Improving the validation of finite element models with ...

Journal of Biomechanics ] (]]]]) ]]]–]]]

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Improving the validation of finite element models with quantitative full-field strain comparisons a,b,n ¨ F. Groning , J.A. Bright c, M.J. Fagan b, P. O’Higgins d a

Department of Archaeology, University of York, York YO10 5DD, UK Department of Engineering, University of Hull, Hull HU6 7RX, UK c Department of Earth Sciences, University of Bristol, Bristol BS8 1RJ, UK d Hull York Medical School, University of York, York YO10 5DD, UK b

a r t i c l e i n f o

abstract

Article history: Accepted 8 February 2012

The techniques used to validate finite element (FE) models against experimental results have changed little during the last decades, even though the traditional approach of using single point measurements from strain gauges has major limitations: the strain distribution across the surface is not captured and the accurate determination of strain gauge positions on the model surface is difficult if the 3D surface topography of the bone surface is not measured. The full-field strain measurement technique of digital speckle pattern interferometry (DSPI) can overcome these problems, but the potential of this technique has not yet been fully exploited in validation studies. Here we explore new ways of quantifying and visualising the variation in strain magnitudes and orientations within and between repeated DSPI measurements as well as between the DSPI measurements and FEA results. We show that our approach provides a much more comprehensive and accurate validation than traditional methods. The measurement repeatability and the correspondence between measured and predicted strains vary to a great degree within and between measurement areas. The two models used in this study predict the measured strain directions and magnitudes surprisingly well considering that homogeneous and isotropic mechanical properties were assigned to the models. However, the full-field comparisons also reveal some discrepancies between measured and predicted strains that are most probably caused by inaccuracies in the models’ geometries and the degree of simplification of the modelled material properties. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Finite element analysis (FEA) Digital speckle pattern interferometry (DSPI) Human Mandible Bone strain

1. Introduction Finite element analysis (FEA) has become a popular tool for studying form-function relationships in bones and, due to recent advances in computer power and the improved availability of high-resolution computed tomography (CT) scans, the resolution and complexity of FE models is continuously increasing. However, the techniques and methods of validating FE models have changed little during the last decades. When validation studies are conducted, bone strain is traditionally measured with strain gauges that provide only single point measurements (e.g. Al-Sukhun et al., 2007; Marinescu et al., 2005; Rayfield, 2011; Vollmer et al., 2000). Therefore, the comparison of measured strains and those predicted by FE models is based on few selected points on the bone surface. This raises two

n Corresponding author at: Department of Engineering, University of Hull, Hull HU6 7RX, UK. Tel.: þ 44 1482 465116. ¨ E-mail address: [email protected] (F. Groning).

main problems: (1) the variation of strain magnitudes and directions across the surface cannot be considered in the comparison and (2) it is difficult to accurately determine the position of the corresponding points on the bone surface and the FE model if the 3D topography of the surface is not measured. The latter is especially problematic when strains vary to a large degree within a small area. These problems can be overcome, at least in in vitro experiments, using full-field strain measurement techniques such as electronic or digital speckle pattern interferometry (ESPI or DSPI, e.g. Yang and Ettemeyer, 2003; Yang and Yokota, 2007). Recently we explored the use of DSPI for the validation of FE models in an ¨ et al., 2009). in vitro experiment with a human mandible (Groning The application of DSPI allowed us to compare our FE models with the measurements based on corresponding strain profiles, which revealed a good correspondence between measured and predicted strains. However, although the comparison of profiles is more informative than single point comparisons, only a small part of the strain field was considered. Therefore, we did not fully exploit the potential of DSPI as a full-field measurement technique.

0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2012.02.009

¨ Please cite this article as: Groning, F., et al., Improving the validation of finite element models with quantitative full-field strain comparisons. Journal of Biomechanics (2012), doi:10.1016/j.jbiomech.2012.02.009

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Here we explore a more comprehensive way of validating FE models against DSPI measurements by taking into account the strains across whole measurement areas. In addition, we try new ways of quantifying and visualising the variation between measurements and the correspondence between measured and predicted strains. This includes the application of principal

components analysis (PCA), a standard statistical method used in biology, but which has only recently been used to assess the ¨ results of FEA (Chalk et al., 2011; Groning et al., 2011; O’Higgins et al., 2011; Reed et al., 2011, see also Ross et al., 2005 for the use of the simliar method of principal coordinates analysis).

Fig. 1. (a) Coronal CT slice showing the resolution of the CT scan used for the FE model creation (voxel size ¼ 0.135 mm in all directions), (b) corresponding section through the model, (c) photo of the experimental setup showing the mandible in the mechanical testing machine, with the two adaptor rings and the DSPI sensor attached to the ring on the mandibular corpus, (d) measurement area on the mandibular corpus, (e) measurement area on the mandibular ramus (scale bar in d and e¼ 1 cm), (f) applied boundary conditions and location of the points (white dots) used for strain comparison between experiment and FEA.

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2. Material and Methods 2.1. Experiment The experimental setup has been explained in detail by Gr¨oning et al. (2009). In summary, a dry human mandible was placed upside down in a Lloyd’s EZ tensile testing machine (Ametek-Lloyd Instruments Inc., UK) and compressive loads were

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applied to the mandibular angles (Fig. 1). The loads were increased in 50 N steps from zero to 250 N and the surface deformations during loading were measured using a Q-100 DSPI measuring system (DANTEC Dynamics GmbH, Germany). Two areas of the bone surface were measured, ca. 25  33 mm in size (Fig. 1). Prior to loading, the 3D surface topography was recorded, and during each load step x-, y- and z-displacements were measured using the software IstraQ100 2.7 (DANTEC Dynamics GmbH, Germany). Based on these displacements and the surface topography measurements, principal strains were calculated. Overall ten measurements were

Fig. 2. Maximum and minimum principal strain contour plots for the two measurement areas (A–F: repeated measurements of the corpus area, G–J: repeated measurements of the ramus area, see Fig. 1 for the location of the areas on the bone surface). The black lines indicate the directions of the principal strains. The contour plots in the bottom row are those from the voxel-based FE model. The solid black lines on the FEA contour plots indicate the borders of the measurement areas.

¨ Please cite this article as: Groning, F., et al., Improving the validation of finite element models with quantitative full-field strain comparisons. Journal of Biomechanics (2012), doi:10.1016/j.jbiomech.2012.02.009

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taken, during which the mandible was loaded as shown in Fig. 1, six measurements of the mandibular corpus (A–F) and four of the mandibular ramus (G–F). The stability of the experimental setup during those measurements was confirmed by the steady increase of mean strain values with increasing loads after an unsteady increase ¨ during the first two load steps (Groning et al., 2009). 2.2. Model creation Prior to mechanical testing, high-resolution CT data were obtained of the specimen with an X-Tek HMX160 mCT system (X-Tek Systems Ltd., UK). The two halves of the specimen were scanned separately, since the mandible was slightly above the size limit for the mCT scanner used. After primary reconstruction with NGI CT Control Software (X-Tek Systems Ltd., UK) two 16-bit TIFF image stacks with a voxel size of 0.12 mm in all three directions were created. All the following image processing was carried out with Amira 4.1.1 (Mercury Computer Systems Inc., USA). Bone and teeth were separated from the surrounding air using the halfmaximum height protocol (Spoor et al., 1993) and the resulting 3D models of the separately scanned left and right halves of the mandible were reconnected by landmark-guided superimposition. In order to compare different FEA approaches, two FE meshes were created. One model with 19.6 million 8-noded linear brick ¨ finite elements was created by direct voxel conversion (for details see Groning et al., 2009). For the second model, the Amira surface was imported into HyperMesh 10.0 (Altair Engineering Inc., USA) and re-meshed with 0.5 mm triangular elements using the ‘‘shrink-wrap’’ function. Surface-based tetrahedral meshing was performed, resulting in a model of 2 million 4-noded linear tetrahedral elements (C3D4 type). This very high number of elements made a convergence study unnecessary, since published convergence tests with similar tetrahedral FE meshes of human mandibles have shown that a much smaller number of elements (o 100,000) is sufficient to reach convergence (Al-Sukhun et al., 2007; Ramos et al., 2011; Zhao et al., 2001).

strain orientations were calculated based on the normal strains and shear strains using the following formula: tan2y ¼

gxy ex ey

where y is the maximum principal strain angle and gxy, ex and ey are the shear and normal strains in the x and y directions, respectively. The strain orientations from the FEA results were calculated based on the eigenvectors of the strain tensors. However, since the coordinate system of the DSPI sensor differed from the coordinate system of the FE models, a coordinate transformation of the strain tensors from the FEA was necessary prior to the calculation of the eigenvectors. The following transformation equation was used: ½ex0 y0 z0 ¼ ½T½exyz ½TT where [e]x0 y0 z0 is the strain tensor in the DSPI coordinate system, [e]xyz is the original strain tensor, [T] is the transformation matrix consisting of the directional cosines of the angles between the x-, y-, and z- and the x0 -, y0 -, and z0 -axes and [T]T is the transpose of the transformation matrix (Young and Budynas, 2002). All strain values, the principal strain magnitudes and directions, were smoothed by calculating the average of the strain values from each point and its closest neighbours. We then used different techniques to analyse and compare the reliability of the DSPI and FEA results. For each of the 1035 sampling points we calculated: (1) the maximum differences in strain directions and magnitudes between repeated measurements (Dyexp and Deexp), (2) the maximum strain differences between different landmark positions on the surface of the voxelbased model (DyFEA and DeFEA), (3) the differences between the experimental means and the strains from the FEA (Dyexp–FEA and Deexp–FEA), and (4) the standard deviations for the measured strain values and whether predicted FEA strain

2.3. FEA To perform the FEA we used the software VOX-FE (Fagan et al., 2007; Liu et al., 2012) for our voxel-based model and Abaqus 6.8.2 (Dassault Syste mes Simulia, Providence RI) for our surface-based tetrahedral model. We assigned homogeneous isotropic material properties of 17 GPa for Young’s modulus and 0.3 for Poisson’s ratio to bone and teeth. Since a range of Young’s modulus values for human mandibular cortical bone is found in the literature (e.g. Arendts and Sigolotto, 1989; Arendts and Sigolotto, 1990; Ashman and van Buskirk, 1987; Dechow et al., 1993; Schwartz-Dabney and Dechow, 2003), we also calculated strains for Young’s moduli of 13, 15 and 19 GPa so that we could estimate the sensitivity of the FEA results to changes in Young’s modulus. This calculation was performed by scaling the strains from the 17 GPa model (e.g. for a Young’s modulus of 15 GPa strain magnitudes were multiplied by 1.13 ( ¼ 17/15)). The experimental loading conditions were simulated by applying constraints in the vertical axis to the tips of the front teeth and the condyles and a vertical compressive force of 125 N to each mandibular angle (Fig. 1). The solution of the voxel-based model was performed on a 32-processor Eagle high-performance cluster. The tetrahedral model was exported to Abaqus and solved on a desktop PC (Windows XP Pro x64, Intel Xeon 5140 2.33 GHz CPU, 16 GB RAM). Both analyses assumed a linear, elastic solution. 2.4. Sampling points for the extraction of strain values In order to identify the areas of the model surfaces that correspond to the measurement areas, 3D surface models based on the DSPI topography measurements were superimposed onto the models using the automatic surface alignment tool in Amira. The coordinates and principal strain values for every eighth point were then extracted from the DSPI measurements. This resulted in 531 and 504 points or landmarks for the areas on the corpus and ramus, respectively (Fig. 1). The landmarks were imported into VOX-FE and HyperMesh and the nearest node for each landmark was selected on the model surfaces. This node selection was performed automatically, using a nearest neighbour algorithm, in VOX-FE, and manually in HyperMesh. Since the superimposition of the DSPI surfaces on the model surfaces and thus the registration of the landmarks on the FE meshes is prone to error, the node selection in VOX-FE was repeated eight times for each measurement area with slightly shifted landmark positions (by moving the landmarks two and four elements upwards and downwards as well as two and four elements more anteriorly and posteriorly). 2.5. Quantitative strain analysis In our analysis we considered maximum (e1) and minimum (e3) principal strain magnitudes and maximum principal strain orientations (y). Principal strain magnitudes (nodal values) were directly extracted from the selected nodes on the FE model surfaces and the corresponding points in the DSPI measurements. The

Fig. 3. Measured and predicted maximum principal strain (e1) directions: (a) measurement area on the corpus, (b) measurement area on the ramus; grey¼ repeated DSPI measurements, green¼ tetrahedral FE model, red ¼ voxelbased FE model. See Fig. 1 for the location of the selected points on the bone surface.

¨ Please cite this article as: Groning, F., et al., Improving the validation of finite element models with quantitative full-field strain comparisons. Journal of Biomechanics (2012), doi:10.1016/j.jbiomech.2012.02.009

F. Gr¨ oning et al. / Journal of Biomechanics ] (]]]]) ]]]–]]] magnitudes fell within or outside two standard deviations of the experimental means. In addition, we performed a PCA based on the strain magnitudes at all sampling points using the statistical computing language and environment R 2.12.2 (R Foundation for Statistical Computing, Austria). Table 1 Differences in maximum principal strain orientation (y) between repeated measurements (Dyexp), between different landmark positions (DyFEA) and between measurements and FE models (Dyexp–FEA). yexp is the measured and yFEA the predicted principal strain orientation at each point (see Fig. 1 for the location and density of these points on the bone surface). All differences are in degrees (1) and shown with 7one standard deviation. Corpus

Ramus

Mean Dyexpa Mean DyFEAb

6 73 24 717

775 23 7 16

Mean Dyexp–FEA values Voxel-based mesh Tetrahedral mesh

13 78 9 79

978 16 7 16

a Mean difference between the lowest and highest yexp value at each point. Number of repeated measurements: six (corpus) and four (ramus). b Mean difference between the lowest and highest yFEA value at each point. Estimated by shifting the landmarks across the surface of the voxel-based model.

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The differences between predicted and measured strain magnitudes were assessed by directly comparing the strains in the models with the measured strains and additionally by using normalised strain values. The strains were normalised by epoint–earea, where epoint is the strain value measured or predicted at a point and earea the mean strain value for the whole measurement area. The comparison of normalised strains allowed us to consider differences in the distribution of high and low strain regions separately from overall magnitude differences.

3. Results Fig. 2 (A–J) shows the maximum and minimum principal strains contour plots for all DSPI measurements. The strain directions and the overall distribution of high and low strain areas show a remarkable consistency between the recordings, and the same pattern of strain directions and high and low strain areas is found in the FE contour plots (bottom row of Fig. 2). The strain direction plots (Fig. 3) confirm that there is very little variation in the principal strain directions between the repeated measurements (see also Table 1). However, the strain magnitudes vary in most areas by more than 30 me, in some areas by more than 50 me (Fig. 4a). On average, Deexp ranges from 37 to 50 me

Fig. 4. Variation of measured and predicted strain magnitudes: (a) difference between the lowest and highest strain value measured at each point, (b) difference between the lowest and highest predicted value at each point when landmarks are shifted across the surface of the voxel-based FE model. See Fig. 1 for the location of the points on the bone surface.

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Table 2 Differences in principal strain magnitudes between repeated measurements (Deexp), different landmark positions (DeFEA) and between measurements and FE models (Deexp–FEA). eexp is the measured and eFEA the predicted principal strain magnitude at each point (see Fig. 1 for the location and density of these points on the bone surface). All differences in me are shown with 7one standard deviation. Corpus

Ramus

Absolute differences in me b 3

a

% of eFEA values within CI

Absolute differences in me

% of eFEA values within CI

e1

e2 and e

e1

e2 and e3

e1

e2 and e3

e1

e2 and e3

50 7 10 40 7 13

37 79 47 724

– –

– –

41 7 14 21 7 13

377 19 247 17

– –

– –

Mean Deexp–FEA values for the voxel-based mesh 16 7 14 Ebone ¼ 13 Gpa Ebone ¼ 15 GPa 20 7 14 25 7 15 Ebone ¼ 17 GPa Ebone ¼ 19 GPa 30 7 16 17 7 14 After normalising the strain valuese

19 713 27 715 35 716 42 717 14 710

91.1 89.8 82.7 74.0 92.8

76.3 52.2 35.8 25.8 85.3

17 7 12 13 7 9 12 7 8 12 7 9 12 7 8

267 26 317 28 367 28 417 28 207 20

80.4 89.1 92.1 93.6 92.3

65.7 50.7 41.1 32.6 73.6

Mean Deexp–FEA values for the tetrahedral mesh 14 7 10 Ebone ¼ 13 GPa Ebone ¼ 15 GPa 11 7 9 16 7 10 Ebone ¼ 17 GPa Ebone ¼ 19 GPa 22 7 11 After normalising the strain values 10 7 9

28 716 30 718 35 720 40 722 22 716

96.0 98.3 95.9 89.6 98.3

53.1 48.2 40.5 32.2 66.3

37 7 29 28 7 22 22 7 16 18 7 13 19 7 12

297 27 337 29 387 29 437 30 227 20

51.2 62.1 74.2 80.4 81.4

58.8 52.0 41.2 31.6 68.7

Mean Deexpc Mean DeFEAd

a

CI ¼confidence interval defined by two standard deviations of the experimental means at each point. Minimum principal strain. In the case of the plane strain values from the experiment the minimum principal strain equals e2, in the 3D FE models it equals e3. Mean difference between the lowest and highest eexp value measured at each point. Number of repeated measurements: six (corpus) and four (ramus). d Mean difference between the lowest and highest eFEA value at each point. Estimated by shifting the landmarks across the surface of the voxel-based model. e Calculated by subtracting the mean strain for each measurement area from the strain value at each measurement point (epoint–earea). b c

(Table 2). These are large differences given that the measured magnitudes are below 150 me in most regions (Fig. 2). The strain differences resulting from different node selections in the voxelbased FE model (DeFEA) are in general smaller than Deexp (Fig. 4b). The largest differences between landmark positions are found where strain magnitudes change to a great extent within a small area (compare Figs. 2 and 4). The comparison between measured and predicted strain values yields a good correspondence of the maximum principal strain directions (Fig. 3, Table 1). Major differences between yFEA and yexp are only found in the bottom left corner of the corpus area and the bottom right corner of the ramus area. The differences between measured and predicted strain magnitudes (Deexp– FEA) vary between the two measurement areas, between different regions within the measurement areas and between the two principal strains (Fig. 5). However, Deexp–FEA tends to be larger for the tetrahedral model than for the voxel-based model (except for maximum principal strain in the corpus area) as well as larger for minimum principal strain than for maximum principal strain (Table 2). Interestingly, there are correlations between the spatial distributions of Deexp–FEA, Deexp and DeFEA values. For instance, the area where the largest differences in minimum principal magnitudes between measurements and landmark positions are observed, the posterior margin of the ramus, is also the area where a low correspondence between measured and predicted strains is found (Figs. 4 and 5). The majority of the eFEA values fall within the confidence interval (CI) that is defined by two standard deviations of the experimental means. Table 2 shows that the percentage of eFEA values within CI is in general higher for maximum principal strain than for minimum principal strain. In most cases these percentages increase when Young’s modulus decreases. However, the opposite, an increase of the percentage with increasing Young’s modulus is found for maximum principal strain in the ramus. The PCA results (Fig. 6) show that for both principal strains and both measurement areas most of the total variance, between 82

and 89%, is explained by the first two components and that PC 1 explains between 60 and 74% (Fig. 5). As seen in the difference plots (Fig. 4), the distance between the FEA results and measurements varies between the two measurement areas, the strain types and the two models. The tetrahedral model falls very close to the measurements for maximum principal strain in the corpus area. In the other PCA plots, however, the voxel-based model is closer to the measurements. In the case of minimum principal strain in the corpus area, the results for different Young’s moduli in the voxel-based model fall on the same line as the measurements, suggesting that the differences between the model and the measurements are mainly magnitude differences. However, in the other cases they do not fall on the same line, which suggests differences in the strain distribution in addition to magnitude differences.

4. Discussion Compared to our previous validation approach, which was based mainly on principal strain magnitudes along lines through ¨ ¨ 2011; Groning DSPI measurement areas (Bright and Groning, et al., 2009), our new approach proves much more informative and accurate. First, the selection of corresponding points in the DSPI measurements and the FEA model is greatly facilitated using the 3D coordinates from the DSPI surface topography measurements, since a direct superimposition onto the model surface is possible and thus the selection of corresponding points in the DSPI measurements and the FE models become more accurate. Second, the number and spatial distribution of the sampling points we used in this study allows a quantitative comparison of the whole measurement areas, and both areas combined represent a large part of the bone surface of the tested specimen, which is a much more comprehensive validation than with single point measurements from strain gauges or with single strain profiles from DSPI measurements. Based on these new results we can refine and elaborate the ¨ results of our previous study (Groning et al., 2009). Although

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Fig. 5. Differences between measured and predicted principal strain magnitudes after normalising the strains by subtracting the mean strain for the measurement area from the strain value for each point (see text for details): (a) voxel-based FE model, (b) tetrahedral FE model.

there is very little variation of the strain directions between repeated measurements, strain magnitudes vary by more than 50 me in several areas, which is relatively high considering the maximum resolution of the DSPI sensor used (5–20 me according to the manufacturer). However, DSPI is known to have a low ¨ signal-to-noise ratio (e.g. Barak et al., 2009; Bright and Groning, 2011) and such a large variation is therefore not surprising. It stresses the importance of a sufficient number of measurement repeats and sufficiently high loads to measure magnitudes larger than the measurement error in order to obtain meaningful results with DSPI. In addition, our results show that the selection of nodes on the model surface is a major source of error and that an accurate identification of the measurement points on the model, e.g. with 3D surface measurements and automatic superimposition on the FE model surface, is crucial. The correspondence between measured and predicted strain directions and magnitudes is relatively good given the simplicity of the models. However, there are also clear differences. In two areas the strain directions differ considerably and on average the magnitudes differ by ca. 10–40 me. Since the measured magnitudes are below 150 me in most areas, this is a relatively large

difference ( 410% of the experimental means in most areas). The two models yielded very similar results apart from the strain directions in two regions. These differences could be due to the differences in resolution and element types (8-noded brick elements vs. 4-noded tetrahedral elements). Some differences between predicted and measured strain directions might be explained by the attachment of the DSPI sensor to the bone surface, which might constrain the bone slightly and such constraints were not considered in the FE models. In addition, the models might behave differently from the real specimen, because some of the modelled teeth were not completely separated from the alveolar bone and adjacent teeth, ¨ which probably resulted in overly stiff models (Groning and ¨ et al., 2011). Apart from that, the use Fagan, in press; Groning of homogeneous and isotropic mechanical properties is a likely explanation, since the differences vary across the surface and between the two principal strains: e.g. both models predict maximum principal strain magnitudes well, but underestimate minimum principal strain magnitudes, and in one area the prediction of maximum principal strain magnitudes improves with increasing Young’s modulus, while in another area it

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Fig. 6. PCA plots for maximum and minimum principal strain magnitudes (A–F: repeated measurements of the corpus area, G–J: repeated measurements of the ramus area, vox13–vox19: FEA results from the voxel-based model for Young’s modulus values ranging from 13 to 19 GPa, tet1–tet19: FEA results from the tetrahedral model for the same range of Young’s modulus values).

improves with decreasing Young’s modulus. Most probably, these problems are caused by the fact that we did not consider the heterogeneity and orthotropy of the mechanical properties of mandibular bone (e.g. Arendts and Sigolotto, 1989; Arendts and Sigolotto, 1990; Dechow et al., 1993). Future studies could explore how the correspondence between experimental and FE results can be further improved by assigning more realistic mechanical properties and by testing mandible replicas with homogeneous isotropic properties. However, determination of the detailed variation of the properties through the mandibular bone at an equivalent scale to this study would itself be challenging, and in reality unfeasible in most FE studies.

Acknowledgements We would like to thank Lee Page and Jia Liu for programming VOX-FE and Andreas Biternas, Gareth Neighbour, Thorsten Siebert, Sue Taft and the workshop of the University of Hull for their technical support. We are also grateful to John Currey for discussions and comments on the manuscript. This work was funded by Marie Curie Action MEST-CT-2005-020601 (PALAEO) and used tools developed with support from BBSRC (BB-E0078131/BB-E009204-1 and BB-E013805-1/BB-E014259-1).

References Conflict of interest statement The authors confirm that there is no conflict of interest in this manuscript.

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¨ Please cite this article as: Groning, F., et al., Improving the validation of finite element models with quantitative full-field strain comparisons. Journal of Biomechanics (2012), doi:10.1016/j.jbiomech.2012.02.009