Finite-element model construction for the virtual synthesis of the skulls in vertebrates: case study of Diplodocus

The measurement of strains in real skulls is an inductive method that yields information about the stresses occurring in the a priori existing shape. In contrast, the approach taken here to determine the relationship between skull function and skull shape applies Wolff's law through a deductive technique of structure synthesis. This article describes the application of this method in the exact virtual synthesis of a sauropod skull, e.g., Diplodocus longus Marsh from Wyoming. An unspecific homogeneous solid is first constructed, giving the stresses ample volume to spread between points of force application and constraint. ANSYS 7.0 is used to form 10-noded tetrahedral finite elements with a maximum of 130,000 nodes. The initial conditions are the functional spaces for the eye openings, muscle forces, and the placement of the dental arcade, including assumed bite forces. Enforcing equilibrium of forces, the primary 3D stress flows in each load case are summarized by a physiological superposition, which accumulates the highest value of stress in each finite element. If the stress free parts are eliminated and the summarized stress flows are maintained, a reduced model appears, which is very similar to the real skull. This reduction of shape can be repeated iteratively and leads to a more exact form. Changes in the form of the dental arcade, its position relative to the braincase, the origins of muscles, or the height of the face lead to models that clearly resemble morphological differences between genera. The synthesis of a skull in this way demonstrates the direct correlation between functional loading and the biological structure and shape and can be used to test hypotheses regarding the relationship between structure and function during skull evolution.