Demo optimization solutions from GENSOLV
Baseline shapes
Here are three select baseline cases to demonstrate shape optimization capabilities of GENSOLV. These are 1. a square plate for which the buckling eigenvalue to mass ratio was to be maximized, 2. a cylindrical tube to which mass and inertia are coupled, to be optimized for highest fundamental frequency to structural mass and finally, 3. a flat annular disk let to evolve into an optimum shell to function as the interface ring adapter of a satellite. All demo solutions were from a 3- machine cluster, adopting elastic and density properties of a standard steel alloy.
Square plate
A thin square plate was considered, hinge- restrained along three edges while of course, free respectively in translation and rotation across and about the loaded edge. The baseline flat surface topology was allowed to morph only normal to plane. The objective norm, to be maximized, was the ratio of buckling eigenvalue to structural mass.
Optimized shape
Seen above, it's easy to see why the square plate has evolved to a corrugated shell as the optimum. Between the geometries of a flat plate and a curved shell, the latter has a lot more (virtual) resistance to load on account of membrane- flexure coupling. The baseline, of size 1000 mm, had a buckling critical load factor of 1.63 for an arbitrarily assumed load. The corresponding mass was 15.6 kg, for a 2 mm wall thickness, also adopted arbitrarily. On optimization, the best design was way ahead, with a load factor of 92.1 or 5551% higher! Although the optimum design's mass was of course higher owing to the increased surface area, that increase was a mere 5.43%.
Cylindrical tube
This baseline, a cantilever deep shell tubular cylinder, is representative of civil engineering structures such as a tower supporting a water tank. The objective was to maximize the ratio of fundamental frequency to structural mass, under a relatively large external mass at one end. Specifically, a 200mm dia cylinder, of length 800 mm was adopted, with a wall thickness of 4 mm. A tip mass of 500 kg was considered on the axis at the end section, coupled to the cylinder wall by rigid elements. This tip mass was significantly higher than that of the cylinder baseline itself, 15.7 kg. The baseline's fundamental frequency (flexural) was determined to be 25.8 Hz. Morphing was allowed only in the cross- sectional plane, while along- axis change was supressed.
Optimized shape
The optimized shape seen above was among the top ones produced by GENSOLV. For this shape, although the structural mass grew by 9.3%, there was a whopping 188% increase in the fundamental frequency, now equal to 74.2 Hz.
Satellite adapter ring
An interface adapter couples a satellite to the launch vehicle, held down by a system of pyro- cut cords. The adapter's flexibility and hence modal characteristics go to determine response of the satellite in what are called coupled load analyses. Here, compared to gain in structural mass, GENSOLV was run hoping to achieve a far higher gain in norm of the modal frequencies. The lowest two frequencies norm was considered for this purpose and the objective was to maximize the ratio of this norm to structural mass.
An annular flat disk was adopted for the baseline, seen here. Outer and inner diameters were 2000 and 1160 mm respectively, somewhat in the neighborhood of their real- world counterparts. Wall thickness, arbitrarily adopted, was 3 mm. A satellite mass of 1410 kg was adopted, 720 mm above the plane of the disk.
For the data adopted, structural mass was determined as 48.8 kg, with a (satellite mass coupled) modal norm equal to 1.49 Hz.
