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
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Cranial design and function in a large theropod dinosaur, Nature, 409:1033–1037,
2001
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Additional
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