lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

a three field stabilized finite element method for fluid structure interaction elastic solid and rigid body limit summary we propose a full eulerian framework for solving fluid structure interaction fsi problems based on a unified formulation in which the fsis are modelled by introducing an extra stress in the momentum equation the obtained three field velocity pressure and stress system is solved using a stabilized finite element method the key feature of this unified formulation is the ability to describe different kind of interactions between the fluid and the structure which can be either elastic or a perfect rigid body without the need of treating this last case via penalization the level set method combined with a dynamic anisotropic mesh adaptation is used to track the fluid solid interface copyright 2015 john wiley sons ltd

ten methods to bound multiple roots of polynomials given a univariate polynomial p with a k fold multiple root or a k fold root cluster near some z we discuss nine different methods to compute a disc near z which either contains exactly or contains at least k roots of p many of the presented methods are known some are new we are especially interested in the behaviour of methods when implemented in a rigorous way that is when taking into account all possible effects of rounding errors in other words every result shall be mathematically correct we display extensive test sets comparing the methods under different circumstances based on the results we present a tenth a hybrid method combining five of the previous methods which for given z i detects the number k of roots near z and ii computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster therefore the resulting discs are numerically nearly optimal

existence of a tree of stieltjes strings corresponding to two given spectra transversal vibrations of a plane rooted tree of stieltjes strings are considered with dirichlet boundary conditions at each pendant vertex continuity and kirchhoff conditions are imposed at each interior vertex except the root the dirichlet problem is that with the dirichlet condition at the root and the neumann problem is that with continuity and kirchhoff conditions at the root it is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of neumann and dirichlet problems generated by stieltjes string recurrent relations on any prescribed tree

bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix we derive some new bounds for the distance between the roots of two polynomials in terms of their coefficients and for the distance between the eigenvalues of two matrices in terms of the norm of their difference

lower bounds for the condition number of a real confluent vandermonde matrix lower bounds on the condition numberp vc of a real con uent vander monde matrix vc are established in terms of the dimension n or n and the largest absolute value among all nodes that deflne the con uent vander monde matrix or n and the interval that contains the nodes in particular it is proved that for any modest kmax the largest number of equal nodes p vc behaves no smaller than on 1 p 2 n and than on 1 p 2 2n if all nodes are nonnegative it is not clear whether those bounds are asymp totically sharp for modest kmax

the condition number of real vandermonde krylov and positive definite hankel matrices we show that the euclidean condition number of any positive definite hankel matrix of order n geq 3 may be bounded from below by gamma n 1 16n with gamma exp 4 cdot it catalan pi approx 3 210 and that this bound may be improved at most by a factor 8 gamma n similar estimates are given for the class of real vandermonde matrices the class of row scaled real vandermonde matrices and the class of krylov matrices with hermitian argument improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval our findings confirm that all such matrices including for instance the famous hilbert matrix are ill conditioned already for moderate order as application we describe implications of our results for the numerical condition of various tasks in numerical analysis such as polynomial and rational i nterpolation at real nodes determination of real roots of polynomials computation of coefficients of orthogonal polynomials or the iterative solution of linear systems of equations

finite elements for fluid structure interaction in ale and fully eulerian coordinates abstract in this work we describe and compare two monolithic models for fluid structure interaction problems first the well established ale model using natural lagrangian coordinates for the structural model and using an artificial coordinate system for the flow problem then a novel approach the fully eulerian coordinates where both subproblems structure and fluid are given in eulerian coordinates the approaches have in common that a closed variational formulation exists this allows the use of implicit solution schemes goal oriented error estimation and gradient based optimization algorithms aim of this work is the introduction and verification of the novel fully eulerian model for stationary fluid structure interaction problems

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

the post processing approach in the finite element method part 1 calculation of displacements stresses and other higher derivatives of the displacements this is the first in a series of three papers in which we discuss a method for post processing a finite element solution to obtain high accuracy approximations for displacements stresses stress intensity factors etc rather than take the values of these quantities directly from the finite element solution we evaluate certain weighted averages of the solution over the entire region these yield approximations are of the same order of accuracy as the strain energy we obtain error estimates and also present some numerical examples to illustrate the practical effectiveness of the technique in the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation

eulerian formulation using stabilized finite element method for large deformation solid dynamics this paper describes an eulerian formulation for large deformation solid dynamics in the present eulerian approach an advective equation is solved using the stream upwind petrov galerkin finite element method the eulerian finite element method is applied to path dependent solid analyses such as impact bar and ductile necking problems these computational results using the eulerian finite element method are compared with the results obtained from using the lagrangian finite element method and an eulerian formulation based on a finite difference method copyright 2007 john wiley sons ltd

on multiplicity of a quantum graph spectrum the sturm liouville equations on the edges of a metric connected graph together with the boundary and matching conditions at the vertices generate a spectral problem for a self adjoint operator it is shown that if the graph is not cyclically connected then the maximal multiplicity of an eigenvalue of such an operator is gt ptin where is the cyclomatic number of the graph and gt and ptin are the number of edges and the number of interior vertices respectively for the tree obtained by contracting all the cycles of the graph into vertices if the graph is cyclically connected then the maximal multiplicity of an eigenvalue is 1

a constructive version of the boyle handelman theorem on the spectra of nonnegative matrices abstract a constructive version of the celebrated boyle handelman theorem on the non zero spectra of nonnegative matrices is presented

rigorous perturbation bounds for eigenvalues and eigenvectors of a matrix tight perturbation bounds are given for the shifts in the eigenvalues and eigenvectors of a matrix the case of simple as well as multiple eigenvalues is treated with the results tested numerically

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

a one field monolithic fictitious domain method for fluid structure interactions abstract in this article we present a one field monolithic fictitious domain fd method for simulation of general fluid structure interactions fsi one field means only one velocity field is solved in the whole domain based upon the use of an appropriate l 2 projection monolithic means the fluid and solid equations are solved synchronously rather than sequentially we argue that the proposed method has the same generality and robustness as fd methods with distributed lagrange multiplier dlm but is significantly more computationally efficient because of one field whilst being very straightforward to implement the method is described in detail followed by the presentation of multiple computational examples in order to validate it across a wide range of fluid and solid parameters and interactions

numerical solution of a secular equation a method is proposed for the solution of a secular equation arising in modified symmetric eigenvalue problems and in several other areas this equation has singularities which make the application of standard root finding methods difficult in order to solve the equation a class of transformations of variables is considered which transform the equation into one for which newton s method converges from any point in a certain given interval in addition the form of the transformed equation suggests a convergence accelerating modification of newton s method the same ideas are applied to the secant method and numerical results are presented

a variational finite element method for stationary nonlinear fluid solid interaction we consider the problem of the interaction of a stationary viscous fluid with an elastic solid that undergoes large displacement the fluid is modeled by the stationary incompressible navier stokes equations in an eulerian frame of reference while a lagrangian reference frame and large displacement small strain theory is used for the solid a variational formulation of the problem is developed that ensures satisfaction of continuity of interface tractions and velocities the variational formulation is approximated by a galerkin finite element method yielding a system of nonlinear algebraic equations in unknown fluid velocities and pressures and solid displacements a newton like method is introduced for solution of the discrete system the method employs a modified jacobian that enables decomposition into separate fluid and solid subdomains this domain decomposition avoids possible ill conditioning of the jacobian as well as the need to compute and store geometric coupling terms between fluid and interface shape the capability of the methodology is illustrated by solution of a problem of the flow induced large displacements of an elastic infinite cylinder

eigenvalue asymptotics for differential operators on graphs we consider the spectral structure for differential equations on graphs in particular we show that self adjointness does not necessarily imply regularity we also show that the algebraic and geometric eigenvalue multiplicities of formally self adjoint differential operators on graphs are equal asymptotic bounds for the eigenvalues are then found

goal oriented error estimation and adaptivity for fluid structure interaction using exact linearized adjoints we develop duality based a posteriori error estimates for functional outputs of solutions of steady fluid structure interaction problems the crucial complication in obtaining these estimates pertains to the derivation of the coupled dual exact linearized adjoint problem owing to the free boundary character of fluid structure interaction we present two approaches to derive the dual problem in the domain map linearization approach the fluid subproblem is first transformed to a fixed reference domain after which one essentially linearizes with respect to the domain transformation map in the shape linearization approach fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape derivative techniques we show that the dual problems correspond to coupled fluid structure problems with nonstandard coupling conditions furthermore we present numerical experiments that demonstrate the consistency of the dual based error estimates and their usefulness in goal oriented adaptive mesh refinement

a fully eulerian formulation for fluid structure interaction problems in this work we present a fully eulerian framework for fluid structure interaction fsi problems coupling the incompressible navier stokes equations with a hyperelastic solid the fully eulerian framework is a monolithic implicit variational formulation for the coupled problem in contrast to the well established arbitrary lagrangian eulerian ale coordinates the fully eulerian framework formulates both subproblems fluid and solid in eulerian coordinates this concept circumvents various difficulties connected to ale coordinates since no artificial domain mapping is used the formulation is an interface capturing method and uses an extension of the solid s deformation the initial point set to detect the interface location by construction very large deformation as well as topology changes like contact of the solid to the domain boundary or other solid parts are possible

fully eulerian fluid structure interaction for time dependent problems abstract a fully eulerian framework formulation for solving time dependent fluid structure interaction problems is proposed in this work although some preliminary work on this approach exists for stationary configurations it remains to validate nonstationary processes the formulation is stated in an implicit monolithic frame of reference a finite difference scheme is used for temporal discretization whereas the spatial discretization is based on a galerkin finite element scheme the nonlinear problem is solved with a newton method and with analytical evaluation of the jacobian matrix in contrast to interface tracking methods for example the arbitrary lagrangian eulerian approach the interface must be captured which is similar to the level set method consequently the interface is allowed to intersect mesh cells which is a crucial difficulty of this method where appropriate treatment must be suggested the proposed formulation is substantiated by three numerical tests in which the performance of fully eulerian fluid structure interaction is demonstrated

inverses of generalized vandermonde matrices on demontre plusieurs equations matricielles comprenant des matrices de vandermonde generalisees qui donnent une information algebrique explicite sur les matrices inverses

approximate boundary flux calculations abstract a technique for determining derivatives fluxes or stresses from finite element solutions is developed the approach is a generalization to higher dimensions of a procedure known to give highly accurate results in one dimension numerical experiments demonstrate that certain difficulties are associated with corners in the higher dimensional extensions and two variants of the method are examined we consider both triangular and quadrilateral elements and observe some interesting differences in the numerical rates of convergence finally this post processing scheme is tested for nonlinear problems

perturbation bounds for polynomials using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of ostrowski acta math 72 99 257 1940 elsner et al linear algebra appl 142 195 209 1990 a comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of beauzamy can math bull 42 1 3 12 1999 using the matrix theoretical approach we also improve the backward stability result of edelman and murakami proceedings of the fifth siam conference on applied linear algebra siam philapdelphia 1994 math comput 64 210 763 1995

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

generalization of the bauer fike theorem the bauer fike theorem for diagonalizable matrices is generalized to cases where i non diagonalizable matrices or ii only part of the spectrum is considered

challenging eigenvalue perturbation problems abstract for estimating error bound of computed eigenvalues of a matrix we need more practical perturbation theory of eigenvalues of matrices in this paper two problems are put forward in such direction

a projection method for coupling two phase vof and fluid structure interaction simulations abstract the study of multiphase fluid structure interaction mfsi is becoming of great interest in many engineering applications in this work we propose a new algorithm for coupling a fsi problem to a multiphase interface advection problem an unstructured computational grid and a cartesian mesh are used for the fsi and the vof problem respectively the coupling between these two different grids is obtained by interpolating the velocity field into the cartesian grid through a projection operator that can take into account the natural movement of the fsi domain the piecewise color function is interpolated back on the unstructured grid with a galerkin interpolation to obtain a point wise function which allows the direct computation of the surface tension forces

motion of an elastic solid inside an incompressible viscous fluid the motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature mathematically such motion is described by a pde system that couples the parabolic and hyperbolic phases the latter inducing a loss of regularity which has left the basic question of existence open until now in this paper we prove the existence and uniqueness of such motions locally in time when the elastic solid is the linear kirchhoff elastic material the solution is found using a topological fixed point theorem that requires the analysis of a linear problem consisting of the coupling between the time dependent navier stokes equations set in lagrangian variables and the linear equations of elastodynamics for which we prove the existence of a unique weak solution we then establish the regularity of the weak solution this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases our functional framework is optimal and provides the a priori estimates necessary for us to employ our fixed point procedure

m matrix asymptotics for sturm liouville problems on graphs we consider a system formulation for sturm liouville operators with formally self adjoint boundary conditions on a graph an m matrix associated with the boundary value problem is defined and related to the matrix prufer angle associated with the system boundary value problem and consequently with the boundary value problem on the graph asymptotics for the m matrix are obtained as the eigenparameter tends to negative infinity we show that the boundary conditions may be recovered up to a unitary equivalence from the m matrix and that the m matrix is a herglotz function this is the first in a series of papers devoted to the reconstruction of the sturm liouville problem on a graph from its m matrix

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

vandermonde matrices on the circle spectral properties and conditioning we study vandermonde matrices whose nodes are given by a van der corput sequence on the unit circle our primary interest is in the singular values of these matrices and the respective spectral condition numbers detailed information about multiplicities and eigenvectors however is also obtained two applications are given to the theory of polynomials

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

goal oriented error estimation for fluid structure interaction problems in this work we present an adaptive finite element method for the numerical simulation of stationary fluid structure interaction problems the coupled system is given in a variational and monolithic arbitrary lagrangian eulerian framework we derive methods for goal oriented error estimation and mesh adaptation with the dual weighted residual method key to applying this error estimator is the underlying canonic variational formulation of the fluid structure interaction problem by mapping the flow problem to ale coordinates the developed method is applied to two and three dimensional stationary benchmark problems coupling the incompressible navier stokes equations with a nonlinear hyper elastic material law

a note on the variation of the spectrum of an arbitrary matrix abstract let a and b be two n n matrices with spectra a 1 n and b 1 n suppose that the nonsingular matrix q satisfies q 1 aq diag j 1 j p where each submatrix j i i 1 p is a jordan block then there exists a permutation of 1 n such that j 1 n j j 2 n 1 n p max q 1 b a q f q 1 b a q f m and for j 1 n j j n 1 n p max n q 1 b a q 2 n q 1 b a q 2 m where m is the order of the largest jordan block of a and f and 2 denote respectively the frobenius norm and the spectral norm

hyman s method revisited the qr algorithm is considered one of the most reliable methods for computing matrix eigenpairs however it is unable to detect multiple eigenvalues and jordan blocks matlab s eigensolver returns heavily perturbed eigenvalues and eigenvectors in such cases and there is no hint for possible principal vectors this paper calls attention to hyman s method as it is applicable for computing principal vectors and higher derivatives of the characteristic polynomial that may help to estimate multiplicity an important information for more reliable computation we suggest a test matrix collection for jordan blocks the first numerical tests with these matrices reveal that the computational problems are deeper than expected at the beginning of this work

a coupled eulerian lagrangian extended finite element formulation for simulating large deformations in hyperelastic media with moving free boundaries we present a coupled eulerian lagrangian cel formulation aimed at modeling the moving interface of hyperelastic materials undergoing large to extreme deformations this formulation is based on an eulerian description of kinematics of deformable bodies together with an updated lagrangian formulation for the transport of the deformation gradient tensor the extended finite element method xfem is used to discretize the mechanical equilibrium and deformation gradient transport equations in a two phase domain a mixed interpolation scheme biquadratic for the velocity and bilinear for the deformation gradient is adopted to improve the accuracy of the numerical formulation the interface describing the deformed shape of the body is represented by the level set function and is evolved using the grid based particle method the performance of the scheme is explored in two dimensions in the compressible regime for an adequate spatial and temporal discretization our numerical results are in good agreement with theory and with numerical results from the traditional lagrangian formulation in abaqus the advantage of the proposed formulation is that material motion is not coupled with that of the mesh this eliminates the issues of mesh distortion and the need for remeshing associated with lagrangian formulations when bodies undergo very large distortions it is therefore well adapted to describe the motion of complex fluids and soft matter whose physical properties are intermediate between conventional liquids and solids

perturbation bounds for the joint spectrum of commuting matrices using the clifford algebra techniques of pryde bhatia and bhattacharyya generalized the classical eigenvalue perturbation bound of henrici to the joint spectrum of commuting tuples of matrices elsner improved on this result using the techniques of pryde and chu we show that for small perturbations a smaller bound may be given than that obtained by elsner a similar result is then obtained using the technique of elsner himself

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

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

mesh deformation using the biharmonic operator the use of the biharmonic operator for deforming a mesh in an arbitrary lagrangian eulerian simulation is investigated the biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh this allows both the position and the normal mesh spacing along a boundary to be controlled which is important for two fluid interfaces and periodic boundaries at these boundaries we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary in addition results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second order partial differential equations a final advantage is that with the biharmonic operator the integrity of a grid in a moving boundary layer can be preserved as the boundary moves the main disadvantage of the approach is its increased computational expense

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

perturbations of invariant subspaces of unreduced hessenberg matrices non structured perturbation of invariant subspaces of unreduced i e nonderogatory hessenberg matrices is considered some perturbation results for the generalized eigenvectors and the characteristic polynomial of unreduced upper hessenberg matrices are given two theorems are on the perturbation of invariant subspaces which are somewhat similar to the sin q theorems of davis and kahan apart from the residual which we do not have here dense perturbations of unreduced hessenberg matrices are also considered finally we prove an invariant subspace perturbation theorem for nonderogatory matrices

long term simulation of large deformation mechano chemical fluid structure interactions in ale and fully eulerian coordinates in this work we develop numerical schemes for mechano chemical fluid structure interactions with long term effects we investigate a model of a growing solid interacting with an incompressible fluid a typical example for such a situation is the formation and growth of plaque in blood vessels this application includes two particular difficulties first growth may lead to very large deformations up to full clogging of the fluid domain we derive a simplified set of equations including a fluid structure interaction system coupled to an ode model for plaque growth in arbitrary lagrangian eulerian ale coordinates and in eulerian coordinates the latter novel technique is capable of handling very large deformations up to contact the second difficulty stems from the different time scales while the dynamics of the fluid demand to resolve a scale of seconds growth typically takes place in a range of months we propose a temporal two scale approach using local small scale problems to compute an effective wall stress that will enter a long scale problem our proposed techniques are substantiated with several numerical tests that include comparisons of the eulerian and ale approaches as well as convergence studies

truly monolithic algebraic multigrid for fluid structure interaction the coupling of flexible structures to incompressible fluids draws a lot attention during the last decade many different solution schemes have been proposed in this contribution we concentrate on strong coupling fluid structure interaction by means of monolithic solution schemes therein a newton krylov method is applied to the monolithic set of nonlinear equations such schemes require good preconditioning to be efficient we propose two preconditioners that apply algebraic multigrid techniques to the entire fluid structure interaction system of equations the first is based on a standard block gauss seidel approach where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem the second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields the resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance

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

a partitioned fully explicit lagrangian finite element method for highly nonlinear fluid structure interaction problems summary in this work a fully explicit partitioned method for the simulation of fluid structure interaction fsi problems is presented the fluid domain is modelled with an explicit particle finite element method pfem based on the hypothesis of weak compressibility the lagrangian description of the fluid is particularly effective in the simulation of fsi problems with free surface flows and large structural displacements since the fluid boundaries are automatically defined by the position of the mesh nodes a distinctive feature of the proposed fsi strategy is that the solid domain is modelled using the explicit integration finite element method fem in a off the shelf commercial software abaqus explicit this allows to perform simulations with a complete and advanced description on the structural domain including advanced structural material models and contact the structure to fluid coupling algorithm is based on a technique derived from the domain decomposition methods namely the gravouil and combescure gc algorithm the method allows for arbitrarily large interface displacements using different time incrementation and non conforming meshes in the different domains which is an essential feature for the efficiency of an explicit solver involving different materials the resulting fully explicit and fully lagrangian finite element approach is particularly appealing for the possibility of its efficient application in a large variety of highly nonlinear engineering problems this article is protected by copyright all rights reserved

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

the m matrix inverse problem for the sturm liouville equation on graphs abstractwe consider an inverse spectral problemfor sturm liouville boundaryvalue problemson a graph with formally self adjoint boundary conditions at the nodes where the giveninformation is the m matrix based on the results found in s currie b a watson m matrix asymptotics for sturm liouville problems on graphs j com appl math doi 10 1016 j cam 2007 11 019 using the green s function we prove that the poles ofthe m matrix are at the eigenvalues of the associated boundary value problem and aresimple located on the real axis and that the residue at a pole is a negative semi de nitematrix with rank equal to the multiplicity of the eigenvalue we de ne the so callednorming constants and relate them to the spectral measure and the m matrix thisenables us to recover from the m matrix the boundary conditions and the potential up to a unitary equivalence for co normal boundary conditions keywords inverse problem di erential operators on graphs m function sturm liouville 2000 msc 34a55 34b45 34b20 34l05 34b27

a secular equation for the eigenvalues of a diagonal matrix perturbation let d denote a diagonal n n complex matrix and suppose x1 xr and w1 wr are complex n vectors it is shown that there is a rational function f such that if is not an eigenvalue for d then is an eigenvalue for p d x 1w1 x rwr if and only if f 0 this generalizes a well known result for the eigenvalues of a rank one self adjoint perturbation an immediate corollary in the rank one self adjoint case is that the eigenvalues of p and d must interlace if the eigenvalues of d are distinct and the perturbation matrix is irreducible it is shown that in the general case the function f also carries information about the eigenvalues of p for example is an eigenvalue of multiplicity m 0 for p if and only if f f f m 1 0 and f m 0 in the self adjoint case a necessary and sufficient condition for the eigenvalues of p and d to interlace is given and the problem of determining the multiplicities of the eigenvalues of d as eigenvalues of p is studied the formula yields a simple algorithm for determining the characteristic polynomial of a tridiagonal matrix

goal oriented error estimation and adaptivity for free boundary problems the domain map linearization approach in free boundary problems the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation we develop duality based a posteriori error estimates for functional outputs of solutions of free boundary problems that include both sources of error the derivation of an appropriate dual problem linearized adjoint is however nonobvious for free boundary problems to derive an appropriate dual problem we present the domain map linearization approach in this approach the free boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map we show for a bernoulli type free boundary problem that this dual problem corresponds to a poisson problem with a nonlocal robin type boundary condition furthermore we present numerical experiments that demonstrate the effectivity of the dual based error estimate and its usefulness in goal oriented adaptive mesh refinement

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