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Example 7.1 A simply supported beam with a single load

A simply supported beam is shown in Figure 7.1. A point load of intensity P=1kN is applied midway down the top side of the beam. The modulus of elasticity_E=105MPa, Poisson’s ratio, v = 0.3 and volume fraction Vs/V=50%.

Figure 7.1 A simply supported beam

The result of using PLATO software (Hassani and Hinton, 1998) with

rectangular microstructure carried out in this research is shown in Figure 7.2.

Figure 7.2 Optimization layout by PLATO software

The layouts of using HDM with nine different material models are shown in Figure 7.3: (a) power-law, (b) triangular, (c) hexagon, (d) cross shape, (e) circular, (f) triangular multi-void, (g) rectangular multi-void, (h) square multi-void, and (i) ranked layered material model.

(a)Power-law one-material model

Figure 7.3 Optimization results for different microstructures by HDM with volume fraction 50% (continued)

(Figure 7.3 continued)

(b)Triangular material model

(c) Hexagon material model

(d) Cross shape material model

(e) Circular material model

(f) Triangular multi-void material model.

Figure 7.3 Optimization results for different microstructures by HDM with volume fraction 50% (continued)

(Figure 7.3 continued)

(g) Rectangular multi-void material model.

(h) Square multi-void material model.

(i) Ranked layered material model.

Figure 7.3 Optimization results for different microstructures by HDM with volume fraction 50%.

Figure 7.3 shows that except for the ranked layered model, all the rest eight one-material models provide optimal shapes similar to that given by the rectangular model using PLATO software. The power-law model (atµ =3) gives the clearest image result. The optimal layout of the ranked layered model is different from the others, in that the material distribution is more complex, the layout is not as sharp and contrast, and material of high density is required at both left and right edges. The reason for this is as discussed in Chapter 6 that

because the optimal microstructure is degenerated in the rank-2 layered material model and the structure cannot sustain a non-aligned shear stress. This will result in the stiffness matrix of the structure becoming singular. The technique used in the thesis to overcome the singularity problem is to use a very soft material instead of voids. However, the minimum strain energy calculated during optimization process (commonly used for objective function which is equivalent to maximum total potential energy) is only modified energy. This leads to the result shown much difference with others.

From the layouts above, we can also see that all the other results from different microstructures have a similar layout, but not exactly the same. The criterion of an optimal structure for this minimum compliance problem is to find the minimum strain energy (maximum total potential energy). Among these microstructure models, the strain energy can be calculated during the optimization process for the triangular, hexagon, cross shape, circular, triangular multi-void, rectangular multi-void and square multi-void material models. But the strain energy is modified during the optimization process for ranked layered and power-law models. For ranked layered and power-law models, Finite Element Method can be used for calculating the true value at the final stage of optimization.

(a)Rectangular model (using PLATO) (b) Power-law model

(c) Triangular material model. (d) Hexagon material model.

(e) Cross shape material model. (f) Circular material model. Figure 7.4 Iteration histories for different models (continued)

(Figure 7.4 continued)

(g) Triangular multi-void model. (h) Rectangular multi-void model.

(i) Square multi-void material model. (j) Ranked layered material model. Figure 7.4 Iteration histories for different material models

In Figure 7.4, (a) rectangular material model by using PLATO software, (b) power-law, (c) triangular, (d) hexagon, (e) cross shape, (f) circular, (g) triangular multi-void, (h) rectangular multi-void, (i) square multi-void and (j) ranked layered material model.

Figure 7.5 shows all the iterations on the same graphic for comparison of the iteration histories given by the microstructure models (except ranked layered)

Figure 7.5 Comparison of iteration histories

Table 7.1 shows iteration numbers and final strain energies at convergence tolerance ∆ =0.050 for the microstructures using HDM.

Microstructure Model Iteration Number Final Strain Energy

Power-law 197 0.000065 Cross shape 233 0.000062 Circular 245 0.000064 Hexagon 267 0.000067 Triangular 229 0.000066 Rectangular Multi-void 220 0.000066 Square Multi-void 243 0.000067 Triangular Multi-void 279 0.000069 Ranked layered 267 0.0019

Table 7.1 Iteration numbers and final strain energies at convergence tolerance

0.05

From the iteration histories illustrated in Figure 7.4, Figure 7.5 and Table 7.1, we can see that all the microstructures give very good convergence, among them the power-law, cross shape and circular models have better convergence. The strain energy of the ranked layered material model is much higher than other material models. This is because the ranked layered model can not resist shear stress; displacement due to loading has to be greater, hence larger strain energy. In this case, the optimum value of strain energy of this material model is much more than those of other models.

For the power-law model, we investigate further the effect of changing the power value µ on the optimum layout. Figure 7.6 shows the optimization layouts for different power values of the power-law models from one to ten.

(a) Power value µ =1

(b) Power value µ =2

(Figure 7.6 continued)

(c) Power value µ =3

(d) Power value µ =4

(e) Power value µ =5

(f) Power value µ =6

(g) Power value µ =7

(Figure 7.6 continued)

(h) Power value µ =8

(i) Power value µ =9

(j) Power value µ =10

Figure 7.6 Optimization results for different power values

From the optimal layout given by Figure 7.6, we can see that for a small value of µ, a grey area is appears in the solution at the central bottom area, however with the value of µ increasing, the supporting from the central bottom area is gradually reduced. When µ >9, all the supports from the

central area disappear. By comparison with other model results, the solutions of the power value between 2 and 7 agree well with other models.

power-law one-material model.

(a) Power value µ =1 (b) Power value µ=2

(c) Power valueµ =3 (d) Power value µ =4

(e) Power value µ =5 (f) Power value µ =6

(Figure 7.7 continued)

(g) Power value µ =7 (h) Power value µ =8

(i) Power value µ=9 (j) Power value µ =10

Figure 7.7 Iteration histories for different power values

Table 7.2 below shows the initial and final strain energy for different power values calculated by HDM.

Power value Initial strain energy Final strain energy 1 0.000091 0.000059 2 0.00018 0.000065 3 0.00036 0.000065 4 0.00073 0.000066 5 0.00146 0.000066 6 0.0029 0.000068 7 0.00583 0.000069 8 0.01166 0.000068 9 0.0233 0.000067 10 0.04667 0.000071 Table 7.1 The initial and final strain energy for different power values. From the results above, we can see that the power-law models have very good convergence character for the power value between 1 and 9. Although the initial strain energy increases very rapidly with the value of power increasing, the final strain energy upon convergence is very similar.

On the simply supported beam problem, we can see that all the microstructure models (apart from the ranked layered model) agree well with the solution using PLATO software and have similar convergence histories. For the concept design stage, we can say the most of the layouts reach similar topological optimal layouts. By comparison of the value of the objective function of

optimization, the cross shape has the lowest strain energy. It also has the best convergence characteristics. From a manufacturing point of view, a solution with contrast and sharp image (black and white image) is easy to implement, their layouts given by power-law model with µ =3 5: are the best. Ranked layered model gives a different layout pattern. The reason for this, as discussed before, is that there is no shear strength in this model and it would lead to a larger displacements between layers.

Example 7.2 Square domain with a single corner load

Figure 7.8 shows a square domain with boundary constraints on the top left corner and the bottom edge. A point load of intensity P = 1kN is applied at the left bottom corner of the square domain. The modulus of elasticity MPa, Poisson’s ratio, v = 0.3 Two cases, Casea : the volume constraint Vs / V = 50% and Caseb : the volume constraint Vs / V = 20%, are considered.

5

10 E=

Casea : Volume constraint Vs / V = 50% :

The optimum layout given by the PLATO software using rectangular material

model calculated by the author is shown in Figure 7.9.

Figure 7.9 Optimum layout of using the PLATO software

The optimum layouts of using HDM with other nine different material models are shown in Figure 7.10: (a) power-law, (b) ranked layered, (c) triangular, (d) hexagon, (e) cross shape, (f) circular, (g) triangular multi-void, (h) rectangular multi-void, and (i) square multi-void material model.

(a) Power-law one-material model µ=3

Figure 7.10 Optimization results for different microstructures for Case a