Filter Scheme and Improved Sensitivity Number

3.3.1

Checkerboard and Mesh-dependency Problems

When a continuum structure is discretized using low order bilinear (2D) or trilinear (3D) finite elements, the sensitivity numbers could become C0discontinuous across element boundaries. This leads to checkerboard patterns in the resulting topologies (Jog and Harber 1996). Figure 3.1 shows a typical checkerboard pattern of a continuum structure from the original ESO method. The presence of checkerboard pattern causes difficulty in interpreting and manufac- turing the ‘optimal’ structure. To suppress the formation of checkerboard patterns in the ESO method, a simple smoothing scheme of averaging the sensitivity numbers of neighbouring elements has been presented by Li et al. (2001). However, this smoothing algorithm cannot overcome the mesh-dependency problem.

The so-called mesh-dependency refers to the problem of obtaining different topologies from using different finite element meshes. When a finer mesh is used, the numerical process of structural optimization will produce a topology with more members of smaller sizes in the final design. Ideally, mesh-refinement should result in a better finite element modelling of the same optimal structure and a better description of boundaries – not in a more detailed or qualitatively different structure (Bendsøe and Sigmund 2003).

Various techniques have been suggested to overcome the mesh-dependency problem such as the perimeter control method (Harber et al. 1996; Jog 2002) and the sensitivity filter scheme (Sigmund 1997; Sigmund and Petersson 1998). The BESO method with perimeter control (Yang et al. 2003) is demonstrated to be capable of obtaining mesh-independent solutions

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20 Evolutionary Topology Optimization of Continuum Structures

Figure 3.1 A typical checkerboard pattern in the ESO method.

because of one extra constraint (the perimeter length) on the topology optimization problem. However, predicting or selecting an appropriate value of the perimeter length for a new design problem can be a difficult task. Thus, the sensitivity filter scheme will be introduced into the new BESO method.

3.3.2

Filter Scheme for BESO Method

Before applying the filter scheme, nodal sensitivity numbers which do not carry any physical meaning on their own are defined by averaging the elemental sensitivity numbers as follows:

αn j = M
i=1 wiαei (3.4)

where M denotes the total number of elements connected to the jth node. wi is the weight

factor of the ith element andiM₌₁wi = 1. wi can be defined by

wi = 1 M− 1 ⎛ ⎜ ⎜ ⎜ ⎝1− ri j M  i=1 ri j ⎞ ⎟ ⎟ ⎟ ⎠ (3.5)

where ri j is the distance between the centre of the ith element and the jth node. The above

weight factor indicates that the elemental sensitivity number has larger effect on the nodal sensitivity number when it is closer to the node.

The above nodal sensitivity numbers will then be converted into smoothed elemental sen- sitivity numbers. This conversion takes place through projecting nodal sensitivity numbers to the design domain. Here, a filter scheme is used to carry out this process. The filter has a length scale rminthat does not change with mesh refinement. The primary role of the scale parameter

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Bi-directional Evolutionary Structural Optimization Method 21

Figure 3.2 Nodes located inside the circular sub-domaini are used in the filter scheme for the ith

element.

rmin in the filter scheme is to identify the nodes that will influence the sensitivity of the ith element. This can be visualized by drawing a circle of radius rmin centred at the centroid of

ith element, thus generating the circular sub-domaini as shown in Figure 3.2. Usually the

value of rmin should be big enough so that i covers more than one element. The size of

the sub-domainidoes not change with mesh size. Nodes located insideicontribute to the

computation of the improved sensitivity number of the ith element as

αi = K  j₌₁ w(ri j)αnj K  j₌₁ w(ri j) (3.6)

where K is the total number of nodes in the sub-domaini,w(ri j) is the linear weight factor

defined as

w(ri j)= rmin− ri j ( j= 1, 2, . . . , K ) (3.7)

It can be seen that the filter scheme smoothes the sensitivity numbers in the whole design domain. Thus, the sensitivity numbers for void elements are automatically obtained. They may have high values due to high sensitivity numbers of solid elements within the sub-domaini.

Therefore, some of the void elements may be changed to solid elements in the next iteration. The above filter scheme is similar to the mesh-independency filter used by Sigmund and Petersson (1998) except that Equation (3.6) uses the nodal sensitivity numbers rather than elemental sensitivities. It is noted that Sigmund and Petersson (1998) include the density of the element in the filter, thus the sensitivity number for void elements will be infinite.

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22 Evolutionary Topology Optimization of Continuum Structures

However, the present filter given in Equation (3.6) does not consider the element status (void or solid), and the initial sensitivity number for void elements is set to zero. Using the above filter technique, nonzero sensitivity numbers for void elements are obtained as a result of filtering the sensitivity numbers of neighbouring solid elements. Using the obtained sensitivity numbers, the void elements can be ranked alongside the solid elements in terms of their structural importance.

The filter scheme is purely heuristic. However, by adopting this simple technique, many numerical problems in topology optimization, such as checkerboard and mesh-dependency, can be effectively overcome. It produces results very similar to those obtained by applying a local gradient constraint (Bendsøe and Sigmund 2003). The filter scheme requires little extra computational time and is very easy to implement in the optimization algorithm.

3.3.3

Stabilizing the Evolutionary Process

As will be demonstrated later in this chapter, the adoption of the above filter scheme can effectively address the mesh-dependency problem. However, the objective function and the corresponding topology may not be convergent. Let us consider a short cantilever example similar to the one shown in Figure 2.7 using a coarse mesh of 32 × 20 elements. With ESO/BESO methods, large oscillations are often observed in the evolution history of the objective function, as illustrated in the Figure 3.3(a). The reason for such chaotic behaviour is that the sensitivity numbers of the solid (1) and void (0) elements are based on discrete design variables of element presence (1) and absence (0). This makes the objective function and the topology difficult to converge. Huang and Xie (2007) has found that averaging the sensitivity number with its historical information is an effective way to solve this problem. The simple averaging scheme is given as

αi = αk i + α k−1 i 2 (3.8)

where k is the current iteration number. Then let αk

i = αi which will be used for the next

iteration. Thus, the updated sensitivity number includes the whole history of the sensitivity information in the previous iterations. Figure 3.3(b) shows the evolution history obtained by adopting the stabilization scheme defined in equation (3.8). Compared to the result in Figure 3.3(a), the new solution is highly stable in both the topology and the objective function (the mean compliance) after the constraint volume fraction (50 %) is achieved. It is worth pointing out that whilst Equation (3.8) affects the search path of the BESO algorithm it has very little effect on the final solution when it becomes convergent. Details of the parameters used in this example can be found in Huang and Xie (2007).