geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

nonlinear vibrations of shallow shells with complex boundary r functions method and experiments geometrically nonlinear vibrations of shallow circular cylindrical panels with complex shape of the boundary are considered the r functions theory and variational methods are used to study the problem the r functions method rfm allows constructing in analytical form the sequence of basis functions satisfying the given boundary conditions in case of complex shape of the boundary the problem is reduced to a single second order differential equation with quadratic and cubic nonlinear terms the method developed has been initially applied to study free vibrations of shallow circular cylindrical panels with rectangular base for different boundary conditions i clamped edges ii in plane immovable simply supported edges iii classically simply supported edges and iv in plane free simply supported edges then the same approach is applied to a shell with complex shape of the boundary experiments have been conducted on an aluminum panel with complex shape of the boundary in order to identify the nonlinear response of the fundamental mode these experimental results have been compared to numerical results

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

the r function method for the free vibration analysis of thin orthotropic plates of arbitrary shape free flexural vibrations of homogeneous thin orthotropic plates of an arbitrary shape with mixed boundary conditions are studied using the r function method the proposed method is based on the use of the r function theory and variational methods in contrast to the widely used methods of the network type finite differences finite element and boundary element methods in the r function method all the geometric information given in the boundary value problem statement is represented in an analytical form this allows one to seek a solution in a form of some formulas called a solution structure these solution structures contain some indefinite functional components that can be determined by using any variational method a method of constructing the solution structures satisfying the required mixed boundary conditions for eigenvalue plate bending problems is described numerical examples for the vibration analysis of orthotropic plates of complex geometry with mixed boundary conditions for illustrating the aforementioned r function method and comparison against the other methods are made to demonstrate its merits

well posedness and smoothing effect of schrodinger poisson equation in this work we take under consideration the cauchy problem for the schrodinger poisson type equation i tu x2u v u u f u 2 u where f represents a local nonlinear interaction we take into account both attractive and repulsive models and v is taken as a suitable solution of the poisson equation v 1 2 x c u 2 c cc is the doping profile or impurities we show that this problem is locally well posed in the weighted sobolev spaces hs hs r 1 x2 1 2 2 with s 1 which means the local existence uniqueness and continuity of the solution with respect to the initial data moreover under suitable assumptions on the local interaction we show the existence of global solutions finally we establish that for s 1 local in time and space smoothing effects are present in the solution more precisely in this problem there is locally a gain of half a derivative

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

solution of vibration problems for shallow shells of arbitrary form by the r function method an effective method for the free vibration of arbitrary plan form shallow shells is proposed the algorithm is based on the r function theory and variational ritz method the effectiveness of the method offered is illustrated by examples of shallow shells of a complex plan form at different boundary conditions three types of curvatures are considered which are the spherical circularly cylindrical and of hyperbolic paraboloid shape a number of the test problems were solved to check the veracity of the proposed method the comparison between the obtained results and those available in the literature was carried out

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

a geometrically exact rod model incorporating shear and torsion warping deformation abstract a fully nonlinear three dimensional rod model is developed that incorporates transverse shear and torsion warping deformation the geometric setting is that of a constrained body model with configuration space modeled on r3 so 3 r a differentiable manifold the proposed model incorporates the classical notion of bi moment and bi shear in a fully nonlinear geometrically exact context explicit properly invariant constitutive equations that generalize those of the linearized theory are developed the underlying variational formulation of the model is discussed and computational procedures employing a galerkin projection are addressed numerical examples are presented that illustrate the performance of the formulation

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

how to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties abstract for a fairly general class of second order accurate algorithms a sub stepping procedure is described which achieves fourth order accuracy while retaining the stability conservation properties and implementation of the underlying second order method two features render the proposed approach especially attractive from a computational perspective over conventional fourth order methods first the improved fourth order accuracy is attained merely by two additional residual evaluations with no changes in the implementation of the underlying second order method in particular no additional storage or extra computations of high order gradients are required second for hamiltonian systems the stability properties and possible conservation properties of the underlying second order method such as energy momentum and symplectic character are preserved the new technique is applied to a number of standard algorithms for elastodynamics structural and rigid body dynamics in particular when applied to the recently proposed second order accurate energy momentum method this approach results in a fourth order accurate energy linear and angular momentum conserving algorithm for general hamiltonian systems

on the existence of ground states for nonlinear schrodinger poisson equation abstract this paper is concerned with the existence of ground states for the schrodinger poisson equation i t u x 2 u v u u f u 2 u where v u is a hartree type nonlinearity stemming from the coupling with the poisson equation which includes the so called doping profile or impurities by means of variational methods in the energy space we show that ground states exist and belong to the schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value it is also shown that for values of the total charge greater than this critical value energy is not bounded from below in addition we show that this critical value is the total charge given by the impurities

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

numerical methods for thermally stressed shallow shell equations abstract we develop efficient and accurate numerical methods to solve a class of shallow shell problems of the von karman type the governing equations form a fourth order coupled system of nonlinear biharmonic equations for the transverse deflection and airy s stress function a second order finite difference discretization with three iterative methods picard newton and trust region dogleg is proposed for the numerical solution of the nonlinear pde system three simple boundary conditions and two application motivated mixed boundary conditions are considered along with the nonlinearity of the system boundary singularities that appear when mixed boundary conditions are specified are the main numerical challenges two approaches that use either a transition function or local corrections are developed to deal with these boundary singularities all the proposed numerical methods are validated using carefully designed numerical tests where expected orders of accuracy and rates of convergence are observed a rough run time performance comparison is also conducted to illustrate the efficiency of our methods as an application of the methods a snap through thermal buckling problem is considered the critical thermal loads of shell buckling with various boundary conditions are numerically calculated and snap through bifurcation curves are also obtained using our numerical methods together with a pseudo arclength continuation method our results are consistent with previous studies

non linear vibration characteristics of clamped laminated shallow shells abstract this paper examines non linear free vibration characteristics of first and second vibration modes of laminated shallow shells with rigidly clamped edges non linear equations of motion for the shells based on the first order shear deformation and classical shell theories are derived by means of hamilton s principle we apply galerkin s procedure to the equations of motion in which eigenvectors for first and second modes of linear vibration obtained by the ritz method are employed as trial functions then simultaneous non linear ordinary differential equations are derived in terms of amplitudes of the first and second vibration modes backbone curves for the first and second vibration modes are solved numerically by the gauss legendre integration method and the shooting method respectively the effects of lamination sequences and transverse shear deformation on the behavior are discussed it is also shown that the motion of the first vibration mode affects the response for the second vibration mode

a comparison of shell theories for large amplitude vibrations of circular cylindrical shells lagrangian approach abstract large amplitude geometrically non linear vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated the lagrange equations of motion are obtained by an energy approach retaining damping through rayleigh s dissipation function four different non linear thin shell theories namely donnell s sanders koiter flugge lur e byrne and novozhilov s theories which neglect rotary inertia and shear deformation are used to calculate the elastic strain energy the formulation is also valid for orthotropic and symmetric cross ply laminated composite shells the large amplitude response of perfect and imperfect simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories numerical responses obtained by using these four non linear shell theories are also compared to results obtained by using the donnell s non linear shallow shell equation of motion a validation of calculations by comparison with experimental results is also performed both empty and fluid filled shells are investigated by using a potential fluid model the effects of radial pressure and axial load are also studied boundary conditions for simply supported shells are exactly satisfied different expansions involving from 14 to 48 generalized co ordinates associated with natural modes of simply supported shells are used the non linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis

nonlinear vibrations of circular cylindrical panels abstract large amplitude geometrically nonlinear vibrations of circular cylindrical panels with rectangular base simply supported at the four edges and subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated two different nonlinear strain displacement relationships from the donnell s and novozhilov s shell theories are used to calculate the elastic strain energy in plane inertia and geometric imperfections are taken into account the solution is obtained by lagrangian approach the nonlinear equations of motion are studied by using i a code based on arclength continuation method that allows bifurcation analysis and ii direct time integration numerical results are compared to those available in the literature and convergence of the solution is shown interaction of modes having integer ratio among their natural frequencies giving rise to internal resonances is also discussed

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

theory and experiments for large amplitude vibrations of circular cylindrical panels with geometric imperfections abstract large amplitude vibrations of circular cylindrical panels open shells subjected to harmonic excitation are numerically and experimentally investigated the donnell nonlinear strain displacement relationships are used to describe the geometric nonlinearity in plane inertia is taken into account specific boundary conditions with zero transverse displacement at the panel edges and free or elastically restrained in plane displacements not previously considered have been introduced in order to model the experimental boundary conditions the nonlinear equations of motion are obtained by the lagrange equations with multi mode approach and are studied by using a code based on the pseudo arclength continuation method two thin circular cylindrical panels of different dimensions and made of stainless steel have been experimentally tested in the laboratory for several excitation amplitudes in order to characterize the nonlinearity the dimensions of the two panels have been chosen in order to have the fundamental mode with one and two circumferential half waves respectively numerical results are able to reproduce the experimental results with high accuracy for both panels the effect of geometric imperfections on the trend of nonlinearity and on natural frequencies is shown convergence of the solution with the number of generalized coordinates is numerically verified

application of the r function method to nonlinear vibrations of thin plates of arbitrary shape in this paper free large amplitude flexural vibrations of thin plates with various planforms and boundary conditions are studied by the r function method this method is based on the joint application of the r function theory and variational methods the main feature of the r function theory is the possibility to present all geometric information given in the boundary value problem in analytical form which allows one to seek a solution in the form of some formula called the solution structure a method of constructing the solution structures for the given nonlinear vibration plate bending problems is developed numerical examples of large amplitude flexural vibrations of thin plates with arbitrary shapes and boundary conditions for illustrating the aforementioned r function method and comparison against the other methods are made to demonstrate its merits and advantages

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells