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.
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.
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.