Elsevier

Computational Materials Science

Volume 67, February 2013, Pages 397-407
Computational Materials Science

Topology optimization of microstructures of cellular materials and composites for macrostructures

https://doi.org/10.1016/j.commatsci.2012.09.018Get rights and content

Abstract

This paper introduces a topology optimization algorithm for the optimal design of cellular materials and composites with periodic microstructures so that the resulting macrostructure has the maximum stiffness (or minimum mean compliance). The effective properties of the heterogeneous material are obtained through the homogenization theory, and these properties are integrated into the analysis of the macrostructure. The sensitivity analysis for the material unit cell is established for such a two-scale optimization problem. Then, a bi-directional evolutionary structural optimization (BESO) approach is developed to achieve a clear and optimized topology for the material microstructure. Several numerical examples are presented to validate the proposed optimization algorithm and a variety of anisotropic microstructures of cellular materials and composites are obtained. The various effects on the topological design of the material microstructure are discussed.

Highlights

► A new BESO algorithm is proposed for a two-scale optimization problem. ► Optimal designs of materials for macrostructures with the maximum stiffness. ► A clear 0/1 topologies for cellular materials/composites are achieved. ► Various interesting anisotropic microstructures of materials are obtained.

Introduction

Many industrial and engineering materials are heterogeneous and consist of dissimilar constituents that are distinguishable at some length scales. The behavior of those heterogeneous materials is determined, on the one hand, by the relevant materials properties of the constituents and, on the other hand, by their topology at the micro-scale level. The theory of homogenization has been recognized as a rigorous modeling methodology for characterizing the mechanical behavior of cellular materials and composites with periodic microstructures [1], [2], [3]. The inverse problem is to design a new microstructure of the periodic representative unit cell (RUC) or representative volume element (RVE) so that the resulting material has desirable physical properties [4], [5]. A systematic and scientific means of microstructural design is formulated as an optimization problem for the parameters that represent the material properties and topology of the material microstructure.

Over the last two decades, various topology optimization algorithms, e.g. homogenization method [6], solid isotropic material with penalization (SIMP) [7], [8], [9], evolutionary structural optimization (ESO) [10], [11], and level set technique [12], [13] have been developed. These topology optimization techniques have been used extensively to solve the design problems not only for macroscopic structures, but also for microstructures of materials/composites in recent years. For instance, tailoring microstructures of materials with prescribed constitutive properties has been investigated using the SIMP method [14], [15], genetic algorithms [16], and the level set method [17]. Some attempts have also been made to design new materials with extraordinary physical properties, e.g., extremal thermal conductivity [18], [19], maximum fluid permeability [20], maximum stiffness and fluid permeability [21], and maximum stiffness and thermal conductivity [22].

Unlike the continuous density-based topology methods, the ESO/BESO methods represent the structural topology and shape with discrete design variables (solid or void) with a clear structural boundary [11], [23]. ESO was originally developed based upon a simple concept of gradually removing redundant or inefficient material from a structure so that the resulting topology evolves towards an optimum. A later version of the ESO method, namely the bi-directional evolutionary structural optimization (BESO) method, allows not only removing materials, but also adding materials to the design domain. It has been demonstrated that the current BESO method is capable of not only generating reliable and practical topologies for macrostructures [23], [24], but also creating optimal microstructures for materials/composites with high computational efficiency [25], [26].

Optimal microstructures obtained by the material design are only optimum in terms of desirable material properties. Hence, the structure constructed from the resulting materials may not be efficient or optimal since the service conditions such as applied loading, boundary condition are varied in practical use. Obviously, topology optimization of materials for structures is in essence a multi-scale problem which should consider the performance of macrostructures and topologies of material microstructures simultaneously. In this context, composite materials of two-dimensional structures are designed using the homogenization design method [27]. Rodrigues et al. [28] proposed a hierarchical computational procedure by integrating the macrostructures with local material microstructures using the SIMP method. Zhuang and Sun [29] studied the integrated optimization of cellular materials for sandwich panels. In order to design lightweight structures, Liu et al. [30] introduced a concurrent topology optimization of materials and structures using the porous anisotropic material with penalization (PAMP) model.

In this paper, a topology optimization approach based on BESO is proposed for designing microstructures of cellular materials and composites for structures. The optimization problem is formulated with the objective at the macro-scale level but with constraints at the micro-scale level. Two-scale finite element analyses of structures and materials are conducted and the sensitivity analysis is established for optimally designing microstructures of materials. Then, the BESO method is applied for iteratively updating the microstructures of the material. Finally, some examples are presented to illustrate the effectiveness of the proposed algorithm for designing cellular materials and composites for structures.

Section snippets

Optimization problem and material interpolation scheme

We consider a macrostructure with the known boundary conditions and external forces as illustrated by Fig. 1a. The macrostructure is composed of cellular material or two-phase composite with uniform microstructures (Fig. 1b) repeated periodically with the base cell (Fig. 1c). The optimization objective is to find the spatial distribution of each phase within the base cell so that the resulting macrostructure has the best load-carrying capability for the given weight or volume fraction of

Finite element analysis and sensitivity analysis

The finite element analysis should be conducted both for the structure at the macro-scale and the material base cell at the micro-scale. Generally, the macro-scale analysis of the structure is to determine the structural deformation and evaluate the objective function. The static behavior of the structure is represented by the following macro FE equilibrium equationKU=Fwhere K is the stiffness matrix of the macrostructure which is assembled by the elemental stiffness matrix, KiKi=ViBDHBdVi

Numerical implementation and BESO procedure

In the BESO method, the sensitivity numbers which denote the relative ranking of the elemental sensitivities will be used to update the design variables of the material unit cell. Therefore, the sensitivity number of the jth element in the material base cell for minimizing the mean compliance of the macrostructure can be defined with the elemental sensitivity multiplying a constant, −1/p, asαj=-1pdCdxj=xjp-12|Y|i=1MUiTViBTYj(I-bu)T(D1-D2)(I-bu)dYjBdViUi

Because the design variable xj is

Results and discussion

In this section, we will present some examples for designing microstructures of cellular materials or two-phase composites for structures. The square base cell is discretized into 100 × 100 4-node quadrilateral elements which represents the microstructure of the materials. To start the optimization procedure, two different initial designs are considered: initial design 1 is full with phase 1 material except for four elements at the center of the base cell with phase 2 material; initial design 2

Conclusions

This paper has developed an optimization approach for a two-scale optimization problem which designs a microstructure of a cellular material or composite for a given volume fraction so that the resulting macrostructure has the maximum stiffness. The effective elastic properties of the material with periodic microstructure are integrated into the finite element analysis of the macrostructure. Then, the resulting displacement field of the structure is integrated into the microstructural design of

Acknowledgement

This research is supported by the Australian Research Council under its Discovery Projects funding scheme (Project Number DP1094403).

References (35)

  • A. Sangani et al.

    J. Mech. Phys. Solids

    (1987)
  • K. Terada et al.

    Comput. Methods Appl. Mech. Eng.

    (2001)
  • B. Hassani et al.

    Comput. Struct.

    (1998)
  • B. Hassani et al.

    Comput. Struct.

    (1998)
  • M.P. Bendsøe et al.

    Comput. Methods Appl. Mech. Eng.

    (1988)
  • Y.M. Xie et al.

    Comput. Struct.

    (1993)
  • J.A. Sethian et al.

    J. Comput. Phys.

    (2000)
  • M.Y. Wang et al.

    Comput. Methods Appl. Mech. Eng.

    (2003)
  • O. Sigmund

    Mech. Mater.

    (1995)
  • O. Sigmund

    Int. J. Solids Struct.

    (1994)
  • J.K. Guest et al.

    Comput. Methods Appl. Mech. Eng.

    (2007)
  • J.K. Guest et al.

    Int. J. Solids Struct.

    (2006)
  • N.E. de Kruijf et al.

    Int. J. Solid Struct.

    (2007)
  • X. Huang et al.

    Topological design of microstructures of cellular materials for maximum bulk or shear modulus

    Comput. Mater. Sci.

    (2011)
  • L. Liu et al.

    Comput. Struct.

    (2008)
  • Y.H. Chen et al.

    Comput. Struct.

    (2010)
  • X. Huang et al.

    Finite Elem. Anal. Des.

    (2007)
  • Cited by (176)

    • Concurrent topology optimization of multiscale piezoelectric actuators

      2024, International Journal of Solids and Structures
    View all citing articles on Scopus
    View full text