rectifiable and unrectifiable oscillations for a class of second order linear differential equations of euler type the main purpose of the paper is to pay attention to a new kind of oscillations so called rectifiable and unrectifiable oscillations which occur in a class of linear second order differential equations including the famous linear euler differential equation it will be shown that this kind of oscillations only depends on a positive real parameter which appears in a singular term of the main equation as well as in a suitable boundary layer condition

on non corotational rates of oldroyd s type and relevant issues in rate constitutive formulations the lower and upper oldroyd rates are two wellknown objective noncorotational rates a remarkable property of these rates is that they establish direct relationships between finite strain measures

complementarity problems this paper provides an introduction to complementarity problems with an emphasis on applications and solution algorithms various forms of complementarity problems are described along with a few sample applications which provide a sense of what types of problems can be addressed effectively with complementarity problems the most important algorithms are presented along with a discussion of when they can be used effectively we also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems finally we provide a brief summary of current research trends

rectifiable oscillations of self adjoint and damped linear differential equations of second order abstract asymptotic and oscillatory behaviours near x 0 of all solutions y y x of self adjoint linear differential equation p p q p y q y 0 on 0 t will be studied where p p x and q q x satisfy the so called hartman wintner type condition we show that the oscillatory behaviour near x 0 of p p q is characterised by the nonintegrability of q p on 0 t moreover under this condition we show that the rectifiable resp unrectifiable oscillations near x 0 of p p q are characterised by the integrability resp nonintegrability of q p 3 4 on 0 t next some invariant properties of rectifiable oscillations in respect to the liouville transformation are proved also sturm s comparison type theorem for the rectifiable oscillations is stated furthermore previous results are used to establish such kind of oscillations for damped linear second order differential equation y g x y f x y 0 and especially the bessel type damped linear differential equations are considered finally some open questions are posed for the further study on this subject

weighted max norms splittings and overlapping additive schwarz iterations summary weighted max norm bounds are obtained for algebraic additive schwarz iterations with overlapping blocks for the solution of ax b when the coefficient matrix a is an m matrix the case of inexact local solvers is also covered these bounds are analogous to those that exist using a norms when the matrix a is symmetric positive definite a new theorem concerningp regular splittings is presented which provides a useful tool for the a norm bounds furthermore a theory of splittings is developed to represent algebraic additive schwarz iterations this representation makes a connection with multisplitting methods with this representation and using a comparison theorem it is shown that a coarse grid correction improves the convergence of additive schwarz iterations when measured in weighted max norm

on constitutive relations at finite strain hypo elasticity and elasto plasticity with isotropic or kinematic hardening abstract the question of generalization to finite strains of the constitutive relations of infinitesimal strain theory of elasticity and classical elasto plasticity with isotropic and kinematic hardening is critically examined simple generalizations which lead to physically plausible material behaviour are presented the current controversies surrounding i the choice of stress rate in the above generalization and ii the anomaly of oscillatory stresses in kinematic hardening plasticity as discussed by e h lee and others are analyzed it is found that these controversies are easily resolvable

minkowski bouligand dimension of solutions of the one dimensional p laplacian the upper minkowski bouligand dimension of the graph of continuous solutions denoted by dimmy is studied for a class of nonlinear one dimensional p laplacian some sufficient conditions on the nonlinearities are given such that dimmy takes the prescribed fractional values next a relation between dimmy and the order of growth for singular behaviour of lp norm of derivatives of solutions is given finally the change and stability of dimmy are considered by means of a double parametric problem

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

on the equivalence between the multiplicative hyper elasto plasticity and the additive hypo elasto plasticity based on the modified kinetic logarithmic stress rate abstract in two theorems presented herein we prove and subsequently demonstrate in several numerical examples involving homogeneous deformation that for isotropic materials hyper elasto plasticity models based on the multiplicative decomposition of the deformation gradient coincide with an additive hypo elasto plasticity model see section 3 2 that employs the spin tensor based on the modified kinetic logarithmic rate in the absence of strain induced anisotropy characterized by kinematic hardening herein this objective stress rate coincides with the kinetic logarithmic rate recently developed by jiao and fish 2017 we also show that other well known additive decomposition models such as those based on the jaumann and logarithmic rates may considerably deviate from the multiplicative model

additive schwarz algorithm for the nonlinear complementarity problem with m function in this paper an additive schwarz algorithm is considered for solving the finite dimensional nonlinear complementarity problem with m function the monotone convergence of the algorithm is obtained with special choices of initial values moreover the weighted max norm bound is obtained for the iterative errors

a unified stress update algorithm for explicit transient shell dynamics with combined isotropic kinematic hardening in eulerian rate type phenomenological finite elasto plasticity models abstract this paper discusses applications of the objective stress rates to a unified stress update algorithm for transient shell dynamic analysis within the context of explicit time integration we propose a unified stress update algorithm via establishing the eulerian rate type constitutive equations of hypoelastic plastic materials the constitutive equations are often given on the additive decomposition of the rate of deformation tensor under the assumption that the elastic part of the rate of deformation tensor may be characterized to be hypoelastic with grade zero i e the fourth rank constant isotropic tensor several objective stress rates are derived through the lie derivative of the cauchy or kirchhoff stress tensor those stress rates cover for instance the jaumann green naghdi truesdell oldroyd and cotter rivlin stress rates further the unified stress update algorithm is implemented along with the so called radial return method and the phenomenological plasticity theory with combined isotropic kinematic hardenings among the objective stress rates employed here we emphasize especially the stress update procedure of the green naghdi stress rate for transient shell dynamics because this algorithm developed in this work is not shown even in the representative commercial codes in the implementation of the unified stress update algorithm we applied our algorithms to the belytschko lin tsay shell theory to validate its accuracy and effectiveness moreover a slightly modified secant algorithm for the zero transverse normal stress of the shell is also developed the ultimate objectivity of this work is to provide a guideline for the best choice of an appropriate objective stress rate for phenomenological elastoplastic models several numerical examples are demonstrated including non contact transient shell dynamics examples and contact impact examples such as a ball plate impact problem by which the accuracy and effectiveness of the unified stress update algorithm developed in this work are addressed

chirp transforms and chirp series abstract motivated by the recent work on the non harmonic fourier atoms initiated by t qian and the non harmonic fourier series which originated from the celebrated work of paley and wiener we introduce an integral version of the non harmonic fourier series called chirp transform as an integral transform with kernel e i t the chirp transform is an unitary isometry from l 2 r d onto l 2 r d and it can be explicitly defined in terms of generalized hermite polynomials the corresponding chirp series take e i n t as a basis which in some sense is dual to the theory of non harmonic fourier series which take e i n t as a basis the chirp version of the shannon sampling theorem and the poisson summation formula are also considered by dealing with sampling points which may non equally distributed since the chirp transform interchanges weighted derivatives into multiplications it plays a role in solving certain differential equations with variable coefficients in addition we extend t qian s theorem on the characterization of a measure to be a linear combination of a number of harmonic measures on the unit disc with positive integer coefficients to that with positive rational coefficients

fractal oscillations of self adjoint and damped linear differential equations of second order abstract for a prescribed real number s 1 2 we give some sufficient conditions on the coefficients p x and q x such that every solution y y x y c 2 0 t of the linear differential equation p x y q x y 0 on 0 t is bounded and fractal oscillatory near x 0 with the fractal dimension equal to s this means that y oscillates near x 0 and the fractal box counting dimension of the graph y of y is equal to s as well as the s dimensional upper minkowski content generalized length of y is finite and strictly positive it verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x 0 like the chirp function y ch x a x s x which often occurs in the time frequency analysis and its various applications furthermore this kind of oscillations is established for the bessel chirp and other types of damped linear differential equations given in the form y x y g x y 0 x 0 t in order to prove the main results we state a new criterion for fractal oscillations near x 0 of real continuous functions which essentially improves related one presented in 1

weighted max norm estimate of additive schwarz iteration scheme for solving linear complementarity problems in this paper we consider an algebraic additive schwarz iteration scheme for solving the finite dimensional linear complementarity problem that involves an m matrix the scheme contains some existing algorithms as special cases we establish monotone convergence of the iteration scheme under appropriate conditions moreover using the concept of weak regular splitting we estimate weighted max norm bounds for iteration errors thereby we show that the sequence generated by the iteration scheme converges to the unique solution of the problem without any restriction on the initial point

rectifiable oscillations in second order linear differential equations abstract we study the linear differential equation p y x f x y x 0 on i 0 1 where the coefficient f x is strictly positive and continuous on i and satisfies the hartman wintner condition at x 0 the four main results of the paper are i a criterion for rectifiable oscillations of p characterized by the integrability of f x 4 on i ii a stability result for rectifiable and unrectifiable oscillations of p in terms of a perturbation on f x iii the s dimensional fractal oscillations for which we assume also f x c x when x 0 2 and s max 1 3 2 2 and iv the co existence of rectifiable and unrectifiable oscillations in the absence of the hartman wintner condition on f x explicit examples related to the above results are given

a novel non linearly explicit second order accurate l stable methodology for finite deformation hypoelastic hypoelasto plastic structural dynamics problems a novel non linearly explicit second order accurate l stable computational methodology for integrating the non linear equations of motion without non linear iterations during each time step and the underlying implementation procedure is described emphasis is placed on illustrative non linear structural dynamics problems employing both total updated lagrangian formulations to handle finite deformation hypoelasticity hypoelasto plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non linear structural dynamics applications

on monotone and geometric convergence of schwarz methods for two sided obstacle problems convergence of schwarz methods is discussed for certain discretizations of two sided obstacle problems monotone and especially geometrical convergence are established if the stiffness matrix is a kind of m matrix and an h independent convergence rate is proved for uniformly overlapping decomposition

fractal oscillations for a class of second order linear differential equations of euler type abstract the s dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied the main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of euler type next it will be shown that the dimensional number s only depends on a positive real parameter appearing in a singular term of the main equation it continues some recent results on the rectifiable and unrectifiable oscillations given in pasic m pasic rectifiable and unrectifiable oscillations for a class of second order linear differential equations of euler type j math anal appl 335 2007 724 738 and wong j s w wong on rectifiable oscillation of euler type second order linear differential equations electron j qual theory differ equ 20 2007 1 12

multigrid algorithms for variational inequalities we present an iterative scheme for obstacle problems described by variational inequalities at each step of the iteration the method leads to a reduced linear resp nonlinear algebraic system whi

spectral residual methods with two new non monotone line searches for large scale nonlinear systems of equations in this study we develop two new non monotone line searches and combine them into a spectral residual method furthermore we apply these methods to complementary problems and discuss global convergence under several suitable conditions numerical experiments support the efficiency of the presented methods

rotations stress rates and strain measures in homogeneous deformation processes abstract the article draws the distinction between homogeneous and pure homogeneous deformation in the latter mode an orthogonal triad can be identified the principal axes which remain orthogonal throughout the deformation an appropriate strain measure in such processes is that of logarithmic strain furthermore its material derivative equals the rate of deformation tensor in homogeneous processes the deformation gradient tensor is unsymmetric and there is no triad which remains orthogonal throughout this often leads to a description of the actual deformation by means of the polar decomposition theorem the material derivative of the tensor logorithm is no longer simply related to the rate of deformation tensor and this is exemplified herein as a result of material rotation the stress components will vary and constitutive equations which involve stress rate must be formulated to compensate for the rotation a number of objective stress rates are examined herein and employed in a constitutive equation describing a hypoelastic solid the effect of stress rate on the evolution of stresses in the deforming solid is demonstrated for the case when the body undergoes simple rectilinear shear

the choice of objective rates in finite elastoplasticity general results on the uniqueness of the logarithmic rate a eulerian rate formulation of finite elastoplasticity is a composite one composed of a rate equation for elastic behaviour and a flow rule for plastic behaviour as well as an evolution equation for hardening behaviour in which objective eulerian tensor rates are used among a large variety of objective rates actually infinitely many how to choose suitable ones has been one of the crucial points in finite elastoplasticity it is realized that the foregoing composite formulation of elastoplasticity should fulfil certain consistency criteria in order to avoid inconsistencies or contradictions these criteria narrow the choice of objective rates these authors bruhns and co workers and xiao and co workers have recently introduced the self consistency criterion in a composite formulation of elastoplasticity the rate equation intended for characterization of elastic behaviour should be exactly integrable to deliver an elastic relation it has been demonstrated in the work of the aforementioned authors that if a composite formulation of elastoplasticity with objective rates is required to fulfil the just stated self consistency criterion as it should be then the newly discovered logarithmic rate is the only possible choice among all objective corotational rates including the zaremba jaumann rate and the green naghdi rate etc in the afore mentioned result non corotational rates including oldroyd rates cotter rivlin rate and truesdell rate etc are not taken into consideration in general it is the main goal of this paper to further establish the uniqueness of the logarithmic rate among all corotational and non corotational objective rates essential to the attainment of this goal is the use of the yielding stationarity criterion the vanishing of the stress rate implies that the yield function is stationary it is shown that the just stated criterion is necessary for a composite eulerian rate formulation of finite elastoplasticity to be consistent and means that the stress rate must be a corotational rate the main goal of this article is thus attained by combining the just stated result and the established result stated before

comparison of some stress rates abstract this paper introduces a new derivation rule by means of which two new stress rates are defined furthermore the paper compares five stress rates jaumann s truesdell s green naghdi s sowerby chu s and durban baruch s for simple shear elastic elastic perfectly plastic and elastic plastic hardening isotropic kinematic and combined material models are considered different solutions have already been published for these cases except for sowerby chu and durban baruch time derivatives using the new derivation rule the new rate form of the hyperelastic doyle ericksen formula is obtained taking advantage of the sowerby chu stress rate a new constitutive equation for hypoelastic material is given

constitutive inequalities for an isotropic elastic strain energy function based on hencky s logarithmic strain tensor hencky s strain energy function for finite isotropic elasticity is obtained by the replacement of the infinitesimal strain measure occurring in the classical strain energy function of infinitesimal

estimating the hausdorff besicovitch dimension of boundary of basin of attraction in helicopter trim helicopter trim involves solution of nonlinear force equilibrium equations as in many nonlinear dynamic systems helicopter trim problem can show chaotic behavior this chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot this study focuses on the boundary of the basin of attraction obtained for a set of control inputs we analyze the boundary by considering it at different magnification levels the magnified views reveal intricate geometries it is also found that the basin boundary exhibits the characteristic of statistical self similarity which is an essential property of fractal geometries these results led the authors to investigate the fractal dimension of the basin boundary it is found that this dimension is indeed greater than the topological dimension from all the observations it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature c 2012 elsevier inc all rights reserved

analyses of large quasistatic deformations of inelastic bodies by a new hybrid stress finite element algorithm abstract a new hybrid stress finite element algorithm suitable for analyses of large quasistatic inelastic deformations is presented the algorithm is based upon a generalization of de veubeke s complementary energy principle the principal variables in the formulation are the nominal stress rate and spin and the resulting finite element equations are discrete versions of the equations of compatibility and angular momentum balance the algorithm produces true rates time derivatives as opposed to increments there results a complete separation of the boundary value problem for stress rate and velocity and the initial value problem for total stress and deformation hence their numerical treatments are essentially independent after a fairly comprehensive discussion of the numerical treatment of the boundary value problem we launch into a detailed examination of the numerical treatment of the initial value problem covering the topics of efficiency stability and objectivity the paper is closed with a set of examples finite homogeneous deformation problems which serve to bring out important aspects of the algorithm

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic