Circular Domain Stability Design Problem
In this second test problem, the circle domain test problem introduced previously by the authors (Rahmatalla and Swan, 2003, 2004) is revisited. A downward acting point-load of magnitude 1.0-109 is applied at the center-node of the circular structural domain of radius R=50. In Fig. 3a, the coarsest mesh features 896 bilinear elements; the next mesh (3c) features 3584 elements; that of (3e) features 14,336 elements; and the finest (3g) features 57,344 elements. The design objective is to find the optimal form of structure that carries the design from the center back to the fixed boundaries. The design problem was first solved on the coarse mesh of 896 elements with a powerlaw value [p=1.75] and a material usage constraint Vmateria/Vstracture < 0.20 and the resulting design is shown in Fig. 3b. This design was then mapped onto the mesh of Fig.3c and the layout optimization continued with [p=2.5] with a tightened material usage constraint of Vmateria/Vstracture < 0.05. The resulting design (Fig. 3d) was then projected onto the mesh of (Fig. 3e), and the optimization process continued with [p=4.0] and an even tighter material usage constraint of Vmaterial/Vstructure < 0.015. The design obtained from Fig. 3f. was projected onto the finest mesh and further optimized with the same powerlaw value and material usage constraint. The buckling eigenvalue on the finest mesh and the material layout are in close agreement with those from the previous mesh, and in this sense have converged.
Size reduction was essential to these computations. As the refinement process utilized finer and finer meshes, the analysis problem size remained very modest in size. For example, even on the finest mesh of 57,344 elements, the reduced problem size involved only 4,464 degrees of freedom, and the actual structural and eigenvalue analysis were performed on an HP J-class workstation in less than 10 cpu – seconds.
Figure 3. a) coarsest mesh of 896 elements; b) design with Vma/VstmcU]re =0.20 and k=2.17T03; c) mesh with 3,584 elements; d) material layout design with Vma/VstmcU]re =0.05 and k=1.20T0, e) mesh with 14,336 elements; f) material layout design with Vmat/Vstracture =0.015 and k=1.00-101; g) mesh with 57,344 elements; h) material layout design with Vmat/Vstructure =0.015 and 7M.05-101;