babuska brezzi condition and thus leads to optimal performance in the incompressible limit this method when combined with the recently proposed linear unconditionally stable algorithms of simo and armero yields a method for solution of the incompressible navier stokes equations ideal for either diffusive or advection dominated flows examples from scalar advection diffusion and the solution of the incompressible navier stokes equations are presented

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

a new higher order family of inclusion zero finding methods starting from a suitable fixed point relation a new one parameter family of iterative methods for the simultaneous inclusion of complex zeros in circular complex arithmetic is constructed it is proved that the order of convergence of this family is four the convergence analysis is performed under computationally verifiable initial conditions an approach for the construction of accelerated methods with negligible number of additional operations is discussed to demonstrate convergence properties of the proposed family of methods two numerical examples results are given

on an improved unusual stabilized finite element method for theadvective reactive diffusive equation an improved unusual finite element method is studied herein for a second order linear scalar differential equation including a zero order term the method consists in subtracting from the standard galerkin method a mesh dependent term suggested by static condensation of the bubbles based on this idea a new stabilized parameter is constructed which improves accuracy of the solution several numerical tests attest the efficiency of the present method and an error analysis is perfomed

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

linear stability analysis in fluid structure interaction with transpiration part i formulation and mathematical analysis the aim of this work is to provide a new linearization principle approach particularly suited for problems in fluid structure stability the complexity here and the main difference with respect to the classical approach comes from the fact that the full non linear fluid equations are written in a moving i e time dependent domain the underlying idea of our approach uses transpiration techniques j fluid mech 4 1958 383 g mortchelewicz application of linearized euler equations to flutter in 85th agard smp meeting aalborg denmark 1997 p raj b harris using surface transpiration with an euler method for cost effective aerodynamic analysis in aiaa 24th applied aerodynamics conference number 93 3506 monterey canada 1993 aiaa 27 6 1989 777 with the formalization and linearization recently developed in rev europeenne elem finis 9 6 7 2000 681 a dervieux ed fluid structure interaction kogan page science london 2003 chapter 3 this allows us to obtain a new grid independent coupled spectral problem involving the linearized navier stokes equations and those of a reduced linear structure the coupling is realized through specific transpiration conditions acting on a fixed interface while keeping a fixed fluid domain we provide a rigorous mathematical treatment of this eigenproblem we prove that the corresponding eigenmodes characterizing the free evolution of the system can be obtained from the characteristic values of a compact operator acting on a hilbert space moreover we localize the eigenfrequencies of the system in a parabolic region of the complex plan centered along the positive real axis

on halley like algorithms for simultaneous approximation of polynomial complex zeros let p z be a polynomial of degree n with n distinct zeros iterative methods for the simultaneous inclusion of k leqq n simple or multiple zeros are given the iterative formulas presented can be regarded as an interval version of halley s iterative formula for multiple zeros realized in circular arithmetic the convergence analysis is done in the presence of multiplicity the convergence order is three for k n and four if k n applying the gauss seidel approach to the basic iterative formula the single step method with accelerated convergence is stated and the lower bound of the r order of convergence of this method is obtained the dependence of convergence rate of the basic methods on the rounding errors is also studied some numerical examples illustrate the theoretical results

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100