topology optimization for effective energy propagation in rate independent elastoplastic material systems abstract a topology optimization framework for effective energy management in dynamically loaded structures with rate independent elastoplastic material behavior is presented the elastoplastic constitutive equations are integrated using an implicit numerical scheme based on a multi level newton procedure this primal analysis procedure requires the computation of the incremental algorithmic consistent tangent operator derived from the mappings that enforce the evolving constitutive relations for our gradient based optimization we develop discrete adjoint sensitivity expressions finite element implementation is discussed and the optimization framework is exemplified via the design of three dimensional structures subjected to dynamic loads

design sensitivity analysis for rate independent elastoplasticity abstract a new incremental direct differentiation method for design sensitivity analysis of structures with rate independent elastoplastic behavior is presented we formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions the sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation combined with the inherent advantages of implicit integration strategies this represents a significant improvement over previous sensitivity formulations for history dependent materials first order sensitivity expressions involving the complete set of design variables including shape design variables are derived for a generic response functional the reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step no iterations are needed in the sensitivity computations a numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem explicit sensitivities from the new method are confirmed by finite difference estimates

conditional stability theorem for the one dimensional klein gordon equation the paper addresses the conditional non linear stability of the steady state solutions of the one dimensional klein gordon equation for large time we explicitly construct the center stable manifold for the steady state solutions using the modulation method of soffer and weinstein and strichartz type estimates the main difficulty in the one dimensional case is that the required decay of the klein gordon semigroup does not follow from strichartz estimates alone we resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces which allows us to close the argument in spaces with very little time decay

decay for the wave and schrodinger evolutions on manifolds with conical ends part ii let c n be a compact imbedded riemannian manifold of dimension d 1 and define the d 1 dimensional riemannian manifold m x r x w x with r 0 and smooth and the natural metric ds 2 1 r x 2 dx 2 r 2 x ds 2 we require that m has conical ends r x x o x 1 as x the hamiltonian flow on such manifolds always exhibits trapping dispersive estimates for the schrodinger evolution e it m and the wave evolution e it m are obtained for data of the form f x y n u x where y n are eigenfunctions of with eigenvalues u 2 n in this paper we discuss all cases d n 1 if n 0 there is the following accelerated local decay estimate with 0 1 e it m y n f l m c n m t d 1 2 w 1 f l 1 m where w x x and similarly for the wave evolution our method combines two main ingredients a a detailed scattering analysis of schrodinger operators of the form 2 v 2 1 4 2 u on the line where u is real valued and smooth with u l 0 3 l for all l 0 as and v 0 in particular we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the wronskian between the outgoing jost solutions as the energy tends to zero in particular the division into part i and part ii can be explained by the former being resonant at zero energy where the present paper deals with the nonresonant case b estimation of oscillatory integrals by non stationary phase

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

on some schrodinger and wave equations with time dependent potentials the existence and uniqueness of solutions in the initial value problem for schrodinger and wave equations in the presence of a large time dependent potential is studied the usual strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials as a byproduct one obtains a counterexample to the two dimensional double endpoint inhomogeneous strichartz estimate

dispersive estimates for schrodinger operators in dimension two we prove l1 2 l 2 for the two dimensional schrodinger operator v with the decay rate t 1 we assume that zero energy is neither an eigenvalue nor a resonance this condition is formulated as in the recent paper by jensen and nenciu on threshold expansions for the two dimensional resolvent

dynamic response sensitivity of inelastic structures abstract a general finite element solution method for the dynamic response sensitivity of inelastic structures is developed employing a direct differentiation method the gradient equation of motion is solved without iteration and by taking advantage of the available solution of the response special attention is given to sensitivities with respect to inelastic material parameters and detailed derivations are made for the j 2 plasticity model with a linear hardening rule the method can be applied to any other inelastic material model that has an analytically defined yield function and flow rule the formulation is easily incorporated in existing finite element codes numerical examples demonstrate the accuracy and efficiency of the method

lp continuity of wave operators in z abstract we recover for discrete schrodinger operators on the lattice z stronger analogues of the results by weder r weder the w k p continuity of the schrodinger wave operators on the line comm math phys 208 1999 507 520 and by d ancona and fanelli p d ancona l fanelli l p boundedness of the wave operator for the one dimensional schrodinger operator comm math phys 268 2006 415 438 on r

lp boundedness of the wave operator for the one dimensional schrodinger operator given a one dimensional perturbed schrodinger operator h d2 dx2 v x we consider the associated wave operators w defined as the strong l2 limits lim s to pm infty e ish e ish 0 we prove that w are bounded operators on lp for all 1 p provided 1 x 2 v x in l 1 or else 1 x v x in l 1 and 0 is not a resonance for p we obtain an estimate in terms of the hilbert transform some applications to dispersive estimates for equations with variable rough coefficients are given

a weighted dispersive estimate for schrodinger operators in dimension two let h v where v is a real valued potential on mathbb r 2 satisfying v x lesssim langle x rangle 3 we prove that if zero is a regular point of the spectrum of h v then w 1 e ith p ac f l infty mathbb r 2 lesssim frac 1 t log 2 t w f l 1 mathbb r 2 t geq 2 with w x log 2 x 2 this decay rate was obtained by murata in the setting of weighted l 2 spaces with polynomially growing weights

first and second order sensitivity analysis of linear and nonlinear structures this paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches the computations required are based on additional load conditions characterized by imposed initial strains body forces or surface tractions as such they are equally applicable to numerical or analytical solution techniques the relative efficiency of various approaches for calculating first and second derivatives is assessed it is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first order sensitivities and adjoint vectors two example problems are used for demonstrating the various approaches

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

a newton schur alternative to the consistent tangent approach in computational plasticity we introduce a newton schur alternative to the consistent tangent approach in plasticity computations and show that the structure of the consistent tangent approach is preserved in the new newton schur approach however the local constitutive iterations used in the conventional computational plasticity analyses are eliminated when using the newton schur approach two numerical examples are presented which illustrate the efficiencies gained by the new approach

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

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

l p boundedness of wave operators for two dimensional schrodinger operators let be the two dimensional schrodinger operator with the real valued potential v which satisfies the decay condition at infinity for we show that the wave operators are bounded in for any 1 p under the condition that h has no zero bound states or zero resonance extending the corresponding results for higher dimensions as w intertwine h 0 and the absolutely continuous part h p ac of h f h p ac w f h 0 w for any borel function f on 1 this reduces the various l p mapping properties of f h p ac to those of f h 0 the convolution operator by the fourier transform of the function f 2

time decay estimates for the wave equation with potential in dimension two abstract we study the wave equation with potential u t t u v u 0 in two spatial dimensions with v a real valued decaying potential with h v we study a variety of mapping estimates of the solution operators cos t h and sin t h h under the assumption that zero is a regular point of the spectrum of h we prove a dispersive estimate with a time decay rate of t 1 2 a polynomially weighted dispersive estimate which attains a faster decay rate of t 1 log t 2 for t 2 finally we prove dispersive estimates if zero is not a regular point of the spectrum of h

dispersive and strichartz estimates for the three dimensional wave equation with a scaling critical class of potentials we consider the three dimensional linear wave equation perturbed by a potential v belonging to a scaling critical class which is the closure of bounded compactly supported functions with respect to the global kato norm without any further smallness assumptions we first prove the dispersive estimate for the solution of the wave equation as a byproduct we prove strichartz estimates under a slightly stronger condition

some remarks on the schrodinger equation with a potential in lrtlsx we study the dispersive properties of the linear schrodinger equation with a time dependent potential v t x we show that an appropriate integrability condition in space and time on v i e the boundedness of a suitable lrtlsx norm is sufficient to prove the full set of strichartz estimates we also construct several counterexamples which show that our assumptions are optimal both for local and for global strichartz estimates in the class of large unsigned potentials v lrtlsx

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

topology and shape optimization for elastoplastic structural response abstract it is a common practice to base both material topology optimization as well as a subsequent shape optimization on linear elastic response however in order to obtain a realistic design it might be essential to base the optimization on a more realistic physical behavior i e to consider geometrically or and materially nonlinear effects in the present paper an elastoplastic von mises material model with linear isotropic hardening softening for small strains is used the objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed with respect to the specific features of either topology or shape optimization for example the number of optimization variables or their local global influence on the structural response different methods are applied for topology optimization problems the gradient of the ductility is determined by the variational adjoint approach in shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach the topology optimization problem is solved by an optimality criteria oc method and the shape optimization problem by a mathematical programming mp method in topology optimization a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities the procedures are verified by 2d design problems under plane stress conditions

nonlinear structural design sensivitity analysis for path dependent problems part 1 general theory abstract two procedures direct differentiation and adjoint structure for design sensitivity analysis of nonlinear structural systems with path dependent effects are described in this paper continuum formulation is used and both geometric and material nonlinearities are included in the derivations incremental nonlinear analysis based on the total lagrangian concept is adopted the reference volume concept is used to unify the shape and nonshape design problems to show the generality of the theory the derived expressions are reduced to some special cases that have been treated previously in the literature in part 2 of the paper a few analytical example problems will be solved to demonstrate the use of the design sensitivity analysis theory the theory has also beendiscretized elsewhere using finite element approximations for implementation into structural analysis programs these developments offer a firm foundation for design optimization of complex structural systems

perturbation expansion and nth order fermi golden rule of the nonlinear schrodinger equations in this paper we consider generalized nonlinear schrodinger equations with external potentials we find the expressions for the fourth and the sixth order fermi golden rules fgrs conjectured in gang and sigal rev math phys 17 1143 1207 2005 geom funct anal 16 no 7 1377 1390 2006 the fgr is a key condition in a study of the asymptotic dynamics of trapped solitons

continuum design sensitivity of transient responses using ritz and mode acceleration methods in this paper a unified continuum based sizing design sensitivity analysis method is developed for the transient dynamic response of built up structures using the direct differentiation method the variational equation of the built up structure is differentiate d with respect to the design variable to obtain a variational design sensitivity equation of the transient response the same finite element analysis model that is used to obtain an approximate solution of the variational equation of the built up structure is also used to obtain an approximate solution of the variational design sensitivity equation for large size built up structures the same superposition method is used both to reduce the dimension of the matrix equation of the built up structure and also to reduce the dimension of the design sensitivity equation the continuum based design sensitivity analysis method can be implemented outside established finite element analysis codes using postprocessing data since it does not require derivatives of the stiffness damping and mass matrices moreover the method is efficient since it does not require derivatives of basis vectors either the ritz or mode acceleration method is used to further improve accuracy and efficiency of both analysis and sensitivity results an example with damping shows that accurate design sensitivity of pointwise transient displacements is obtained using only 10 of basis vectors this example indicates that in the continuum based design sensitivity analysis method the same basis vectors that are used for analysis of the built up structure are suitable for analysis of the design sensitivity equation

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

endpoint strichartz estimates for the magnetic schrodinger equation abstract we prove strichartz estimates for the schrodinger equation with an electromagnetic potential in dimension n 3 the decay and regularity assumptions on the potentials are almost critical i e close to the coulomb case in addition we require repulsivity and a nontrapping condition which are expressed as smallness of suitable components of the potentials while the potentials themselves can be large the proof is based on smoothing estimates and new sobolev embeddings for spaces associated to magnetic potentials

shape and non shape structural sensitivity analysis for problems with any material and kinematic non linearity abstract the general problem of sizing material selection load parameter and shape sensitivity of non linear structural response is considered the non linear sensitivity path is followed by an incremental strategy employing the continuum based updated lagrangian description the resulting finite element equations are derived both the direct differentiation and adjoint structure strategies are considered the merits of using the consistent tangent matrix for sensitivity calculations are indicated the formulation employs the so called reference volume concept and may be seen as a further attempt to unify the shape and non shape sensitivity analysis procedures

time decay of scaling critical electromagnetic schrodinger flows we obtain a representation formula for solutions to schrodinger equations with a class of homogeneous scaling critical electromagnetic potentials as a consequence we prove the sharp l 1 to l infty time decay estimate for the 3d inverse square and the 2d aharonov bohm potentials

a counterexample to dispersive estimates for schrodinger operators in higher dimensions in dimension n 3 we show the existence of a compactly supported potential in the differentiability class c alpha alpha frac n 3 2 for which the solutions to the linear schrodinger equation in mathbb r n i partial t u delta u vu quad u 0 f fail to satisfy an evolution estimate of the form u t cdot infty leq c t frac n 2 u 0 cdot 1 this contrasts with known results in dimensions n 3 where a pointwise decay condition on v is generally sufficient to imply dispersive bounds

nonlinear structural design using multiscale topology optimization part i static formulation we present a hierarchical multiscale design framework that couples computational homogenization with topology optimization to design a composite structure s microstructure to optimize its nonlinear elastostatic behavior to generate a well posed macroscopic topology optimization problem we use relaxation which requires homogenization to relate the macroscopic homogenized response to its microstructure and because closed form expressions for homogenized properties generally do not exist for materials with nonlinear response we rely on computational homogenization to evaluate them to optimize the homogenized properties of the unit cell we again use topology optimization and to make this unit cell optimization problem well posed we use restriction and thereby obtain a minimum microstructural length scale the coupled nonlinear analyzes and optimization problems are computationally intensive tasks that we resolve with a scalable parallel framework based on a single program multiple data programming paradigm numerical implementation is discussed and examples are provided

dispersive estimates for higher dimensional schrodinger operators with threshold eigenvalues i the odd dimensional case we investigate l1 rn l rn dispersive estimates for the schrodinger operator h v when there is an eigenvalue at zero energy and n 5 is odd in particular we show that if there is an eigenvalue at zero energy then there is a time dependent rank one operator ft satisfying ft l1 l t 2 n2 for t 1 such that eithpac ft l1 l t 1 n2 for t 1 with stronger decay conditions on the potential it is possible to generate an operator valued expansion for the evolution taking the form eithpac h t 2 n2a 2 t 1 n2a 1 t n2a0 with a 2 and a 1 finite rank operators mapping l1 rn to l rn while a0 maps weighted l1 spaces to weighted l spaces the leading order terms a 2 and a 1 vanish when certain orthogonality conditions between the potential v and the zero energy eigenfunctions are satisfied we show that under the same orthogonality conditions the remaining t n2a0 term also exists as a map from l1 rn to l rn hence eithpac h satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero

dispersive estimates for schrodinger equations with threshold resonance and eigenvalue let h v x be a three dimensional schrodinger operator we study the time decay in lp spaces of scattering solutions e ithpcu where pc is the orthogonal projection onto the continuous spectral subspace of l2 r3 for h under suitable decay assumptions on v x it is shown that they satisfy the so called lp lq estimates e ithpcu p 4 t 3 1 2 1 p u q for all 1 q 2 p with 1 p 1 q 1 if h has no threshold resonance and eigenvalue and for all 3 2 q 2 p 3 if otherwise

on the lp boundedness of wave operators for two dimensional schrodinger operators with threshold obstructions abstract let h v be a schrodinger operator on l 2 r 2 with real valued potential v and let h 0 if v has sufficient pointwise decay the wave operators w s lim t e i t h e i t h 0 are known to be bounded on l p r 2 for all 1 p if zero is not an eigenvalue or resonance we show that if there is an s wave resonance or an eigenvalue only at zero then the wave operators are bounded on l p r 2 for 1 p this result stands in contrast to results in higher dimensions where the presence of zero energy obstructions is known to shrink the range of valid exponents p

asymptotic stability of ground states in 2d nonlinear schrodinger equation including subcritical cases abstract we consider a class of nonlinear schrodinger equations in two space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l 2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a carefully chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

sensitivity analysis for non linear constrained elastostatic systems adjoint and direct differentiation methods are used to formulate design sensitivities for non linear constrained elastostatic systems variations of a general response functional are expressed in explicit form with respect to all design field variations including shape the response functional is expressed over the undeformed configuration however the displacement gradient is included so that field quantities that are defined over the deformed configuration such as cauchy stress may be modelled the finite element method is used to evaluate the sensitivities in an example problem

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

nonlinear structural design sensitivity analysis for path dependent problems part 2 analytical examples abstract in part 1 of this paper a general theory of design sensitivity analysis for nonlinear structures with history dependent effects is given the theory is also reduced to several special cases to show its generality in the present paper three examples are used to analytically verify the theory both the adjoint structure method asm and the direct differentiation method ddm are demonstrated the examples are 1 reaction force sensitivity for an indeterminate beam 2 displacement sensitivity for several cases of an axial force element and 3 stress and displacement sensitivities for a visco elastic problem these examples provide insights for numerical implementation of the theory for more complex design problems

weighted energy decay for 3d klein gordon equation abstract we obtain a dispersive long time decay in weighted energy norms for solutions of the 3d klein gordon equation with generic potential the decay extends the results obtained by jensen and kato for the 3d schrodinger equation for the proof we modify the spectral approach of jensen and kato to make it applicable to relativistic equations

topology optimization of energy absorbing structures with maximum damage constraint summary a novel density based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed this framework enables topologies to absorb large amount of energy via plastic work before failure occurs to account for the plasticity and damage during the energy absorption a coupled elastoplastic ductile damage model is incorporated with topology optimization appropriate material interpolation schemes are proposed to relax the damage in the low density regions while still ensuring the convergence of newton raphson solution process in the nonlinear finite element analyses an effective method for obtaining path dependent sensitivities of the plastic work and maximum damage via adjoint method is presented and the sensitivities are verified by the central difference method the effectiveness of the proposed methodology is demonstrated through a series of numerical examples copyright 2017 john wiley sons ltd

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

resolvent expansion and time decay of the wave functions for two dimensional magnetic schrodinger operators we consider two dimensional schrodinger operators h b v given by eq 1 1 below we prove that under certain regularity and decay assumptions on b and v the character of the expansion for the resolvent h b v 1 as 0 is determined by the flux of the magnetic field b through mathbb r 2 subsequently we derive the leading term of the asymptotic expansion of the unitary group e ith b v as t and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ith 0 v

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

tangent operators and design sensitivity formulations for transient non linear coupled problems with applications to elastoplasticity tangent operators and design sensitivities are derived for transient non linear coupled problems the solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods the derived formulations are suitable for finite element implementations analyses of systems with materials that exhibit history dependent response may be obtained directly by applying the analyses of transient non linear coupled systems rate independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

an adjoint variable method for sensitivity analysis of non linear elastic systems summary an adjoint variable method for design sensitivity analysis of non linear elastic systems is presented the method uses domain parameterization and a mutual form of the hu washizu energy principle and extends results reported in a recent work for linear elastic systems to non linear elasticity non linearities due to finite deformations and non linear hyperelastic constitutive models are considered in contrast to other methods for non linear sensitivity analysis the present formulation can be applied with force displacement or mixed approximate solution methods the mutual energy expression used in the adjoint sensitivity derivation is developed from a non linear extension of the hu washizu energy functional and yields a linear governing equation for the adjoint system this has important ramifications for the computational cost of a sensitivity analyses of non linear systems excluding the cost of determining the response of the system the cost of a sensitivity analysis for a non linear system is essentially the same as that for a linear system finite element implementation of the resulting sensitivity expressions is discussed and two numerical examples are presented the first example involves large deformations of a mooney rivlin body while the second involves design sensitivity analysis for mixed solution methods

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

weighted dispersive estimates for two dimensional schrodinger operators with aharonov bohm magnetic field abstract we consider two dimensional schrodinger operators h with an aharonov bohm magnetic field and an additional electric potential we obtain an explicit leading term of the asymptotic expansion of the unitary group e i t h for t in weighted l 2 spaces in particular we show that the magnetic field improves the decay of e i t h with respect to the unitary group of non magnetic schrodinger operators and that the decay rate in time is determined by the magnetic flux

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds

on the spectral theory and dispersive estimates for a discrete schrodinger equation in one dimension based on the recent work komech et al dispersive estimates for 1d discrete schrodinger and klein gordon equations appl anal 85 1487 2006 for compact potentials we develop the spectral theory for the one dimensional discrete schrodinger operator h v n 1 n 1 2 n vn n we show that under appropriate decay conditions on the general potential and a nonresonance condition at the spectral edges the spectrum of h consists of finitely many eigenvalues of finite multiplicities and the essential absolutely continuous spectrum while the resolvent satisfies the limiting absorption principle and the puiseux expansions near the edges these properties imply the dispersive estimates eithpa c h l 2 l 2 t 3 2 for any fixed 52 and any t 0 where pa c h denotes the spectral projection to the absolutely continuous spectrum of h in addition based on the scattering theory for the discrete jost solutions and the previous results by stefanov and kevrekidis asymptotic behaviour of small

optimal design of multiphase composites under elastodynamic loading abstract an algorithm is proposed to optimize the performance of multiphase structures composites under elastodynamic loading conditions the goal is to determine the distribution of material in the structure such that the time averaged total stored energy of structure is minimized a penalization strategy is suggested to avoid the checkerboard instability simultaneously to generate near 0 1 topologies as a result of this strategy the solutions of presented algorithm are sufficiently smooth and possess the regularity of h 1 function space a simple method for the continuum adjoint sensitivity analysis of the corresponding pde constrained optimization problem is presented it is general and can be easily applied to a wide range of alternative problems the success of the introduced algorithm is studied by numerical experiments on two dimensional model problems for different numbers of phases according to numerical results the objective functional is reduced monotonically with iterations moreover the final topologies at the optimal solutions are near 0 1 the dynamic behavior of optimal designs is compared to that of initial ones to show the impact of optimization on the performance of structures

an algorithm for the design of group sequential triangular tests for single arm clinical trials with a binary endpoint consider the problem of testing h0 p p0 vs h1 p p0 where p could for example represent the response rate to a new drug the group sequential tt is an efficient alternative to a single stage test as it can provide a substantial reduction in the expected number of test subjects whitehead provides formulas for determining stopping boundaries for this test existing research shows that test designs based on these formulas wtts may not meet type i error and or power specifications or may be over powered at the expense of requiring more test subjects than are necessary we present a search algorithm with program available from the author which provides an alternative approach to triangular test design the primary advantage of the algorithm is that it generates test designs that consistently meet error specifications in tests on nearly 1000 example combinations of n group size p0 p1 and the algorithm determined triangular test att design met specified type i error and power constraints in every case considered whereas wtt designs met constraints in only 10 cases actual type i error and power values for the atts tend to be close to specified values leading to test designs with favorable average sample number performance for cases where the wtt designs did meet type i error and power constraints the corresponding att designs also had the advantage of providing on average a modest reduction in average sample numbers calculated at p0 p1 and p0 p1 2 copyright 2010 john wiley sons ltd