In order to reduce requirements in computational ressources, efficient Phase Field formulations should be derived in such a way that the evolving interfaces are resolved with ultimately a single grid point, such as within the Sharp Phase Field Method (A. Finel et al, PRL, 2018). However, an immediate consequence is that, when elastic fields are involved, elastic material properties (eigenstresses, stiffness tensors) display strong discontinuities which, in turn, generates mathematical problems when mechanical equilibrium is identified using standard mechanical solvers.
Despite their popularity, most available Fast Fourier Transform (FFT) solvers, such as the original continuous Khachaturyan and Moulinec-Suquet solvers (A. Khachaturyan et al, Phys. Rev. B, 1995, H. Moulinec and P. Suquet, Comp. Meth. Appl. Mech. Engrg, 1998) as well as the discrete “rotated scheme” (F. Willot, C. R. Mécanique, 2015), suffer from these mathematical difficulties, such as the presence of unphysical stress overshoots, oscillations and checkerboarding close to and away from the interfaces. Also, simple fixed point algorithms, whose convergence rate is often inversely proportional to material contrasts, converge slowly when contrast is high and do not converge at all in presence of voids or rigid inclusions.
We present here a new discrete mechanical solver based on FFT that is simple, fast and robust. It is, by construction, mathematically stable and therefore free of any unphysical patterns, such as oscillations, ringing or checkerboarding, even in the presence of infinite elastic contrasts or stress singularities.  We illustrate the ability of this solver to accurately reproduce mechanical fields in a variety of situations, including voids, cracks, precipitates and composite materials. We also demonstrate its numerical efficiency in terms of convergence rate and show in particular that, even with the simplest iterative scheme, the number of iterations to reach mechanical equilibrium is always moderate and stays finite when the elastic contrast diverges.