[SY-F11] Efficient FFT-based Homogenisation without Linear Reference Medium
In the field of computational homogenisation of periodic representative volume elements (RVE), over the last two decades, fast Fourier transform (FFT)-based spectral solvers have emerged as a promising alternative to the finite element method (FE).
Most spectral methods are based on work of Moulinec and Suquet [1] and split an RVE's response into that of a linear reference medium and a periodic fluctuation due to heterogeneities. The main advantage of this formulation over FE is that it can be both significantly faster and memory-saving. The two main problems are 1) the choice of the reference medium, which is typically based on heuristics, non-trivial and has a strong impact on the method's convergence (A bad choice can render the method non-convergent), and 2) convergence is not uniform. Numerous studies have suggested mitigations to both of these problems (e.g. [2]), but they have remained substantial disadvantages compared to the more expensive, but also more robust FE.
Recent work by Zeman et al. [3] proposes a new formulation for spectral solvers which dispenses with the linear reference problem and converges unconditionally. We present µSpectre, an open implementation of this novel method and use it to show that the new approach is more computationally efficient than its linear reference medium-based predecessors, converges in the presence of arbitrary phase contrast - including porosity - and eliminates or drastically reduces Gibbs ringing.
[1] H. Moulinec and P. Suquet. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering, 157(1):69-94, 1998
[2] M. Kabel, et al. Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations. Computational Mechanics, 54:1497-1514, 2014
[3] J. Zeman, et al. A finite element perspective on non-linear FFT-based micromechanical simulations. Int. J. Num. Meth. Eng., 2016
Most spectral methods are based on work of Moulinec and Suquet [1] and split an RVE's response into that of a linear reference medium and a periodic fluctuation due to heterogeneities. The main advantage of this formulation over FE is that it can be both significantly faster and memory-saving. The two main problems are 1) the choice of the reference medium, which is typically based on heuristics, non-trivial and has a strong impact on the method's convergence (A bad choice can render the method non-convergent), and 2) convergence is not uniform. Numerous studies have suggested mitigations to both of these problems (e.g. [2]), but they have remained substantial disadvantages compared to the more expensive, but also more robust FE.
Recent work by Zeman et al. [3] proposes a new formulation for spectral solvers which dispenses with the linear reference problem and converges unconditionally. We present µSpectre, an open implementation of this novel method and use it to show that the new approach is more computationally efficient than its linear reference medium-based predecessors, converges in the presence of arbitrary phase contrast - including porosity - and eliminates or drastically reduces Gibbs ringing.
[1] H. Moulinec and P. Suquet. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering, 157(1):69-94, 1998
[2] M. Kabel, et al. Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations. Computational Mechanics, 54:1497-1514, 2014
[3] J. Zeman, et al. A finite element perspective on non-linear FFT-based micromechanical simulations. Int. J. Num. Meth. Eng., 2016