j spectral factorization and equalizing vectors abstract this note presents some basic results on j spectral factorization and its connection with state space realizations the assumptions needed are mild and the proofs are given in detail

an efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities the finite element method is commonly used for simulations of dynamic plate bending problems a major disadvantage of this method is its practical frequency limitation in that the computational loads become prohibitively large at higher frequencies a newly developed wave based prediction technique aims to relax this frequency limitation through an enhanced computational efficiency this paper discusses the application of the wave based method for the particular case where stress singularities appear in one or more corners of a polygonal plate domain in this case the conventional set of field variable expansion functions is extended with some special purpose functions which incorporate the corner point singularities the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples

a general solution on stress singularities in an anisotropic wedge abstract this paper presents a general solution for determining the stress singularity order in an anisotropic wedge the order depends on the wedge angle boundary conditions and material properties the characteristic equation which governs the eigenvalues is established from the lekhnitskii s complex function method a domain of one quarter circle is proposed in which the contours of singularity order are plotted for all fiber orientations and a certain wedge angle the numerical results agree well with the open literature for special cases to reduce to strength of stress singularity at the wedge corner the fiber direction corresponding to minimum singularity order can be determined

h controller synthesis by j lossless coprime factorization this paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the mathcal h infty norm of the closed loop is strictly less than a given bound by the use of coprime factorizations it is shown that the mathcal h infty control problem is fundamentally related to the so called analytic systems considered by helton et al regional conference series in mathematics 68 1987 such problems admit a solution if and only if a certain j lossless factorization exists interpreted in the mathcal h infty control context this means that for a controller of the requisite type to exist the plant must admit a certain j lossless coprime factorization the full mathcal h infty synthesis problem requires that two nested j lossless factorizations exist the results are independent of whether discrete time or continuous time systems are being considered it is then shown that j lossless factorization is equivalent to the existen

asymptotic approach to analyze singular stress state in anisotropic multi material application to the rivets the problem of stress singularities due to multi material junctions is described in this paper a semi analytical approach is proposed based on an asymptotic model in the case of anisotropic linear elasticity in three dimensions the advantage of the present method is the quasi explicit knowledge of the stress and displacement fields around the junctions thus after a brief explanation of the approach leading to the mechanical fields around the singular line at the junction between different materials numerical results are presented concerning various configurations of an assembly with rivets included in a bi layer composite that illustrate this method

sub optimal hankel norm approximation problem a frequency domain approach we obtain a simple solution for the sub optimal hankel norm approximation problem for the wiener class of matrix valued functions the approach is via j spectral factorization and frequency domain techniques

from a standard factorization to a j spectral factorization for a class of infinite dimensional systems matrix valued functions in the wiener class on the imaginary line are considered in this note this class of functions is large enough to be suitable for many applications in systems and control of infinite dimensional systems for this class of functions three kinds of factorization are discussed right standard factorization canonical factorization and j spectral factorization in particular we focus on an algorithmic procedure to find a right standard factorization and a j spectral factorization for matrix valued functions in the wiener class under the assumption that such factorizations exists in practice the j spectral factors for irrational functions are usually calculated using rational approximations we show that approximation using rational functions may be achieved in the wiener norm

stress singularities in anisotropic multi material wedges and junctions abstract the general problem of stress singularities in anisotropic multi material wedges and junctions due to geometric and material discontinuities is investigated the anisotropic wedges and junctions discussed here comprise two or three materials with different geometric configurations and different material properties the interfaces in multi material junctions are either all perfectly bonded or a disbond is introduced along one interface the order of stress singularities is obtained by solving the characteristic equations that are developed from the boundary conditions of the problem numerical examples for the order of stress singularities are given for the general problem including two and three material wedges a crack terminating at an interface between two materials bonded and disbonded two and three material junctions as well an extra root of the characteristic equations in addition to three roots corresponding to higher singular stresses is observed for the cases of disbonded anisotropic two and three material junctions the stress singularities affected by geometric configurations and material properties of anisotropic materials are also investigated

boundary layers in wedges of laminated composite strips under generalized plane deformation part i asymptotic solutions abstract based upon lekhnitskii s formulation and the stroh formalism the structure of the asymptotic solutions has been examined for the boundary layer on the wedge type cross section of a laminated composite strip the composite strip is assumed under the so called generalized plane deformation which includes tension bending and or torsion by the terminal tractions as well as the generalized plane strain problem the solution structures are obtained with the aid of numerical calculation for various kinds of wedge geometry including the free edge and the delamination cracks with the crack faces opened or closed finally the nature of the asymptotic solutions is discussed including the mode mixity of singular stress field ahead of the wedge tip it is found that for a free edge problem the mode mixity of the singular asymptotic traction vector on the interfacial plane near the free edge remains invariant under varying types of remote loadings once a pair of adjacent materials or ply orientations is given and that accordingly one single scaling parameter governs the near field response

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

efficient computation of order and mode of three dimensional stress singularities in linear elasticity by the boundary finite element method the boundary finite element method bfem a novel semi analytical boundary element procedure solely relying on standard finite element formulations is employed for the investigation of the orders and modes of three dimensional stress singularities which occur at notches and cracks in isotropic halfspaces as well as at free edges and free corners of layered plates after a comprehensive literature review and a concise introduction to the standard three dimensional bfem formulation for the static analysis of general unbounded structures we demonstrate the application of the bfem for the computation of the orders and modes of two dimensional and three dimensional stress singularities for several classes of problems within the framework of linear elasticity special emphasis is placed upon the investigation of stress concentration phenomena as they occur at straight free edges and at free corners of arbitrary opening angles in composite laminates in all cases the bfem computations agree excellently with available reference results the required computational effort is found to be considerably lower compared to e g standard finite element method fem computations in the case of free laminate corners numerous new results on the occurring stress singularities are presented it is found that free corner problems generally seem to involve a more pronounced criticality than the corresponding free edge situations

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

fracture initiation at sharp notches correlation using critical stress intensities abstract in the context of linear elasticity a stress singularity of the type k n r k n the magnitude of the intensity fully characterizes the stress state in the region of the corner a critical value of k n may be appropriate as a failure criterion in situations where the region around the corner dominated by the singular field is large compared to intrinsic flaw sizes inelastic zones and process zone sizes we determined k n for notched mode i three point flexure specimens using a combination of the williams williams m l 1952 stress singularities resulting from various boundary conditions in angular corners of plates in extension journal of applied mechanics 74 526 528 asymptotic method dimensional considerations and detailed finite element analysis we carried out an experimental study of the feasibility of using a critical value of k n to correlate failure with a series of notched polymethyl methacrylate pmma three point flexure specimens with notch angles of 60 90 and 120 using the measured failure loads and the finite element solutions for k n we infer the critical notch stress intensity k cr n for sharp notched pmma the data show that excellent failure correlation is obtained through the use of the single parameter k cr n furthermore simple estimates of the size of the inelastic zone for the notched pmma specimens show it to be small relative to the singularity dominated zone this supports the applicability of linear elastic notch mechanics lenm and the idea that a critical value of k n can be used to correlate fracture initiation

the semi infinite intersection cohomology sheaf abstract we introduce the semi infinite category of sheaves on the affine grassmannian and construct a particular object in it which we call the semi infinite intersection cohomology sheaf we relate it to several other entities naturally appearing in the geometric langlands theory

j spectral factorization and equalizing vectors for the wiener class of matrix valued functions we provide necessary and sufficient conditions for the existence of a j spectral factorization one of these conditions is in terms of equalizing vectors the second one states that the existence of a j spectral factorization is equivalent to the invertibility of the toeplitz operator associated to the matrix to be factorized our proofs are simple and only use standard results of general factorization theory clancey and gohberg factorization of matrix functions and singular integral operators operator theory advances and applications vol 3 birkhauser basel 1981 note that we do not use a state space representation of the system however we make the connection with the known results for the pritchard salamon class of systems where an equivalent condition with the solvability of an algebraic riccati equation is given

analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation abstract the antiplane strain problem of two dissimilar anisotropic composite wedges of arbitrary angles that are bonded together along a common edge is considered the surfaces of the wedges can be subjected to traction traction traction displacement or displacement displacement boundary conditions the dependence of the order of the stress singularity on the wedge angles and material constants is studied the angular distribution of stresses at the apex and the exact full field stress solutions are also investigated explicit solutions for the order of stress singularity are obtained for some special cases it is found that the order of the stress singularity is always real for the antiplane dissimilar anisotropic wedge problem this is quite different from the in plane case in which the complex type of stress singularity might exist

stress singularities in an anisotropic body of revolution abstract to fill the gap in the literature on the application of three dimensional elasticity theory to geometrically induced stress singularities this work develops asymptotic solutions for williams type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials the cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution so the problems under consideration are very complicated the eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three dimensional equilibrium equations in terms of the displacement components the correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed form characteristic equations for an isotropic body of revolution and using the commercial finite element program abaqus for orthotropic bodies of revolution thereafter the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries made of a single material or bi materials under different boundary conditions

localization of g modules on the affine grassmannian we consider the category of modules over the affine kac moody algebra of critical level with regular central character in our previous paper we conjectured that this category is equivalent to the category of hecke eigen d modules on the affine grassmannian g t g t this conjecture was motivated by our proposal for a local geometric langlands correspondence in this paper we prove this conjecture for the corresponding i 0 equivariant categories where i 0 is the radical of the iwahori subgroup of g t our result may be viewed as an affine analogue of the equivalence of categories of g modules and d modules on the flag variety g b due to beilinson bernstein and brylinski kashiwara

edge singularities in anisotropic composites abstract the stress singularity at the vertex of an anistropic wedge has the form r f r as r 0 where 0 f is a real function of the polar coordinates r in many cases f is independent of r the explicit form of f r depends on the eigenvalues of the elasticity constants called p here and on the order of singularity k when k is real k if k is complex is the real part of k the p s are all complex and consist of 3 pairs of complex conjugates which reduce to i when the material is isotropic the function f depends not only on p and k it also depends on whether p and k are distinct roots of the corresponding determinants in this paper we present the function f r in terms of p and k for the cases when p and k are single roots as well as when they are multiple roots the relationship between the complex variable z introduced in the analysis and the polar coordinates r is interpreted geometrically after presenting the form of f for individual cases a general form of f is given in eqn 74 we also show that the stress singularity at the crack tip of general anisotropic materials has the order of singularity 1 2 which is at least a multiple root of order 3 the implication of this on the form f r is discussed

effects of change of reference coordinates on the stress analyses of anisotropic elastic materials abstract when the displacements are independent of the x 3 coordinate the eigenvalues and eigenvectors of the anisotropic elasticity constants depend on the orientation of the x 1 x 2 axes it is shown that the components of the eigenvectors which are complex values transform according to the law of transformation for tensors of order one the transformation of the eigenvalues is more complicated the effects of change of the reference coordinates on the form of general solution are discussed also discussed is the form of general solution when the eigenvalue p is a multiple root finally we show that as the angle of rotation of the coordinate axes varies from 0 to 2 each p traverses a circle in the complex plane which is orthogonal to the unit circle with center at the origin a graphical solution of the eigenvalue p for a given is presented some functions of p which are invariant to the rotation of the coordinate axes are obtained

nehari problems and equalizing vectors for infinite dimensional systems for a class of infinite dimensional systems we obtain a simple frequency domain solution for the suboptimal nehari extension problem the approach is via j spectral factorization and it uses the concept of an equalizing vector moreover the connection between the equalizing vectors and the nehari extension problem is given

enrichment functions for weak singularities in 2d elastic problems with isotropic and orthotropic materials abstract in this paper a new enrichment technique is presented for weak singularities in isotropic and orthotropic 2d linear elasticity problems with absolutely no information of the singularity order in a novel numerical approach the enriching bases are constructed through a weighted residual imposition of the partial differential equation conceptually the bases can be categorized as equilibrated basis functions as another advantage of the proposed technique it will be shown that the enriched solution may be extracted for either isotropic or orthotropic materials in a similar manner noting that singularity effect in orthotropic materials has not been as widely referred to as for isotropic materials in the literature the presented examples selected from the well known literature reveal the accuracy and applicability of the enrichment in the framework of a boundary type method the method may also be found useful for the researchers focusing on the extended finite element method or other similar mesh based or mesh less methods

singular functions for heterogeneous composites with cracks and notches the use of equilibrated singular basis functions abstract in this paper a novel method is presented to solve problems with weak singularities in two dimensional heterogeneous media using equilibrated singular basis functions this especially includes crack problems in composites of functionally graded material types the method is mainly presented in a boundary formulation although it may be found quite useful in other mesh based or mesh less approaches as the extended finite element method xfem the present paper considers harmonic and elasticity problems the most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem to this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach after developing the formulation in a mapped polar co ordination some primary basis functions generated from chebyshev polynomials and trigonometric functions along with corresponding weight functions are employed the numerical examples either selected from the well known literature or solved by well established techniques will demonstrate the capability of the method in problems related to composite materials

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

reconciling nu gap metric and iqc based robust stability analysis this technical note elaborates on the flexibility of an approach that combines gap metric and integral quadratic constraint iqc based analysis in the study of uncertain feedback interconnections of distributed parameter transfer functions it is established that a standard gap ball robust stability result can be recovered within the blended iqc gap framework which only requires the existence of gap continuous paths within the uncertainty set of interest this is achieved in part by showing that sufficiently small gap balls are pathwise connected in the graph topology a linear fractional characterisation of the gap is a key ingredient this characterisation is underpinned by a certain j spectral factorisation also shown to exist herein

corner stress singularities in an fgm thin plate abstract describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities this analysis derives asymptotic solutions for functionally graded material fgm thin plates with geometrically induced stress singularities the classical thin plate theory is used to establish the equilibrium equations for fgm thin plates it is assumed that the young s modulus varies along the thickness and poisson s ratio is constant the eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner the characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner the non homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support finally the effects of material non homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle