A class of general algorithms for multi-scale analyses of heterogeneous media

Abstract

A class of computational algorithms for multi-scale analyses is developed in this paper. The two-scale modeling scheme for the analysis of heterogeneous media with fine periodic microstructures is generalized by using relevant variational statements. Instead of the method of two-scale asymptotic expansion, the mathematical results on the generalized convergence are utilized in the two-scale variational descriptions. Accordingly, the global–local type computational schemes can be unified in association with the homogenization procedure for general nonlinear problems. After formulating the problem in linear elastostatics, that with local contact condition and the elastoplastic problem, we present representative numerical examples along with the computational algorithm consistent with our two-scale modeling strategy as well as some direct approaches.

Introduction

It is known that the mathematical theory of homogenization, a term coined by Babuska [1], provides the asymptotic behavior of the differential equations that govern the original boundary value problem when a parameter ε, characterizing the differential operators, goes to zero. In other words, the homogenization theorems tell us the limit points of a series of differential equations along with the field variables according to the variations of the parameter ε. Although various theorems hold without any restriction on the geometry of microstructure, most of the developments often assume that the parameter ε is taken as the repesentative size of microstructures that are periodically arranged and negligibly small relative to the overall dimensions of the structure. In this context, the mathematical method has been extensively studied in the 1970s in the framework of mathematical physics for heterogeneous media by using the method of two-scale expansions; see the textbooks of Benssousan et al. [2], Sanchez-Palencia [3], Lions [4], Kalamkarov and Kolpakov [5], and others.

The theoretical studies focus on the mechanical behavior of heterogeneous media with a periodic or quasi-periodic microstructure, assuming that the solution of the problem takes an asymptotically expanded form; see, e.g., [6] and references therein. These theoretical developments involve functional analysis in the applied mathematics area and the resulting variational statements are accompanied with an appropriate functional setting. Accordingly, the resulting governing equations define the well-posed initial/boundary value problems that can easily be incorporated with numerical analyses by the finite element method (FEM), in particular, to determine the homogenized elasticity constants for composite materials; see, e.g., [7], [8]. In addition, the FEM was utilized to evaluate the actual stress field in a microscale from the macroscopic responses [9]. This distinctive approach is referred to as the localization process and provides a great advantage over various counterparts in theoretical mechanics.

A class of the homogenization methods based on the two-scale asymptotic expansions has recently been infiltrated into engineering research and used to analyze the thermomechanical behavior of composite materials. Since the method enables one not only to systematically derive both the microscopic and macroscopic governing equations but also evaluate the localized behavior for fine-scale heterogeneity, it has become a multi-scale modeling tool in the area of computational mechanics. The computational significance of the localization process was demonstrated in some pioneering work in computational mechanics [10], [11], [12] and is extensively utilized to analyze specific composites along with the justification of various computational methods for microstructural analyses; see, e.g., [13], [14], [15], [16], [17].

The ability to evaluate the microscopic variables appears to be amenable, especially to the computational technique for nonlinear multi-scale analysis and, indeed, has been applied to various challenging problems: the evaluation of local damage [18], [19], [20], [21], small deformation elastoplastic problems [22], [23], [24], [25], [26], their extension to the finite deformation regime [27], and to compute so-called nonlocal effects in macroscopic point of view [28], [29], [30]. Among these, some provide their own interpretations of mathematical homogenization and sometimes combine the ideas in theoretical mechanics and the large-scale computational techniques. In spite of the successful computations of the nonlinear mechanical behavior of heterogeneous media, less attention has been paid to the mathematical theory of nonlinear homogenization. In fact, most of the computational developments are based on the method of two-scale asymptotic expansions and can be viewed as the direct extension of linear counterparts by means of rate or incremental form of the governing equations. However, the ansaz is known to be only formal and might not provide an acceptable consequence of the expanded form of nonlinear micro–macro relationships.

In this context, abstract notions of functional or generalized convergence arguments in mathematical homogenization theory, such as Γ-, G-, H- or two-scale convergence, would provide insight into a general class of nonlinear problems. The concept of Γ-convergence, which was introduced by De Giorgi [31], is not restricted to homogenization and has many applications in calculus of variations, such as singular perturbation problems. Elaborate discussions of Γ-convergence and several applications may be found in [32], [33], [34], [35], in which we notice that the pure mathematical results have an intimate relationship with nonlinear mechanics. The H-convergence of Tartar [36] and Murat and Tartar [37] is a generalization of G-convergence of Spagnolo [38] to nonsymmetric, second-order, elliptic operators and also helps us to understand the mathematical significance of the generalized convergence; see, e.g., [39]. It should be remarked that, in these mathematical developments, no special assumptions, like periodicity or stationarity, are placed on the sequence of functionals or differential operators. Moreover, in association with the two-scale representation of variables, the concept of two-scale convergence by Alliare [40], [41] provides the clear implication of the method of two-scale asymptotic expansions for periodic homogenization problems. Accordingly, the results of these theoretical developments could be applicable to multi-scale modeling for some classes of general nonlinear problems.

In the area of applied or computational mechanics, the relevant theoretical developments based upon the mathematical homogenization can be found in a series of work of Suquet [42], [43], [44] and Michel et al. [45]. In their theoretical descriptions, the homogenization method for a class of nonlinear or inelasticity problems is developed without using the asymptotic expansions in terms of two spatial scales. Instead of discussing asymptotic behavior or convergence of the solution, their attention is focussed on the characterization of macroscopic mechanical behavior by posing the governing equations in a microscale. While these have been followed by some analytical developments in the area of theoretical mechanics [46], [47], only a few discussions are addressed to the computational aspects in the literature [45], [46], [47], [48], [49]. The essential feature of the alternative mathematical homogenization has not been studied in the spirit of multi-scale modeling in computational mechanics.

In this paper, a class of computational algorithms for multi-scale analyses is proposed. Our ultimate goal is to generalize the multi-scale problems for heterogeneous media with fine periodic microstructures by the relevant variational statements so that the computational treatments would be unified. Regardless of the concrete information of macroscopic constitutive equations, the mathematical results on the generalized convergence provide us with the clear understanding of the homogenized variational equations. That is, the governing equations for general nonlinear problems can be incorporated with two-scale representation of the variables by using the convergence results in the mathematical theory of homogenization. While the method of two-scale asymptotic expansions may fail to give the appropriate multi-scale governing equations for general nonlinear problems, the algorithm implicated by the generalized variational statement could be applied to broad class of nonlinear problems. For example, only the convexity of the stored energy is needed to homogenize the functional defining the total potential energy and the multi-scale variational problems leads to one analogous to the linear counterparts. In such a formulation, an appropriate functional setting is crucial in both the two-scale modeling and the computational implementations.

An outline of this paper is as follows. In Section 2, we start out by presenting the basic notation and the assumptions that will be employed in the subsequent sections and provide the variational statements with appropriate functional setting. We shall state the convergence results in the mathematical homogenization such as Γ- or G-convergence and two-scale convergence in terms of the representative two scales, namely, the micro- and macroscales. In order to understand the multi-scale nature of the analysis, the generalized variational principle of Hu–Washizu, which has the three independent primal variables, is utilized in the framework of nonlinear elasticity. The resulting formulation reveals the intimate relationship between governing equations derived in both micro- and macroscales. Section 3 is devoted to a discussion of various implications of the variational problems posed in Section 2. Three typical static equilibrium equations of the heterogeneous body are considered; the one for elastostatics with linear kinematics, the one with local contact conditions that leads to nonlinear kinematics in a microscale and the other is for elastoplasticity with the two-scale variational formulation. Several mathematical features of multi-scale modeling are reviewed for the development of the computational algorithms and the numerical analyses in 4 Multi-scale computational strategy with successive iteration schemes, 5 Multi-scale computational strategy with tangent moduli. The algorithms presented in this paper may involve intuitive ones as well as elaborate computational techniques. Thus, our presentation employing the generalized variational principle with suitable functional setting provides a new insight into the fundamental ideas of multi-scale modeling by the homogenization theory in the context of computational inelasticity.