Finite‐element model construction for the virtual synthesis of the skulls in vertebrates: Case study of Diplodocus

Abstract

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.

Predicting Skull Loading: Applying Multibody Dynamics Analysis to a Macaque Skull

“…In ANSYS, isotropic properties were defined with a Young's modulus of 17GPa and Poisson's ratio of 0.3. The Young's modulus and Poisson's ratio values are comparable to those used in other studies (e.g., Strait et al, 2005;Witzel and Preuschoft, 2005); however, the specific values were not critical to the present research, because directly comparable models were being assessed relative to each other and not with anatomical specimens. The loading data were taken directly from the MDA simulations and included bite forces, joint forces, muscle forces, and in the case of muscle wrapping, temporal muscle contact forces.…”

“…In FEA of the cranium in which masticatory function is modeled, forces, either in terms of force vectors or force inducting elements, are applied to represent the action of muscles of mastication (e.g., Witzel et al, 2004;Ross et al, 2005;Witzel and Preuschoft, 2005;Strait et al, 2007;Wroe et al, 2007). The application of loading is usually in the form of single force components representing entire muscle groups.…”

“…The complexity of skull models is continually increasing, with some models now including a degree of variation in material properties across the skull, muscle groups divided into different segments and the effects of different loading conditions being assessed (e.g., Ross et al, 2005;Strait et al, 2005;Witzel and Preuschoft, 2005;Wroe et al, 2007). MDA offers an opportunity for a further improvement in the biofidelity of these simulations.…”

“…Most studies load anatomically accurate models and assess the resulting deformations in what is termed an inductive approach; however, an alternative deductive method is described in work by Witzel and Preuschoft (e.g., Preuschoft and Witzel, 2004;Witzel et al, 2004;Witzel and Preuschoft, 2005). In this deductive approach extremely simplified general skull forms are created and then loaded with muscle and bite forces.…”

The Mechanical Significance of the Temporal Fasciae in Macaca fascicularis: An Investigation Using Finite Element Analysis

“…The meshed geometry was imported into commercially available FEA software (ANSYS version 12, ANSYS), and all muscle and bone structures were defined with six or eight noded (plane strain) second-order elements. Bone was specified with a Young's modulus of 17 GPa and a Poisson's ratio of 0.3 (consistent with direct measurements and within the ranges applied by others; Strait et al, 2005;Witzel and Preuschoft, 2005;Dumont et al, 2009;Wang et al, 2010), and the temporalis muscle with a Young's modulus of 10 MPa and a Poisson's ratio of 0.3. Notional thermal expansion properties were additionally assigned to the muscle elements so that their expansion (bulging) could be simulated, and a solution reference temperature of zero and an expansion coefficient of 0.07 (/ C) were assigned.…”