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

optimal fractional integration preconditioning and error analysis of fractional collocation method using nodal generalized jacobi functions in this paper a nonpolynomial based spectral collocation method and its well conditioned variant are proposed and analyzed first we develop fractional differentiation matrices of nodal jacobi polyfractonomials m zayernouri and g e karniadakis j comput phys 252 2013 pp 495 517 and generalized jacobi functions s chen j shen and l l wang math comp 85 2016 pp 1603 1638 on jacobi gauss lobatto jgl points we show that it suffices to compute the matrix of order mu in 0 1 to compute that of any order k mu with integer k geq 0 with a different definition of the nodal basis our approach also fixes a deficiency of the polyfractonomial fractional collocation method in m zayernouri and g e karniadakis siam j sci comput 38 2014 pp a40 a62 second we provide explicit and compact formulas for computing the inverse of direct fractional differential collocation matrices at interior points by virtue of fractional jgl birkhoff interpolation this leads t

error estimates of a spectral petrov galerkin method for two sided fractional reaction diffusion equations abstract we study regularity and the spectral method for two sided fractional diffusion equations with a reaction term we show that the regularity of the solution in weighted sobolev spaces can be greatly improved compared to that in standard sobolev spaces with this regularity we prove an optimal error estimate for the spectral petrov galerkin method numerical results are presented to verify our theoretical convergence orders

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

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

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

a high order numerical method for the nonlinear helmholtz equation in multidimensional layered media we present a novel computational methodology for solving the scalar nonlinear helmholtz equation nlh that governs the propagation of laser light in kerr dielectrics the methodology addresses two well known challenges in nonlinear optics singular behavior of solutions when the scattering in the medium is assumed predominantly forward paraxial regime and the presence of discontinuities in the optical properties of the medium specifically we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole the key components of the methodology are a semi compact high order finite difference scheme that maintains accuracy across the discontinuities and enables sub wavelength resolution on large domains at a tolerable cost a nonlocal two way artificial boundary condition abc that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves and a nonlinear solver based on newton s method the proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1d and for multi d it facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the augmented paraxial models in particular it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar nlh model that includes the nonparaxial effects it also enables simulation of the wavelength width spatial solitons as well as of the counter propagating solitons

high order numerical solution of the nonlinear helmholtz equation with axial symmetry the nonlinear helmholtz nlh equation models the propagation of intense laser beams in a kerr medium the nlh takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear schrodinger model in g fibich s tsynkov high order two way artificial boundary conditions for nonlinear wave propagation with backscattering j comput phys 171 2001 632 677 and g fibich s tsynkov numerical solution of the nonlinear helmholtz equation using nonorthogonal expansions j comput phys 210 2005 183 224 a novel high order numerical method for solving the nlh was introduced and implemented in the case of a two dimensional cartesian geometry the nlh was solved iteratively using the separation of variables and a special nonlocal two way artificial boundary condition applied to the resulting decoupled linear systems in the current paper we propose a major improvement to the previous method instead of using lu decomposition after the separation of variables we employ an efficient summation rule that evaluates convolution with the discrete green s function we also extend the method to a three dimensional setting with cylindrical symmetry under both dirichlet and sommerfeld type transverse boundary conditions

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

recovery of high order accuracy in jacobi spectral collocation methods for fractional terminal value problems with non smooth solutions abstract an open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high order accuracy for non smooth solutions the limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes in this paper we derive and analyze an exponentially accurate jacobi spectral collocation method for the numerical solution of nonlinear terminal value problems involving the caputo fractional derivative of rational order 0 1 the main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate we systematically describe the necessary steps in the implementation process additionally a comprehensive numerical analysis including error estimates under the l and weighted l 2 norms is derived the extensive numerical examples that accompany our analysis confirm our theoretical estimates as well as give additional insights into the convergence behavior of our method for problems with smooth and non smooth solutions

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

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

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

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

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

fractional spectral and pseudo spectral methods in unbounded domains theory and applications abstract this paper is intended to provide exponentially accurate galerkin petrov galerkin and pseudo spectral methods for fractional differential equations on a semi infinite interval we start our discussion by introducing two new non classical lagrange basis functions nlbfs 1 and nlbfs 2 which are based on the two new families of the associated laguerre polynomials galfs 1 and galfs 2 obtained recently by the authors in 28 with respect to the nlbfs 1 and nlbfs 2 two new non classical interpolants based on the associated laguerre gauss and laguerre gauss radau points are introduced and then fractional pseudo spectral differentiation and integration matrices are derived convergence and stability of the new interpolants are proved in detail several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives and integrals of some functions moreover the pseudo spectral galerkin and petrov galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones some useful comments from the numerical point of view on galerkin and petrov galerkin methods are listed at the end

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

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

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

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

high order numerical method for the nonlinear helmholtz equation with material discontinuities in one space dimension the nonlinear helmholtz equation nlh models the propagation of electromagnetic waves in kerr media and describes a range of important phenomena in nonlinear optics and in other areas in our previous work we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity the method enabled a direct simulation of nonlinear self focusing in the nonparaxial regime and a quantitative prediction of backscattering however our simulations showed that there is a threshold value for the magnitude of the nonlinearity above which the iterations diverge in this study we numerically solve the one dimensional nlh using a newton type nonlinear solver because the kerr nonlinearity contains absolute values of the field the nlh has to be recast as a system of two real equations in order to apply newton s method our numerical simulations show that newton s method converges rapidly and in contradistinction with the iterations based on freezing the nonlinearity enables computations for very high levels of nonlinearity in addition we introduce a novel compact finite volume fourth order discretization for the nlh with material discontinuities our computations corroborate the design fourth order convergence of the method the one dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future

numerical solution of the nonlinear helmholtz equation using nonorthogonal expansions in j comput phys 171 2001 632 677 we developed a fourth order numerical method for solving the nonlinear helmholtz equation which governs the propagation of time harmonic laser beams in media with a kerr type nonlinearity a key element of the algorithm was a new nonlocal two way artificial boundary condition abc set in the direction of beam propagation this two way abc provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain altogether the algorithm of j comput phys 171 2001 632 677 has allowed for a direct simulation of nonlinear self focusing without neglecting nonparaxial effects and backscattering to the best of our knowledge this capacity has never been achieved previously in nonlinear optics in the current paper we propose an improved version of the algorithm the principal innovation is that instead of using the dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation we now introduce sommerfeld type local radiation boundary conditions which are constructed directly in the discrete framework numerically implementation of the sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non hermitian matrix subsequently the separation of variables which is a key building block of the aforementioned nonlocal abc is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance as well as in the range of physical phenomena that it is capable of simulating

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

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

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

an existence theorem for some semilinear elliptic systems abstract an existence theorem is obtained for a class of semilinear second order uniformly elliptic systems obtained formally from a variational principle and modeled on nonlinear helmholtz systems superlinear growth of the nonlinear term precludes application of standard methods to these systems indeed we permit very rapid growth of the nonlinear term so the underlying functional is not defined on the hilbert space within which a solution is naturally sought mollification of the nonlinear term nonetheless results in the resulting functional satisfying the palais smale condition critical points are determined by solution of a dynamical system the limit of vanishing mollification then produces a weak solution of the original problem

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