an asymptotic analysis of composite beams with kinematically corrected end effects a finite element based beam analysis for anisotropic beams with arbitrary shaped cross sections is developed with the aid of a formal asymptotic expansion method from the equilibrium equations of the linear three dimensional 3d elasticity a set of the microscopic 2d and macroscopic 1d equations are systematically derived by introducing the virtual work concept displacements at each order are split into two parts such as fundamental and warping solutions first we seek the warping solutions via the microscopic 2d cross sectional analyses that will be smeared into the macroscopic 1d beam equations the variations of fundamental solutions enable us to formulate the macroscopic 1d beam problems by introducing the orthogonality of asymptotic displacements to six beam fundamental solutions the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method the boundary conditions obtained herein are applied to composite beams with solid and thin walled cross sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method based beam analysis famba presented in this paper the numerical results are compared to those reported in literature as well as 3d fem solutions

towards the implementation of the singular function method for singular perturbation problems abstract the numerical solution of singular perturbation problems spps is delicate because the perturbation parameter e and the mesh size h cannot vary independently of one another in the extended abstract j m s lubuma k c patidar finite element methods for self adjoint singular perturbation problems in t e simos g maroulis eds iccmse 2005 advances in computational methods in sciences and engineering lecture series on computer and computational sciences vol 4 vsp international science publishers the netherlands 2005 pp 344 347 j m s lubuma k c patidar reliable finite element methods for self adjoint singular perturbation problems university of pretoria technical report upwt 2005 11 the authors proposed the singular function method sfm to solve a self adjoint singular perturbation problem the sfm is a variant of the finite element method fem where the space of trial and test functions is enriched by the singular functions of the spp we proved in the above mentioned work that the sfm is e uniformly convergent of optimal order in appropriate norms however the numerical implementation of the sfm based on numerical integration did not provide satisfactory results in this paper we present an alternative way of implementing the sfm that is we avoid numerical integration by making use of the explicit form of the singular functions in the computations of the stiffness matrix and the load vector we obtain improved results these results are compared with those obtained with numerical integration the new results confirm theoretical order of convergence

nested newton strategies for energy corrected finite element methods energy corrected finite element methods provide an attractive technique for dealing with elliptic problems in domains with re entrant corners optimal convergence rates in weighted l 2 norms can be fully recovered by a local modification of the stiffness matrix at the re entrant corner and no pollution effect occurs although the existence of optimal correction factors is established it remains open how to determine these factors in practice first we show that asymptotically a unique correction parameter exists and that it can be formally obtained as the limit of level dependent correction parameters which are defined as roots of an energy defect function second we propose three nested newton type algorithms using only one newton step per refinement level and show local or even global convergence to this asymptotic correction parameter

isogeometric collocation methods with generalized b splines we introduce isogeometric collocation methods based on generalized b splines and we analyze their performance through numerical examples for univariate and multivariate scalar and vector valued problems in particular advection diffusion and linear elasticity model problems are addressed the resulting method combines the favorable properties of isogeometric collocation and the geometrical and analytical problem oriented advantages of generalized b splines

a higher order beam finite element for bending and vibration problems abstract the finite element equations for a variationally consistent higher order beam theory are presented for the static and dynamic behavior of rectangular beams the higher order theory correctly accounts for the stress free conditions on the upper and lower surfaces of the beam while retaining the parabolic shear strain distribution the need for a shear correction coefficient is therefore eliminated full integration of the shear stiffness terms is shown to result in the recovery of the kirchhoff constraint for thin beams without introducing spurious locking constraints the accuracy of this formulation is demonstrated by using several numerical examples for the cases of small and large displacements for a hinged hinged beam the linear thickness shear mode frequency can be matched with the timoshenko frequency to yield a shear coefficient of 0 824 matching the bending frequencies between the two theories indicates a shear coefficient for the timoshenko theory that changes with mode number and slenderness ratio the influence of in plane inertia and slenderness ratio on the non linear frequency is examined for beams with a number of different support conditions

an investigation concerning optimal design of solid elastic plates abstract we consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory assuming the material volume and plate domain to be given we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account on the basis of a convenient tensorial formulation of the problem where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained these numerical results enable us to investigate the optimization problem itself in terms of its major parameters particularly the maximum and minimum values specified for the plate thickness for problems associated with large ratios between these constraint values plate designs with significant integral stiffeners are obtained we find however that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities in order to determine a global optimal solution associated with given thickness constraint values it is therefore necessary to change the optimal design formulation with implications for a number of similar two dimensional optimization problems our results offer valuable indications of the lines along which such changes should be performed

on the asymptotic boundary conditions of an anisotropic beam via virtual work principle in this research an efficient and effective method is proposed to derive the boundary conditions of an anisotropic beam in the asymptotic sense we first set up the constrained virtual work by introducing the lagrange multiplier on the displacement prescribed boundary the macroscopic beam and microscopic cross section equations with the boundary conditions are simultaneously obtained by taking the asymptotic expansion on the displacement vector in this way the three dimensional characteristics of the beam are asymptotically smeared into the macroscopic beam equations and the beam boundary conditions the boundary conditions obtained are then compared to those from the decay analysis method the beam bending slope boundary condition obtained in the frame work of variational principle is different from the well known average condition this new boundary condition is more accurate than the average one for a sandwich beam this is further demonstrated and discussed via the examples of a cantilever beam loaded at the end

green s operator for a periodic medium with traction free boundary conditions and computation of the effective properties of thin plates this paper presents a new green s operator gamma operator for mixed periodic and stress free boundary conditions the solution is split into a classical periodic solution and a complementary solution to account for stress free boundary conditions this operator is used to obtain the effective properties of thin heterogeneous plates made of periodic cells using the fast fourier transform the periodic thin plate theory and the computation of the strain energy within the cell numerical applications are provided including comparisons with closed form solutions and finite element solutions

effective models of composite periodic plates ii simplifications due to symmetries abstract this paper is aimed at investigating how material and geometrical symmetries of a composite plate simplify the formulae derived in part i by the asymptotic method it occurs that the symmetry with respect to the middle plane results in splitting the subsequent overall and local problems into membrane and bending problems additional assumptions of orthotropy of the material and of certain symmetries of the periodicity cells imply far reaching simplifications e g the vanishing of some terms in the first order correctors for displacements and the cancelation of discrepancies in the formulation of some boundary conditions in the last section a computational algorithm for evaluating the effective stiffnesses is suggested

an asymptotic analysis of end effects in the axisymmetric deformation of elastic tubes weak in shear higher order shell theories are inadequate and unnecessary abstract this paper specializes to a semi infinite tube morgan s int j solids structures10 837 852 1974 two stress function formulation of the equations for the axisymmetric deformation of a linearly elastic transversely isotropic cylindrical body free of surface tractions the ratio of the tube s shear modulus to its radial transverse extensional modulus is taken to be of the order of magnitude of the square root of its thickness to its mean radius the equations are solved by formal asymptotic expansion in fractional powers of the thickness to radius ratio for four canonical sets of end conditions a axisymmctric equilibrated tractions b and c two different combinations of tractions and displacements and d axisymmctric radial and axial displacements the solutions exhibit interior i e shell like parts and wide and narrow boundary or edge layers the latter containing components that vary extremely rapidly through the thickness of the tube the analysis focuses on computing the lowest order correction both in the interior and in the boundary layers to classical shell theory it is shown that in cases a c the interior correction to classical shell theory that is those effects so called higher order shell theories attempt to capture can ultimately be determined directly in terms of the edge data but that in case of prescribed displacements d the computation of three dimensional boundary layer effects is essential these conclusions are consistent with those for elastically isotropic shells found by gregory and wan 1992 who used ingenious arguments based on the betti reciprocity principle

energy corrected finite element methods for corner singularities it is well known that the regularity of solutions of elliptic partial differential equations on domains with re entrant corners is limited by the maximal interior angle this results in reduced convergence rates for finite element approximations on families of quasi uniform meshes following an idea of zenger and gietl we show that it is possible to regain the full order of convergence by a local modification of the bilinear form in a vicinity of the singularity and thus to overcome the pollution effect a complete convergence analysis in weighted sobolev spaces is presented and we also show that the stress intensity factors can be computed with optimal accuracy the theoretical results are illustrated by numerical tests that demonstrate second order convergence of linear finite elements on families of quasi uniform meshes

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

some estimates for h p k refinement in isogeometric analysis in this paper we propose a theoretical study of the approximation properties of nurbs spaces which are used in isogeometric analysis we obtain error estimates that are explicit in terms of the mesh size h the degree p and the global regularity measured by the parameter k our approach covers the approximation with global regularity from c 0 up to c k 1 with 2k 1 p notice that the interesting case of higher regularity up to k p is still open however our results give an indication of the role of the smoothness k in the approximation properties and offer a first mathematical justification of the potential of isogeometric analysis based on globally smooth nurbs

effective stiffnesses of transversely non homogeneous plates with unidirectional periodic structure abstract this paper presents a derivation and analysis of closed formulae for the membrane bending and reciprocal effective stiffnesses of elastic orthotropic and transversely asymmetric plates with microstructures periodic in one direction the derivation is based on the concept of imposing hencky type displacement constraints on the solutions to the caillerie kohn vogelius local homogenization problems reduction of the dimension of the problems by one makes it possible to solve the new approximate local problems exactly thus enabling one to find the relevant formulae in closed form ready for engineering applications as well as for optimization and sensitivity studies this paper generalizes the previous results of the author int j solids structures 29 309 326 1992 concerning the bending of transversely symmetric periodic plates

multi iterative methods abstract a general class of iterative methods is introduced for solving positive definite linear systems ax b these methods use two or more different iterative techniques and each of them reduces the error by a constant factor in a different subspace of r n the multigrid as an example of a multi iterative method combines these partial results and we reach a reduction of the error by at each step where is the number of applications of the first scheme and gt 1 is a constant independent of n in the second part we consider the prolongation operator p with a special antidiagonal structure and we show that the multigrid method for toeplitz matrices 1 is an example of a method of this class in a third part we extend the multigrid method to block toeplitz matrices obtaining a block antidiagonal structure for toeplitz and block toeplitz cases the choice of p which maximizes convergence speed simply depends on the coefficients of matrix a the numerical results on 1 d and 2 d elliptic differential problems confirm the effectiveness of this multi iterative method in fact its spectral radius is very small and independent of the mesh spacing h

higher order effective modeling of periodic heterogeneous beams i asymptotic expansion method this paper is concerned with the elastostatic behavior of heterogeneous beams with a cross section and elastic moduli varying periodically along the beam axis by using the two scale asymptotic expansion method the interior solution is formally derived up to an arbitrary desired order in particular this method is shown to constitute a systematic way of improving bernoulli s theory by including higher order terms without any assumption in contrast to timoshenko s theory or other refined beam models moreover the incompatibility between the interior asymptotic expansions and the real boundary conditions is emphasized and the necessity of a specific treatment of edge effects is thus underlined

unified and extended form of three types of splines the three types refer to polynomial trigonometric and hyperbolic splines in this paper we unify and extend them by a new kind of spline ue spline for short defined over the space cos wt sin wt 1 t t l where l is an arbitrary nonnegative integer w is a frequency sequence w i a i a i r existing splines such as usual polynomial b splines cb splines hb splines nuat splines ah splines fb splines and the third form fb splines etc are all special cases of ue splines ue splines inherit most properties of usual polynomial b splines and enjoy some other advantageous properties for modelling they can exactly represent classical conics the catenary the helix and even the eight curve a kind of snake like curves etc

on the analysis of prismatic beams using first order warping functions abstract a one dimensional theory for isotropic homogeneous beams with a general cross section is developed that utilizes both deformations in the cross section plane and warping out of the cross section plane where these deformation functions are known from saint venant s elasticity solutions for bending and flexure of a prismatic beam first order warping functions in contrast to existing technical beam theories which assume that local cross section deformations and some stresses can be neglected the present theory makes no assumption the finite element model is developed using the resulting differential equations along with a partially weak form formulation full three dimensional constitutive equations are used solutions are found using the finite element method numerical results which are presented for a wide variety of loadings and boundary conditions show that the calculated stresses and displacements are in excellent agreement with existing elasticity solutions

an effective model for lipschitz wrinkled arches abstract within the framework of the koiter s linear elastic shell theory we study the limit model of a lipschitz curved arch whose mid surface is periodically waved the magnitude and the period of the wavings are of the same order to achieve the asymptotic analysis we consider a mixed formulation for which we perform a two scale homogenization technique we prove the convergence of the displacements the rotation of the normal and the membrane strain from the limit formulation we derive an effective model for curved critically wrinkled arches it introduces two membrane strain functions instead of one in the classical case and exhibits a corrector membrane term to the coupling between the rotation of the normal and the membrane strain

asymptotic analysis of linearly elastic shells iii justification of koiter s shell equations we consider as in part i a family of linearly elastic shells of thickness 2 all having the same middle surfaces r3 where r2 is a bounded and connected open set with a lipschitz continuous boundary and l3 r3 the shells are clamped on a portion of their lateral face whose middle line is 0 where 0 is any portion of withlength 0 0 we make an essential geometrical assumption on the middle surfaces and on the set 0 which states that the space of inextensional displacements begin gathered v f omega eta eta i in h 1 omega times h 1 omega times h 2 omega hfill eta i partial v eta 3 0 on gamma 0 gamma alpha beta eta 0 in omega hfill end gathered where gamma alpha beta are the components of the linearized change is metric tensor ofs contains non zero functions this assumption is satisfied in particular ifs is a portion of cylinder and 0 is contained in a generatrix ofs

least weight design of perforated elastic plates ii abstract the paper investigates implications of some recent mathematical developments in the fields of shape optimization and relaxation of variational problems considering the least weight design of perforated elastic plates in either flexure or plane stress for a prescribed compliance it is shown that at low rib densities microstructures consisting of a combination of first and second order infinitesimal ribs is superior to those consisting of purely first order infinitesimal ribs moreover it is indicated that thin ribs of infinite length thickness ratio do not contribute significantly to the stiffness in a direction normal to their plane on the basis of this conclusion a simple specific cost function is derived and then it is used in the design of circular uniformly loaded perforated plates with zero value of poisson s ratio as a basis for comparison several intuitively selected topographies are optimized for the case of simply supported plates and in part ii of this study a variational analysis is used to obtain the optimal solutions for plates with simply supported clamped or loaded edges

on the general solutions of transversely isotropic elasticity abstract in this paper we give the generalized boussinesq galerkin general solution of transversely isotropic elasticity as well as its simplified forms in two special cases and we prove the completeness of the lekhnitskii hu nowacki solution and the elliott lodge solution in such cases that s 0 2 s 1 2 and s 2 2 are possibly equal to each other

average of the navier s law on the rapidly oscillating boundary we study the flow of newtonian fluid in a domain with periodically wrinkled boundary with slip navier s boundary condition the goal of this paper is to replace a microscopic boundary condition posed on the rough boundary by some macroscopic boundary condition posed on the middle surface of the oscillating boundary depending on the shape of wrinkles and the friction coefficient we get four different effective models

a bending gradient model for thick plates part i theory abstract this is the first part of a two part paper dedicated to a new plate theory for out of plane loaded thick plates where the static unknowns are those of the kirchhoff love theory 3 in plane stresses and 3 bending moments to which six components are added representing the gradient of the bending moment the new theory called the bending gradient plate theory is described in the present paper it is an extension to arbitrarily layered plates of the reissner mindlin plate theory which appears as a special case of the bending gradient plate theory when the plate is homogeneous however we demonstrate also that in the general case the bending gradient model cannot be reduced to a reissner mindlin model in part two lebee and sab 2011 the bending gradient theory is applied to multilayered plates and its predictions are compared to those of the reissner mindlin theory and to full 3d pagano s exact solutions the main conclusion of the second part is that the bending gradient gives good predictions of both deflection and shear stress distributions in any material configuration moreover under some symmetry conditions the bending gradient model coincides with the second order approximation of the exact solution as the slenderness ratio l h goes to infinity

homogenization of rough boundaries and interfaces homogenization is used to analyze a partial differential equation in a domain with a very rough boundary or interface following the procedure of kohler papanicolaou and varadhan boundary and interface problems in regions with very rough boundaries in multiple scattering and waves in random media p chow w kohler and g papanicolaou eds north holland amsterdam 1981 pp 165 197 and brizzi and chalot homogeneisation de frontiere ph d thesis department of mathematics universite de nice nice france 1978 it is shown that such a boundary or interface can be replaced by an equivalent layer within which a modified differential equation holds the coefficients in this new equation are certain effective parameters such as the effective conductivity the effective dielectric constant the effective refractive index etc these coefficients are determined by the solutions of certain special problems which involve the detailed shape of the boundary or interface this analysis is applied to

sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates abstract this paper deals with optimal design of solid elastic axisymmetric plates performing free transverse vibrations it is the objective to determine the plate thickness distribution from the condition that the plate volume is minimized for a given value of the fundamental natural frequency or for a given higher order natural frequency that corresponds to a vibration mode with a prescribed number of nodal diameters it is found that the weierstrass necessary condition for optimality is generally not satisfied for a traditional formulation of this problem and that the optimal design is characterized by a sliding regime of control where the plate thickness exhibits an infinite number of discontinuities as a system of infinitely thin circumferential stiffeners are formed on the optimal axisymmetric plate this inherent anisotropy of the optimal design is taken into account in a regularization of the initial optimization problem by establishing the tensorial character of the plate bending rigidity and using the concentration of thin circumferential stiffeners as a new design variable control it is shown that the new formulation of the problem can be solved numerically and examples of optimal designs are presented in the paper

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

the rate of convergence of toeplitz based pcg methods for second order nonlinear boundary value problems in previous works 21 23 we proposed the use of tau 5 and band toeplitz based preconditioners for the solution of 1d and 2d boundary value problems bvp by means of the preconditioned conjugate gradient pcg methods as tau and band toeplitz linear systems can be solved 4 by using fast sine transforms 8 these methods become especially attractive in a parallel environment of computation in this paper we extend this technique to the nonlinear nonsymmetric case and in addition we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear therefore these methods work much finer than those based on separable preconditioners 18 45 on incomplete lu factorizations 36 13 27 and on circulant preconditioners 9 30 35 since the latter two techniques do not assure a linear rate of convergence on the other hand the proposed technique has a wider range of application since it can be naturally used for nonlinear nonsymmetric problems and for bvp in which the coefficients of the differential operator are not strictly positive and only piecewise smooth finally the several numerical experiments performed here and in 22 23 confirm the effectiveness of the theoretical analysis

gb splines of arbitrary order explicit formulae and recurrence relations for the calculation of generalized b splines gb splines of arbitrary order are given we derive main properties of gb splines and their series i e partition of unity shape preserving properties invariance with respect to ane transformations etc it is shown that such splines have the variation diminishing property and are chebyshevian splines c 1999 elsevier science b v all rights reserved

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

further analysis of end effects for plane deformations of sandwich strips abstract saint venant end effects for plane deformations of sandwich strips are investigated for two phase composite strips with homogeneous isotropic layers that are subjected to self equilibrated end loads the exponential decay of end effects is characterized in terms of eigenvalues analogous to the fadle papkovich eigenvalues for the homogeneous strip an earlier study of this problem was carried out by choi and horgan 1978 saint venant end effects for plane deformation of sandwich strips int j solids structures 14 187 195 here we simplify and extend this analysis by using the results of dundurs on reduced dependence on the elastic constants the decay rate is shown to depend quadratically on the two dundurs constants numerical results yield graphs for the decay rate in the parallelogram which are immediately applicable for design purposes for any combination of materials in the two phase sandwich strip these plots allow one to determine the saint venant decay length for the case of a relatively compliant inner core an asymptotic result shows that the characteristic decay length is much larger than that for a homogeneous isotropic strip

on a class of weak tchebycheff systems in this paper we study the approximation power the existence of a normalized b basis and the structure of a degree raising process for spaces of the form requiring suitable assumptions on the functions u and v the results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of cagd

effective models of composite periodic plates i asymptotic solution abstract this paper is concerned with the statical problem of plates with oscillating material properties and rapidly varying face shapes both the material moduli and the geometrical data vary periodically with the same period by using the asymptotic method of caillerie 1982 c r acad sci paris 294 ii 159 the higher order terms of the expansion for deformations and stresses are arrived at these terms are determined by a subsequent homogenized problem for new auxiliary functions of a local variable appropriate basic cell problems are established proof that the conventional asymptotic expansion does not comprise boundary effects at the clamped edge is also given

a new formulation of the problem of optimum reinforcement of reissner mindlin plates abstract a new formulation of the problem of computing the layout of reinforcement of reissner mindlin plates for minimum compliance is presented bending and transverse shear stiffness properties of plates with an arbitrary but finite number of local rib directions and widths are computed using standard homogenization techniques these properties are then expressed in terms of only four variables that fully describe the anisotropy of the plate the optimization problem is solved in two stages consisting of a local four dimensional maximization problem where a given amount of material is optimally allocated into an arbitrary number of fine scale stiffeners of different widths and orientations and a global one dimensional optimization problem where the optimum spatial layout of material is determined a discussion of implementation issues is included and an example is solved to illustrate the approach

decay of stresses induced by sele equilibrated end loads in a multilayered composite abstract the axial decay of stresses induced by self equilibrated end loads in a multilayeredcomposite is investigated in the context of the plane problem here the multilayered composite is composed of alternating layers of two dissimilar isotropic materials by adopting appropriate boundary conditions it is sufficient to consider a cell which consists of three layers an airy stress function approach is utilized to obtain the characteristic equation where the non zero roots correspond to the decay rates the dominant exponential decay rate which corresponds to the smallest positive real part of the roots is presented in the dundurs parallelogram

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

general solution of the problem of design of laminated plates possessing the given stiffnesses abstract the paper deals with the problem of design laminated plates possessing the given stiffnesses the following two design problems are considered 1 the continuous design problem when one can use materials with any elastic characteristics to manufacture the plate 2 the discrete design problem when one can use a finite set of materials it is known that design problems are closely related with convex analysis problems it is shown that the laminated plate design problems are related with the convex combinations problem ccp using the ccp technique one can analyze the laminated plate design problems in detail this paper is concentrated on the general solution of the design problem i e the set of all solutions of the problem the general solutions are constructed for both the continuous and the discrete design problems the methods developed in the paper can be used to solve the design problem numerically

analysis of preconditioning strategies for collocation linear systems in the past few years several authors have studied the preconditioning of collocation matrices by finite differences fds matrices arising from the associated collocation points here we discuss how to solve in an efficient way nonuniform grid fd linear systems including those related to a generic fd collocation preconditioner the main idea is based on a further step of preconditioning defined in terms of diagonal and toeplitz matrices first we identify the limit spectral distributions of the involved fd collocation matrix sequences and then we prove that the proposed toeplitz based preconditioners assure a clustering at the unity with respect to the eigenvalues in the 1d case in the 2d case the situation is different so that more appropriate strategies are discussed a wide numerical experimentation emphasizing the correctness of the theoretical results is also reported

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

a new rectangular beam theory abstract a new theory for beams of rectangular cross section which includes warping of the cross sections is presented in the present work by satisfying the shear free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required it is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in timoshenko beam theory provided that for the timoshenko equation the shear coefficient is taken to be 5 6 this value lies within the range of values 0 822 0 870 appearing in the literature for the beam of rectangular cross section results for two typical static examples are given for both the new theory and timoshenko beam theory these results are compared with the solutions of the comparable problems in the linear theory of elasticity for the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while timoshenko beam theory underestimates the shear correction term by 20 for the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution

homogenizing stiffnesses of plates with periodic structure abstract the paper develops the method of homogenization of elastic properties of reissner hencky plates with periodic structure the method is based on the new scaling which for each e preserves the shape of the three dimensional periodicity cell of the original plate structure the homogenization formulae obtained in this manner turn out to coincide with those found previously by the hencky reissner approximation of the three dimensional caillerie kohn vogelius local problems the proposed method has made it possible to derive a new closed form formula for evaluating the effective torsional stiffness of plates with one family of stiffeners the formulae for remaining stiffnesses have been known contrary to the hitherto used formula of duvaut and to the one followed from a direct homogenization of the reissner hencky equations the derived new formula is sensitive to the thickness to period ratio and hence much more realistic the formulae used previously turn out to be the upper and lower bounds for the derived expression

regularized formulation for optimal design of axisymmetric plates abstract this paper presents a regularized mathematical formulation necessary conditions of optimality and solutions of problems of optimal design of solid elastic axisymmetric plates of prescribed total material volume single purpose design criteria of minimum static compliance maximum stiffness and maximum frequency of free transverse vibrations are considered the regularization which alleviates some anomalies and difficulties encountered earlier in plate optimization problems is based on a new compound plate model with two simultaneous design variables namely variable thickness of a solid part of the plate and variable concentration of a dense system of thin integral stiffeners attached to the solid plate part a numerical optimality criterion based method of solution is developed for the problem and several optimal designs are presented the results are compared with results obtained from optimal design formulations applied heretofore and substantiate the superiority of the new regularized formulation

symbol based multigrid methods for galerkin b spline isogeometric analysis we consider the stiffness matrices arising from the galerkin b spline isogeometric analysis discretization of classical elliptic problems by exploiting their specific spectral properties compactly described by a symbol we design an efficient multigrid method for the fast solution of the related linear systems the convergence rate of general purpose multigrid methods based on classical stationary smoothers is optimal i e bounded independently of the matrix size but it also worsens exponentially with respect to the spline degree the symbol allows us to give a detailed theoretical explanation of this exponential worsening in the case of the two grid scheme in addition thanks to a specific factorization of the symbol we are able to design an ad hoc multigrid method with an effective preconditioned cg or gmres smoother at the finest level in the spirit of the multi iterative idea the convergence rate of this multi iterative multigrid method is not only optimal but also robust i e bounded su

two level multiscale enrichment methodology for modeling of heterogeneous plates a new two level multiscale enrichment methodology for analysis of heterogeneous plates is presented the enrichments are applied in the displacement and strain levels the displacement field of a reissner mindlin plate is enriched using the multiscale enrichment functions based on the partition of unity principle the strain field is enriched using the mathematical homogenization theory the proposed methodology is implemented for linear and failure analysis of brittle heterogeneous plates the eigendeformation based model reduction approach is employed to efficiently evaluate the non linear processes in case of failure the capabilities of the proposed methodology are verified against direct three dimensional finite element models with full resolution of the microstructure copyright 2009 john wiley sons ltd

asymptotic analysis of an end loaded transversely isotropic elastic semi infinite strip weak in shear abstract we use linear elasticity to study a transversely isotropic or specially orthotropic semiinfinite slab in plane strain free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction the governing equations which are identical to those for a strip in plane stress free of traction on its long sides and at infinity and subject to tractions or displacements on its short side are reduced in the standard way to a fourth order partial differential equation with boundary conditions for a dimensionless airy stress function we study the asymptotic solutions to this equation for four sets of end conditions traction mixed two displacement as g3 the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli approaches zero in all cases the solutions for consist of a wide boundary layer that decays slowly in the axial direction over a distance that is long compared to the width of the strip plus a narrow boundary layer that decays rapidly in the axial direction over a distance that is short compared to the width of the strip moreover we find that the narrow boundary layer has a sinuous part that varies rapidly in the transverse direction but which to lowest order does not enter the boundary conditions nor affect the transverse normal stress or the displacements because the exact biorthogonality condition for the cigenfunctions associated with can be replaced by simpler orthogonality conditions in the limit as e b 0 we are able to obtain to lowest order explicit formulae for the coeflicients in the eigenfunction expansions of for the four different end conditions

anti plane shear deformations in linear and nonlinear solid mechanics the intent of this expository paper is to draw the attention of the applied mathematics community to an interesting two dimensional mathematical model arising in solid mechanics involving a single second order linear or quasi linear partial differential equation this model has the virtue of relative mathematical simplicity without loss of essential physical relevance anti plane shear deformations are one of the simplest classes of deformations that solids can undergo in anti plane shear or longitudinal shear generalized shear of a cylindrical body the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate thus anti plane shear with just a single scalar axial displacement field may be viewed as complementary to the more complicated yet perhaps more familiar plane strain deformation with its two in plane displacements in recent years considerable attention has been paid to the analysis of anti plane shear deformations within the context of variou

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

heterogeneous thin films of martensitic materials we study the effective behavior of heterogeneous thin films with three competing length scales the film thickness and the length scales of heterogeneity and material microstructure we start with three dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der waals type and derive the effective energy density as all length scales tend to zero with given limiting ratios we do not require any a priori selection of asymptotic expansion or ansatz in deriving our results depending on the dominating length scale the effective energy density can be identified by three procedures averaging homogenization and thin film limit we apply our theory to martensitic materials with multi well energy density and use a model example to show that the shape memory behavior can crucially depend on the ratios of these length scales we comment on the effective conductivity of linear composites and also on multilayers made of shape memory and elastic materials

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

on non smoothness in optimal design of solid elastic plates abstract in the present paper minimum compliance design problems of thin solid elastic plates are discussed we follow up indications provided by numerical results obtained in a recent paper 1 and propose a new plate model where a uniform plate is equipped with an infinite number of infinitely thin integral stiffeners on the basis of this new plate model numerical and analytical investigations of a new formulation of the minimum compliance design problem are carried out it is demonstrated that the new formulation is under certain conditions superior to the corresponding traditional formulation for determining a possible global optimal solution for solid plates since both the new formulation and the traditional formulation are based on classical thin plate theory the superiority of the new plate model over the corresponding smooth one directly implies that optimal thickness distributions for thin solid elastic plates are generally not smooth functions but functions with an infinite number of discontinuities

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

superlinear convergence of conjugate gradients we give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other krylov subspace methods we present a new bound on the relative error after n iterations this bound is valid in an asymptotic sense when the size n of the system grows together with the number of iterations the bound depends on the asymptotic eigenvalue distribution and on the ratio n n under appropriate conditions we show that the bound is asymptotically sharp our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials an important tool in our investigations is a constrained energy problem in logarithmic potential theory the new asymptotic bounds for the rate of convergence are illustrated by discussing toeplitz systems as well as a model problem stemming from the discretization of the poisson equation

refined one dimensional formulations for laminated structure analysis this paper proposes one dimensional formulations based on hierarchical expansions of the unknown displacement variables for the analysis of multilayered structures made of anisotropic composite layers the hierarchical technique shows variable kinematic properties and it is based on the carrera unified formulation two different classes of refined theories are proposed the first expands the unknown variables in terms of power polynomials of the cross sectional coordinates it consists of a taylor like expansion the second class of onedimensional theories uses lagrange polynomials lagrange expansion and subdomain discretizations of the cross section and it leads to only pure displacements as the unknown variables taylor like expansion is used to develop equivalent single layer formulations and lagrange expansion is used to construct both equivalent single layer and layerwise descriptions the finite element method is employed to develop numerical applications using the carrera unified formulation finite element matrices are obtained in terms of a few fundamental nuclei that are formally independent of all the considered one dimensional formulations a number of numerical examples are given concerning on beams plates and more complex structures comparisons with results from plate and solid models are provided the following has been concluded 1 the proposed formulation represents a reliable compact and accurate method to develop refined one dimensional models 2 the present one dimensional models are very effective at detecting both global and local responses of composite structures 3 shell and solidlike results are obtained with a significant reduction in the computational costs

generalized b splines as a tool in isogeometric analysis the concept of isogeometric analysis has been proposed in 13 where nurbs are considered as basis of the analysis thanks to their ability to construct an exact geometric model in several practical applications and to their popularity in commercial cad systems in this paper we propose an alternative to the rational model presenting an isogeometric analysis approach based on generalized b splines geometric models exactly represented by generalized b splines include those generated by nurbs moreover generalized b splines possess all fundamental properties of algebraic b splines and nurbs including classical refinement processes as h p k refinements finally since generalized b splines are not confined to rational functions they behave completely similar to algebraic b splines with respect to differentiation and integration this seems to be of interest in the treatment of some relevant problems

stiffened plates subject to transverse loading abstract the generalized conditions under which stiffened plates can be approximated as a two dimensional continuum are investigated and identified a simplified set of constitutive equations defining the full class of problems described above subject to elastic behaviour are then obtained by making suitable assumptions the significance of these assumptions is investigated under a wide range of conditions and the results calculated for typical problems using the simple theory compared with experimentally obtained values the simple theory is then amended to allow consistently for the effects due to poisson s ratio in the stiffeners a study of the resulting equations indicates that whilst the normal assumption of centroidal neutral axes is justified under all conditions the additional strain energy generated in the stiffeners can be significant and that under the assumption of centroidal neutral axes this is readily included in the formulation without penalty shear effects other than those due to st venant torsion are not considered in the detailed analysis

toward a new engineering theory of bending fundamentals the classical engineering theory of bending due to bernoulli and euler serves as a cornerstone for structural analysis and design limitations of this theory however become apparent in flexural wave propagation studies it predicts infinite phase velocity as the wavelength becomes shorter this theoretical deficiency is corrected by timoshenko theory which accounts for transverse shear deformation a thorough study of exact elasticity solutions reveals that there are two additional effects that are the same order as transverse shear in bending behavior a new theory accounting for them is presented along with several applications the new equations are no more complicated than those of timoshenko ty pe theory yet they yield solutions which are exact or indistinguishable from exact in the examples studied

on the accuracy of one dimensional models for multilayered composite beams abstract the accuracy of 1 d models for composite beams made of linearly elastic orthotropic layers is estimated by means of the prager synge hypercircle method statically admissible and kinematically admissible stress fields are derived and asymptotic forms for the error bounds of o h l and o h l 2 for h l approaching zero are obtained for displacement based laminated beam theories where the axial displacement is represented in terms of a given set of coordinate functions defined over the beam height the condition of vanishing of relative mean square error for h l 0 is used to derive the constitutive law for the 1 d model explicit forms for the error bounds are given for the classical lamination theory first order shear deformation theory and two higher order theories quantitative error bounds are calculated for simply supported multilayered beams under sinusoidal transverse loading it is shown that starting from a wise choice of coordinate functions the accuracy of higher order theories can be almost independent of the beam lamination and up to 150 and 80 times higher than clt and fsdt respectively

completeness and nonuniqueness ofgeneral solutions of transversely isotropic elasticity abstract in this paper we give general solutions of transversely isotropic elasticity their completenessand nonuniqueness are proved we point out that famous lekhnitskii hu nowacki solutions and elliott lodge solutions are complete if the elastic region is z convex

zeroth order shear deformation micro mechanical model for composite plates with in plane heterogeneity abstract this article introduces a new model for investigating the mechanical behavior of heterogeneous plates which are composed of periodically repeated microstructures along the in plane directions we first formulate the original three dimensional problem in an intrinsic form for implementation into a single unified formulation and application to geometrically nonlinear problem taking advantage of smallness of the plate thickness to length parameter and heterogeneity and performing homogenization along dimensional reduction simultaneously the variational asymptotic method is used to rigorously construct an effective zeroth order plate model which is similar to a generalized reissner mindlin model the first order shear deformation model capable of capturing the transverse shear deformations but still carries out the zeroth order approximation which can maximize simplicity and promote efficiency this present approach is incorporated into a commercial analysis package for the purpose of dealing with realistic and complex geometries and constituent materials at the microscopic level a few examples available in literature are used to demonstrate the consistence and efficiency of this new model especially for the structures in which the effects of transverse shear deformations are significant

finite element and nurbs approximations of eigenvalue boundary value and initial value problems abstract we study the spectral approximation properties of finite element and nurbs spaces from a global perspective we focus on eigenfunction approximations and discover that the l 2 norm errors for finite element eigenfunctions exhibit pronounced spikes about the transition points between branches of the eigenvalue spectrum this pathology is absent in nurbs approximations by way of the pythagorean eigenvalue error theorem we determine that the squares of the energy norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the l 2 norm eigenfunction errors the spurious behavior of the higher eigenvalues for standard finite elements is well known and therefore inherited by the energy norm errors along with the spikes in the l 2 norm of the eigenfunction errors the eigenvalue pathology is absent for nurbs the implications of these results to the corresponding elliptic boundary value problem and parabolic and hyperbolic initial value problems are discussed

computational homogenization for heterogeneous thin sheets in this paper a computational homogenization technique for thin structured sheets is proposed based on the computational homogenization concepts for first and second order continua the actual three dimensional 3d heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element rve incorporating the full thickness of the sheet and an in plane representative cell of the macroscopic structure at an in plane integration point of the macroscopic shell the generalized strains i e the membrane deformation and the curvature are used to formulate the boundary conditions for the microscale rve problem at the rve scale all microstructural constituents are modeled as an ordinary 3d continuum described by the standard equilibrium and the constitutive equations upon proper averaging of the rve response the macroscopic generalized stress and the moment resultants are obtained in this way an in plane homogenization is directly combined with a through thickness stress integration from a macroscopic point of view a numerical generalized stress strain constitutive response at every macroscopic in plane integration point is obtained additionally the simultaneously resolved microscale rve local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design

decay rate of saint venant end effects for multilayered orthotropic strips saint venant decay rate of end effects is investigated for generally laminated orthotropic strips under self equilibrated end loads the problem is governed by a fourth order partial differential equation for the airy stress function with discontinuous coefficients at the layer interfaces where displacements and traction continuities are imposed the solution is found in the form of product of an exponentially decaying function along the strip and an unknown function defined over its total height external face and interface conditions are used to obtain the characteristic equation for the eigenvalues governing the decay rate of end effects along the strip for orthotropic and strongly orthotropic sandwich strips the transcendental eigenvalue equation is explicitly given the case of laminated strips with periodic layout is finally considered making use of the homogenization method both effective elastic constants and expressions for the local stress variation at the layer level are obtained numerical calculations confirm that the eigenvalues of the homogenized material are the asymptotic values of those of periodically laminated strips when the number of layers increases moreover homogenization method is shown to be very powerful also in the calculation of local stress distributions

homogenization and dimensional reduction of composite plates with in plane heterogeneity the variational asymptotic method is used to construct a new model for composite plates which could have in plane heterogeneity due to both geometry and material we first formulate the original three dimensional problem in an intrinsic form which is suitable for geometrically nonlinear analysis taking advantage of smallness of the plate thickness and heterogeneity we use the variational asymptotic method to rigorously construct an effective plate model unifying a homogenization process and a dimensional reduction process this approach is implemented in the computer code vapas using the finite element technique for the purpose of dealing with realistic heterogeneous plates a few examples are used to demonstrate the capability of this new model

on the spectrum of stiffness matrices arising from isogeometric analysis we study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems motivated by the applicative interest in the fast solution of the related linear systems we are looking for a spectral characterization of the involved matrices in particular we investigate non singularity conditioning extremal behavior spectral distribution in the weyl sense as well as clustering of the eigenvalues to a certain compact subset of mathbb c c all the analysis is related to the notion of symbol in the toeplitz setting and is carried out both for the cases of 1d and 2d problems

effective models of composite periodic plates iii two dimensional approaches abstract this paper is concerned with two dimensional approximations of the three dimensional local problems of caillerie 1984 math meth appl sci 6 159 191 and kohn and vogelius 1984 int j solids structures 20 333 350 the solutions to these problems make it possible to evaluate the effective stiffnesses of periodic plates the kirchhoff type approximation results in the formulae shown by duvaut 1976 in theoretical and applied mechanics edited by w t koiter pp 119 132 north holland amsterdam by imposing hencky reissner type constraints one is led to new formulae which have a wider range of applicability the paper also discusses the formulae which result from homogeneizing two dimensional kirchhoff s reissner hencky s and reddy s equations of plates with periodic structure

extrapolation techniques for reducing the pollution effect or reentrant corners in the finite element method standard finite element schemes applied to boundary value problems on domains with reentrant corners suffer from a global loss of accuracy caused by the influence of the corner singularities for a model situation it is shown that this pollution effect can be completely described in terms of an asymptotic error expansion with respect to fractional powers of the mesh size parameter this provides the basis of an extrapolation procedure which recovers the full order of accuracy for the solution as well as for the corresponding stress intensity factors

a new model for thin plates with rapidly varying thickness abstract we study the bending of a thin plate with rapidly varying thickness for example one with rib like stiffeners or perforated by small holes we obtain a fourth order equation for the midplanc displacement using an asymptotic analysis based on three dimensional linear elasticity the coefficients of this equation represent the constitutive law relating bending moments to midplane curvature they are explicitly determined by the plate geometry our analysis distinguishes between three different cases in which the thickness varies on a length scale longer than on the order of or shorter than the mean thickness

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

spectral analysis of matrices in galerkin methods based on generalized b splines with high smoothness we present a first step towards the spectral analysis of matrices arising from iga galerkin methods based on hyperbolic and trigonometric gb splines second order differential problems with constant coefficients are considered and discretized by means of sequences of both nested and non nested spline spaces we prove that there always exists an asymptotic eigenvalue distribution which can be compactly described by a symbol just like in the polynomial case there is a complete similarity between the symbol expressions in the hyperbolic trigonometric and polynomial cases this results in similar spectral features of the corresponding matrices we also analyze the iga discretization based on trigonometric gb splines for the eigenvalue problem related to the univariate laplace operator we prove that for non nested spaces the phase parameter can be exploited to improve the spectral approximation with respect to the polynomial case as part of the analysis we derive several fourier properties of cardinal gb splines

beam theory for strongly orthotropic materials abstract this paper presents a displacement based model for orthotropic beams under plane linear elasticity based on the only kinematic assumption of transverse inextensibility any given axial and transverse loading as well as boundary conditions at the beam ends are considered the solution is decomposed into the principal and the residual part corresponding to the interior and the boundary problems which are obtained by series expansions of polynomial functions and eigenfunctions respectively it is proved that the proposed one dimensional theory gives both interior and boundary exact two dimensional elasticity solutions for strongly orthotropic materials i e for ratio between shear modulus and axial young modulus approaching zero for isotropic and orthotropic materials the accuracy of the beam model is also analysed and compared with that of other theories in particular the complementary energy error of the interior solution with respect to two dimensional elasticity is evaluated the asymptotic estimate of the characteristic decay length of end effects given in choi and horgan j appl mech asme 44 424 430 1977 by two dimensional analysis is reobtained and the range of validity of boundary solution is discussed the numerical results presented agree very well with exact and finite element solutions even in the neighbourhood of clamped cross sections where the solution is mainly governed by the boundary problem

optimality conditions for mass design problems and applications to thin plates we derive necessary and sufficient optimality conditions for a quite large class of structural design problems which can be formulated as follows under a given load and a total volume constraint minimize a suitable notion of compliance among all admissible mass distributions represented by positive measures with prescribed integral mean as a special case we focus attention on the optimization of thin plates we detail the corresponding optimality conditions and we show how they can be handled in order to determine analytically some optimal plates

generalized beam theory applied to shear stiffness there is no consensus of opinion on the correct expressions for shear stiffness even in the apparently simple case of rectangular sections made from homogeneous isotropic material a general beam theory has been proposed which is applicable to all regular prismatic systems this has been used to find the appropriate beam like flexibilities for trusses the same approach can be used for normal beams giving values for the shear stiffnesses of various cross sections as particular results of a general theory embracing torsion binding extension and shear of regular prismatic systems

isogeometric analysis in advection diffusion problems tension splines approximation we present a novel approach within the new paradigm of isogeometric analysis introduced by hughes et al 2005 6 to deal with advection dominated advection diffusion problems the key ingredient is the use of galerkin approximating spaces of functions with high smoothness as in iga based on classical b splines but particularly well suited to describe sharp layers involving very strong gradients