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Checkerboard instability refers to the formation of checkerboard-like patterns

made of alternating solid and void (or hard and soft) elements. This problem

was investigated by D´ıaz and Sigmund (1995) and Jog and Haber (1996).

They demonstrated that the reason for checkerboard formation is that these

patterns gain artificially high stiffness due to numerical problems in the mixed

formulation of the finite element discretisation. An example of checkerboard

patterns is given in Figure 2.23. This topology is obtained by the SIMP

method solving exactly the same problem already solved in section 2.5.3.

All the parameters have been selected similar to the original problem, but

instead of high order 9-node elements, here, 4-node elements have been used.

The homogenisation, the SIMP, the ESO, and the original BESO methods

Figure 2.23 The checkerboard instability in a result obtained by the SIMP method for the SCB problem. Instead of 9-node elements, 4-node elements have been used here.

element formulation which is used in all of these methods. However, there

are several ways to prevent formation of checkerboard patterns. Here some

of the well-known techniques will be introduced.

Using stable elements

Jog and Haber (1996) proposed a patch test to identify unstable elements

which can cause the checkerboards formation in topology designs. Also D´ıaz

and Sigmund (1995) presented some guidelines for choosing stable elements.

It has been shown in both papers that higher order finite elements can prevent

checkerboard formation. In the examples already presented in this chapter,

9-node and 8-node high-order finite elements have been used and no checker-

board pattern has been found in the final topologies. However in the SIMP

method this approach does not guarantee a checkerboard-free result if one

applies big penalty values (D´ıaz and Sigmund 1995). Also this approach

might not work with the ESO and the BESO methods. Furthermore, us-

effort compared to 4-node elements.

Special types of non-conforming finite elements may also help preventing

checkerboard formation. Jang et al. (2003) reported checkerboard-free results

using non-conforming 4-node finite elements with their nodes located in the

middle of the edges rather than corners.

filtering techniques

The use of filtering techniques to overcome checkerboard formation has been

first studied by Sigmund (1994). Although no rigorous proof was ever pro-

posed for this approach, it can successfully prevent checkerboard formation

with low computational effort. The filtering scheme is similar to filtering

techniques used in image processing. This technique was originally used in

the SIMP method. However Li et al. (2001) successfully applied a similar

technique to the ESO method. The linear filter used in the BESO algorithm

(2.34) is also capable of preventing checkerboard formation. As mentioned

before, in the BESO method, the filtering scheme plays an additional role of

assigning sensitivity numbers to voids as well.

The short cantilever beam problem is solved here using the SIMP method

with 4-node finite elements but sensitivities are filtered using a radius of twice

of the size of the elements. In the SIMP method the filtered sensitivities are

calculated using the following formula

d∂c ∂ρi = PN j=1ρj ∂c ∂ρjwij ρi PN j=1wij (2.38)

in Figure 2.24. Note the blurred boundaries in this figure. This is due to

the applied filtering scheme which assures a smooth transition from solids to

voids exists.

Figure 2.24 Preventing formation of checkerboard patterns using filtering techniques. The filtering radius is twice of the elements’ size.

An approach similar to filtering was proposed by Ghabraie (2005) to

overcome checkerboards instability in the SIMP method. In this approach

the sensitivities at nodes are calculated by averaging the sensitivities of the

elements connected to them. Then the sensitivities of the elements are re-

calculated by averaging the resulted nodal sensitivities.

Perimeter control

Another way to control checkerboards is to impose a constraint on the perime-

ter. Between two topologies with the same amount of material the one with

less holes has a shorter perimeter. By setting an upper limit on the perimeter,

one can thus control the complexity of the resulted topologies and prevent

checkerboards. This approach was first used by Haber et al. (1996). Perime-

Other techniques

Apart from the abovementioned techniques, several other approaches have

been proposed that can deal with the checkerboards problem. Guest et al.

(2004), Rahmatalla and Swan (2004), and Matsui and Terada (2004) used

nodal sensitivities instead of elemental ones and reported checkerboard-free

topologies using the SIMP and the homogenisation methods. Another ap-

proach was proposed by Poulsen (2002) in which an additional constraint

is added to prevent one-node connected hinges. A simple method was pro-

posed by Kim et al. (2000) which can control the number of holes in the

final topology produced by the ESO method and also can solve the checker-

boards problem. Using fixed-grid finite element in the ESO method has been

investigated by same researchers (Kim et al. 2003) and it was shown that

this approach is capable of preventing checkerboard patterns. To overcome

the checkerboards problem in the homogenisation method Fujii and Kikuchi

(2000) introduced a gravity control function to be added to the objective

function. This function is similar to the perimeter control function, however

it penalises the intermediate densities as well. It was shown through exam-

ples that this approach can solve the mesh dependency and local minima

problems as well.

It should be noted here that most of the techniques which are used to

deal with the mesh dependency problem can also overcome the checkerboards