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

Abstract

We present an inverse method to identify the spatially varying stiffness distributions in 3 dimensions. The method is an extension of the classical Virtual Fields Method—a numerical technique that exploits information from full‐field deformation measurements to deduce unknown material properties—in the spatial frequency domain, which we name the Fourier‐series‐based virtual fields method (F‐VFM). Three‐dimensional stiffness distributions, parameterised by a Fourier series expansion, are recovered after a single matrix inversion. A numerically efficient version of the technique is developed, based on the Fast Fourier Transform. The proposed F‐VFM is also adapted to deal with the challenging situation of limited or even non‐existent knowledge of boundary conditions. The three‐dimensional F‐VFM is validated with both numerical and experimental data. The latter came from a phase contrast magnetic resonance imaging experiment containing material with Poisson's ratio close to 0.5; such a case requires a slightly different interpretation of the F‐VFM equations, to enable the application of the technique to incompressible materials.

On improving the accuracy of nonhomogeneous shear modulus identification in incompressible elasticity using the virtual fields method

“…We observed from both simulations and experiments that the proposed virtual fields performed much better in identifying the shear moduli of incompressible solids than the conventional virtual fields. This reveals that the conventional approach, though prevalent in solving parameter identification problems [20,[27][28][29], leads to errors in the identified shear modulus values. Conversely, the proposed virtual fields are capable of identifying shear moduli with very high precision for incompressible and nonhomogeneous solids.…”

“…Thereby, it is necessary to make assumptions about the hydrostatic pressure values but if these assumptions are not satisfied, this will induce significant errors on the estimated mechanical properties of solids. To address this issue, a conventional choice of virtual fields which were previously considered [29,30] was based on null displacements on the boundary 0 on    u (4) where  is the boundary of the problem domain. In this case,…”

“…( 5)) [20]. This approach has been commonly used to identify mechanical parameters of incompressible materials when the VFM was adopted [20,[27][28][29].…”

Stabilized finite elements for time‐harmonic waves in incompressible and nearly incompressible elastic solids

“…Elastomeric materials, for example, are often accurately modeled as incompressible, and modeling their dynamical response is of current interest in understanding elastomeric actuators . Biological tissues are also often modeled as incompressible elastic or viscoelastic materials, and propagation of shear elastic waves in such media is of interest in a new medical imaging field called elastography . Therefore, it is of interest to have accurate, efficient, and stable methods to compute elastic wave fields in incompressible and nearly incompressible materials.…”

“…[1][2][3] Biological tissues are also often modeled as incompressible elastic or viscoelastic materials, and propagation of shear elastic waves in such media is of interest in a new medical imaging field called elastography. [4][5][6][7][8][9][10][11][12][13][14][15] Therefore, it is of interest to have accurate, efficient, and stable methods to compute elastic wave fields in incompressible and nearly incompressible materials. Unfortunately, classical Galerkin discretization struggles with both incompressibility and wave dispersion.…”