Chapter 2 Literature review
2.2 Topology optimization for continuum periodic structures
While most of the researches done in this area deal with material microstructure design, it is reasonable to start this section with periodic structure design in the micro scale and then make a transition to the less researched periodic optimal design for macro- structures that this thesis is mainly concerned with.
• Micro scale
The basic idea of designing material microstructures is that the periodic stress and strain fields are assumed due to micro level of problems, then it is possible to analyse a single cell of the material by the finite element analysis and perform topology optimi- zation on the single cell (usually termed the periodic base cell or PBC) instead of the whole structure. In the design problem of materials, the mathematical homogenization as a technique to smear out complicated microstructural behaviours is usually used to determine the PBC’s macroscopic properties such as in (Benssousan et al. 1978; Bakhvalov and Panasenko 1989; Aboudi 1991). For more complicated microstructures, analytical determination of material properties is extremely difficult, therefore often finite element based numerical homogenization methods are used to obtain the mate- rial properties such as in (Bourgat 1977; Guedes and Kikuchi 1990), where several independent loading cases are applied on PBC, such as simulation of the shearing and tension in both directions in two dimensional cases.
Sigmund (1994a; 1994b; 1995) proposed a method for designing materials with pre- scribed elastic/thermoelastic properties by topology optimization of a periodic material
microstructure represented by a base cell (PBC). The PBC was modelled by truss, frame or continuum elements. This method used the weight as the objective function and the cross-sectional sizes or densities for corresponding elements as the design variable. The prescribed elastic properties acted as the optimization constraint. The topology optimization problem was solved by mathematical homogenization method that made the material design problem defined as an inverse homogenization problem. In Bendsøe and Sigmund’s work (Bendsøe and Sigmund 1999) of exploring various material interpolation schemes in topology optimization, the SIMP material model was applied to inverse homogenization for microstructure design of materials with specific material properties.
Sigmund and Torquato (1997) proposed an approach for designing materials with ex- treme thermal expansion through a three-phase topology optimization method. Com- posites of two solid phases and a void phase were dealt with within a PBC where the distribution of solid phases are sought for optimization of thermo-elastic properties such as maximizing directional thermal expansion, zero isotropic thermal expansion and the negative isotropic thermal expansion, through the mathematical homogeniza- tion method and the sequential linear programming.
Neves et al. (2002) presented a two-scale asymptotic method for the linearized elastic stability analysis. An average strain at the macro scale level is considered and an ei- genvalue criterion at the micro level is used to assess the PBC subject to highly local- ized and periodic buckling modes. Topology optimization of the periodic microstruc- tures is carried out based on the local buckling instabilities in the PBC. Combining the linearized elastic buckling model with the inverse homogenization and an eigenvalue buckling analysis with Floquet-bloch wave theory (Neves 2006), the minimum critical
buckling strain is obtained and maximized with the PBC having a constant volume fraction of materials.
Guest and Prévost (2007) proposed a method for designing material considering the permeability property. Microstructures of two-phases for solid and void/fluid were considered and the layout for both phases was determined in this work. The objective is to maximize the permeability of the material with prescribed flow symmetries in the bulk material. The permeability is computed with finite element based numerical ho- mogenization method on the PBC. Solutions obtained are found to be characterized with simply connected pore spaces that are closely similar to minimal surfaces.
In the recent research by Zhou and Li (2008) proposed an approach for designing graded two-phase microstructures by an inverse homogenization method. This work presented the design of solid/void functionally graded material microstructures with PBCs varying in a direction parallel to the property gradient and periodically repeating in the perpendicular direction. As the PBCs are not identical in the gradient direction, the design problem may be based on each individual PBC or across a limited umber of PBCs as a whole. The effectiveness of this approach was demonstrated through exam- ples with constant positive or negative Poisson’s ratios.
• Macro scale
Compared with the research on topology optimization of cellular microstructures, ex- isting research on topology optimization of continuum periodic structures in the macro scale is very limited. The essential difference between periodic structures in micro scale and macro scale is that microstructures have periodic stress and strain distribu- tion that is not guaranteed in macrostructures. Therefore it is not possible to directly
extract a single base cell from the whole periodic macrostructure for analysis and to- pology optimization.
Moses and co-workers carried out the work bridging the topology optimization for infinitely small microstructures and topology optimization for macrostructures with infinitely periodicity. First Moses and Fuchs et al. (Fuchs and Ryvkin 2001; Moses et
al. 2001) proposed a finite element methodology for analysing infinite periodic struc-
tures. Unlike periodic microstructures of materials, the macrostructures are subject to general loadings, i.e. no assumption for periodic stress and strain distribution is made. Using the discrete Fourier transformation, loadings and boundary conditions are trans- formed onto a single representative cell, based on which the displacements and stresses are to be calculated. For obtaining the real system response, stresses are com- puted through an inverse Fourier transformation back to the real structure. Based on this FE methodology, Moses et al. (2003) proposed a method for topology optimiza- tion of infinite periodic structures under arbitrary loadings. Combing the SIMP method with the FE analysis on the representative cell constructed by the discrete Fourier transformation, topology optimization on the module (representative cell) is realized. Then duplicating the module can produce the real structure in macro scale. This method was applied to stiffness optimization and the influence of the periodic con- straints on the performances of the optimal design is examined through a disc example under a single point load. This method is able to reduce optimization of the whole structure to optimization of a single module but confined to the scope of infinite struc- tures.
Acknowledging that the optimal solution relies a lot on the scale effect modelling of the periodic microstructure of material unit cell and that in the asymptotic homogeni-
zation method effective predicted material properties can give rise to limit values de- pending only on the volume faction, properties and spatial distribution of constituents regardless of scale effect, Zhang and Sun (2006) proposed an approach integrating the optimization of lightweight cellular materials and macrostructures. The design element (DE) was proposed to deal with optimal design problems for both materials and struc- tures in a unified way. The procedure consists of two phases: the preliminary design in the macro scale and the design for material microstructures. The design problem will become a macrostructure topology optimization when there is only one cell in the structure, and a pure material microstructure optimization when the periodicity ap- proaches to infinity. This approach was applied to stiffness optimization and several promising numerical results were obtained.
Huang and Xie (2008) proposed an approach to topology optimization of general con- tinuum periodic structures in the macro scale using BESO. The algorithm is quite straightforward and simple to implement. Similar with Moses’s approach (Moses et al. 2003), the basic idea is to reduce the real structure to a module where topology optimi- zation is performed. An imaginary representative unit cell (RUC, the “module”) is created and discretized such that the finite element mesh is exactly the same as that in all the single unit cells of the real structure. The static analysis is performed and the sensitivities are calculated based on the real structure. The sensitivities from all the unit cells are then averaged and assigned to the RUC. Element removal and addition are executed based on the sensitivity ranking and thus an iterative topology optimiza- tion process is able to be realized on the RUC. The final optimal design of the whole structure is constructed by duplicating the optimal RUC. Unlike the method of Moses
structures. Huang and Xie (2008) applied this approach to stiffness optimization and results from numerical examples compared well with those from Zhang and Sun’s work (Zhang and Sun 2006).
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