Fourier-series-based Virtual Fields Method for the identification of 3-D stiffness distributions

The virtual fields method (VFM) is a powerful technique for determining stiffness distributions of a sample, based on measured full-field strain data. The advantage of the VFM over many other methods is its ability to solve inverse problems of this type without any iteration. A key step in any application of the VFM is the selection of the virtual fields. Several techniques are based on the use of polynomials of spatial variables (either on the whole domain or in a piecewise form), and the material properties are considered as having single values (homogeneous) within the domain. The first attempt to parameterise the material properties as a function of spatial variables was proposed in for reconstruction of the stiffness map of a plate with impact damage. In this paper, we retain the basic concepts underlying the VFM but approach the parameterisation of the material properties in the spatial frequency, rather than spatial, domain by performing a 3-D Fourier series expansion of the stiffness distribution over the region of interest. Furthermore, the virtual fields are not selected as polynomials of spatial variables as in the previous VFM literature, but from a set of simple cosine or sine functions of different spatial frequencies. The abbreviation F-VFM will be used to denote the VFM in which both a Fourier series is used for the material property parameterization, and cosine/sine functions for the virtual fields. The F-VFM was developed originally for 2-D geometries; here it is extended to volumetric datasets resulting, for example, from measurements with Digital Volume Correlation or Phase Contrast Magnetic Resonance Imaging.