FEA for "dummies"

So a lot of my previous posts have talked about finite element analysis or FEA, but I realise some people will not know what the heck I'm talking about unless they've gone and Googled it afterwards. I've ripped huge chunks of the following from my PhD thesis and hopefully made it less academic, as well as interspersing some thoughts throughout.

Finite element analysis (FEA) is a commonly used method in biology and palaeontology (see below) even though it is still quite a recent development (for a historical review see Oden, 1987). Most biological structures are too complex to analyse biomechanically as a whole, so FEA works by breaking these structures (e.g. a skull) into a series of discrete units called finite elements (FE). These can be triangles, cubes, tetrahedra etc. and are interconnected by the corners (nodes). A force replicating a biological load (e.g. a muscle force) can be applied to nodes, and other nodes are constrained (e.g. by a joint) (see figure below). The elements have material properties applied: of particular importance are the Young’s modulus (the elastic modulus - the stress [force per unit area] divided by the strain [the deformation - change in length that occurs under loads]), which determines a materials rigidity and the Poisson’s ratio (the negative of the transverse strain [change in width under load]/axial strain [change in length under load]). With all the complexities of preprocessing done, the whole model is then run in a program that solves a series of complex differential equations for each node, and stress/strain/displacement patterns and orientations can be recovered (Zienkiewicz et al., 2005).

Simplified finite element model. Red arrows are loads, blue triangles are constraints. Each brick (cube) is connected to its neighbours by nodes (black dots).

How do we know it works?
This is one of the most annoying questions you can ever ask someone working on FEA, and likely they will give you some flippant answer. However, I get the chance to explain it with lots of pretty pictures of cool applications. As this method has been used in some form or another for getting on to a century (I don't envy the poor people who had to do the calculations by hand in the time before computers), it's known to be very effective with respect to man-made materials that we know their material properties and structures. Basically what I am saying is Google finite element analysis:

 

Images from: http://www.stressebook.com/finite-element-analysis/ and 

http://www.directindustry.com/prod/msc-software/non-linear-fea-software-finite-element-analysis-6042-493371.html

Instantly the flippant answer becomes apparent. FE modelling is used in all sorts of applications that play vital roles in building everything from planes, to cars and bridges. The design phase of pretty much everything engineering based now goes through a virtual FE phase that is the starting point for in-silico (virtual) testing. When the designs are optimised on computers, the models are made before the final products produced.

That's all well and good, and FEA has worked remarkably well in predicting mechanical performance of engineering structures but there has been some controversy in the predictive/modelling abilities of FEA in the far more complex biological systems that are modelled. Which leads to the next question:

How well does it work on biological material?
The big problem with biological material is often associated with the material properties. Many man-made materials are isotropic (where the properties are the same in every direction) e.g. glass and metals. Indeed many objects are also homogenous with the same material making up the entire structure, thus making modelling easy. However, biological structures are fundamentally difficult. I'm going to focus on bones, as that's what I work on. Most bones are made up of multiple types of "bone" (cortical - that hard stuff on the outside, and cancellous/trabecular - bone that looks like honeycomb on the inside of bones)

Vertebrae, with one sectioned to show internal structure (http://galvacell.wordpress.com/category/uncategorized/)

That's before considering any of the soft tissues (blood vessels, nerves, cartilage, cellular matrix etc.). As bone is mainly hydroxyapatite, it might be assumed it should be easy to model. But biology isn't nice. Under force/loads bones will increase thickness in regions of high stress and strain, and remove bone from low stress and strain. However, osteoblasts (the bone thickening cells) do not lay down the new bone the same way in the entire structure so the structure is not homogeneous, and the material properties for the bone is anisotropic (it bends differently depending on the orientation of the stress/strain). As such there is no quick and easy model for bones, but it is possible to measure the material properties. using using micro- or nanoindentation (Panagiotopoulou et al., 2010; Rayfield, 2011; Soons et al., 2012). These methods basically take a small triangular or diamond-shaped tipped device that "pokes" the bone with a certain force. This leaves an indent of a measurable size (normally micron scale) to give hardness values. The clever ones now actually time the length of time for the bone to "rebound"which is how we calculate the Young's modulus.

The wise amongst you will point out that using micron sized indents to get material properties is probably not an accurate assessment of the whole structure if it is much bigger. Indeed this is an issue, and one way to deal with it is to take many samples across the entire region and take an average. Another option is to model different regions with different properties which might work best for skulls (where there are many bones, so each bone has an average), and to model cortical bone and trabecular bone separately. The problem with doing this is how you divide regions/material properties and it becomes computationally expensive quickly.

But despair not, we can still carry out analyses, but we have to perform validations of our models to check how close models come to real values, either in-vivo (living animals) or ex-vivo (dead material) (Zapata et al., 2010). How this done will vary from study to study, but strain gauges are the most commonly used method for obtaining the in- or ex-vivo strains. However, there are new methods using image correlation or speckle interferometry which work roughly the same as each other - a pattern is created on the bone and the change measured during load application (its roughly the same as INSAR for measure the deformation of Earth before and after earthquakes/volcanic eruptions). This change can be used to calculate strain and patterns of strain. The computer model strains are then taken from the same regions to compare how the digital strains match the experimental ones. So far validation experiments have assessed the fit between FE-model and experimentally derived strains in: birds (beaks: e.g. Soons et al., 2012); mammals (skulls: e.g. Bright and Rayfield, 2011; jaws: e.g. Panagiotopoulou et al., 2010 etc.); and reptiles (skulls: Metzger et al., 2005; jaws: e.g. Porro et al. 2013). With most of the published studies the FE models closely match strain patterns and strain orientations, some struggle to replicate strain magnitudes (especially in complex regions like zygomatic arches): most recent studies match strain magnitudes much closer than older studies.

Porro et al. 2013.

FEA in palaeontology
Whilst now fairly common in palaeontology, it was not until the landmark paper on the cranial function of Allosaurus (Rayfield et al., 2001) that finite element modelling was used on vertebrate palaeontological material. Ever since, it has been used to study broader patterns of theropod cranial evolution (Rayfield, 2005; Rayfield, 2011a), as well as specimen specific questions for feeding in both 2D (e.g. Rayfield, 2004) and 3D (e.g. Degrange et al., 2010). In addition, tests of convergence of rostra (e.g. Rayfield et al., 2007), effects of sutures (e.g. Rayfield, 2004), and deductive methods where heavily simplified models were loaded biologically and destressed regions removed to obtain a structure representing that being studied (e.g. a Diploducus skull: Witzel and Preuschoft, 2005) (finite element structure synthesis or FESS) have also been used. In addition there have been studies on the postcrania: limbs or parts thereof (e.g. Snively and Russell, 2002; Manning et al., 2009), tail clubs of ankylosaurids (Arbour and Snively, 2009), trackways (e.g. Falkingham et al., 2009) and teeth (e.g. Macho et al., 2005)

Rayfield et al. 2001

Again many of you will question what we can garner from fossil stuff as it is not possible to directly test the material properties of extinct animals bone and other tissues, so direct validations are not possible. This poses an issue, but phylogenetic bracketing - where you find the closest living relatives (e.g. crocodiles and birds for dinosaurs) offers an option as it is reasonable to assume that the material properties should fall close to or between the living relatives. Failing that bones can be studied under the microscope and similar structures might be expected to have similar material properties (e.g. cow bone is similar to Allosaurus bone when sectioned - Rayfield et al., 2001).

The result of all of this is that we can often gain insights into stress and strain distributions within fossil taxa fairly reliably. The magnitudes of stress/strain may be off based on material property estimations, but this does not affect our ability to study the patterns within taxa or between taxa or to gain an insight into the function of the fossils. The one major caveat is to do with the effects of soft tissues (mainly because they don't preserve in most fossils), so structures like claws and beaks which are covered with a keratin sheath will inevitably be tough to model, although again estimates can be made if reconstructions are carried out using phylogenetic bracketing.

My dealings with FEA
So where do I fit into this exciting pretty pictured world of stress (and strain)? My PhD relied heavily on the method, so I have been working on a range of studies. I spent the vast majority of the first year trying to validate an ostrich skull (bird skulls have received very little research into them despite being the most closely related to dinosaurs). With much help from Jen Bright during the experimental setup we managed to obtain data from strain gauges on the ostrich skull that was being loaded by an artificial tendon (it was all very cool involving carbon fibre and resin). Strangely, no matter how many materials were included in the digital model (cortical and trabeculllar bone, sutures, beak), we just could not get the data to match up between the two. Despite being an interesting result, the reviewers gave some interesting advice but pointed out that it wasn't validated and thus really wasn't worth of publication at its current stage. So that project has died for the minute but I am hoping that a grant application from Bristol may one day be successful in getting a student/postdoc to pick up the ashes of that work, and make it rise like the phoenix and solve all the birdy mysteries about why their skulls are weird and don't match FE models (or at least why mine didn't - hopefully it isn't because I messed up). It may have insights into modelling dinosaur skulls and be of vital importance.

Screwy ostrich skull. Cuff, 2014 thesis.

After that, I spent spent a chunk of my 2nd and 3rd years carrying out FE studies on ornithomimosaur skulls (will probably end up doing a post about this crazy group shortly). There appear to be differences between some of the taxa depending on feeding methods (pecking or biting), and interestingly there is a cool strain pattern change within the ornithomimsaurs where strains shift from the braincase to the snout. This pattern is also seen in theropods (the meat eating clade of dinosaurs with T. rex and Velociraptor) in general. But, I remain sufficiently vague here because it is still very much work in progress/work in prep for publication so would hate to scoop myself on a blog when publications are all a postdoc really worries about for their career.

I have also had the privilege of working with several students in Bristol during my time there. I supervised a project on a crazy phytosaur (crocodile looking reptiles that aren't closely related at all) which remains a work in progress but hopefully the results of that are published soon. The stand-out project for me (probably because it was my first project of my own) is the one that I initiated at an outreach event (will do a blog on outreach at some point too), and it spiralled into a full on MSc project when Emily Rayfield told me I wasn't allowed to do it in my own time with my PhD being ongoing. That one is on a massive pliosaur (marine reptile from the Jurassic - at least this one was), that was found in Weymouth Bay, UK. Its on display in Dorchester at the museum with a cool display if anyone is keen to see it. Davide Foffa was the MSc student, and reconstructed the skull and muscles and carried out some FE studies on the snout and lower jaw (Foffa et al. 2014) showing that the skull was a relatively weak shape (probably limited by its need for streamlining in the water), but by being massive it compensated for any weakness.

Foffa et al. 2014.

Since then there has been a new MSc at Bristol who has carried out some FE studies on a a crurotarsan archosaur (crocodilian-like reptiles) that looks like the ornithomimosaurs I mentioned earlier. Andrew Jones is now doing a PhD but will hopefully be publishing on this screwy convergence soon with me as a co-author.

I am not personally done with FE just yet for my own. I have some bone validations of my own to carry out on modern cat bones (probably a domestic cat and a lion or tiger) to see how they behave under loading and how accurately we can model them, and that will start soon (maybe early 2015) to continue my yearly battles with this technique. Hopefully that explains what FE is, and what (palaeo)biologists have been up to with it, but feel free to leave comments/questions about any of it.

References
Arbour VM, Snively E, 2009. Finite element analyses of ankylosaurid dinosaur tail club impacts. The Anatomical Record 292, 1412-1426.

Bright JA, Rayfield EJ, 2011. Sensitivity and ex-vivo validation of finite element models of the domestic pig cranium. Journal of Anatomy 219, 456-471.

Cuff AR, 2014. Functional mechanics of ornithomimosaurs. PhD Thesis, University of Bristol

Degrange FJ Tambussi CP, Moreno K, et al., 2010. Mechanical analysis of feeding behaviour in the extinct “terror bird” Andalgalornis steulleti (Gruiformes: Phorusrhacidae). PLoS ONE 5(8): e11856.

Falkingham PL, Margetts L, Smith IM, et al., 2009. Reinterpretation of palmate and semi-palmate (webbed) fossil tracks; insights from finite element modelling. Palaeogeography, Palaeoclimatology, Palaeoecology 271, 69-76.

Foffa D, Cuff AR, Sassoon J, et al., 2014. Functional anatomy and feeding biomechanics of a giant Upper Jurassic pliosaur (Reptilia: Sauropterygia) from Weymouth Bay, Dorset, UK. Journal of Anatomy 225, 209-219.

Macho GA, Shimizu D, Jiang Y, et al., 2005. Australopithecus anamensis: a finite element approach to studying the functional adaptations of extinct hominins. The Anatomical Record 283, 310-318.

Manning PL, Margetts L, Johnson MR, et al., 2009. Biomechanics of Dromaeosaurid Dinosaur Claws: Application of X-Ray Microtomography, Nanoindentation, and Finite Element Analysis. The Anatomical Record 292, 1397-1405.

Metzger KA, Daniel WJT, Ross CF, 2005. Comparison of beam theory and finite-element analysis with in vivo bone strain data from the alligator cranium. The Anatomical Record 283A, 331-348.

Oden JT, 1987. Historical comments on finite elements, in: Proceedings of the ACM Conference on History of Scientific and Numeric Computation. Princeton, New Jersey, USA, pp. 125-130.

Panagiotopoulou O, Curtis N, O’Higgins P, et al., 2010. Modelling subcortical bone in finite element analyses: a validation and sensitivity study in the macaque mandible. Journal of Biomechanics 43, 1603-1611.

Porro LB, Metzger KA, Iriarte-Diaz J, et al., 2013. In vivo bone strain and finite element modeling of the mandible of Alligator mississippiensis. Journal of Anatomy 223, 195-227.

Rayfield EJ, 2004. Cranial mechanics and feeding in Tyrannosaurus rex. Proceedings of the Royal Society B 271, 1451-1455.

Rayfield EJ, 2011. Strain in the ostrich mandible during simulated pecking and validation of specimen-specific finite element models. Journal of Anatomy 218, 47-58.

Rayfield EJ, Milner AC, Xuan VB, et al., 2007. Functional morphology of spinosaur ‘crocodile-mimic’dinosaurs. Journal of Vertebrate Paleontology 27, 892-901.

Rayfield EJ, Norman DB, Horner CC, et al., 2001. Cranial design and function in a large theropod dinosaur. Nature 409, 1033-1037.

Snively E, Russell AP, 2002. The tyrannosaurid metatarsus: bone strain and inferred ligament function. Senckenbergiana lethaea 82, 35-42.

Soons J, Herrel A, Aerts P, et al., 2012. Determination and validation of the elastic moduli of small and complex biological samples: bone and keratin in bird beaks. Journal of the Royal Society Interface 9, 1381-1388.

Witzel U, Preuschoft H. 2005. Finite-element model construction for the virtual synthesis of the skulls of vertebrates: case study of Diplodocus. The Anatomical Record 283A, 391-401.

Zapata U, Metzger K, Wang Q, et al., 2010. Material properties of mandibular cortical bone in the American alligator, Alligator mississippiensis. Bone 46, 860-867.

Zienkiewicz OC, Taylor RL, Zhu JZ, 2005. The Finite Element Method: Its Basis and Fundamentals (Sixth ed.). Butterworth-Heinemann. ISBN 0 7506 6320 0