out of plane vibrations of thick rings abstract depending on its cross sectional geometry a ring can be modelled by at least three different theories kirkhope developed a simple ring theory which is applicable to out of plane vibrations of rings with small cross sectional aspect ratio i e compact rings and shell like rings for plate like rings the present analysis uses a levinson type plate theory to analyze the out of plane vibrations also a three dimensional finite element analysis using nastran shows better agreement than results of kirkhope s ring theory and mindlin s thick plate theory

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

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

exact theory for a linearly elastic interior beam abstract in this paper an elasticity solution for a two dimensional 2d plane beam is derived and it is shown that the solution provides a complete framework for exact one dimensional 1d presentations of plane beams first an interior solution representing a general state of any 2d linearly elastic isotropic plane beam under a uniform distributed load is obtained by employing a stress function approach the solution excludes the end effects of the beam and is valid sufficiently far away from the beam boundaries then three kinematic variables defined at the central axis of the plane beam are formed from the 2d displacement field using these central axis variables the 2d interior elasticity solution is presented in a novel manner in the form of a 1d beam theory by applying the clapeyron s theorem it is shown that the stresses acting as surface tractions on the lateral end surfaces of the interior beam need to be taken into account in all energy based considerations related to the interior beam finally exact 1d rod and beam finite elements are developed by the aid of the axis variables from the 2d solution

another second order beam vibration theory explicit bending warping flexibility and restraint abstract a new symmetric beam bending vibration theory which explicitly includes bending warping is presented the governing differential equations are of sixth order and the boundary conditions allow enforcement of total fixity of the end of a beam the motivation for the study was the desire to determine if consideration of bending warping restraint in bending theory is as important as is torsion warping restraint in torsion theory natural frequency predictions are given and compared with other theories for constant cross section cantilevered beams of two types short deep rectangular beams and short thin shallow circular cylindrical shells the accuracy of the new theory is comparable with murty s levinson s bickford s and rehfield s theories for beams with rectangular cross sections it is also comparable with cowper s theory for general symmetric cross sections

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

phase field computations of single needle crystals crystal growth and motion by mean curvature the phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase or order parameter and temperature respectively the parameters in the equations are related directly to the physical observables including the interfacial width epsilon which we can regard as a free parameter in computation the phase field equations can be used to compute a wide range of sharp interface problems including the classical stefan model its modification to incorporate surface tension and or surface kinetic terms the cahn allen motion by mean curvature the hele shaw model etc also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists by adjusting the parameters the computations can be varied continuously from single needle dendritic to faceted crystals the computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field e

monge ampere simulation of fourth order pdes in two dimensions with application to elastic electrostatic contact problems abstract we present an efficient moving mesh method for the simulation of fourth order nonlinear partial differential equations pdes in two dimensions using the parabolic monge ampere pma equation pma methods have been successfully applied to the simulation of second order problems but not on systems with higher order equations which arise in many topical applications our main application is the resolution of fine scale behavior in pdes describing elastic electrostatic interactions the pde system considered has multiple parameter dependent singular solution modalities including finite time singularities and sharp interface dynamics we describe how to construct a dynamic mesh algorithm for such problems which incorporates known self similar or boundary layer scalings of the underlying equation to locate and dynamically resolve fine scale solution features in these singular regimes we find a key step in using the pma equation for mesh generation in fourth order problems is the adoption of a high order representation of the transformation from the computational to physical mesh we demonstrate the efficacy of the new method on a variety of examples and establish several new results and conjectures on the nature of self similar singularity formation in higher order pdes

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

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

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

simulations of a fractional rate type nanofluid flow with non integer caputo time derivatives abstract unsteady developed flow of a rate type anomalous nanofluid with non integer caputo fractional derivatives is studied numerically in this article mixed convection and diffusion are taken into account while analyzing transport phenomena in the flow field thermophoresis and pedesis effects are also incorporated fluid is confined between nonisothermal parallel plates and flows by the moving lower plate variable concentration is assumed at both the plates in literature no such result exists with non integer caputo fractional derivatives boundary layer flow is modeled with the help of fractional calculus approach governing flow partial differential equations with appropriate conditions are solved by finite difference finite element scheme given scheme is flexible for the solution of non linear flow problems local nusselt and sherwood numbers are computed for the fractional model flow behavior is presented for various values of involved parameters influence of different dimensionless quantities on the nusselt and sherwood numbers is discussed by tabular results the acquired results revealed that fractional exponents have opposite effects on the velocity profiles it is also noted that thermophoresis and pedesis parameters have similar effects on heat flux while opposite effects are observed for mass flux at both the plates various stretching flows particularly in paper polymeric and food production processes can be modeled in a similar manner

lagrange type formulation for finite element analysis of non linear beam vibrations abstract a lagrange type formulation for finite element analysis of non linear vibrations of immovably supported beams is presented two equations of motion coupled in axial and transverse displacements are derived by using lagrange s equations by neglecting the in plane inertial effects these equations are written in terms of the transverse displacement alone upon defining certain properties for the non linear oscillatory behaviour of the transverse displacement the governing equation is reduced to an equation in space alone from which the eigenvalue like quantity is computed the governing equation is solved in two ways a direct iteration technique is used in the first method to compute a numerically exact mode shape and the corresponding frequency a rayleigh quotient type of formulation similar to linear vibration analysis is used in the second approach to evaluate the frequency of vibration for a fundamental mode which is determined from a linear fem model and is maintained constant at all amplitudes numerical results are compared with available results and they corroborate the observations of earlier research workers

relationships between bending solutions of classical and shear deformation beam theories the exact relationships between the deflections slopes rotations shear forces and bending moments of a third order beam theory and those of the euler bernoulli theory and the timoshenko beam theory are developed the relationships enable one to obtain the solutions of the third order beam theory from any known euler bernoulli or timoshenko beam theory solutions of beams for any set of boundary conditions and transverse loads the relationships can also be used to develop finite element models of the timoshenko and third order beam theories and determine numerical solutions from the finite element model of the euler bernoulli beam theory the finite element models are free of the shear locking that is found in the conventional shear deformable finite elements

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

advanced 3d beam element of arbitrary composite cross section including generalized warping effects summary in this paper an advanced 20 20 stiffness matrix and the corresponding nodal load vector of a member of arbitrary composite cross section is developed taking into account shear lag effects due to both flexure and torsion the case of the three dimensional beam element of arbitrary homogeneous cross section is treated as a special one the composite member consists of materials in contact each of which can surround a finite number of inclusions nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions ten boundary value problems with respect to the kinematical components are formulated and solved employing the analog equation method a bem based technique the aforementioned boundary value problems are formulated employing either an improved stress field arising from the correction of shear stress components or the stress field arising directly from displacement considerations the warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure bem approach that is only boundary discretization of the cross section is used numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy the deviations arising from the use of the advanced 20 20 stiffness matrix and the classical 12 12 or 14 14 ones employed in commercial software packages are illustrated through examples of great practical interest moreover the influence of nonuniform warping effects necessitating the use of the aforementioned advanced stiffness matrix is also demonstrated copyright 2015 john wiley sons ltd

saint venant solutions for prismatic anisotropic beams abstract an analytical model is presented for determining the displacement and stress distributions of the saint venant extension bending torsion and flexure problems for a homogeneous prismatic beam of arbitrary section and rectilinear anisotropy the determination of the complete displacement field requires solving a coupled two dimensional boundary value problem for the local in plane deformations and warping out of the section plane the principle of minimum potential energy is applied to a discretized representation of the cross section ritz method to calculate solutions to this problem the behavior of an anisolropic beam is studied in detail using the resulting displacement and stress solutions where definitions are presented for the shear center center of twist torsion constant and a new geometric parameter the line of extension bending centers two sets of numerical results are presented to illustrate how section geometry beam length and material properties affect the behavior of a homogeneous anisotropic cantilever beam

generalized warping effect in the dynamic analysis of beams of arbitrary cross section abstract in this paper a general formulation for the nonuniform warping dynamic analysis of beams of arbitrary simply or multiply connected cross section under arbitrary external loading and general boundary conditions is presented taking into account the effects of rotary and warping inertia the nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions respectively which are obtained by solving the corresponding boundary value problems formulated exploiting the longitudinal local equilibrium equation a shear stress correction is also performed in order to improve the stress field arising from the employed kinematical considerations ten initial boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameters and solved using the analog equation method a boundary element method based technique in combination with an appropriate time integration scheme the warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure bem approach

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

nondestructive determination of buckling for plates and bars including shear it will be shown that the generic integral equation for nondestructive determination of the buckling loads obtained previously baruch m integral equations for nondestructive determination of buckling loads for elastic bars and plates israel journal of technology vol 11 no 1 2 1973 pp 1 8 includes the influence of the transverse shear strains on the buckling load this can not be said for the integral equations obtained by differentiation of the generic integral equation

a modified fourier series method for the torsion analysis of bars with multiply connected cross sections a modified fourier series method is proposed for the torsion analysis of prismatic bars with multiply connected cross sections the key feature in the present approach is the combined use of polynomials and fourier series solutions unlike in the existing approaches which use the fourier series only the replacement of the zeroth harmonic terms in the fourier series solutions by carefully selected polynomials resolves the major problem of functional dependence which the direct fourier series method may pose the polynomials and fourier series solutions are selected to satisfy the governing equation exactly so that the numerical calculation is minimal the effectiveness and generality of the present method is verified through numerical examples

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

high order theory of homogeneous plate flexure the need for higher order models to obtain accurate estimates to transverse shear and normal stresses and strains is well known in this paper a higher order shear deformation theory with provision for cubic variation of in plane displacements and parabolic variation of the normal displacement has been considered for detailed study numerical results have been obtained for infinite plate strip simply supported on opposite edges and infinitely long in the other direction subjected to sinusoidal static load displacements strains and stresses are compared with the exact solution results by the classical plate theory levinson s theory and a higher order theory based on partial deflections are also included this study shows that the present model can predict all displacements strains and stresses reasonably accurately further it is also noted that the statically equivalent estimates to transverse stresses from classical theory and the corresponding strains also agree closely with the exact solution

reduced integration and the shear flexible beam element the timoshenko beam element is studied in an investigation of shear locking in the development of c 0 continuous elements using shear flexible or penalty function type formulations a penalty type argument is used to degenerate thick elements to thin elements and the various approximations of the shear related energy terms act as different types of constraints depending on the formulation two types of constraints may emerge which are classified as true or spurious formulations that ensure only true constraints in the extreme penalty limit cases display superior performance in the thick element situation the argument is extended to a shallow curved timoshenko beam and another mechanism inplane locking arises if all energy terms corresponding to the membrane energy are exactly integrated

the second frequency spectrum of timoshenko beams abstract a second spectrum of frequencies was reported in early analytical work on the vibrations of timoshenko beams however in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded in fact from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies in this paper a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately guidelines are provided which help to identify and classify the frequencies into two separate spectra

estimates for eigenvalues of the paneitz operator abstract for an n dimensional compact submanifold m n in the euclidean space r n we study estimates for eigenvalues of the paneitz operator on m n our estimates for eigenvalues are sharp

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

stability of the reeb vector field of h contact manifolds it is well known that a hopf vector field on the unit sphere s 2n 1 is the reeb vector field of a natural sasakian structure on s 2n 1 a contact metric manifold whose reeb vector field is a harmonic vector field is called an h contact manifold sasakian and k contact manifolds generalized k spaces and contact metric three manifolds with strongly normal are h contact manifolds in this paper we study in dimension three the stability with respect to the energy of the reeb vector field for such special classes of h contact manifolds and with respect to the volume when is also minimal in terms of webster scalar curvature finally we extend for the reeb vector field of a compact k contact 2n 1 manifold the obtained results for the hopf vector fields to minimize the energy functional with mean curvature correction

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

a generalization of shear deformation theories for axisymmetric multilayered shells a generalization of geometrically linear shear deformation theories for small elastic strains is presented for multilayered axisymmetric shells of general shape without any assumption other than neglecting the transverse normal strain the shear is taken into account by using a function f which is introduced in the assumed kinematics the boundary value problem is derived from the principle of virtual power with the shear function f in the kinematics all equations are directly applicable to kirchhoff love first order shear deformation third order shear deformation theories and obviously the proposed generalized shear deformation theory by using a certain sine shear function no shear correction factors are needed with the proposed generalization of shear deformation theories a numerical evaluation of the new theory is presented and compared with the above classic theories for a simply supported thick laminated cylindrical shell under an internal pressure

warping displacement of timoshenko beam model abstract a refined timoshenko beam model which takes into account warping of the cross sections is presented the model extends saint venant s theory of uniform torsion to the generic loading of a beam in a coherent manner the use of kinetic and kinematic assumptions virtual work expression of full elasticity problem and the principle of virtual work in derivation brings the presentation to the usual context of engineering models a new definition of the warping displacement in terms of a variational problem is one of the outcomes boundary value problems for the stretch shear bending and torsion loading modes are also given analytic and numeric solutions to warping displacements refined constitutive equations and correction factors for circular rectangle triangle and open annular cross sections of isotropic material are compared with expressions in literature as an interesting detail shear correction factors are found to be purely geometrical quantities which contradicts many findings in literature

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

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

a finite element model for a higher order shear deformable beam theory the theory for a higher order shear deformable beam model is first developed it is based on a higher order displacement model and incorporates linear and quadratic variation of transverse normal strain and transverse shearing strain respectively through the beam thickness the effects of the transverse normal and shear stresses are included in the definition of the material s constitutive law the warping of the transverse normal cross section of the beam is automatically incorporated in the mathematical model the question of selecting a shear correction coefficient as in a first order shear deformable timoshenko theory does not arise a linear two noded finite element model of this theory is introduced and developed next both static and free vibration results of this theory are presented and compared with those of euler and timoshenko theories for various boundary and loading conditions

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

a simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method a size dependent finite element model is developed nonlocal elasticity theory is used finite element solution is presented for buckling of microtubules the effect of elastic matrix surrounded by microtubules is modeled a simple nonlocal beam model is proposed to study buckling response of protein microtubules the size effect for buckling model of microtubules is considered by using the nonlocal continuum theory finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability the influence of the small length scale on the buckling value is examined for different geometric parameters the effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented

conformal energy in four dimension the conformal energy for 4 manifolds using the paneitz operator is introduced in this article the conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume we also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator

second eigenvalue of paneitz operators and mean curvature for n 7 we give the optimal estimate for the second eigenvalue of paneitz operators for compact n dimensional submanifolds in an n p dimensional space form in terms of the mean curvature and the q curvature

relationships between classical and shear deformation theories of axisymmetric circular plates the exact relationships between the deflections slopes rotations shear forces and bending moments of a third order theory and those of the classical theory and the first order shear deformation theory for axisymmetric bending of isotropic circular plates are developed the relationships enable one to obtain the solutions of the third order theory from any known classical or first order theory solutions of axisymmetric circular plates for any set of boundary conditions and transverse loads the relationships may also be used to develop finite element models of axisymmetric bending of circular plates according to the first order and third order theories

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

a moving mesh method based on the geometric conservation law a new adaptive mesh movement strategy is presented which unlike many existing moving mesh methods targets the mesh velocities rather than the mesh coordinates the mesh velocities are determined in a least squares framework by using the geometric conservation law specifying a form for the jacobian determinant of the coordinate transformation defining the mesh and employing a curl condition by relating the jacobian to a monitor function one is able to directly control the mesh concentration the geometric conservation law an identity satisfied by any nonsingular coordinate transformation is an important tool which has been used for many years in the engineering community to develop cell volume preserving finite volume schemes it is used here to transform the algebraic expression specifying the jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy it is shown that the resulting method bears a close relation with the lagrangian method advantages of the new approach include the ease of controlling the cell volumes and therefore mesh adaption and a theoretical guarantee for existence and nonsingularity of the coordinate transformation it is shown that the method may suffer from the mesh skewness a consequence resulting from its close relation with the lagrangian method numerical results are presented to demonstrate various features of the new method

computational solution of two dimensional unsteady pdes using moving mesh methods numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two dimensional partial differential equations pdes here we are concerned with algorithms based on moving mesh methods proposed by w huang and r d russell siam j sci comput 20 998 1999 we show that the accuracy of the computations is strongly influenced by the choice of monitor function and we present a monitor function which yields a higher rate of convergence than those that are commonly used in an earlier paper g beckett j a mackenzie a ramage and d m sloan j comput phys 167 372 2001 we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical pde and the time step is controlled automatically the present work extends this algorithm to deal with problems in two space dimensions

optimal mass transport based adaptive mesh method for phase field models of two phase fluid flows in this paper we present an efficient adaptive mesh method for solving phase field model of a mixture of two phase incompressible fluid flows the adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation also known as mongekantorovich problem mkp the numerical solution of the mkp is computed by taking the gradient of the steady state solution of a parabolic mongeampre equation pmae several numerical experiments are conducted to demonstrate the performance of the pmae method for solving phase field models the numerical results illustrate that the adaptive mesh computed with pmae method captures well the moving interface of the phase field function

vibration and buckling of deep beam columns on two parameter elastic foundations natural frequencies and buckling stresses of a deep beam column on two parameter elastic foundations are analyzed by taking into account the effect of shear deformation depth change the transverse displacement w can vary in the depth direction of beam columns and rotatory inertia by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a one dimensional higher order theory for thin rectangular beam columns subjected to axial stress is derived through hamilton s principle several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported deep elastic beam column in order to assure the accuracy of the present theory convergence properties of the minimum natural frequency and the buckling stress are examined in detail it is noted that the present approximate theories can predict the natural frequencies and buckling stress of deep beam columns on elastic foundations accurately compared with the timoshenko beam theory and the classical beam theory

vibration of a rectangular beam with a deforming cross section abstract levinson recently proposed a new theory for rectangular beam vibrations in which an inherent contradiction in conventional timoshenko beam theory is avoided by relaxing the assumption that plane sections remain plane in this paper this theory is reformulated in energy minimization terms it is found that the energy expressions are similar to those found for a timoshenko beam with a shear coefficient of 5 6 but they are not identical the two theories are therefore not equivalent at variance with levinson s conclusion it is shown that considerable care is needed to deal with a clamped boundary

bending solutions of levinson beams and plates in terms of the classical theories using the mathematical similarity of the governing equations of the classical beam and plate theories and the levinson beam and plate theories and the basis of load equivalence exact relationships between the bending solutions of the two theories for beams and plates are derived these relationships enable the conversion of the well known classical euler bernoulli beam and kirchhoff plate solutions to their shear deformable counterparts using the levinson beam and plate theories examples are given to illustrate the use of these relationships

minimal immersions of surfaces by the first eigenfunctions and conformal area let 0 m s be a minimal immersion of a compact surface into a unit sphere then the linear functions of 0 are eigenfunctions for the laplacian of m corresponding to the eigenvalue 2 2 the main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non zero eigenvalue of its laplacian this kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems as li and yau have shown in 6 the methods that we use in this paper are based for the most part on 6 it is known that the only metric on a 2 dimensional sphere admitting a minimal immersion into s by the first eigenfunctions is the standard one this follows for example from the fact that the multiplicity of the first eigenvalue for such a metric is at most three see the cheng work 3 our first result shows that it is possible to extend this property for an arbitrary compact surface in the following way

deformation of short composite beam using refined theories high order flexural theories for short laminated composite beams subjected to mechanical and thermal loading are presented the formulation allows for warping of the cross section of the beam and eliminates the need for using arbitrary shear correction coefficients as in other theories based on higher order shear deformation theories the governing equations are obtained using the principle of virtual work pvw the justification for use of higher order shear deformation theories is established for short and composite beams where cross sectional warping is predominant

free vibrations of a simply supported rectangular plate an exact elasticity solution abstract an exact three dimensional solution for the free vibrations of simply supported rectangular plates of arbitrary thickness within the linear theory of elastodynamics is given in this paper the solution obtained in a semi inverse fashion as was the solution of the elastostatic problem for such plates satisfies all of the boundary conditions of the problem in a pointwise manner it is found that there are two types of modes of oscillation possible which are consistent with the kinematic assumptions made to find the semi inverse solution other modes of oscillation may exist in the three dimensional theory of elastodynamics for such plates but our kinematic assumptions would not be consistent with such modes the two types of modes found are analogous to the flexural modes of classical plate theory and the thickness twist modes here called breathing modes of mindlin plate theory some numerical results are given which indicate that the predictions of mindlin plates are uncannily good approximations to the flexural frequencies given by the present three dimensional analysis even for very thick plates however the predictions of mindlin plate theory for the thickness twist or breathing frequencies are not nearly so good these discrepancies are discussed briefly in an appendix

generalized variational principle on nonlinear theory of naturally curved and twisted beams the nonlinear behavior for naturally curved and twisted beams undergoing large displacements and rotations but small strains has been investigated in this paper an incomplete generalized variational functional for the beams with complete constrained boundaries at two ends is established by means of the lagrange multiplier method a set of equilibrium equations which takes into account the effects of transverse shear deformations as well as torsion related warping has been derived from the stationary conditions of the functional the various elastic couplings of bending twisting and or extension twisting are also included in this analysis the present theory can also be easily extended to other various incomplete constrained boundaries in case of small displacement the equilibrium equations obtained by the present theory reduce to the well known results in literature

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

dynamic response of thin walled structures by variable kinematic one dimensional models abstract this paper investigates the accuracy capabilities of using variable kinematic modeling in compact and thin walled beam like structures with dynamic loadings carrera unified formulation cuf is employed to introduce refined one dimensional 1d models with a variable order of expansion for the displacement unknowns over the beam cross section classical euler bernoulli and timoshenko beam theories are obtained as particular cases of these variable kinematic models while a higher order expansion permits the detection of in plane cross section deformation since it leads to shell like solutions finite element fe method is used to provide numerical results and the newmark method is implemented as a time integration scheme some assessments with closed form solutions are discussed and comparisons with shell type results obtained with commercial fe software are made further analyses address both compact and thin walled cross sections in particular the case of a deformable thin walled cylinder loaded by time dependent internal forces is discussed the results clearly show that finite elements which are formulated in the cuf framework do not introduce additional numerical problems with respect to classical beam theories comparisons with elasticity solutions prove that the present 1d cuf model offers an accuracy in analyzing thin walled structures which is typical of shell or three dimensional models with a remarkable reduction in the computational cost required

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

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

analogy between refined beam theories and the bernoulli euler theory abstract some refinements of the classical bernoulli euler theory of the bending of beams are shown to be completely analogous to the effects produced by sources of self stress in the bernoulli euler beam this is true for deflections bending moments and shearing forces they determine rotations stresses and strains according to the specific refined theory under consideration thus by analogy differences between the results of various refined theories having been the object of discussions in recent literature become accessible to a systematic classification in an appendix this strategy of treating refined theories from the point of view of the classical one is put into the more general context of the theory of science