review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

minimax principle for the dirac equation the problem of calculating the eigenvalues of the dirac equation by the finite basis expansion method is studied bounds for the eigenvalues are obtained explaining the numerical results on the spectrum that have been observed previously it is argued that the problem of variational collapse can be avoided by finding the minimum over the wave function large component of the maximum over the wave function small component of the energy functional a numerical example is discussed

self adjointness for dirac operators via hardy dirac inequalities distinguished self adjoint extensions of dirac operators are constructed for a class of singular potentials including coulombic ones up to and including the critical case x 1 the method uses hardy dirac inequalities and quadratic form techniques

equivariant and fractional index of projective elliptic operators in this note the fractional analytic index for a projective elliptic operator associated to an azumaya bundle of 5 is related to the equivariant index of 1 6 for an associated transversally elliptic operator

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

a new approach to spectral approximation a new technique for approximating eigenvalues and eigenvectors of a self adjoint operator is presented the method does not incur spectral pollution uses trial spaces from the form domain has a self adjoint algorithm and exhibits superconvergence

variational methods in relativistic quantum mechanics this review is devoted to the study of stationary solutions of lin ear and nonlinear equations from relativistic quantum mechanics involving the dirac operator the solutions are found as critical points of an energy func tional contrary to the laplacian appearing in the equations of nonrelativistic quantum mechanics the dirac operator has a negative continuous spectrum which is not bounded from below this has two main consequences first the energy functional is strongly indefinite second the euler lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap between the negative and positive continuous spectra moreover since we work in the space domain r 3 the palais smale condition is not satisfied for these reasons the problems discussed in this review pose a challenge in the calculus of variations the existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems in the first part we consider the fixed eigenvalue problem for models of a free self interacting relativistic particle they allow to describe the localized state of a spin 1 2 particle a fermion which propagates without changing its shape this includes the soler models and the maxwell dirac or klein gordon dirac equations the second part is devoted to the presentation of min max principles al lowing to characterize and compute the eigenvalues of linear dirac operators with an external potential in the gap of their essential spectrum many con sequences of these min max characterizations are presented among them a new kind of hardy like inequalities and a stable algorithm to compute the eigenvalues in the third part we look for normalized solutions of nonlinear eigenvalue problems the eigenvalues are lagrange multipliers lying in a spectral gap we review the results that have been obtained on the dirac fock model which is a nonlinear theory describing the behavior of n interacting electrons in an external electrostatic field in particular we focus on the problematic definition of the ground state and its nonrelativistic limit in the last part we present a more involved relativistic model from quan tum electrodynamics in which the behavior of the vacuum is taken into ac count it being coupled to the real particles the main interesting feature of this model is that the energy functional is now bounded from below providing us with a good definition of a ground state

the spectrum of relativistic one electron atoms according to bethe and salpeter bethe and salpeter introduced a relativistic equation different from the bethe salpeter equation which describes relativistic multi particle systems here we will begin some basic work concerning its mathematical structure in particular we show self adjointness of the one particle operator which will be a consequence of a sharp sobolev type inequality yielding semi boundedness of the corresponding sesquilinear form moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum

superlinear schrodinger operators abstract we find nontrivial and ground state solutions for the nonlinear schrodinger equation under conditions weaker than those previously assumed

on the eigenvalues of operators with gaps application to dirac operators this paper is devoted to a general min max characterization of the eigenvalues in a gap of the essential spectrum of a self adjoint unbounded operator we prove an abstract theorem then we apply it to the case of dirac operators with a coulomb like potential the result is optimal for the coulomb potential

localization of the riemann roch character abstract we present a k theoretic approach to the guillemin sternberg conjecture v guillemin and s sternberg invent math 67 1982 515 538 about the commutativity of geometric quantization and symplectic reduction which was proved by e meinrenken j amer math soc 9 1996 373 389 adv math 134 1998 240 277 and tian zhang y tian and w zhang invent math 132 1998 229 259 besides providing a new proof of this conjecture for the full non abelian group action case our methods lead to a generalization for compact lie group actions on manifolds that are not symplectic these manifolds carry an invariant almost complex structure and an abstract moment map

a quadratically convergent unstructured remeshing strategy for shape optimization a novel unstructured remeshing environment for gradient based shape optimization using triangular finite elements is presented the remeshing algorithm is based on a truss structure analogy in solving for the equilibrium position of the truss system the quadratically convergent newton s method is used exact analytical sensitivity information is therefore made available to the shape optimization algorithm the overall computational efficiency in gradient based shape optimization is very high in solving the truss structure analogy we compare our quadratically convergent newton solver with a previously proposed forward euler solver this includes notes regarding mesh uniformity element quality convergence rates and efficiency we present three numerical examples it is then shown that remeshing may introduce discontinuities and local minima we demonstrate that the effects of these on gradient based algorithms are alleviated to some extent through mesh refinement and may largely be overcome with a simple multi start strategy copyright 2005 john wiley sons ltd

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

on approximation of the eigenvalues of perturbed periodic schrodinger operators this paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic schrodinger operator perturbed by a fast decreasing potential we use a recently developed technique the so called quadratic projection method in order to achieve convergence free from spectral pollution we describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

a fictitious energy approach for shape optimization this paper deals with shape optimization of continuous structures as in early works on shape optimization coordinates of boundary nodes of the fe domain are directly chosen as design variables convergence problems and problems with jagged shapes are eliminated by a new regularization technique an artificial inequality constraint added to the optimization problem limits a fictitious total strain energy that measures the shape change of the design with respect to a reference design the energy constraint defines a feasible design space whose size can be varied by one parameter the upper energy limit by construction the proposed regularization is applicable to a wide range of problems although in this paper the application is restricted to linear elastostatic problems

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

a computational methodology for shape optimization of structures in frictionless contact this paper presents a computational methodology for shape optimization of structures in frictionless contact which provides a basis for developing user friendly and efficient shape optimization software for evaluation it has been implemented as a subsystem of a general finite element software the overall design and main principles of operation of this software are outlined the parts connected to shape optimization are described in more detail the key building blocks are analytic sensitivity analysis an adaptive finite element method an accurate contact solver and a sequential convex programing optimization algorithm results for three model application examples are presented in which the contact pressure and the effective stress are optimized

shape optimization with p adaptivity the p method of analysis when used with shape optimization has certain advantages over the h method the method of h adaptivity results in fine meshes with small elements in areas of stress concentration the presence of these small elements prevents the shape from changing significantly during optimization due to mesh distortion the p adaptive method uses relatively large sized elements and adaptivity is achieved by increasing the polynomial order of the shape functions rather than by introducing small elements as in the h method thus the p adaptive method allows for greater shape changes without sacrificing the accuracy of the analysis a methodology is developed in this paper to integrate p adaptive analysis with shape optimization for general velocity fields the key contributions lie in an easily implementable shape update strategy and an adaptive scheme to locally increase the polynomial order based on error indicators numerical results show that significant shape change is achievable prior to serious mesh distortion in addition need for an h p adaptive strategy rather than a pure p adaptive strategy is also highlighted

shape sensitivity analysis in linear elastic fracture mechanics shape sensitivity analysis of an elastic solid in equilibrium with a known load system applied over its boundary is presented in this work the domain and boundary integral expressions of the first and second order shape derivatives of the total potential energy are established by using an arbitrary change of the domain characterized by a velocity field defined over the initial body configuration in these expressions we recognize free divergence tensors that are denoted in this paper as energy shape change tensors next shape sensitivity analysis is applied to cracked bodies for that purpose a suitable velocity distribution field is adopted to simulate the crack advance of a unit length in a two dimensional body finally the corresponding domain and the equivalent path independent integral expressions of the first and second order potential energy release rate of fracture mechanics are also derived

hypersingular residuals a new approach for error estimation in the boundary element method summary this paper presents a new approach for a posteriori pointwise error estimation in the boundary element method the estimator relies upon evaluation of the residual of hypersingular integral equations and is therefore intrinsic to the boundary integral equation approach a methodology is developed for approximating the error on the boundary as well as in the interior of the domain extensive computational experiments have been performed for the two dimensional laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve moreover a reasonable estimate of the magnitude of the actual error is also predicted

variational principles and eigenvalue estimates for unbounded block operator matrices and applications in this paper we establish variational principles eigenvalue estimates and asymptotic formulae for eigenvalues of three different classes of unbounded block operator matrices the results allow to characterise eigenvalues that are not necessarily located at the boundary of the spectrum applications to an example from magnetohydrodynamics and to dirac operators on certain manifolds are given

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

state space newton s method for topology optimization abstract we introduce a new algorithm for solving certain classes of topology optimization problems which enjoys fast local convergence normally achieved by the full space methods while working in a smaller reduced space the computational complexity of newton s direction finding subproblem in the algorithm is comparable with that of finding the steepest descent direction in the traditional first order nested reduced space algorithms for topology optimization that is the space reduction is computationally inexpensive and more importantly it does not ruin the sparsity of the full space system of optimality conditions the fast local convergence of the algorithm allows one to efficiently solve a sequence of optimization problems for varying parameters numerical continuation this can be utilized for eliminating the errors introduced by the approximate enforcement of the boundary conditions or 0 1 type constraints on the design field through penalties in many topology optimization approaches we test the algorithm on the benchmark problems of dissipated power minimization for stokes flows and in all cases the algorithm outperforms the traditional first order reduced space nested approaches by a factor varying from two to almost twenty in terms of the number of iterations while attaining an almost unprecedented accuracy in solving the discretized topology optimization problem finally we present a few extensions to the algorithm one involving computations on adaptively refined meshes and another related to solving topology optimization problems for non newtonian fluids

a general methodology for structural shape optimization problems using automatic adaptive remeshing this methodology is based on the use of the adaptive mesh refinement techniques in the context of 2d shape optimization problems a technique for the parametrization of the optimization problem using b splines to define the boundary is first presented then mesh generation using the advancing frontal method the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems in particular the analytical sensitivity analysis of the different items ruling the problem b splines finite element mesh structural behaviour and error estimator is studied in detail

on efficient solutions to the continuous sensitivity equation using automatic differentiation shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation pde these shape sensitivities are described by a continuous sensitivity equation cse automatic differentiation ad can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation the approximate solution of the pde uses a spatial discretization mesh that often depends on the shape parameters therefore the straightforward application of ad introduces derivatives of the mesh there are two drawbacks to this approach first extra computational effort especially memory is used in these calculations due to mesh sensitivities second this mesh sensitivity information needs to be computed in order to obtain accurate results in this work we provide a methodology that avoids mesh sensitivities and their drawbacks by defining a modified pde on a fixed domain i e independent of the shape parameter such that ad provides the desired approximation of the cse using two examples we demonstrate significant improvement in the computational effort both in terms of floating point operations and memory requirements we explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code these changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation

an analytical proof of hardy like inequalities related to the dirac operator we prove some sharp hardy type inequalities related to the dirac operator by elementary direct methods some of these inequalities have been obtained previously using spectral information about the dirac coulomb operator our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

shape design optimization of hyperelastic structures using a meshless method ashapedesignoptimizationprocedureforhyperelasticstructuresisdevelopedusing ameshlessmethodforanalysis and a continuum based design sensitivity analysis dsa method the meshless method greatly reduces the mesh distortion or entanglement encountered in using the e nite element method for large deformation nonlinear analysis and structural shape design optimization the dsa method of grindeanu et al grindeanu i chang k h choi k k and chen j s design sensitivity analysis of hyperelastic structures using a meshless method aiaa journal vol 36 no 4 1998 pp 618 627 is extended by using a pressure projection method to avoid volumetric locking for nearly incompressible materials without the need for large support sizes for the meshless shape functions and to reduce the cpu time the lagrange multiplier method is used to impose the essential boundary conditions an engine mount is employed as an example to demonstrate the feasibility of the proposed optimization method the mass is minimized subject to constraints on hydrostatic pressure and stiffness characteristics of the component the design velocity e elds corresponding to theshape design parameters are obtained using the design sensitivity analysis and optimization tool shape design optimization is carried out using the modie ed feasible direction of the design optimization tool

analytical integral formulae related to convex quadrilateral finite elements abstract analytical quadrature formulae for rational functions integrated over finite elements with quadrilateral geometry are presented definitively for second order differential equations these formulae produce algebraic finite element relations associated with the isoparametrical or subparametrical rectangular interpolations

shape variable definition with c0 c1 and c2 continuity functions the present paper proposes a new technique for the definition of the shape design variables in 2d and 3d optimisation problems it can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the cad defining surfaces the proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori this capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process structural analysis fluid dynamic analysis crash analysis aerodynamic analysis etc

design sensitivity analysis for shape optimization based on the lie derivative abstract the paper presents a theoretical framework for the shape sensitivity analysis of systems governed by partial differential equations the proposed approach based on geometrical concepts borrowed from differential geometry shows that sensitivity of a performance function i e any function of the solution of the problem with respect to a given design variable can be represented mathematically as a lie derivative i e the derivative of that performance function along a flow representing the continuous shape modification of the geometrical model induced by the variation of the considered design variable theoretical formulae to express sensitivity analytically are demonstrated in detail in the paper and applied to a nonlinear magnetostatic and a linear elastic problem following both the direct and the adjoint approaches following the analytical approach one linear system of which only the right hand side needs be evaluated the system matrix being known already has to be solved for each of the design variables in the direct approach or for each performance functions in the adjoint approach a substantial gain in computation time is obtained this way compared to a finite difference evaluation of sensitivity which requires solving a second nonlinear system for each design variable this is the main motivation of the analytical approach there is some freedom in the definition of the auxiliary flow that represents the shape modification we present a method that makes benefit of this freedom to express sensitivity locally as a volume integral over a single layer of finite elements connected to both sides of the surfaces undergoing shape modification all sensitivity calculations are checked with a finite difference in order to validate the analytic approach convergence is analyzed in 2d and 3d with first and second order finite elements

eigenvalue excluding for perturbed periodic one dimensional schrodinger operators subject of investigation in this paper is a one dimensional schrodinger equation where the potential is a sum of a periodic function and a perturbation decaying at graphic 1 it is well known that the essential spectrum consists of spectral bands and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands while enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations e g by d weinstein s bounds there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps i e which identifies subregions of a gap which contain no eigenvalues here we propose such a method it makes heavy use of computer assistance nevertheless the results are completely rigorous in the strict mathematical sense because all computational errors are taken into account 1 embed inline graphic 1 gif

a physics based parameterization method for shape optimization a physical approach is presented to provide shape modification of nurbs curve the movement of the control polygon of nurbs is simulated by a physical frame structure and the curve deformation is then carried out through changes of the mode coordinates which are derived from the eigenvectors of the finite element equations of the physical frame system the shape of the design boundaries is described by a linear combination of the mode shape vectors based on the presented design variable representation and p version finite element analysis an integrated approach to structural shape optimization is discussed a method is developed that enables the description of the shape to be adaptively refined during the shape optimization process numerical examples are used to demonstrate the applicability of this approach to structural shape optimization

hardy carleman type inequalities for dirac operators general hardy carleman type inequalities for dirac operators are proved new inequalities are derived involving particular traditionally used weight functions in particular a version of the agmon inequality and treve type inequalities is established the case of a dirac particle in a potential magnetic field is also considered the methods used are direct and based on quadratic form techniques

spectral pollution and how to avoid it this paper devoted to the study of spectral pollution contains both abstract results and applications to some self adjoint operators with a gap in their essential spectrum occuring in quantum mechanics first we consider galerkin basis which respect the decomposition of the ambient hilbert space into a direct sum h ph oplus 1 p h given by a fixed orthogonal projector p and we localize the polluted spectrum exactly this is followed by applications to periodic schrodinger operators pollution is absent in a wannier type basis and to dirac operator several natural decompositions are considered in the second part we add the constraint that within the galerkin basis there is a certain relation between vectors in ph and vectors in 1 p h abstract results are proved and applied to several practical methods like the famous kinetic balance of relativistic quantum mechanics

the chern character of a transversally elliptic symbol and the equivariant index let g be a compact lie group acting on a compact manifold m in this article we associate to a g transversally elliptic symbol on m a g invariant generalized function on g constructed in terms of equivariant closed differential forms on the cotangent bundle t m

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

estimates for the optimal constants in multipolar hardy inequalities for schrodinger and dirac operators by expanding squares we prove several hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities these inequalities involve the l2 norm such results are generalized to an arbitrary number of singularities and compared with standard results given by the ims method the generalized version of hardy inequalities with several singularities is equivalent to some spectral information on a schrodinger operator involving a potential with several inverse square singularities we also give a generalized hardy inequality for dirac operators in the case of a potential having several singularities of coulomb type which are critical for dirac operators

galerkin method for unsplit 3 d dirac equation using atomically kinetically balanced b spline basis a galerkin method is developed to solve the time dependent dirac equation in prolate spheroidal coordinates for an electron molecular two center system the initial state is evaluated from a variational principle using a kinetic atomic balanced basis which allows for an efficient and accurate determination of the dirac spectrum and eigenfunctions b spline basis functions are used to obtain high accuracy this numerical method is used to compute the energy spectrum of the two center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field

review of localization in geometry review of localization in geometry equivariant cohomology characteristic classes atiyah bott berline vergne formula atiyah singer equivariant index formula mathai quillen formalism and application to path integrals

on the minimax principle for coulomb dirac operators let b be a self adjoint operator constructed as a form perturbation of a self adjoint operator q assuming that b has a gap a b subset mathbb r in the essential spectrum we prove a minimax principle for eigenvalues of b in a b using a suitable orthogonal decomposition of the form domain of q this allows us to justify two minimax characterisations of eigenvalues in the gap of three dimensional dirac operators with potentials which may have a strong coulomb singularity

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

shape optimal design using fictitious loads implementation aspects of the natural velocity field approach for shape optimal design are discussed the problem of concern is to find the optimum shape of an elastic body that requires minimizing an objective function subject to stress displacement frequency and manufacturing constraints in this paper fictitious loads are applied at certain control nodes on an auxiliary structure and the deformation produced by these loads is used to update the shape the use of beam stiffeners in the auxiliary structure leads to smoother shapes furthermore softer material properties in the auxiliary structure result in faster convergence when iterating through the feasibile region the method is applied in conjunction with both discrete and continuum methods of design sensitivity analysis

a meshless compositional approach to shape optimal design meshless computational strategies for shape optimal design through the composition of behavioral fields are developed in this paper the composition of behavioral fields is carried out in a manner that is procedurally similar to composition of primitive geometrical entities through set theoretic boolean operations in constructive solid geometry the methodology is shown to lead to a tighter integration between design and analysis and is termed constructive solid analysis a meshless approximation using non uniform rational b spline basis functions is used to discretize the behavioral fields defined over the geometrical primitives remeshing which is a critical bottleneck in shape optimization using conventional finite element analysis is either eliminated or limited to those primitives that are modified further it is demonstrated that the reconstruction of stiffness matrices is required only for those geometrical primitives that are affected by the design changes design sensitivity analysis techniques are developed for the discretized solution system arising out of the compositional analysis an adaptive numerical integration scheme is developed and shown to be necessary to accurately compute the effect of small domain perturbations on the problem functions some of the common challenges in conventional shape optimization problems such as difficulty in creating and merging multiple holes are naturally resolved using the developed compositional approach several representative problems are solved to validate and to demonstrate the methodology

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

high order sensitivity analysis in shape optimization problems abstract a formulation for high order directional differentiation of functions defined through integration is presented the formulation has been developed initially for sensitivity analysis in optimal shape design problems by the finite elements method however its application range is much wider explicit analytical expressions are given for sequential directional derivatives special emphasis has been made in holding recurrence and simplicity of intermediate operations the proposed scheme does not depend strongly on any particular form of the state equations and its numerical implementation in finite elements method programs can be considered easy and efficient as an example up to second order sensitivity techniques are applied to the optimization of a concrete roof structural analysis has been performed in linear elastic conditions with quadratic three dimensional elements for the discretization of the roof