structural optimization a new dual method using mixed variables a new and powerful mathematical programming method is described which is capable of solving a broad class of structural optimization problems the method employs mixed direct reciprocal design variables in order to get conservative first order approximations to the objective function and to the constraints by this approach the primary optimization problem is replaced with a sequence of explicit subproblems each subproblem being convex and separable it can be efficiently solved by using a dual formulation an attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs examples of application to real life aerospace structures are offered to demonstrate the power and generality of the approach presented

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

a unified approach for parameter identification of inelastic material models in the frame of the finite element method this work is concerned with identification of parameters for inelastic material models in order to account for possible non uniformness of stress and strain distributions the identification is performed in the frame of the finite element method in particular linearization procedures are described in a systematic manner for the case of complex material models within a geometric linear theory this unified approach allows one to apply the newton method for solving the associated direct problem and to apply gradient based methods for solving the associated inverse problem which is considered as an optimization problem two numerical examples demonstrate the versatility of our approach firstly we consider cooks membrane problem based on simulated data for re identification of material parameters for a viscoplastic power law furthermore material data for j2 flow theory are determined based on experimental data obtained by a grating method for a compact specimen and we will investigate the results by using different starting values and stochastic perturbation of the experimental data

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

asymptotic behaviour of optimal spectral planar domains with fixed perimeter we consider the problem of minimizing the kth dirichlet eigenvalue of planar domains with fixed perimeter and show that as k goes to infinity the optimal domain converges to the ball with the same perimeter we also consider this problem within restricted classes of domains such as n polygons and tiling domains for which we show that the optimal asymptotic domain is that which maximises the area for fixed perimeter within the given family i e the regular n polygon and the regular hexagon respectively physically the above problems correspond to the determination of the shapes within a given class which will support the largest number of modes below a given frequency

ultraconvergence of the patch recovery technique ii the ultraconvergence property of a gradient recovery technique proposed by zienkiewicz and zhu is analyzed for the laplace equation in the two dimensional setting under the assumption that the pollution effect is not present or is properly controlled it is shown that the convergence rate of the recovered gradient at an interior node is two orders higher than the optimal global convergence rate when even order finite element spaces and local uniform rectangular meshes are used

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

enhancing finite element approximation for eigenvalue problems by projection method abstract this paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems the projection method is a postprocessing procedure that constructs a new approximation by using the least squares method the results are based on some regularity assumption for the elliptic problem and are applicable to the finite element approximations of self adjoint elliptic eigenvalue problems with general quasi regular partitions therefore the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

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

shapes of drums with lowest base frequency under non isotropic perimeter constraints we study the minimizers of the sum of the principal dirichlet eigenvalue of the negative laplacian and the perimeter with respect to a general norm in the class of jordan domains in the plane this is equivalent modulo scaling to minimizing the said eigenvalue or the base frequency of a drum of this shape subject to a hard constraint on the perimeter we show that for all norms a minimizer exists is unique up to spatial translations and is convex but not necessarily smooth we give conditions on the norm that characterize the appearance of facets and corners we also demonstrate that near minimizers have to be close to the optimal ones in the hausdorff distance our motivation for considering this class of variational problems comes from a study of random walks in random environment interacting through the boundary of their support 1 main results the aim of this paper is to analyze the planar domains that minimize the principal harmonic frequency i e the lowest dirichlet eigenvalue of the negative laplacian or the base frequency of the drum of this shape subject to a hard constraint on the perimeter defined using a general norm let us introduce the problem and state our results on it first and leave the discussion of our motivation as well as connections to existing literature to a later section 1 1 the model and main theorems we will consider the following class of admissible domains u u r2 jordan domain 0 u 1 1 note that each u u is open bounded connected and simply connected with u being the trace of a simple closed curve the principal dirichlet eigenvalue of the negative laplacian is for any open u r2 given by u inf g 2 g c r2 supp g u g 2 1 1 2 where 2 denotes the l2 norm on r2 with respect to the lebesgue measure next fix a norm on r2 and denote by 0 1 r2 the curve constituting u where 0 1 then we c 2016 m biskup e b procaccia reproduction by any means of the entire article for non commercial purposes is permitted without charge

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

a stabilization procedure by line search computation for first order approximation strategies in structural optimization the paper is concerned with first order approximation methods to solve non linear optimization problems as they arise in structural optimization a modification of the original method of moving asymptotes mma is presented to prevent oscillation or divergence by incorporating a line search into the global iteration algorithm in this line search a merit function of augmented lagrangian type is decreased by using a backtracking strategy a proof is presented to show the descent property of the proposed merit function for mma numerical examples demonstrate how the new technique stabilizes the iteration process

superconvergence of numerical gradient for weak galerkin finite element methods on nonuniform cartesian partitions in three dimensions abstract a superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak galerkin finite element scheme on the uniform and non uniform cubic partitions due to the loss of the symmetric property from two dimensions to three dimensions this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions li et al 0000 from rectangular partitions to cubic partitions the error estimate for the numerical gradient in the l 2 norm arrives at a superconvergence order of o h r 1 5 r 2 when the lowest order weak galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed a series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

a local superconvergence analysis of the finite element method for the stokes equations by local projections abstract in this paper we focus on a local superconvergence analysis of the finite element method for the stokes equations by local projections the local and global superconvergence results of finite element solutions are provided for the stokes problem under some corresponding regularity assumptions conclusion can be drawn that the local superconvergence has advantages over the global superconvergence in two important aspects on the one hand it offsets theoretical limitation in practical applications on the other hand interior estimates are derived on the base of local properties of the domain without global smoothness for the exact solution and prior regularity of the problem globally over the whole domain

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

superconvergence two grid scheme based on shifted inverse power method for eigenvalue problems by function value recovery abstract in the paper an improved two grid scheme based on shifted inverse power method is proposed to solve the elliptic eigenvalue problems with this new scheme the solution of the elliptic eigenvalue problem on a fine grid t h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid t h and the solution of an elliptic boundary value problem and the recovered solution on the fine grid t h theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate finally some numerical experiments are carried out to confirm the theoretical analysis

brunn minkowski inequalities for variational functionals and related problems abstract recently several inequalities of brunn minkowski type have been proved for well known functionals in the calculus of variations e g the first eigenvalue of the laplacian the newton capacity the torsional rigidity and generalizations of these examples in this paper we add new contributions to this topic in particular we establish equality conditions in the case of the first eigenvalue of the laplacian and of the torsional rigidity and we prove a brunn minkowski inequality for another class of variational functionals moreover we describe the links between brunn minkowski type inequalities and the resolution of minkowski type problems

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

an interior estimate of superconvergence for finite element solutions for second order elliptic problems on quasi uniform meshes by local projections this paper establishes some superconvergence estimates for finite element solutions of second order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution the projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares in particular some local superconvergence estimates in the l2 and l infty norms are derived for the local projections of the galerkin finite element solution the results have two prominent features first they are established for any quasi uniform meshes which are of practicalimportance in scientific computation second they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem therefore the result of this paper can be employed to provide useful a posteriori error estimators in practical computing

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives

the size of the first eigenfunction of a convex planar domain the goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains in particular we will locate the place where u achieves its maximum to within a distance comparable to the inradius uniformly for arbitrarily large diameter in addition we will estimate the location of other level sets of u by showing that u is well approximated by the first eigenfunction of a naturally associated ordinary differential schrodinger operator we intend to show in a separate paper that the estimates here are best possible in order of magnitude the present paper depends on the ideas and results of our earlier work j and gj where detailed estimates for the zero set of the second eigenfunction or first nodal line are obtained the paper j also contains some estimates for the first eigenfunction and lowest eigenvalue but the techniques of gj and new techniques introduced here are essential to the best possible estimates for the first eigenfunction presented here the maximum of the first eigenfunction occurs at the point of largest displacement of a vibrating drum with fixed edges when it vibrates at its fundamental or first resonant frequency the first nodal line is the stationary curve of the drum at the second resonant frequency the maximum is harder to find experimentally than the nodal line because it is a single point its location has less influence on the eigenvalue or dirichlet integral so it is also harder to locate mathematically another way to describe the difficulty is as follows to find the maximum of the first eigenfunction we will need to estimate its first directional derivative derivatives of the first eigenfunction are roughly speaking analogous to the second eigenfunction because they are solutions to an eigenfunction equation moreover convexity properties of the first eigenfunction imply that the zero set of the derivative divides the region into two connected components but the derivatives are harder to estimate than a second eigenfunction because they do not vanish at the boundary