a multi level method for transmission eigenvalues of anisotropic media in this paper we propose a multi level finite element method for the transmission eigenvalue problem of anisotropic media the problem is non standard and non self adjoint with important applications in inverse scattering theory we employ a suitable finite element method to discretize the problem the resulting generalized matrix eigenvalue problem is large sparse and non hermitian to compute the smallest real transmission eigenvalue which is usually an interior eigenvalue we devise a multi level method using arnoldi iteration at the coarsest mesh the eigenvalue is obtained using arnoldi iteration with an adaptive searching technique this value is used as the initial guess for arnoldi iteration at the next mesh level this procedure is then repeated until the finest mesh level numerical examples are presented to show the viability of the method

finite element methods for maxwell s transmission eigenvalues the transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory and more importantly are parts of inverse algorithms for estimating material properties in this paper we propose two finite element methods to compute a few lowest maxwell s transmission eigenvalues which are of interest in applications since the discrete matrix eigenvalue problem is large sparse and in particular non hermitian due to the fact that the problem is neither elliptic nor self adjoint we devise an adaptive method which combines the arnoldi iteration and estimation of transmission eigenvalues exact transmission eigenvalues for balls are derived and used as a benchmark numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues

a robust numerical algorithm for computing maxwell s transmission eigenvalue problems we study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three dimensional maxwell s transmission eigenvalue problem the discretized governing equations by the nedelec edge element result in a large scale quadratic eigenvalue problem qep for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form xy 1 z both of which prevent existing eigenvalue solvers from being efficient to remedy these difficulties we rewrite the qep as a particular nonlinear eigenvalue problem and develop a secant type iteration together with an indefinite locally optimal block preconditioned conjugate gradient lobpcg method to sequentially compute the desired positive eigenvalues furthermore we propose a novel method to solve the linear systems in each iteration of lobpcg intensive numerical experiments show that our proposed method is robust although the desired real eigenvalues are surrounded by complex eigenv

stress distribution at sharp and rounded v notches in quasi orthotropic plane abstract the eigensolution of the quasi orthotropic wedge problem is presented the degenerate orthotropic material of the wedge is characterized by doubled roots of the characteristic equation using the method of singular integral equations the distribution of the stress field in the quasi orthotropic plane weakened by semi infinite rounded v notch was obtained the edge of the notch was free of load and the asymptotic stress field defined by a stress intensity factor was set at the infinity on this basis the relationship between stress intensity factor at the sharp notch vertex and the normal stress at the vertex of corresponding rounded v notch with the same opening angle was established as it was shown in the example this relationship is asymptotic for finite notched bodies and it can be used to calculate a notch stress intensity factor based on a stress concentration results obtained for a notch rounded with small radius of curvature

a two grid discretization scheme of non conforming finite elements for transmission eigenvalues abstract in this paper for the helmholtz transmission eigenvalue problem we propose a two grid discretization scheme of non conforming finite elements with this scheme the solution of the transmission eigenvalue problem on a fine grid h is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid h and the solutions of two linear algebraic systems with the same positive definite hermitian and block diagonal coefficient matrix on the fine grid h we prove the resulting solution still maintains an asymptotically optimal accuracy and we report some numerical examples in two dimension and three dimension on the modified zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues

a new class of zienkiewicz type non conforming element in any dimensions in this paper a new class of zienkiewicz type non conforming finite element in n spatial dimensions with n 2 is proposed the new finite element is proved to be convergent for the biharmonic equation

singularities of the quad curl problem abstract we consider the quad curl problem in smooth and non smooth domains of the space we first give an augmented variational formulation equivalent to the one from 25 if the datum is divergence free we describe the singularities of the variational space which correspond to the ones of the maxwell system with perfectly conducting boundary conditions the edge and corner singularities of the solution of the corresponding boundary value problem with smooth data are also characterized we finally obtain some regularity results of the variational solution

a type of adaptive c0 non conforming finite element method for the helmholtz transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering in this paper for the helmholtz transmission eigenvalue problem in r d d 2 3 we discuss the a posteriori error estimates and adaptive algorithm of a type of c 0 non conforming finite elements including the morley zienkiewicz element and the modified zienkiewicz element we give the a posteriori error estimators for approximate eigenpair and prove their reliability and efficiency for eigenfunctions we also analyze their reliability for eigenvalues numerical results not only in r 2 but also in r 3 indicate that the a posteriori error estimators are sharp and the adaptive algorithm is efficient for computing real and complex transmission eigenvalues as expected

hadamard type operators on spaces of real analytic functions in several variables abstract we consider multipliers on the space of real analytic functions of several variables a r d open i e linear continuous operators for which all monomials are eigenvectors if zero belongs to these operators are just multipliers on the sequences of taylor coefficients at zero in particular euler differential operators of arbitrary order are multipliers we represent all multipliers via a kind of multiplicative convolution with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals moreover we represent multipliers via suitable holomorphic functions with laurent coefficients equal to the eigenvalues of the operator we identify in some standard cases what topology should be put on the suitable space of analytic functionals in order that the above mentioned isomorphism becomes a topological one when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on a we also characterize in the same cases when the discovered topology coincides with the classical topology of bounded convergence on the space of analytic functionals we provide several examples of multipliers and show surjectivity results for multipliers on a if r d

non conforming finite element methods for transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory in this paper for the helmholtz transmission eigenvalue problem we give a weak formulation which is a nonselfadjoint linear eigenvalue problem based on the weak formulation we first discuss the non conforming finite element approximation and prove the error estimates of the discrete eigenvalues obtained by the adini element the morley zienkiewicz element the modified zienkiewicz element et al and we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem

mixed hp finite element method for linear elasticity with weakly imposed symmetry abstract the paper presents a generalization of arnold falk winther elements to meshes with elements of variable order the formulation is straightforward but the stability analysis involves a non trivial modification of interpolation operators the analysis addresses only h convergence but the presented numerical results indicate p convergence as well

a c0ipg method and its error estimates for the helmholtz transmission eigenvalue problem abstract the interior penalty methods using c 0 lagrange elements c 0 ipg developed in the last decade for the fourth order problems are an interesting topic in academia at present in this paper we apply the methods to the helmholtz transmission eigenvalue problem which is quadratic and non self adjoint and has important applications in the inverse scattering theory we propose a discrete scheme present a complete error analysis and first prove the c 0 ipg method can capture smooth eigenpairs efficiently using the argument in this paper we can prove that the c 0 ipg methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently numerical examples confirm our theoretical analysis

a mixed fem for the quad curl eigenvalue problem the quad curl problem arises in the study of the electromagnetic interior transmission problem and magnetohydrodynamics in this paper we study the quad curl eigenvalue problem and propose a mixed method using edge elements assuming stringent regularity of the solution of the quad curl source problem we prove the convergence and show that the divergence free condition can be bypassed

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

multigrid methods for a quad curl problem based on c0 interior penalty method abstract in this paper we study the multigrid algorithm for a quad curl problem based on c 0 interior penalty method on bounded polygonal domains it is shown that the contraction numbers of w cycle multigrid algorithm for the c 0 interior penalty method converge uniformly with rate of m where m is the number of pre smoothing and post smoothing steps is the index of elliptic regularity the performance of the multigrid fast solver is illustrated by numerical results

mixed methods for the elastic transmission eigenvalue problem abstract the elastic transmission eigenvalue problem is quadratic in the eigenvalue parameter nonselfadjoint and of fourth order in this paper we apply the ciarlet raviart mixed method to this problem give a mixed variational form and establish two mixed methods using the classical lagrange finite element and the spectral element respectively we deduce the error estimates of the discrete eigenpairs theoretical analysis and numerical experiments show that these two methods are simple and easy to implement and can efficiently compute real and complex elastic transmission eigenvalues

computation of the stress intensity factors of sharp notched plates by the fractal like finite element method the fractal like finite element method ffem is an accurate and efficient method to compute the stress intensity factors sifs of different crack configurations in the ffem the cracked notched body is divided into singular and regular regions both regions are modelled using conventional finite elements a self similar fractal mesh of an infinite number of conventional finite elements is used to model the singular region the corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co ordinates after performing a global transformation by using global interpolation functions in this paper we extend this method to analyse the singularity problems of sharp notched plates the exact stress and displacement fields of a plate with a notch of general angle are derived for plane stress strain conditions these exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the sifs the use of the global interpolation functions reduces the computational cost significantly and neither post processing technique to extract sifs nor special singular elements to model the singular region are needed the numerical examples demonstrate the accuracy and efficiency of the ffem for sharp notched problems copyright 2008 john wiley sons ltd

singular solutions of a transmission problem in plane linear elasticity for wedge shaped regions in plane elasticity when two different wedge shaped elastic materials isotropic homogeneous are bonded together along a common edge and subject to tractions on the boundary the stress field will become infinite at the apex in fact asymptotically the displacementu satisfies u approx sum limits j 1 n r i lambda j sum limits k 0 eta j 1 ln k r phi jk theta asr to 0 where r are the polar coordinates with origin at the wedge apex and the complex numbers j are the roots with positive imaginary part and multiplicity j of an explicitly given transcendental equation g 0 for applications it is desirable to understand how the number of terms that give rise to unbounded stresses in the above expansion depends on the elasticity constants and on the wedge angles setting n q sum limits 0 operatorname im lambda j q eta j the problem is then to studyn 1 as a function of the material constants and wedge angles a rigorous analysis is presented here for the case in which the wedge angles are equal symmetric domains using the fact thatg is an entire function of which depends continuously on the parameters of the problem we develop a continuity method which enables us to evaluate the functions n left frac pi a right and n left frac pi 2a right wherea is the measure of each wedge angle this yields bounds onn 1 as well as some information on the zeros ofg that correspond to higher order singularities

a nontrivial lower bound for an np problem on automata an np problem l is linearly np complete if each operatorname ntime n problem is reducible to l in linear time on a deterministic turing machine this implies that l notin operatorname dtime cn for each c geqq 1 let r i s a reduction of incompletely specified automata be the following np complete problem quoted al7 in the classical book m r garey and d s johnson computers and intractability a guide to the theory of np completeness w h freeman san francisco 1979 instance a positive integer k and an incompletely specified deterministic finite state automaton a q sigma delta q 0 f where delta is a partial transition function from q times sigma into q and q sigma q 0 f are defined as usual question can delta be extended to a total transition function from q times sigma into q in such a way that the resulting completely specified automaton has an equivalent reduced automaton with k or fewer states it is proved that problem r i s a is

the dpg methodology applied to different variational formulations of linear elasticity abstract the flexibility of the dpg methodology is exposed by solving the linear elasticity equations under different variational formulations including some with non symmetric functional settings different infinite dimensional trial and test spaces the family of formulations presented is proved to be mutually ill or well posed when using traditional energy spaces on the whole domain moreover they are shown to remain well posed when using broken energy spaces and interface variables four variational formulations are solved in 3d using the dpg methodology numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed

asymptotic description of the logarithmic singular stress field and its application abstract in a joint of two dissimilar materials under mechanical or thermal loading the stresses at the intersection of the edges and the interface are very high or singular for elastic materials behaviour for most joint geometries and material combinations there is a type of r singularity however for some joint geometries and material combinations there is a type of ln r singularity in this paper the type of ln r stress singularity for a two dissimilar materials joint under thermal loading is treated by the mellin transform method emphasis is placed on the asymptotical description of the stress distribution near the singular point the angular functions used to describe the stresses are given in a general form which are different to those of the same joint under edge tractions for a quarter planes joint angular functions are given in an explicit form finally examples are presented to show when the asymptotical solution for the type of ln r stress singularity can be used to describe stresses near the singular point in the practical relevant joints for a finite joint the unknown factor k used to describe the stress is given as well

a coupled bem and fem for the interior transmission problem in acoustics abstract the interior transmission problem itp is a boundary value problem arising in inverse scattering theory and it has important applications in qualitative methods in this paper we propose a coupled boundary element method bem and a finite element method fem for the itp in two dimensions the coupling procedure is realized by applying the direct boundary integral equation method to define the so called dirichlet to neumann dtn mappings we show the existence of the solution to the itp for the anisotropic medium numerical results are provided to illustrate the accuracy of the coupling method

a locking free hp dpg method for linear elasticity with symmetric stresses we present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p this is achieved within the recently developed discontinuous petrov galerkin dpg framework in this framework both the stress and the displacement approximations are discontinuous across element interfaces we study locking free convergence properties and the interrelationships between the two dpg methods

stress singularities in a dissimilar materials joint with edge tractions under mechanical and thermal loadings in a two dissimilar materials joint the stresses at the intersection of the edges and the interface are singular for elastic material behaviour for a joint with edge tractions the stresses near the singular point are the sum of singular terms and regular terms earlier investigations have shown that the singular stress exponents are the same for a joint with free edge and edges with tractions in the literatures only the singular term has been studied the emphasis in this paper is placed on giving an explicit form of the regular stress terms as a function of the edge tractions the material properties and the geometry of the joint it is shown that the regular terms are important also for the stress distribution near the singular point

the approximation of the maxwell eigenvalue problem using a least squares method in this paper we consider an approximation to the maxwell s eigenvalue problem based on a very weak formulation of two div curl systems with complementary boundary conditions we formulate each of these div curl systems as a general variational problem with different test and trial spaces i e the solution space is l 2 l 2 3 and components in the test spaces are in subspaces of h 1 the sobolev space of order one on the computational domain q a finite element least squares approximation to these variational problems is used as a basis for the approximation using the structure of the continuous eigenvalue problem a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div curl systems we give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values finally some results of numerical experiments are given

hadamard multipliers on spaces of real analytic functions abstract we consider multipliers on spaces of real analytic functions of one variable i e maps for which monomials are eigenvectors we characterize sequences of complex numbers which are sequences of eigenvalues for some multiplier we characterize invertible multipliers in particular we find which euler differential operators of infinite order have global analytic solutions on the real line we present a number of examples where our theory applies in some cases we give algorithms for solving the respective equations perturbation results for solvability are presented

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

fracture analysis of v notched components effects of first non singular stress term the effect of the first non singular stress term on the fracture behavior of notched structures was investigated under symmetric geometry and loading conditions according to the williams series expansion for a large domain of notch angles the non singular stress terms of sharp notches are functions of complex eigenvalues and their corresponding complex coefficients hence a new representation of stress field near the notch tip was developed in which the higher order terms are expressed as several explicit functions of real and imaginary parts of both the complex eigenvalues and their complex coefficients a critical stress criterion was then applied to the new stress formulations to assess the influence of the first non singular stress term on the apparent fracture toughness several finite element analyses were also performed on two laboratory specimens in order to show the effects of first non singular term on the near field stress distribution of notched specimens the results demonstrated that neglecting the first non singular stress term could lead to significant errors in predicting the apparent fracture toughness of notched components

layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains the symbolic calculus of pseudodifferential operators of mellin type is applied to study layer potentials on a plane domain q whose boundary 9q is a curvilinear polygon a singularity type is a zero of the determinant of the matrix of symbols of the mellin operators and can be used to calculate the bad values of p for which the system is not fredholm on lp 9q using the method of layer potentials we study the singularity types of the system of elastostatics lu uau au vdivu 0 in a plane domain q whose boundary 9k2 is a curvilinear polygon here u 0 and u a oo when a oo the system is the stokes system of hydrostatics for the traction double layer potential we show that all singularity types in the strip 0 re z i lie in the interval 1 1 so that the system of integral equations is a fredholm operator of index 0 on lp aq for all p 2 p oo the explicit dependence of the singularity types on a and the interior angles 0 of aq is calculated the singularity type of each corner is independent of a if the corner is nonconvex introduction recently there has been considerable interest in using layer potentials to solve lp boundary value problems for elliptic operators and systems on a lipschitz domain q in r for the systems of elastostatics dkv and hydrostatics fkv dahlberg fabes kenig and verchota have used rellich type identities to prove that the double layer potential integral equations yield a fredholm operator of index 0 on l 2 q for p 2 only limited information is available on the boundary integral equations for general lipschitz domains in rn the general problem of the notion of symbol on the boundary of a general lipschitz domain is still very much open in this paper we treat a very special case a curvilinear polygonal domain in r in this 2 dimensional case a precise symbolic calculus of pseudodifferential operators of mellin type is available we show that certain double layer boundary integral equations yield operators which for all p 2 p 00 are received by the editors december 14 1988 1980 mathematics subject classification 1985 revision primary 35j25 45e05 research partially supported by the italian cnr 1990 american mathematical society 0002 9947 90 1 00 25 per page

first order spectra with one binary predicate the spectrum sp of a sentence is the set of cardinalities of finite structures which satisfy we prove that any set of integers which is in func 1 i e in the class of spectra of first order sentences of type containing only unary function symbols is also in bin1 i e in the class of spectra of first order sentences of type involving only a single binary relation

efficient boundary element analysis of sharp notched plates the present paper further develops the boundary element singularity subtraction technique to provide an efficient and accurate method of analysing the general mixed mode deformation of two dimensional linear elastic structures containing sharp notches the elastic field around sharp notches is singular because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields these singularities are subtracted out of the original elastic field using the first term of the williams series expansion this regularization procedure introduces the stress intensity factors as additional unknowns in the problem hence extra conditions are required to obtain a solution extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the williams solution the procedure can take into account any number of terms of the series expansion the standard boundary element method is modified to handle additional unknowns and extra boundary conditions analysis of plates with symmetry boundary conditions is shown to be straightforward with the modified boundary element method in the case of non symmetrical plates the singular tip tractions are not primary boundary element unknowns the boundary element method must be further modified to introduce the boundary integral stress equations of an internal point approaching the notch tip as primary unknowns in the formulation the accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed mode problems new results are presented for the mixed mode analysis of a non symmetrical configuration of a single edge notched plate

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

linear time and the power of one first order universal quantifier the aim of this paper is to give two new logical characterizations of nlin nondeterministic linear time improving significantly a previous characterization of e grandjean 1994 siam j comput 23 573 597 and f olive 1998 comput complexity 7 54 97 it is known that nlin coincides with the class of problems definable by formulas of the prenex form f1 fk where the fi are unary function symbols and is first order prenex with only one universal quantifier we show that the characterization remains true in the two following cases a unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h b when only ordered structures are considered unary functions are restricted to be bijective

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

machine independent characterizations and complete problems for deterministic linear time this article presents two algebraic characterizations and two related complete problems for the complexity class dlin that was introduced in e grandjean ann math artif intell 16 1996 pp 183 236 dlin is essentially the class of all functions that can be computed in linear time on a random access machine which uses only numbers of linear value during its computations the algebraic characterizations are in terms of recursion schemes that define unary functions one of these schemes defines several functions simultaneously while the other one defines only one function from the algebraic characterizations we derive two complete problems for dlin under new very strict and machine independent affine reductions

a numerical procedure for the determination of certain quantities related to the stress intensity factors in two dimensional elasticity abstract this work is devoted to the investigation of corner and interface singularities in linear elasticity and to the development of a numerical code that computes certain quantities associated with those singularities we compute several terms of the asymptotic expansion of the solution of the elasticity system near the corner or interface which is of the form knr nfn in particular the n and fn we treat our problem as a non linear eigenvalue problem and we use an approach based on the shooting method for boundary value problems to create an analytic function whose zeros are the n above the zeros of this function are then found using an iteration method and a global method to make sure that no zero was overlooked the program works for the general case of anisotropic non homogeneous wedge with general boundary conditions as well as in the case of crossing interfaces in an internal point we test the convergence accuracy robustness efficiency and reliability of the code from those tests it can be seen that the code works quite well for the majority of the applications

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

a new error analysis of a mixed finite element method for the quad curl problem abstract in this paper we study a new numerical approach for a quad curl model problem which arises in the inverse electromagnetic scattering problems and magnetohydrodynamics mhd we first split the quad curl problem with homogeneous boundary conditions into a system of second order equations and then apply a mixed finite element method to solve the resulting system the perturbed mixed finite element method is constructed by using edge elements the well posedness of the numerical scheme is derived the optimal error estimates in h curl and l2 norms for the primal and auxiliary variables are obtained respectively the theoretical results are verified by numerical experiments

shifted inverse iteration based multigrid methods for the quad curl eigenvalue problem abstract this paper is devoted to multgrid methods for solving the quad curl eigenvalue problem the multgrid methods based on the rayleigh quotient iteration and the inverse iteration with fixed shift are proposed we prove the error estimates for both methods numerical experiments confirm our theoretical analysis and validate the efficiency of both methods

a mixed finite element method for the biharmonic equation we consider the mixed finite element approximation of the biharmonic problem on a smooth domain with either clamped or simply supported boundary conditions error estimates are proved for the ciarlet raviart formulation of the clamped plate problem and for a similar method for the simply supported plate problem

crack tip interpolation revisited it is well known that the near tip displacement eld on a crack surface can be represented in a power series in the variable p r where r is the distance to the tip it is shown herein that the coecients of the linear terms on the two sides of the crack are equal equivalently the linear term in the crack opening displacement vanishes the proof is a completely general argument valid for an arbitrary e g multiple nonplanar crack conguration and applied boundary conditions moreover the argument holds for other equations such as laplace a limit procedure for calculating the surface stress in the form of a hypersingular boundary integral equation is employed to enforce the boundary conditions along the crack faces evaluation of the nite surface stress and examination of potentially singular terms lead to the result inclusion of this constraint in numerical calculations should result in a more accurate approximation of the displacement and stress elds in the tip region and thus a more accurate evaluation of stress intensity factors