The morphology of a skeletal structure is driven in part by the forces it endures over evolutionary time. We demonstrate that generative design, an iterative engineering tool, can yield insight into the loading regimes of complex skeletal elements. With only a simple model of the spatial constraints and potential forces seen by a skeletal element, we generated analogous structures. We used this technique to provide insights into the specialization of the radial elements of batoid pectoral fins, a case that would be challenging to analyze by conventional means. Particular configurations of generative designs resulted in structures with a morphology remarkably similar to that of real radials. We suggest that these cases reveal the loading configurations that the real radials have evolved to withstand.

We imaged material from two batoid fish species from the collection of the Biomechanics Lab at Friday Harbor Labs. The longtailed river stingray, Plesiotrygon iwamae, is an undulatory swimmer with a rounded pectoral fin profile producing multiple wave periods along the wing per flap. The banded eagle ray, Aetomylaeus nichofii, is an oscillatory swimmer with a triangular fin profile generating a half wave period per flap. We dissected the skeleton of the pectoral fin from both species and micro-CT imaged the samples on a Bruker Skyscan 1173. We produced volume and surface renderings of radials with the free, open source software 3D Slicer.

Fusion 360 (Autodesk, San Francisco), was used under educational license for computer aided design (CAD) and solving the generative design cases. The simulation requires geometric dimensions, loadings, and material properties. We elected to use a model with biologically based aspect ratio and arbitrary size, loading, and properties. We specified geometric constraints - a pair of circular end-caps with a central cylindrical obstruction, to simulate the stingray radial. The cartilage of the radials only calcifies at the outer surface (perichondrium), and this was captured in the generative design model by requiring supporting material to lie outside a central core.

Histology shows this relationship between the mineralized tesserae and the unmineralized core. This informed our positioning of a ‘forbidden’ zone for the generative designs.

We constrained one end cap position in all three directions and three rotations, then added various combinations of compressive, tensile, and torsional loads to the other end. The length between the ends was 300 mm, the ends were each 5 mm thick, the diameter of the ends was 50 mm, and the diameter of the cylindrical obstruction was 24 mm. For small structures like radials, we recommend increasing the size of the model as we have done. The program works at a resolution defined on a unitless Low(0)-High(10) scale in the study settings. Models which are too small end up lacking in complexity as the program is unable to generate any smaller geometry. Compressive loads ranged from 100 N-10.000 N, and the torsional loads from 0.1 Nm-100 Nm.

For all trials, gravity was suppressed and the objective was set to ‘Minimize Mass’ with a safety factor of 1.00 and the displacement option deselected, as in our case we were not defining a maximum allowed deflection under loading. The ‘manufacturing method’ was set as 'Additive' only and was oriented in the Z+ Axis with 45° overhang and 3 mm allowed minimum thickness. We posit that using an additive procedure to set a minimum thickness simulates the "building blocks" of a structure. Radials are made up of mineralized tesserae meaning that a minimum thickness larger than cell-size exists. We used HP 3D CB PA 12 plastic from the Fusion 360 Additive Material Library. We added a slotted cylinder around the central obstacle body defined as a “Starting Shape” to speed up the solving process. With no starting shape, the initial iteration is determined based on the position and geometry of the preserved bodies. A starting shape circumvents that first step. We ran the program under compression, and torsion individually to establish the effect of each loading regime on geometry. 

We tested a quantitative method for comparing the results of a generative study to a morphology using unitless volume fraction representing space filled.

For the generative models, the total volume is defined to be that of the entire cylindrical region including the end caps. For the radials, an total volume was approximated as the cylinder found using the averages of the major and minor axes of both end caps.