2.1. Structural Topology Optimization
2.1.6 Bi-directional Evolutionary Structural Optimization (BESO)
With the goal of improving the search capability of the original ESO, the Bi-directional Evolutionary Structural Optimization (BESO) aims at simultaneously adding or removing elements from the finite element model of the structure. In the ESO, because the inefficient elements are completely removed from the structure, there is no information about the effects
of these elements on the objective function, in later stages of optimization. The general idea of the BESO is to devise a scheme to restore the deleted elements, if necessary. The BESO approach can be seen as a significant development that has resulted from studies on the ESO.
All of the BESO schemes that have been introduced so far apply the idea of ground structure (Dorn et al., 1964), in which its elements covers the whole design domain including solid and void regions. The BESO turns these elements on and off, but keeps the record of their geometrical information through the whole optimization procedure. The primary schemes on the improved ESO algorithm were suggested by Querin et al. (1998, 2000a, 2000b) and Yang et al.(1999) and further improvement by introducing the enhanced hard-kill (Huang and Xie, 2007b) and soft-kill BESO methods (Huang and Xie, 2009a).
2.1.6.1. Hard-kill BESO
In contrast to the ESO which gradually removes the inefficient elements from the finite element model of the structure, the “Additive Evolutionary Structural Optimization” (AESO) has been developed with the aim of generating optimum structures by starting from a minimum ground structure and gradually adding elements to it (Querin et al., 1998,2000a). In this method, the elements are added around the free edges surrounding the most efficient elements. The most efficient elements are selected among the elements with highest stress or sensitivity numbers (Querin et al., 2000a). ESO has been combined with AESO in order to develop a BESO (Querin et al., 2000a). In each iteration, the numbers of added or removed elements are controlled by two given parameters, namely, the inclusion ratio (IR) and rejection ratio (RR) respectively.
In another BESO that has been introduced by Yang et al. (1999), the criteria for adding or removing elements was based on their effects on the variation of the objective functions. As mentioned before such effects are expressed by sensitivity numbers. For solid elements, the sensitivity number is calculated based on the results of structural analysis. For void elements the nodal displacement is calculated by extrapolating the nodal displacements of their surrounding solid elements. The sensitivity number of the void elements can then be calculated using these extrapolated nodal displacements. The procedure follows by the ranking of elements based on the magnitude of their sensitivities and switching to solid for elements with higher sensitivities and to void for elements with lower sensitivity numbers. Similar to the previous method, the numbers of removed and added elements are treated with different criteria, through introducing the rejection ratio and an inclusion ratio.
As it was discussed earlier, the optimization with a solid-void material distribution is an ill- posed problem. Such an optimization is dependent on the selection of the elements’ sizes and discretization mesh (Bendsøe and Kikuchi, 1988). One drawback of these early approaches is that the numerical instability is not addressed properly and computational efficiency is low, due to the convergence problem (Rozvany, 2009, Huang and Xie, 2010b, 2010c). It has also been noticed that the best solution needs to be selected among several topologies that can be generated by varying RR and IR (enumeration method) (Huang and Xie, 2010b, Rozvany, 2009).
In 2007, Huang and Xie (2007b) developed a new algorithm for the hard-kill BESO, in which many issues such as a proper statement of optimization problem and numerical instability (see section 2.1.7) of the procedure has been addressed (Huang and Xie, 2010b). Suppose that the
aim of optimization is to find the stiffest structure under volumetric constraint. In the hard-kill BESO setting the optimization problem statement is defined as:
Minimize: f(x)=K (2.20.a)
Subject to: (2.20.b)
(2.20.c)
in which the design variable indicates the absence (0) or presence (1) of the element in the model. In contrast to the SIMP approach here, the elements itself is considered as the design variable. Huang and Xie (2007b) have used a filtering scheme to extrapolate the sensitivity number of voids. The filtering is performed by using the following weighting equation
∑
∑
= = = N j ij N j i ij i w w 1 1 ˆ α α (2.21)in which is the total number of finite elements in structural model and is the calculated sensitivity number. The weight factor of is defined as:
(2.22) 0 1 *−
∑
= = N i i ix V V 1 or 0 = i x i x Nα
i wij − < = otherwise 0 if min min r r r r wij ij ijin which denotes the distance between element and element centres. The filter radius of rmin is to identify the neighbouring elements that affect the sensitivity of element i.
The sensitivity numbers of void elements are set to be zero initially, and then modified through the filter scheme. The addition and removal of elements is based on the ranking of elements, followed by switching elements to void for elements with lower sensitivity numbers and solid for elements with higher sensitivity numbers.
The above mentioned filtering scheme, together with incorporating the historical information of elemental sensitivities, has shown to be able to overcome a great deal of the numerical instabilities, which had been a controversial problem of the original versions (Huang and Xie, 2010b, Zhou and Rozvany, 2001, Rozvany et al., 2006, Rozvany, 2009). On the other hand the unified criterion for adding and removing elements, offers an explicit control over the volumetric constraint. The new hard-kill BESO also have a very high computational efficiency, as the results of the mentioned improvements, as well as the fact that the eliminated elements are not involved in finite element analysis (Huang and Xie, 2010b).
2.1.6.2. Soft-kill BESO
In the hard-kill BESO, solid elements can only grow in the proximity of the existing elements, which in some cases may failed in rectifying the incorrect elemental rejection (Rozvany, 2001b, Zhu et al., 2007, Zhou and Rozvany, 2001). The complete removal of elements also may cause some theoretical predicaments, especially in multi-physics problems (Sigmund, 2001, Zhu et al., 2007, Huang and Xie, 2010b). An alternative approach can be the assigning of very small density for the void elements (Hinton and Sienz, 1995). The strain values of
these elements can then be directly calculated; hence the solid elements can grow in the desired regions of the structure away from existing solid regions (Zhu et al., 2007, Rozvany, 2001b).
Hinton and Sienz (1995) devised a fully stressed Bi-directional approach based on ESO, in which, instead of completely removing elements, they are replaced by elements with lower elastic modulus of the order 10-6. Zhu et al. (2007) developed a sensitivity based BESO method, in which the void elements are replaced by a microstructural system known as Orthotropic Cellular Microstructure (OCM). The OCM has a low density and in this approach for adding or removing elements, they are assigned as OCM’s or solid elements respectively. The numerical stability is addressed through a scheme, to limit the number of connected solid elements along each principal direction (Zhu et al., 2007). However the convergence of both approaches encounter difficulties (Huang and Xie, 2010b).
In the soft-kill BESO proposed by Huang and Xie (2009a), the design variable is limited to a minimum value (e.g. 0.001). That means the void elements are not completely removed from the structure. Therefore the equation (2.20.c) is replaced by:
(2.23)
The Optimality Criterion for stiffness optimization is applied based on the sensitivity of elements with respect to the objective function. To improve the convergence of the procedure the effective property (for example Young’s modulus in stiffness optimization) is determined though a power-low material interpolation scheme (Bendsøe, 1989):
i x min x 1 or min i x x =
(2.24)
in which is the Young’s modulus of the solid material. It is shown that by selecting the penalty exponent large enough the convergence of the procedure considerably improves. The results obtained from the soft-kill BESO shows similarities to those of the hard-kill BESO (Huang and Xie, 2007b) which can be considered as the justification of the validity of hard-kill methodology (Huang and Xie, 2010b).
Although the new soft-kill BESO has been introduced very recently (Huang and Xie 2009a; 2010b), it has shown its capability for solving a wide range of shape and topology optimization problems with high computational efficiency. In the following chapters this methodology will be extended into the design of microstructures for materials.