Structural Optimization using Finite Elements

Today's engineering structures are often analyzed using the finite element method (FEM). At ADOPTECH, we can perform analyses, sensitivity studies, and optimization using existing or custom FE models. With powerful design packages such as GENESIS^TM (Vanderplaats Research & Development, Inc.), optimization of large-scale structures with thousands of elements is possible. The two examples provided below demonstrate these capabilities for a small design problem.

The stiffened panel shown in Figure 1 consists of a 100" x 100" square Aluminum plate with clamped edges, one longitudinal stiffener, and three lateral stiffeners. For the nominal dimensions shown in Figure 1, the original design has a fundamental frequency of 22.4 Hz and weighs 568.8 lbs. The fundamental mode shape is illustrated in Figure 2.

Figure 1 - Finite element model of the stiffened panel used for Examples 1 and 2. Dimensions shown are for the original design.

Figure 2 - Contours of total displacement for the fundamental mode shape are shown on the deformed mesh.

Example 1 - Minimize the weight of the stiffened panel while maintaining the fundamental frequency of the original panel.

One of the motivations in optimizing structures is to reduce the overall weight, which often results in reduced material cost. Weight reduction is also highly desirable for air and ground vehicles since lighter vehicles have improved range, fuel savings, and increased payload. Decreased weight, on the other hand, can have undesirable effects on structural performance. A lighter structure is likely to be more flexible, and a more flexible structure will have a reduced natural frequency that is more easily excited to vibrate with large amplitude. Therefore, weight minimization problems are often posed along with a lower-bound constraint on performance. For this example, the weight is to be minimized while the natural frequency is constrained to be at least that of the original structure.

In order to achieve a weight savings, the four design parameters (t_p, t_lat, t_long, h) defined in Figure 1 are allowed to change during the optimizer's search for the best design. Table 1 illustrates the changes in these design parameters as well as the reduction in total weight. Inspection of this table indicates that the optimization of Example 1 successfully reduces the weight from 568.8 lbs. to 433.7 lbs., or about 24%. It should also be noted that the fundamental frequency of the optimal design for Example 1 is at least that of the original design.

Example 2 - Maximize the fundamental frequency of the stiffened panel while maintaining the weight of the original panel.

This second example is a dual formulation of the first example. In this case, an increase in structural performance (higher natural frequency) is sought while constraining the overall weight to be no more than that of the original design. The optimization results for this example are also displayed in the Table 1. It can be seen that a 27% increase in the fundamental frequency is achieved while maintaining the weight of the original panel.

Table 1 - Comparison of original and optimal panel weights, frequencies, and dimensions for the two examples.