Finite element analysis (paleontology)

Biologists have long commented on how the vertebrate skeleton appears to be designed in accordance with engineering principles. Bones and their associated soft tissues are adapted to transmit, resist, and take advantage of the many forces the skeleton experiences during normal function. How bones respond to forces, that is, how they are stressed and strained, is therefore linked to their function. A great deal of functional information can be gained from examining how a skeleton responds to stresses and strains; yet paleontologists are limited because fossil bones are petrified (and thus have different properties than biomaterials in living animals), and skulls in particular are complex shapes not applicable to straightforward mathematical analysis. Recently, however, biologists and paleontologists have begun to borrow from the sophisticated design toolkit of engineers in order to examine skeletal construction.

Finite element analysis (FEA) is one such tool. FEA is a means by which stress, strain, and displacement in a two- or three-dimensional structure may be deduced using mathematical principles. FEA can be performed using predesigned computer software or specific user-written code. The exponential increase in computing power over the past 10–20 years has increased the accessibility of FEA to researchers who traditionally lie outside the engineering sphere. By creating digital models of skeletons, FEA can potentially offer a new avenue of inquiry to vertebrate paleontologists who are interested in the function of extinct animal skeletons and the mechanical principles and evolutionary pressures underlying their construction.

 

How FEA works

 

In the first stage of FEA, a digital two- or three-dimensional model of the structure to be tested is divided into a finite number of simple geometric shapes called elements. Elements are joined to each other at discrete points called nodes, and the element and node construct is known as a mesh (Fig. 1). Specific material properties that approximate those of the actual structure, such as stiffness and density, are applied to the element mesh. Boundary conditions are then applied to the mesh: constraints are applied to prevent the structure from moving in a particular direction; and loadings are applied as forces, pressures, or accelerations, which mimic loads the structure would experience during life or use. This process of model creation is known as preprocessing.

 

 

Fig. 1  Finite element model of Allosaurus fragilis skull. Elements are represented by each individual triangle. Geometric equations can be used to calculate strain and stress in simple structures. FEA permits the application of these geometric equations to complex, nongeometric shapes by calculating stress/strain in each individual element. (Reprinted with permission, © Emily Rayfield)

 

 

 

fig 1

 

 

 

The next stage is analysis, which involves the calculation of force vectors and displacements at each individual node, taking into account the material properties of the structure. Stress and strain at each nodal point are subsequently calculated to provide a composite picture of the mechanical behavior of the structure. Mathematical solvers integrated into FE software perform this stage of the analysis.

Finally, during postprocessing, results are visualized and interpreted (Fig. 2). Emphasis is placed upon the checking of errors and refinement of the original mesh and boundary conditions to ensure the model represents the original structure as accurately as possible.

 

 

Fig. 2  FEA-generated stress plot of skull in Fig. 1. A color-coordinated map of stress distribution within the structure may be produced from a FEA. Strain maps, vector plots, and displacement values are produced in a similar fashion. Here color and white areas of the mesh represent increased compressional stress. (Reprinted with permission, © Emily Rayfield)

 

 

 

fig 2

 

 

 

 

FEA in zoology and paleontology

 

The basic principles of FEA were originally derived by engineers in the late 1950s and early 1960s. Since the early 1970s, this method has been used widely in orthopedic medicine and bioengineering; however, there have been only a handful of studies utilizing FEA in zoology and paleontology. In 2003, the application of this technique to these fields was still in its infancy. Interestingly, an offshoot of FEA known as finite element scaling analysis (FESA), which is concerned with the calculation of displacements only, and not stress and strain, has been used since the mid-1970s to quantitatively examine shape change (such as comparing the prominence of facial features in early hominoids).

A 200-element FE model of the bill of a shoebill (a type of stork), published in the mid-1980s, appears to represent the first application of FEA to zoology or paleontology. Stress plots and displacements were displayed for two bill-loading regimes; however, actual loads and material properties were not specified. Not until the late 1990s did further zoological studies appear, mainly focused on primate lower jaws and teeth and horses' hooves.

 

Ammonites

 

The first published application of FEA in paleontology was an investigation into the shell strength of ammonites (an extinct group of mollusks) in the late 1990s. FEA showed that increasingly complex septal (internal shell wall) construction weakened ammonite shells (with weakening indicated by higher stress magnitudes). FEA models used in this analysis consisted of around 10,000 elements subject to hydrostatic pressure. A more recent analysis of the problem involved the creation of more morphologically accurate septal models, created from 20,250 realistic eight-node curved elements with the strength and stiffness of Nautilus nacre (the shell lining of a genus of mollusks), rather than the four-noded flat plate elements used in the initial study. In contrast to the previous analysis, the new study discovered complex septal morphology was in fact stronger than simple septal morphology. These results highlight accuracy problems in model geometry and element choice, a situation all users of FEA must face.

 

Evolutionary questions

 

General questions in cranial morphology have begun to be addressed using FEA. From the late 1990s to the present day, a few simple FEA models of three-dimensional beam and structural models and flat two-dimensional planes containing holes have been used to investigate the adaptive and mechanical significance of fenestra (openings) in the skulls of amniotes and the effect of increased nasal and braincase size in mammals. FEA has also been used to examine the ossification of limb bones during growth and to investigate why separate ossification centers are found at the ends of limb bones in some vertebrates but are absent from the limbs of others. Since the analysis was placed within an evolutionary framework, the results have implications for extinct and living animals.

 

 

Extinct animal models

 

Currently, the use of FEA in vertebrate paleontology has generally focused upon the mechanical behavior and function of skulls.

 

Snouts: basal synapsids

 

In the late 1990s, two three-dimensional FE models of the snout of a gorgonopsid and a therocephalian synapsid (mammal-like reptiles) were created. Models were geometric approximations of snout morphology, but for the first time an attempt was made to apply realistic bite forces to a FE vertebrate model in numerous directions, representing bilateral, unilateral, or shear biting. Actual material properties were not estimated; however, a close association of model stress distribution to actual cranial buttressing and mobile joint articulation was discovered. Predictions on the role of such predatory animals in Permian ecosystems have been made based partly on these results.

 

Snouts: archosaurs

 

Other snout models have investigated stress within the anterior skull of the theropod dinosaur Megalosaurus and the archosauriform Proterosuchus. The Megalosaurus model utilized appropriate material properties and investigated the effect on stress distribution of introducing kinetic (mobile) joints into the skull.

 

Whole-skull models: Allosaurus

 

Snout models are useful indicators of cranial stress within a particular region of the skull. However, errors may occur where the connection of the snout to the back of the skull must be estimated. It is, therefore, advantageous to model complete crania, and indeed this has been achieved on two occasions. In 2001, a 200,000-element model of the skull and lower jaw of the theropod dinosaur Allosaurus fragilis was completed (Figs. 1 and 2). This is currently the most complete and complex FE vertebrate model. Material properties, jaw muscular forces, bite force, and condylar (jaw joint) forces were accurately estimated. Stiffness and density values of structurally similar bovine bone were taken as an estimate of allosaur bone material properties. Size and force production of jaw adductors was estimated and used to calculate bite and jaw joint force, and all forces were applied to the model in the correct anatomical position. The model was constrained at insertion points of neck musculature and vertebral elements on the posterior surface of the skull. Numerous biting regimes were modeled, including bilateral, unilateral, and tearing bites. FEA revealed that the skull is extremely strong, and appears designed to accommodate high-magnitude stresses produced during prey capture and killing.

 

Whole-skull models: pterosaurs

 

Recently, a FE model of a pterosaur skull was created. The model is a reasonably accurate geometric representation of skull form incorporating bony material properties. The pterosaur model and a FE tooth model with properties of dentin (bonelike tissue) and enamel are currently part of an investigation into element formation and material property determination. Bite force analysis is being undertaken; yet the results are currently not available in the literature.

 

Foot: Gorgosaurus libratus

 

Finally, a recent FEA of the long bones in the foot of Gorgosaurus libratus elucidated the dynamic strengthening function of foot bones and associated ligaments. Researchers examined strain energy in the long bones of the foot of G. libratus during locomotion. Their results supported the idea that associated ligaments helped transfer footfall energy along the long axis of the splintlike middle foot bone and prevented damage from bending.

 

 

Problems

 

The above examples provide a review of the current status of FEA in vertebrate paleontology. Use of the technique will surely increase, as FEA has the potential to address both specific and wide-ranging questions concerning the functional morphology and evolution of fossil animals. However, a number of technical and theoretical problems face future FEA users.

First, experimental evidence from living animals has shown that not all structures are adapted to the functions they undertake or that functional signatures are muddled in bones that undertake numerous functional tasks. With structure decoupled from function, the elucidation of skeletal stress patterns may not yield satisfactory hypotheses of function. Nevertheless, FEA still bears the potential to test predetermined hypotheses of function and adaptation, and the strengths of the technique lie in this particular area.

Second, creation of model geometry is difficult, and problems are faced deciding how abstract a model should be when created. Material properties of bone and other tissues must be estimated from analogs in living animals. Boundary conditions (constraints and loading forces) must be estimated if not absolutely, then relatively. Moreover, there are technical issues involving element choice, mesh size, and position of constraints that could potentially influence the output of a FEA.

 

Conclusion

 

Taking these cautionary notes into account, the importance of FEA as a tool to investigate the mechanical behavior and function of extinct animal skeletons is evident. Paleobiologists will be able to utilize FEA, a technique relatively new to paleontology, in order to address old, fundamental questions such as why the skeletons of extinct animals were shaped the way they were.

 See also: Dinosauria; Finite element method; Fossil; Paleontology; Skeletal system; Wind stress; Synapsida

Emily Rayfield

 

Bibliography

 

 

R. M. Alexander, Bones: The Unity of Form and Function, Macmillan, New York, 1994

M. Fastnacht et al., Finite element analysis in vertebrate palaeontology, Senckenbergiana lethaea, 82(1):195–206, 2002

C. McGowan, A Practical Guide to Vertebrate Mechanics, Cambridge University Press, 1999

E. J. Rayfield et al., Cranial design and function in a large theropod dinosaur, Nature, 409:1033–1037, 2001

E. F. Weibel, C. R. Taylor, and L. Bolis, Principles of Animal Design, Cambridge University Press, 1998

 

Additional Readings

 

 

T. L. Daniel et al., Septal complexity in ammonoid cephalopods increased mechanical risk and limited depth, Paleobiology, 24(4):470–481, 1997

M. A. Hassan et al., Finite-element analysis of simulated ammonoid septa (extinct Cephalopoda): Septal and sutural complexities do not reduce strength, Paleobiology, 28(1):113–126, 2002

I. Jenkins, J. J. Thomason, and D. B. Norman, Primates and engineering principles: Applications to craniodental mechanisms in ancient terrestrial predators, Senckenbergiana lethaea, 82(1):223–240, 2002

H. Preuschoft and U. Witzsel, Biomechanical investigations on the skulls of reptiles and mammals, Senckenbergiana lethaea, 82(1):207–222, 2002

E. Snively and A. P. Russell, The Tyrannosaurid metatarsus: Bone strain and inferred ligament function, Senckenbergiana Lethaea, 82:35–42, 2002