[SY-F11] Development of a new consistent discrete Green operator for FFT-based methods to solve heterogeneous problems with eigenstrains
A new expression of the periodized discrete Green operator using the Discrete Fourier Transform method and consistent with the Fourier grid is derived from the classic ''Continuous Green Operator'' in order to avoid the problem refered to as ''aliasing'' inherent to Discrete Fourier Transform methods. It is shown that the easy use of the conventional continous Fourier transform of the modified Green operator for heterogeneous materials with eigenstrains leads to spurious oscillations when computing the local responses of composite materials close to materials discontinuities like interfaces, dislocations, edges...We also focus on the calculation of the displacement field and its associated discrete Green operator which may be useful for materials characterisation methods like diffraction techniques.
The development of these new consistent discrete Green operators in the Fourier space allows to eliminate oscillations while retaining similar convergence capability. For illustration, the new discrete Green operators are implemented in a fixed-point algorithm for heterogeneous periodic composites knows as the Moulinec and Suquet (1994,1998) ''basic scheme'' that we extended to consider eigenstrain fields, as in Anglin, Lebensohn and Rollett(2014). Numerical examples are reported, such as the computation of the local stresses and displacement of composite materials with homogeneous or heterogeneous elasticiy combined with dilatational eigenstrain or representing prismatic dislocation loops. The mechanical fields obtained for cubic-shaped inclusion, spherical Eshelby and inhomogeneity, are compared with analytical solutions and the discretization method using the classic Green operator.
The development of these new consistent discrete Green operators in the Fourier space allows to eliminate oscillations while retaining similar convergence capability. For illustration, the new discrete Green operators are implemented in a fixed-point algorithm for heterogeneous periodic composites knows as the Moulinec and Suquet (1994,1998) ''basic scheme'' that we extended to consider eigenstrain fields, as in Anglin, Lebensohn and Rollett(2014). Numerical examples are reported, such as the computation of the local stresses and displacement of composite materials with homogeneous or heterogeneous elasticiy combined with dilatational eigenstrain or representing prismatic dislocation loops. The mechanical fields obtained for cubic-shaped inclusion, spherical Eshelby and inhomogeneity, are compared with analytical solutions and the discretization method using the classic Green operator.