The files in this tutorial can be analyzed with FEBio 1.5.2 onward.

FEBio can be a very effective tool for curve-fitting material properties from experimental measurements. For example, indentation stress-relaxation of a biological tissue with a spherical indenter is a relatively easy experiment to perform, but curve-fitting the resulting load response to an analytical expression is difficult (since analytical solutions are available only for highly specialized conditions, such as Hertzian contact of half-spaces under infinitesimal strains). Since FEBio can solve this contact problem under finite deformation for arbitrary dimensions of the indenter and tissue, it can be used effectively for curve-fitting experimental data.

To illustrate this process, consider a rigid spherical indenter contacting a cylindrical tissue specimen consisting of a viscoelastic solid whose elastic response is neo-Hookean (SphericalIndenter-axisym.feb). The bottom side of the cylindrical specimen adheres to a rigid substrate. For computational efficiency, since the problem is axisymmetric, only a wedge section is used for the analysis (see the tutorial on Axisymmetric Analyses in FEBio):

SphericalIndenter-axisym.jpg

The optimization is performed on three of the four parameters that govern this material: The relaxation time constant t1, the corresponding coefficient g1 in the Prony series of the relaxation function, and Young's modulus E for the elastic part. (Only Poisson's ratio is kept fixed.) The optimization file (SphericalIndenter-axisym-opt.feb) defines these parameters and provides an initial guess, a range (min and max) and a representative scale for each of them. This file also includes the experimental stress-relaxation load response which will be fitted with the optimization analysis. (For this example, the 'experimental' response was generated with FEBio.)

Note that the experimental load response needs to be adjusted since the actual mesh only uses a thin wedge to represent the true geometry. In this example, the wedge angle is 7.5 degrees, which means that the reaction force predicted from the finite element model is only 1/48 of the actual force (7.5/360). So the experimental reaction force provided in the optimization file needs to be 1/48th of the actual force (in this example), to yield an accurate curve-fitting procedure. Thinner wedges may be used to produce a more faithful axisymmetric analysis, in which case the correction factor for the experimental reaction force should be modified accordingly.

The command to run this optimization analysis is

febio -s SphericalIndenter-axisym-opt.feb

The analysis file SphericalIndenter-axisym.feb must reside in the same directory.

The optimization analysis for this example converges with 6 major iterations and 28 minor iterations. Each minor iteration takes about 10 s on a Mac laptop, so that the entire optimization analysis completes in just a few minutes. The resulting fit is very faithful,

SphericalIndenter-axisym-Fz.jpg

and the fitted parameters (g1=0.9929, t1=99.88, E=1.001) agree very well with the actual parameters used to generate the 'experimental' data (g1=1, t1=100, E=1).

Gerard