آنالیز عناصر محدود در دیرینه شناسی-Finite element analysis (paleontology
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)

 
 
 
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)

 
 
 
 
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, 
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, 
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, 
 
Additional 
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
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