homogenization of two phase elasto plastic composite materials and structures study of tangent operators cyclic plasticity and numerical algorithms we develop homogenization schemes and numerical algorithms for two phase elasto plastic composite materials and structures a hill type incremental formulation enables the simulation of unloading and cyclic loadings it also allows to handle any rate independent model for each phase we study the crucial issue of tangent operators elasto plastic or continuum versus algorithmic or consistent and anisotropic versus isotropic we apply two methods of extraction of isotropic tangent moduli we compare mathematically the stiffnesses of various tangent operators all rate equations are discretized in time using implicit integration we implemented two homogenization schemes mori tanaka and a double inclusion model and two plasticity models classical j 2 plasticity and chaboche s model with non linear kinematic and isotropic hardenings we consider composites with different properties and present several discriminating numerical simulations in many cases the results are validated against finite element fe or experimental data we integrated our homogenization code into the fe program abaqus using a user material interface umat a two scale procedure allows to compute realistic structures made of non linear composite materials within reasonable cpu time and memory usage examples are shown c 2002 elsevier science ltd all rights reserved

a genetic search for optimal multigrid components within a fourier analysis setting in this paper fourier analysis is used for finding efficient multigrid components the individual multigrid components for several discrete partial differential operators are chosen automatically by a genetic optimization method from a set of multigrid components such as different smoothers coarse grid correction components cycle types number of smoothing iterations and relaxation parameters an optimal three grid fourier convergence factor corrected for computational complexity is obtained by the genetic search the resulting methods can be tuned for optimal efficiency or toward robustness the analysis results are verified by numerical experiments

objective evaluation of linearization procedures in nonlinear homogenization a methodology and some implications on the accuracy of micromechanical schemes a systematic methodology for an accurate evaluation of various existing linearization procedures sustaining mean fields theories for nonlinear composites is proposed and applied to recent homogenization methods it relies on the analysis of a periodic composite for which an exact resolution of both the original nonlinear homogenization problem and the linear homogenization problems associated with the chosen linear comparison composite lcc with an identical microstructure is possible the effects of the sole linearization scheme can then be evaluated without ambiguity this methodology is applied to three different two phase materials in which the constitutive behavior of at least one constituent is nonlinear elastic or viscoplastic a reinforced composite a material in which both phases are nonlinear and a porous material comparisons performed on these three materials between the considered homogenization schemes and the reference solution bear out the relevance and the performances of the modified second order procedure introduced by ponte castaneda in terms of prediction of the effective responses however under the assumption that the field statistics first and second moments are given by the local fields in the lcc all the recent nonlinear homogenization procedures still fail to provide an accurate enough estimate of the strain statistics especially for composites with high contrast

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

adaptive smoothed aggregation alpha sa substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice these systems often arise from discretizing partial differential equations over unstructured meshes and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver algebraic multigrid amg and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry these methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error for smoothed aggregation sa methods applied to discretizations of elliptic problems these assumptions typically consist of knowledge of the near nullspace of the weak form this paper introduces an extension of the sa method in which good convergence properties are achieved in situations where explicit knowledge of the near nullspace components is unavailable this extension is accomplished by using the method itself to determine near nullspace components and adjusting the coarsening processes accordingly

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

bootstrap amg we develop an algebraic multigrid amg setup scheme based on the bootstrap framework for multiscale scientific computation our approach uses a weighted least squares definition of interpolation based on a set of test vectors that are computed by a bootstrap setup cycle and then improved by a multigrid eigensolver and a local residual based adaptive relaxation process to emphasize the robustness efficiency and flexibility of the individual components of the proposed approach we include extensive numerical results of the method applied to scalar elliptic partial differential equations discretized on structured meshes as a first test problem we consider the laplace equation discretized on a uniform quadrilateral mesh a problem for which multigrid is well understood then we consider various more challenging variable coefficient systems coming from covariant finite difference approximations of the two dimensional gauge laplacian system a commonly used model problem in amg algorithm development for linear systems arising in lattice field theory computations

new explicit expressions of the hill polarization tensor for general anisotropic elastic solids abstract except for particular cases the classical expressions of the eshelby or hill polarization tensors depend respectively on a simple or double integral for a fully anisotropic two dimensional or three dimensional elastic body when the body is two dimensional we take advantage of cauchy s theory of residues to derive a new explicit expression which depends on the two pairs of complex conjugate roots of a quartic equation if the body exhibits orthotropic symmetry these roots are explicitly given as a function of the independent components of the elasticity tensor similarly the double integral is reduced to a simple one when the body is three dimensional the corresponding integrand depends on the three pairs of complex conjugate roots of a sextic equation which reduces to a cubic one for orthotropic symmetry this new expression improves significantly the computation times when the degree of anisotropy is high for both two and three dimensional bodies degenerate cases are also studied to yield valid expressions in any events

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

adaptive algebraic multigrid efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort overcoming the limitations of geometric multigrid methods to simple geometries and differential equations algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix while this allows for efficient black box solution of the linear systems associated with discretizations of many elliptic differential equations it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices this paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive definite linear systems

black box learning of multigrid parameters abstract this paper studies the optimality of the restriction and prolongation operators in the geometric multigrid method gmg gmg is used in solving discretized partial differential equation pde and it relies greatly on the restriction and prolongation operators many methods to find these operators were proposed but most of them have limited optimality proofs to study their optimality we introduce stochastic convergence functional which estimates the spectral radius of the iteration matrix for given gmg parameters we implement the gmg method in a modern machine learning framework that can automatically compute the gradients of the introduced convergence functional with respect to restriction and prolongation operators therefore we can minimize the proposed functional starting from some initial parameters and get better ones after some iterations of stochastic gradient descent to illustrate the performance of the proposed approach we carry out experiments on the discretized poisson equation helmholtz equation and singularly perturbed convection diffusion equation and demonstrate that proposed approach gives operators which lead to faster convergence

a hybrid multigrid method for convection diffusion problems this paper presents an advanced performance study of a multigrid method designed for convection diffusion problems developed in khelifi et al 2011 24 the proposed scheme with the separation of the operators enables an individual treatment for each operator while the piecewise constant operator is used for the convective part each off diagonal entry of the coarse diffusion operator is scaled by a geometric factor numerical examples illustrate the fast convergence and the outstanding robustness of the proposed method compared to other known methods

smoothing technique and its applications in semidefinite optimization in this paper we extend the smoothing technique nesterov in math program 103 1 127 152 2005 nesterov in unconstrained convex mimimization in relative scale 2003 onto the problems of semidefinite optimization for that we develop a simple framework for estimating a lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices using this technique we can justify the lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices we analyze the efficiency of the special gradient type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix which depends linearly on the design variables we show that in the first case the number of iterations of the method is bounded by o 1 epsilon where epsilon is the required absolute accuracy of the problem in the second case the number of iterations is bounded by 4 delta sqrt 1 delta r ln r where is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality thus the latter method is a fully polynomial approximation scheme

a multigrid method for the helmholtz equation with optimized coarse grid corrections we study the convergence of multigrid schemes for the helmholtz equation focusing in particular on the choice of the coarse scale operators let g rm c denote the number of points per wavelength at the coarse level if the coarse scale solutions are to approximate the true solutions then the oscillatory nature of the solutions implies the requirement g rm c 2 however in examples the requirement is more like g rm c gtrsim 10 in a trade off involving also the amount of damping present and the number of multigrid iterations we conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators standard 5 point finite differences in two dimensions are our first example a new coarse scale 9 point operator is constructed to match the fine scale phase speeds we then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator the required g rm c is reduced from about 10 to about 3 5 with less dam

homogenization models for predicting local field statistics in ellipsoidal particles reinforced composites comparisons and validations abstract this paper validates the performance of analytical homogenization models at predicting the local field statistics in randomly distributed and oriented ellipsoidal particles reinforced composites the numerical validation was based on a newly introduced load independent metric which allowed to formulate general conclusions a large validation campaign was conducted and the representative volume element rve was rigorously determined for each combination of phases mechanical properties particles volume fraction and aspect ratio the load independent properties computed numerically were compared to those predicted by a range of analytical models the study revealed that the original benveniste s interpretation of the mori tanaka scheme led to the most accurate first order moments and intra matrix second order moments provided that the mechanical properties contrast between the particles and the matrix are low for higher contrasts lielens model delivers the most accurate estimates the self consistent scheme is the most suited model to predict the intra particle second order moments when the particles aspect ratio is low for particles with high aspect ratio lielens model could be an alternative to the self consistent scheme furthermore an interpolation model has been developed and it has been shown to predict accurately the mean and covariance tensors of the intraphase stress fields

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3d convection diffusion equation an improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection diffusion equation is presented the novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation the new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method wang and zhang 2010 16 to obtain high order solutions since the proposed method needs a fourth order compact scheme with unequal meshsize grids a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection diffusion equation numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure

randomized algorithms for estimating the trace of an implicit symmetric positive semi definite matrix we analyze the convergence of randomized trace estimators starting at 1989 several algorithms have been proposed for estimating the trace of a matrix by 1 m i 1m zit azi where the zi are random vectors different estimators use different distributions for the zis all of which lead to e 1 m i 1m zit azi trace a these algorithms are useful in applications in which there is no explicit representation of a but rather an efficient method compute ztaz given z existing results only analyze the variance of the different estimators in contrast we analyze the number of samples m required to guarantee that with probability at least 1 the relative error in the estimate is at most we argue that such bounds are much more useful in applications than the variance we found that these bounds rank the estimators differently than the variance this suggests that minimum variance estimators may not be the best we also make two additional contributions to this area the first is a specialized bound for projection matrices whose trace rank needs to be computed in electronic structure calculations the second is a new estimator that uses less randomness than all the existing estimators

a general interpolation strategy for algebraic multigrid using energy minimization algebraic multigrid methods solve sparse linear systems ax b by automatic construction of a multilevel hierarchy this hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method we propose a methodology to improve grid transfers through energy minimization the proposed strategy is applicable to hermitian non hermitian definite and indefinite problems each column of the grid transfer operator p is minimized in an energy based norm while enforcing two types of constraints a defined sparsity pattern and preservation of specified modes in the range of p a krylov based strategy is used to minimize energy which is equivalent to solving a p j boldsymbol 0 for each column j of p with the constraints ensuring a nontrivial solution for the hermitian positive definite case a conjugate gradient cg based method is utilized to construct grid transfers while methods based on generalized minimum residual gmres and cg on the normal equations cgnr are explored for the general case the approach is flexible allowing for arbitrary coarsenings unrestricted sparsity patterns straightforward long distance interpolation and general use of constraints either user defined or auto generated we conclude with numerical evidence in support of the proposed framework

on accuracy of approximation of the spectral radius by the gelfand formula abstract the famous gelfand formula a limsup n a n 1 n for the spectral radius of a matrix is of great importance in various mathematical constructions unfortunately the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities a n 1 n to a in the paper this deficiency is made up to some extent by using the bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities a n 1 n to a the obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets

a nearly optimal multigrid method for general unstructured grids in this paper we develop a multigrid method on unstructured shape regular grids for a general shape regular unstructured grid of mathcal o n o n elements we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique such a construction is realized by a cluster tree which can be obtained in mathcal o n log n o nlogn operations for a grid of n elements this tree structure in turn is used for the definition of the grid hierarchy from coarse to fine for the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned cg for an elliptic pde is 1 mathcal o 1 log n 1 o 1 logn numerical experiments confirm the theoretical bounds and show that the total complexity is in mathcal o n log n o nlogn

transformation operators for spherical schrodinger operators abstract the present work aims at obtaining estimates for transformation operators for one dimensional perturbed radial schrodinger operators it provides more details and suitable extensions to already existing results that are needed in other recent contributions dealing with these kinds of operators