Jan Zeman

Short note: Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients

In this short note, we present a new technique to accelerate the convergence of a FFT-based solver for numerical homogenization of complex periodic media proposed by Moulinec and Suquet [1]. The approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to Vainikko [2], to give a linear system which can be efficiently solved by conjugate gradient methods. Computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high-contrast coefficients at a low overhead per iteration.

From random microstructures to representative volume elements

A unified treatment of random microstructures proposed in this contribution opens the way to efficient solutions of large-scale real world problems. The paper introduces a notion of statistically equivalent periodic unit cell (SEPUC) that replaces in a computational step the actual complex geometries on an arbitrary scale. A SEPUC is constructed such that its morphology conforms with images of real microstructures. Here, the appreciated two-point probability function and the lineal path function are employed to classify, from the statistical point of view, the geometrical arrangement of various material systems. Examples of statistically equivalent unit cells constructed for a unidirectional fibre tow, a plain weave textile composite and an irregular-coursed masonry wall are given. A specific result promoting the applicability of the SEPUC as a tool for the derivation of homogenized effective properties that are subsequently used in an independent macroscopic analysis is also presented.

Numerical evaluation of effective elastic properties of graphite fiber tow impregnated by polymer matrix

Abstract Homogenized elastic material properties are found for a fibrous graphite–epoxy composite system with fibers randomly distributed within a transverse plane section of the composite aggregate using the finite element method. To enhance efficiency of the numerical analysis the real microstructure is replaced by a material representative volume element, represented here by a periodic unit cell consisting of a small number of particles, which statistically resembles the actual composite. Such a unit cell is derived from a simple optimization procedure formulated in terms of various statistical descriptors characterizing the microstructure of the random medium. In the present approach the two-point probability and the second order intensity functions are employed. The upper bound on the macroscopic elastic stiffnesses then follows from the principle of minimum potential energy. The Finite Element Method (FEM) is used to carry out the numerical analysis. Results derived herein confirm applicability of the present approach and suggest that the unit cell, which effectively exploits the knowledge of the materials statistics of the composite, is more reliable then the one constructed simply as a cut of a small part of the real microstructure.

Applying genetic algorithms to selected topics commonly encountered in engineering practice

A carefully selected group of optimization problems is addressed to advocate application of genetic algorithms in various engineering optimization domains. Each topic introduced in the present paper serves as a representative of a larger class of interesting problems that arise frequently in many applications such as design tasks, functional optimization associated with various variational formulations, or a number of problems linked to image evaluation. No particular preferences are given to any version of genetic algorithms, but rather lessons learnt up-to-date are effectively combined to show the power of the genetic algorithm in effective search for the desired solution over a broad class of optimization problems discussed herein.

A competitive comparison of different types of evolutionary algorithms

Abstract This paper presents comparison of several stochastic optimization algorithms developed by authors in their previous works for the solution of some problems arising in civil engineering. The introduced optimization methods are: the integer augmented simulated annealing (IASA), the real-coded augmented simulated annealing (RASA) [Comp. Meth. Appl. Mech. Eng. 190 (13–14) (2000) 1629], the differential evolution (DE) in its original fashion developed by Storn and Price [R. Storn, On the usage of differential evolution for function optimization, NAPHIS, 1996] and simplified real-coded differential genetic algorithm (simplified atavistic differential evolution, SADE) [O. Hrstka, A. Kucerova, Search for optimization methods on multi-dimensional real domains, Contributions to Mechanics of Materials and Structures, CTU Reports 4, 2000, pp. 87–104]. Each of these methods was developed for some specific optimization problem; namely the Chebychev trial polynomial problem, the so called type 0 function and two engineering problems––the reinforced concrete beam layout and the periodic unit cell problem, respectively. Detailed and extensive numerical tests were performed to examine the stability and efficiency of proposed algorithms. The results of our experiments suggest that the performance and robustness of RASA, IASA and SADE methods are comparable, while the DE algorithm performs slightly worse. This fact together with a small number of internal parameters promotes the SADE method as the most robust for practical use.

An FFT-based Galerkin method for homogenization of periodic media

In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann-Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic problems, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature.

Novel anisotropic continuum‐discrete damage model capable of representing localized failure of massive structures: Part II: identification from tests under heterogeneous stress field

Purpose – The purpose of this paper is to discuss the identification of the model parameters for constitutive model capable of representing the failure of massive structures, from two kinds of experiments: a uniaxial tensile test and a three‐point bending test. Design/methodology/approach – A detailed development of the ingredients for constitutive model for failure of massive structures are presented in Part I of this paper. The salient feature of the model is in its ability to correctly represent two different failure mechanisms for massive structures, the diffuse damage in so‐called fracture process zone with microcracks and localized damage in a macrocrack. The identification of such model parameters is best performed from the tests under heterogeneous stress field. Two kinds of tests are used: the simple tension test and the three‐point bending test. The former allows us illustrate the non‐homogeneity of the strain field at failure even under homogeneous stress, whereas the latter provides a very good illustration for the proposed inverse optimization problem for which the specimen is subjected to a heterogeneous stress field. Findings – Several numerical examples are presented in order to illustrate a very satisfying performance of the proposed methodology for identifying the corresponding material parameters of the constitutive model for failure of massive structures. Originality/value – The paper confirms that one can make a very good use of the proposed identification procedure for estimating the corresponding parameters of damage model for localized failure of massive structure, and the advantages to using the experimental results obtained by testing under heterogeneous stress field.

Stochastic Modeling of Chaotic Masonry via Mesostructural Characterization

The purpose of this study is to explore three numerical approaches to the elastic homogenization of disordered masonry structures with moderate meso/macrolengthscale ratio. The methods investigated include a representative of perturbation methods, the Karhunen-Lo eve expansion technique coupled with Monte-Carlo simulations and a solver based on the Hashin-Shtrikman variational principles. In all cases, parameters of the underlying random eld of material properties are directly derived from image analysis of a real-world structure. Added value as well as limitations of individual schemes are illustrated by a case study of an irregular masonry panel.

Mori-Tanaka Based Estimates of Effective Thermal Conductivity of Various Engineering Materials

The purpose of this paper is to present a simple micromechanics-based model to estimate the effective thermal conductivity of macroscopically isotropic materials of matrix-inclusion type. The methodology is based on the well-established Mori-Tanaka method for composite media reinforced with ellipsoidal inclusions, extended to account for imperfect thermal contact at the matrix-inclusion interface, random orientation of particles and particle size distribution. Using simple ensemble averaging arguments, we show that the Mori-Tanaka relations are still applicable for these complex systems, provided that the inclusion conductivity is appropriately modified. Such conclusion is supported by the verification of the model against a detailed finite-element study as well as its validation against experimental data for a wide range of engineering material systems.

Macroscopic constitutive law for Mastic Asphalt Mixtures from multiscale modeling

A well established framework of an uncoupled hierarchical modeling approach is adopted here for the prediction of macroscopic material parameters of the Generalized Leonov (GL) constitutive model intended for the analysis of exible pavements