THE VIRTUAL FIELDS METHOD
Association des Parents d'Elèves du Conservatoire de musique agréé et de danse de Châlons en Champagne

General presentation

A recent review paper presents the basics of the method as well as its current state-of-the-art.
Grédiac M., Pierron F., Avril S., Toussaint E., The virtual fields method for extracting constitutive parameters from full-field measurements: a review Strain: an International Journal for Experimental Mechanics, vol. 42, pp. 233-253, 2006.
This paper can be downloaded here courtesy of Blackwell publishing.

Theoretical developments

Choice of the virtual fields in elasticity

In the initial work on the VFM, the virtual fields were selected manually. Usually, rather simple polynomial virtual fields were selected but it was not easy to ensure that the resulting equations were independent. Also, some parameters proved to be relatively unstable depending on the choice of the virtual field
(Pierron F., Grédiac M., Identification of the through-thickness moduli of thick composites from whole-field measurements using the Iosipescu fixture Composites Part A: Applied Science and Manufacturing, vol. 31, n° 4, pp 309-318, 2000.).

Some improvement was achieved when "special" virtual fields were introduced
(Grédiac M., Toussaint E., Pierron F., Special virtual fields for the direct determination of material parameters with the virtual fields method. 1- Principle and definition, International Journal of Solids and Structures, vol. 39, n° 10, pp. 2691-2705, 2002 - Grédiac M., Toussaint E., Pierron F., Special virtual fields for the direct determination of material parameters with the virtual fields method. 2- Application to in-plane properties, International Journal of Solids and Structures, vol. 39, n° 10, pp. 2707-2730, 2002. - Grédiac M., Toussaint E., Pierron F., Special virtual fields for the direct determination of material parameters with the virtual fields method. 3- Application to the bending rigidities of anisotropic plates, International Journal of Solids and Structures, vol. 40, n° 10, pp. 2401-2419, 2003).
With this procedure, it was possible to select virtual fields in an automated way ensuring both the independence of the different equations and the best robustness to noise for a given basis of functions over which the virtual fields were projected. Initially, these were polynomials.
With the decisive contribution of S. Avril, a faster and mathematically sounder procedure was devised to select the best virtual fields over a basis of functions.
Avril S., Grédiac M., Pierron F., Sensitivity of the virtual fields method to noisy data, Computational Mechanics, vol. 34, n° 6, pp. 439-452, 2004.

Piecewise virtual fields

Although initially, polynomials were used to expand the virtual fields, experience has shown that piecewise virtual fields were more flexible, in particular when when conditions have to be introduced over curved edges.
Toussaint E., Grédiac M., Pierron F., The virtual fields method with piecewise virtual fields, International Journal of Mechanical Sciences, vol. 48, n° 3, pp. 256-264, 2006.

More recently, it was shown that it was convenient to use the same mesh for both expanding the virtual fields and smoothing the measured displacements. An integrated procedure was therefore set up that is now the basis of the CamFit software.
Avril S., Pierron F., General framework for the identification of elastic constitutive parameters from full-field measurements, International Journal of Solids and Structures, vol. 44, pp. 4978-5002, 2007.

Comparison between the VFM and finite element model updating

The general method used in the literature to extract mechanical parameters from kinematic full-field measurements is finite element model updating (FEMU, see the references). Recently, it was shown that in the case of linear elasticity (and some cases of non-linear elasticity) and when full-field measurements were available, finite element model updating was equivalent to the VFM. In fact, selecting a cost function in FEMU is equivalent to applying the VFM with a particular set of virtual fields. The consequence is that it is not necessary to use iterative procedures to solve this problem. The demonstration is given in the following paper.
Avril S., Pierron F., General framework for the identification of elastic constitutive parameters from full-field measurements, International Journal of Solids and Structures, vol. 44, pp. 4978-5002, 2007.