Non-periodic structures

The first example is a volume minimization problem for a 2D double clamped beam. The design domain with the boundary and loading conditions is shown in Fig. 6.2. Note that only one concentrate load is acting in this example. The model properties and the optimization parameters are found in Tab. 6.1.

Fig. 6.2 A double-clamped beam: design domain

Tab. 6.1 Optimization parameters for the double clamped beam

BESO parameters Model property

d* (m) 2.0×10-5 Thickness (m) 1.0

Evo. ratio 2.0% Elem. type Plane stress, 4 node

Filter radius (m) 1.0 Number of elements 4800 E (pa) 2.0×1011

v 0.3

The evolutionary history for the maximum absolute displacement and the volume frac- tion is plotted in Fig. 6.3. It is seen that the evolution process is very smooth, and the final convergence procedure is highly stable. No jumps of the maximum displacement are identified in the history. The smoothness of evolution is caused by the fact that the maximum absolute displacement happens always at the load point as there is only one concentrate load in this model. The virtual load is then always determined not only at the same node, but also in the same direction. No change of the control point has ever happened through the history at all. In this case, the global displacement control prob- lem is equivalent to the local displacement control problem.

0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05 2.50E-05 1 4 7 10 13 16 19 22 25 28 31 34 37 40 Iteration A b so lu te d is p lacemen t ( m ) 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% V o lu me F ract io n

dmax Volume Fraction

Fig. 6.3 Evolutionary history of the beam with displacement constraint

The final topology is shown in the Fig. 6.4. A minimized volume fraction of 67.29% is obtained while the maximum displacement of 1.99921×10-5m satisfies the absolute displacement constraint of 2.0×10-5m.

Fig. 6.4 Final topology of the beam with displacement constraint: V=67.29%,

dmax=1.99921×10-5m

A comparison is made with the final optimal stiffness design shown in Fig. 6.5. This rival is produced by a conventional stiffness optimization with the volume constraint of v=67.25% (to compete with the displacement design above), for the same model. One finds easily that these two solutions are actually the same (except the small vol- ume difference). Comparing the sensitivity calculation for these two problems, one can easily identify a unique multiple between the sensitivities of the same element for

stiffness and displacement, 1 , , _P i i T ik i i T i = = u K u u K u

λ , where P is the actual load and 1 denotes

the virtual unit load. Therefore the sensitivity ranking of all elements is the same for these two problems and the consequence is that they evolve to the same final topology.

Fig. 6.5 A comparison from conventional stiffness optimal design: V=67.25%,

dmax=2.00049×10-5m

It concludes that when the structure is single point loaded, the global displacement control problem can be realized by the conventional stiffness optimization, which is much more computation-efficient while only one FEA on the real system is needed.

• A cantilever with distributed loading

The second example is a volume minimization problem for a 2D cantilever shown in Fig. 6.6. Note that a uniform distributed load is acting on the top non-designable layer (here termed the deck) marked in grey. The model properties and the optimization pa- rameters are found in Tab. 6.2.

Fig. 6.6 A cantilever with non-designable deck: design domain Tab. 6.2 Optimization parameters for the cantilever

BESO parameters Model property

d* (m) 1.48×10-6 Thickness (m) 1.0

Evo. ratio 2.0% Elem. type Plane stress, 4 node

Filter radius (m) 1.0 Number of elements 2400 E (pa) 2.0×1011

v 0.3

The evolutionary history for the maximum absolute displacement and the volume frac- tion is plotted in the next figure. Several jumps of the maximum displacement are identified during the evolution process. The instability of the maximum displacement is caused by switching the local controlled node. The final convergence procedure is not as smooth as in last example. However due to the present convergence strategy, the change of the volume gets smaller adaptively and the maximum displacement still converges into a very small range below the allowable value.

0.00E+00 2.00E-07 4.00E-07 6.00E-07 8.00E-07 1.00E-06 1.20E-06 1.40E-06 1.60E-06 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 Iteration Ma x imu m d is p la c e me n t ( m ) 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% 120.0% V o lu m e Fr act ion

dmax Volume Fraction

Fig. 6.7 Evolutionary history of the cantilever with displacement constraint

The final topology is shown in Fig. 6.8. A minimized volume fraction of 49.8% is ob- tained while the maximum displacement of 1.47×10-6m satisfies the absolute dis- placement constraint of 1.48×10-6m.

(a)

Fig. 6.8 Final topology of the cantilever with displacement constraint (V=49.8%,

dmax=1.47×10-6m): (a) undeformed view; (b) deformed view.

A comparison is made with the final optimal stiffness design shown Fig. 6.9. This rival is produced by a conventional stiffness optimization with the volume constraint of v=49.8% (to compete with the displacement design above), for the same model. The difference between these two final topologies is obvious by surveying the deformed views. The displacement design has a strong bar supporting the end of the non- designable deck while no bar is left at the middle, in order to reduce the deflection of the deck end point which is the critical place for the maximum displacement to happen. On the other hand, as the stiffness design considers the structural overall stiffness, the supporting bars are positioned relatively evenly under the non-designable deck, com- plying with the distributed loading condition. The two final solutions have a signifi- cant difference in the maximum displacement as the structural performance: while the volume fraction remains the same, the stiffness design has the maximum displacement 24.6% higher than that of the displacement design.

(b)

Fig. 6.9 A comparison from conventional stiffness optimal design (V=49.8%,

dmax=1.834×10-6m): (a) undeformed view; (b) deformed view.

It concludes that when the structure is not single point-loaded, the optimal design with global displacement constraint is different from that of conventional stiffness optimal design. In this case, to control the maximum global displacement, a virtual system in each iteration must be accommodated in the optimization procedure, which however increases the computation cost.