# Stability Design of Sparse Structure under Fixed-Fixed End Conditions

In this problem, a point load is applied to the top central portion of a structural region with fixed supports at both lateral edges of the domain. A sparse, stable structural design is sought that carries the applied load back to the supports. This problem is solved here by optimizing the material layout such that the minimum buckling eigenvalue is maximized. Details on the formulation of such problems were presented in references 6 and 8. The problem has also been solved using different formulations and objective functions by Buhl et al (2000) and Gea and Luo (2001). In this specific example (Figure 2a), a point load of magnitude 1.0105 is applied to the domain as shown which has relative dimensions of 100 by 50. The optimization problem is solved to find the constrained material layout under the applied loading that maximizes the minimum buckling eigenvalue (see Eq. 14).

 Fig. 2. Stability design of sparse structure for fixed-fixed end conditions. a) shows the design domain, support conditions, and loading; b) design obtained on a mesh of 100 x 50 bilinear elements with material usage constraint of 0.10; b) preceding design mapped onto mesh of 200 x 100 elements and further optimized while imposing a tightened material usage constraint of 0.05, and a perimeter constraint of 5000; d) preceding design mapped onto mesh of 400 x 200 elements and further optimized with material usage constraint of 0.03 and perimeter constraint of 3750.

The material layout optimization problem was first solved on a relatively coarse mesh of 100 by 50 bilinear quadrilateral continuum elements. The material layout shown in Figure 2b was obtained from a starting “design” of structural material completely occupying the entire the structural region, followed by imposition of a constraint on structural material: V, . , < 0.20V where V. represents the volume

of the 100 by 50 structural domain. The design solution shown was obtained using a powerlaw parameter [p=1.75]. The material layout in Fig. 2b is both heavy and somewhat difficult to interpret. Consequently, the design shown was then projected onto a uniformly refined mesh of 200 by 100 bilinear continuum elements. With this refine mesh, a reduced material usage constraint of V, ., < 0.080V, . was

imposed in addition to a perimeter constraint P < 2 (l + k) where i and k are the lateral and vertical

dimensions of the structural region in Fig. 2a. The material layout was then optimized on the refined mesh for 100 SLP (sequential linear programming) iterations with the result being as shown in Fig. 2c. To obtain a very clear, yet sparse, structural design, the design of Fig. 2c. was then mapped onto a mesh of 400 by 200 bilinear continuum elements. The material usage constraint was tightened to Vmaterial < 0.030Vdomain, and a mixing rule powerlaw parameter of [p=4] was employed along with a tightened perimeter constraint p <1 (i + h). The final, optimal material layout design achieved is as shown in Fig. 2d. This final design shown in Fig. 2d. is similar those obtained in the previous study by the authors using this same problem(6).