Elsevier

Applied Mathematics and Computation

Volume 224, 1 November 2013, Pages 450-462
Applied Mathematics and Computation

Numerical material representation using proper orthogonal decomposition and diffuse approximation

https://doi.org/10.1016/j.amc.2013.08.052Get rights and content

Abstract

From numerical point of view, analysis and optimization in computational material engineering require efficient approaches for microstructure representation. This paper develops an approach to establish an image-based interpolation model in order to efficiently parameterize microstructures of a representative volume element (RVE), based on proper orthogonal decomposition (POD) reduction of density maps (snapshots). When the parameters of the RVE snapshot are known a priori, the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters. Otherwise, a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients. We showcase the approach and its potential applications by considering a set of two-phase composite snapshots. The choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization.

Introduction

The constant increase of computing power coupled with easier-than-ever access to high performance computing platforms enables the computational investigation of materials at the microscopic level: microstructure generation and modeling [1], [2], material property prediction and evaluation [3], [4], [5], multi-scale analysis [6], [7], [8], [9], and within a stochastic framework to include the effects of the input uncertainties at the material level [10]. At the same time, the progress made in the field of material science allows us to control the material microstructure composition to an unprecedented extent [11], [12].

Recently, image-based microstructure modeling and analysis have attracted the interest of more and more researchers. One category of research employs voxel-based finite element models using a mesh that is automatically built by converting each voxel into a finite element [13], [14], [15]. Given the high computational cost of a voxel-based approach, another theme of research generates image-based microscopic models by incorporating level-sets and the extended finite element method (XFEM) [16], [17]. A comparison of the two approaches has been made in a recent work [18]. Thanks to the proposed analysis approaches, the material constitutive behavior can be predicted by imaged-based numerical models. However, access to microstructural images is economically expensive by experiments or time-consuming by numerical simulations [19], [20], [21]. Therefore, there is a great demand for an economical and efficient approach to generate microstructure images.

Techniques of model reduction have been widely used in the fields computational mechanics and multidisciplinary optimization [22], [23]. The surrogate models have also been applied in complex structural and shape designs [24], [25], [26]. Recently [27], [28] used surrogate modeling in structural shape optimization using the coefficient manifolds to reduce the input space. To the knowledge of the authors, the literature reveals little investigation into developing microstructure image generation approach using model reduction techniques. In [29], [30], principal component analysis (PCA) has been applied to reduce the parametric space constructed by a large-dimensional data set using the so called method of snapshots [31], [32]. Each snapshot may be represented as a combination their retained number of eigen-images.

Suppose we have means to generate microstructure representations for instance by varying processing parameters (numerical or experimental, etc.), the goal of this work is to establish a parameterized geometrical description from learning the set of given instances. Using the method of snapshots, microstructure snapshots are represented in terms of a POD basis, where the number of modes retained is decided after taking both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization into consideration. The geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients using the method of moving least squares (MLS) [33], also called diffuse approximation [34]. Two cases are considered:

  • When the parameters v=[v1,v2,]T of the RVE are known a priori as shown in Fig. 1, the parameterized microstructure representation is given by a series of response surfaces of the projection coefficients in terms of v.

  • When the parameters are unknown a priori, a set of pseudo parameters η=[η1,η2,]T corresponding to the detected dimensionality of the set of M snapshots (Fig. 1) are locally taken from learning the manifolds of the projection coefficients, and then represent the parametric space using the approximated manifolds with respect to η.

The ultimate goal of our work is to propose an unified approach for reduced order material representation and behavior; the latter aspect is object of active research [35], [36], [37]; in the present article we concentrate on the material representation part of the work. In this paper, the proposed material representation model is applied to numerical homogenization analysis of a set of two-phase composite materials snapshots. The model can also be further applied to analysis such as localized failure in heterogeneous materials [38] and optimal design of the nonlinear behavior of the considered composite [39].

The remainder of this paper is organized in the following manner: Section 2 presents the POD-based interpolation approach for the RVE snapshots. Numerical homogenization in the framework of periodic boundary conditions (PBC) is introduced in Section 3. We showcase the approach and its potential applications by considering a series of two-phase composite RVEs with numerical homogenization in Section 4. The paper ends with concluding comments and suggestions for future work.

Section snippets

POD-based RVE interpolation

Without loss of generality, consider a 2D real-valued continuous or discrete material phase indicator function s=s(x,y,v) depending on parameters vRp, where x and y are the coordinates. Given an N×N grid of sampling points by the indicator function [S(v)]i,j=s(x(i),y(j),v), i=1,,N, j=1,,N, continuous representation could be discretized giving a representation matrix S(v), whose precision depends on the resolution, as shown in Fig. 2.

Numerical homogenization

In this work, we assume that the appropriate RVE has been already selected so as to satisfy the conditions: large enough to be considered in the framework of continuum mechanics, and at the same time small enough to be considered surrounded by its own pattern ordered periodically, as illustrated in Fig. 6. Based on [45], the strain energy predicted by the different boundary conditions satisfy the following inequality if the average strain ε¯ for each case is assumed to be the same.ε¯:Etrh:ε¯ε¯:

Numerical test case

In this section, we consider a commonly analyzed periodic two-phase microstructure pattern as shown in Fig. 7. The data table gives the parametric information of the seven elliptical inclusions. Microstructures of this form may be used to model various types of materials, such as fiber composites, reinforced alloys, porous aluminum, and quasi-brittle materials [49], [50], [8], [51]. By relating the parameters of the inclusions to two random variables v=[v1,v2]T, v1,v2[0,1], 400 binary

Conclusion and perspectives

In this work, we have proposed a POD-based parameterization approach for material microstructures RVE interpolation. Using this approach, snapshots of the microstructure RVE can be interpolated for an arbitrary given value of the parameters or pseudo parameters with a sufficient accuracy of representation. The numerical tests have demonstrated the effectiveness and potential applications of the proposed method. As has been demonstrated by the application to numerical homogenization, this

Acknowledgements

This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program Investments for the future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02) and funded by the China Scholarship Council (CSC). The fourth author is supported by the 973 Project (2012CB025904) and National Natural Science Foundation of China (51221001).

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