A Fourier‐series‐based virtual fields method for the identification of 2‐D stiffness distributions

Abstract

The Virtual Fields Method (VFM) is a powerful technique for the calculation of spatial distributions of material properties from experimentally-determined displacement fields. A Fourier-series-based extension to the VFM (the F-VFM) is presented here, in which the unknown stiffness distribution is parameterised in the spatial frequency domain rather than in the spatial domain as used in the classical VFM. We present in this paper the theory of the F-VFM for the case of elastic isotropic thin structures with known boundary conditions. An efficient numerical algorithm based on the 2-D fast Fourier transform is presented, which reduces the computation time by 3-4 orders of magnitude compared to a direct implementation of the F-VFM for typical experimental dataset sizes. Artefacts specific to the F-VFM (ringing at the highest spatial frequency near to modulus discontinuities) can be largely removed through the use of appropriate filtering strategies. Reconstruction of stiffness distributions with the F-VFM has been validated on three stiffness distribution scenarios under varying levels of noise in the input displacement fields. Robust reconstructions are achieved even when the displacement noise is higher than in typical experimental fields.

Fast Fourier Virtual Fields Method for Determination of Modulus Distributions from Full-Field Optical Strain Data

“…For a thin 2-D sample made of an isotropic material, subject to known traction distributions around its boundary and negligible volume forces, the fundamental equation underlying the VFM may be written [1,7]: …”

“…The fact that both the expansion of Q xx and the virtual fields are represented as sine and cosine functions means that the integrals can be e xpressed as 2-D Fourier coefficients of a linear combination of the experimental strain fields. It can be shown [7] that a total of only four 2-D Fast Fourier Transforms (FFTs) are required to assemble all the terms in M. For large matrix sizes, the computational effort becomes essentially independent of the resolution of the experimental strain fields, with a theoretical reduction in computational effort by a factor of N x N y by using the fast algorithm over the direct (i.e., element by element) method of assembling the matrix M.…”

“…This Fourier version of the VFM will be denoted the F-VFM. An example of its successful application to the identification of an unknown stiffness distribution under known boundary conditions is summarized here; further details are given in [7]. …”

A Fourier‐series‐based virtual fields method for the identification of three‐dimensional stiffness distributions and its application to incompressible materials

“…The F‐VFM was developed originally for 2D geometries and extended to the case of incomplete knowledge of the boundary value distributions. In the current paper, it is extended for the first time to volumetric data resulting from, for example, measurements with digital volume correlation or phase contrast MRI.…”

On the identifiability of Anand visco-plastic model parameters using the Virtual Fields Method

“…where ε f and ε exp correspond to fitted and experimentally computed strains respectively, W i are 44 weighting factors, t f and t exp correspond to predicted and experimental lifetimes, and N t and N e 45 refer to the number of creep curves and the number of points per curve respectively. This cost func- incur high computational expense due to the large number of finite-element analyses required.…”