the low energy scattering for slowly decreasing potentials for the radial schrodinger equation with a potentialq x decreasing at infinity asq0q 0 2 the low energy asymptotics of spectral and scattering data is found in particular it is shown that forq0 0 the spectral function vanishes exponentially as the energyk2 tends to zero on the contrary there is always a zero energy resonance forq0 0 t in the case 1 2 it is shown that for q0 0 the phase shift tends to ask 0 and its asymptotics is evaluated

strain and stress based continuum damage models ii computational aspects abstract in part i of this work we developed continuum isotropic and anisotropic elastoplastic damage models formulated either in strain space on the basis of the effective stress concept or in stress space and employing the dual notion of effective strain in part ii we consider in detail the variational formulation and subsequent numerical implementation of these models the former relies crucially on the notions of maximum plastic and maximum damage dissipation the latter makes systematic use of the operator splitting methodology to derive unconditionally stable algorithms for the numerical integration of the elastoplastic damage equations of evolution appropriate extensions to treat the proposed rate dependent viscous damage are also presented for stress based damage models our numerical treatment relies on a new three step operator split the algorithms developed lead to simple and efficient stress update procedures suitable for large scale finite element calculations application is made to a class of inviscid and rate dependent cap models with an isotropic strain based damage mechanism remarkably good agreement with existing experimental data that includes complicated stress paths is obtained numerical examples are also presented that demonstrate the good performance of the proposed algorithms

numerical analysis of an elasto piezoelectric problem with damage in this paper we analyze an algorithm for the quasistatic evolution of the mechanical state of an elasto piezoelectric body with damage both damage caused by the development and the growth of internal microcracks and piezoelectric effects are included in the model the mechanical problem is expressed as an elliptic system for the displacement field coupled with a non linear parabolic partial differential equation for the damage field and a linear partial differential equation for the electric potential the variational formulation leads to a coupled system composed of two linear variational equations for the displacement field and the electric potential and a non linear parabolic variational equation for the damage field the existence of a unique weak solution is stated then a fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an euler scheme to discretize the time derivatives error estimates are derived on the approximate solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions finally some numerical simulations are performed in one two and three dimensions to demonstrate the accuracy of the scheme and the behaviour of the solution copyright 2008 john wiley sons ltd

analysis of two frictional viscoplastic contact problems with damage in this work we study two quasistatic frictional contact problems arising in viscoplasticity including the mechanical damage of the material caused by excessive stress or strain and modelled by an inclusion of parabolic type the variational formulation is provided for both problems and the existence of a unique solution is proved for each of them then a fully discrete scheme is introduced using the finite element method to approximate the spatial domain and the euler scheme to discretize the time derivatives error estimates are derived and under suitable regularity assumptions the linear convergence of the algorithm is deduced finally some numerical examples are presented to show the performance of the method

elastoplastic damage modelling including the gradient of damage formulation and computational aspects a framework for continuum elastoplastic damage modelling which employs irreversible thermodynamics and internal state variables is investigated the damage part of the modelling involves the gradient of damage quantity which together with the equations of motion are issued from a new formulation of the principle of virtual power recently proposed it is shown how the plastic part of the modelling takes its standard format next we consider in detail the variational formulation and subsequent numerical implementation of the elastoplastic damage models the development of an algorithm consistent with the present formulation is given where for the plastic part it leaves the standard return mapping algorithms unchanged application is made to different classical rate independent and rate dependent plastic models with the presence of a damage mechanism the numerical implementation in the context of the finite element method is discussed in detail and a set of representative applications and numerical examples is given

micromorphic approach for finite gradient elastoplasticity fully coupled with ductile damage formulation and computational aspects abstract it is well established that the use of inelastic constitutive equations accounting for induced softening leads to pathological space mesh and time discretization dependency of the numerical solution of the associated initial and boundary value problem ibvp to avoid this drawback many less or more approximate solutions have been proposed in the literature in order to regularize the ibvp and to obtain numerical solutions which are at convergence much less sensitive to the space and the time discretization the basic idea behind these regularization techniques is the formulation of nonlocal constitutive equations by introducing some effects of characteristic lengths representing the materials microstructure in this work using the framework of generalized nonlocal continua a thermodynamically consistent micromorphic formulation using appropriate micromorphic state variables and their first gradients is proposed in order to extend the classical local constitutive equations by incorporating appropriate characteristic internal lengths the isotropic damage the isotropic and the kinematic hardenings are supposed to carry the targeted micromorphic effects first the theoretical aspects of this fully coupled micromorphic formulation is presented in details and the proposed generalized balance equations as well as the fully coupled micromorphic constitutive equations deduced the associated numerical aspects in the framework of the classical galerkin based fe formulation are briefly discussed in the special case of micromorphic damage specifically the formulation of 2d finite elements with additional degrees of freedom d o f the dynamic explicit global resolution scheme as well as the local integration scheme to compute the stress tensor and the state variables at each integration point of each element are presented application is made to the typical uniaxial tension specimen under plane strain conditions in order to chow the predictive capabilities of the proposed micromorphic model particularly against its ability to give at convergence a mesh independent solution even for high values of the ductile damage i e the macroscopic cracks

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

an elastic viscoplastic quasistatic contact problem existence and uniqueness of a weak solution in this paper the contact between an elastic viscoplastic body and a deformable obstacle is studied the effect of the damage due to internal tension or compression and caused by the opening and growth of micro cracks and micro cavities is also considered the variational formulation leads to a coupled system of evolutionary equations an existence and uniqueness result is established by using approximate problems the pseudomonotone operators theory schauder fixed point theorem and a comparison result

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

algorithms for gradient damage models based on a semi smooth newton method in this paper two different algorithms are applied to a model of brittle damage including the gradient of the damage variable both algorithms are based on a modified newton method in the first algorithm the newton method is applied directly to the whole system of equations while in the second algorithm the equilibrium equations and the damage evolution problem are solved uncoupled from each other in a gauss seidel scheme the algorithms are applied to two different formulations of the problem in the first formulation the thermodynamic force work conjugate to the damage variable is kept as an auxiliary variable while in the second formulation this variable is eliminated it turns out that the formulation using the extra variable is more robust in the sense that the algorithms converge even for very large load increments on the other hand the formulation where the extra variable is eliminated is more efficient for small time steps since a smaller system of equations is solved for each increment furthermore it is found that the direct approach is more efficient than the decoupled approach for this problem

phase field modeling of ductile fracture at finite strains a robust variational based numerical implementation of a gradient extended theory by micromorphic regularization summary this work provides a robust variational based numerical implementation of a phase field model of ductile fracture in elastic plastic solids undergoing large strains this covers a computationally efficient micromorphic regularization of the coupled gradient plasticity damage formulation the phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics it has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture specific computational structures such as finite element design of crack discontinuities or intricate crack tracking algorithms the proposed gradient extended plasticity damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities this ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post critical ranges the proposed setting is rooted in a canonical variational principle the coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture the latter represents a coupled resistance to plasticity and damage depending on the gradient extended internal variables that enter plastic yield functions and fracture threshold functions with this viewpoint on the generalized internal variables at hand the thermodynamic formulation is outlined for gradient extended dissipative solids with generalized internal variables that are passive in nature it is specified for a conceptual model of von mises type elasto plasticity at finite strains coupled with fracture the canonical theory proposed is shown to be governed by a rate type minimization principle which fully determines the coupled multi field evolution problem this is exploited on the numerical side by a fully symmetric monolithic finite element implementation an important aspect of this work is the regularization towards a micromorphic gradient plasticity damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms this enhances the robustness of the finite element implementation in particular on the side of gradient plasticity the performance of the formulation is demonstrated by means of some representative examples copyright 2016 john wiley sons ltd

asymptotically near optimal blind estimation of multipath cdma channels in this paper correlation matching techniques are applied to estimate multipath code division multiple access cdma channels we arrange unknown multipath parameters for each of j active users in a vector then the output correlation matrix is parameterized by j unknown rank one matrices with each one formulated from the corresponding channel vector this correlation matrix is further compared with its sample average the resulting error can be first minimized to obtain unbiased estimates of j unknown rank one matrices in closed forms thus our estimator for each channel vector is derived by singular value decomposition svd on the associated rank one matrix within a scalar ambiguity it turns out that the performance of our estimator can be improved by introducing an asymptotically optimal weighting matrix in our cost function this weighting matrix can be estimated directly from data samples only with a small penalty on the asymptotic performance the asymptotic covariance of our estimator is also derived and can be compared with the cramer rao lower bound both in closed forms simulation results show the applicability of the proposed methods and consistency with our theoretical analysis

a quasistatic contact problem for viscoelastic materials with friction and damage abstract we consider a mathematical model which describes the frictional contact between a deformable body and a foundation the process is quasistatic the material is assumed to be viscoelastic with long memory and the frictional contact is modelled with subdifferential boundary conditions the mechanical damage of the material is described by the damage function which is modelled by a nonlinear partial differential equation we derive the variational formulation of the problem which is a coupled system of a hemivariational inequality and a parabolic equation then we prove the existence of a unique weak solution to the model the proof is based on arguments of abstract stationary inclusion and a fixed point theorem

threshold properties of matrix valued schrodinger operators ii resonances we present some results on the perturbation of eigenvalues embedded at a threshold for a matrix valued hamiltonian with three dimensional dilation analytic schrodinger operators as entries and with a small off diagonal perturbation the main result describes how a threshold eigenvalue generates resonances that is poles of the meromorphic continuation of the perturbed hamiltonian

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

numerical analysis and simulations of a dynamic frictionless contact problem with damage the process of dynamic frictionless contact between a viscoelastic body and a reactive foundation which includes material damage is modelled numerically analyzed and simulated contact is modelled with the normal compliance condition the damage of the material resulting from tension or compression is described by the damage field which measures the pointwise fractional decrease in the load carrying capacity of the material the evolution of the damage is governed by a parabolic differential inclusion a fully discrete scheme for the problem is proposed and error estimates on the numerical solutions derived the scheme has been implemented and numerical simulations of the evolution of the mechanical state of the system and its material damage are presented

a direct finite element implementation of the gradient dependent theory the enhanced non local gradient dependent theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical theories and the classical local continuum they are successful in explaining the size effects encountered at the micron scale and in preserving the well posedeness of the i bvp governing the solution of material instability triggering strain localization this is due to the incorporation of an intrinsic material length scale parameter in the constitutive description however the numerical implementation of these theories is not a direct task because of the higher order of the governing equations in this paper a direct computational algorithm for the gradient approach is proposed this algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches int j solids struct 1988 24 581 597 int j numer meth engng 1992 35 521 539 eng comput 1993 10 99 121 dissertation 1994 int j numer meth engng 1996 39 2477 2505 int j numer meth engng 1996 39 3731 3755 comput meth appl mech eng 1998 163 11 32 comput meth appl mech eng 1998 163 33 53 euro j mech a solids 1999 18 939 962 int j solids struct 2000 37 7481 7499 a predictor corrector scheme is proposed for the solution of the non linear algebraic problem from the fem the expressions of the continuum and consistent tangent matrices are provided the method is validated by conducting various numerical tests as a result pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered copyright 2005 john wiley sons ltd

perturbation of eigenvalues embedded at a threshold results are obtained on perturbation of eigenvalues and half bound states zero resonances embedded at a threshold the results are obtained in a two channel framework for small off diagonal perturbations the results are based on given asymptotic expansions of the component hamiltonians

a geometrically regularized gradient damage model with energetic equivalence abstract aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids this paper addresses a novel geometrically regularized gradient damage model with energetic equivalence for cracking evolution with the free energy potential defined as usual in terms of the local strain and damage fields the constitutive relations are derived consistently from the standard framework of thermodynamics upon the sharp crack topology geometrically regularized by the damage localization band with a length scale the ensuing energetic equivalence naturally yields the damage evolution law of gradient type and the associated boundary condition of neumann type compared to other gradient damage models no extra assumptions like the nonlocal energy residual and insulation condition are introduced moreover the damage gradient physically accounting for microscopic nonlocal interactions is fully dissipative as expected in line with the unified phase field theory recently proposed by the author wu 2017 during failure processes the material behavior is uniquely characterized by two constitutive functions i e the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology in particular optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated with the involved parameters calibrated from standard material properties the proposed model is numerically implemented into the multi field finite element method and applied to several benchmark tests of concrete under mode i and mixed mode failure it is found that the incorporated length scale can be regarded either a numerical parameter or a material property for the former considered in this work the length scale has negligible if not no effects on the global responses so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved furthermore the localization bandwidth does not exhibit spurious widening but rather it approaches to a finite value proportional to the length scale comparison between the numerical and experimental results regarding the curve of load versus displacement and crack path confirms validity of the proposed gradient damage model for characterizing localized failure in quasi brittle solids

on bound states for systems of weakly coupled schrodinger equations in one space dimension we establish the birman schwinger relation for a class of schrodinger operators d2 dx2 1h v on l2 r h where h is an auxiliary hilbert space and v is an operator valued potential as an application we give an asymptotic formula for the bound states which may arise for a weakly coupled schrodinger operator with a matrix potential having one or more thresholds in addition for a two channel system with eigenvalues embedded in the continuous spectrum we show that under a small perturbation such eigenvalues turn into resonances

theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage this paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background the main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results we propose a formulation in terms of the helmholtz free energy incorporating the gradient of the damage field a dissipation potential and the postulate of maximum dissipation as a result the driving force conjugated to damage evolution is identified as the quasi nonlocal energy release rate which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate on the numerical side besides balance of linear momentum the algorithmic consistency condition must be solved in weak form thereby the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming in the examples we compare the behavior in local damage with the performance of the gradient formulation

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

threshold properties of matrix valued schrodinger operators we present some results on the perturbation of eigenvalues embedded at a threshold for a two channel hamiltonian with three dimensional schrodinger operators as entries and with a small off diagonal perturbation in particular we show how the threshold eigenvalue gives rise to discrete eigenvalues below the threshold and moreover we establish a criterion on existence of half bound states associated with embedded pseudo eigenvalues

spectral properties at a threshold for two channel hamiltonians i abstract theory spectral properties at thresholds are investigated for two channel hamiltonians in various mostly fairly singular settings in an abstract framework we deduce asymptotic expansions of the resolvent as the spectral parameter tends to a threshold the results are based on given asymptotic expansions of the component hamiltonians applications to scattering theory are given in a companion paper

spectral properties at a threshold for two channel hamiltonians ii applications to scattering theory spectral properties and scattering theory in the low energy limit are investigated for two channel hamiltonians with schrodinger operators as component hamiltonians in various mostly fairly singular settings asymptotic expansions of the resolvent are deduced as the spectral parameter tends to the threshold zero furthermore scattering theory for pairs of two channel hamiltonians is established as an application of the expansions of the resolvent asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the threshold zero

effects of imperfect blind channel estimation on performance of linear cdma receivers in a code division multiple access cdma system signal detection under multipath distortion typically requires estimation of unknown channel parameters first in such a scenario performance of receivers highly relies on the accuracy of channel estimates in this paper effects of channel estimation errors on the performance of linear cdma receivers due to finite data samples are studied when channel parameters are estimated blindly by a recently proposed covariance matching technique those receivers include zero forcing zf direct matrix inversion dmi minimum mean square error mmse subspace mmse and rake receivers their output signal to interference plus noise ratios sinrs and bit error rates bers are adopted for performance measures expressions for performance indicators under such an imperfect condition are derived from a perturbation perspective and verified by simulation examples

damageable contact between an elastic body and a rigid foundation in this work the contact problem between an elastic body and a rigid obstacle is studied including the development of material damage which results from internal compression or tension the variational problem is formulated as a first kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field the existence of a unique local weak solution is stated then a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an euler scheme to discretize the time derivatives error estimates are derived on the approximate solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions finally three two dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme

an energetic formulation of a gradient damage model for concrete and its numerical implementation abstract the energetic formulation of a rate independent system assumes that the evolution of the system is driven by two scalar valued functions the storage energy functional and the dissipation pseudo potential the evolution of the material system is then characterized by two energetic principles the global stability condition and the energy balance between stored and dissipated energies with the work of external loading rate independent generalized standard materials are endowed with such structure whereas the existence of such energetic structure is not so apparent in materials with non associated flow rule in this paper we consider a rate independent gradient damage model for concrete where the evolution of the damage does not follow the normality rule we will show that such model can be nevertheless derived by a dissipation potential at the expense of having a state variable dependent potential and therefore the energetic formulation can be obtained also in this case after introducing the incremental minimization problem consistent with such formulation we obtain a discrete version of the stability condition and establish lower and upper a priori energy bounds met by the energetic solution these are fundamental results for the analysis of the formulation the actual numerical solution of the incremental minimization problem is realized by considering first a variable splitting in order to treat the gradient of the damage field as an independent variable and then by applying the augmented lagrangian method to tackle with the resulting constrained optimization problem we solve the first order stationarity conditions of the augmented lagrangian functional by a path following newton s method based on the energy dissipation rate control we show that we are able to describe highly non linear responses of the material such as softening branches and snap back responses several numerical tests are performed to verify the objectivity of the formulation and of the proposed numerical method details of the numerical implementation are also given