\begin{alignat}{1} \varepsilon_\text{eq} \ge \frac{\operatorname{RMS}(\operatorname{Div} \textbf{P}(\textbf x)) }{|| \bar{\textbf{P}}||} \times \text{m} \label{eq:convergenceCriterion} \end{alignat}

based on the the root mean square (RMS) value of $\operatorname{Div}(\tnsr P)$, which can be conveniently calculated in Fourier space employing Parseval's theorem.

- as before (uncorrected)
- 1 meter / grid point number (fixed dimension)
- 1 meter (fixed grid point distance,
**default**) - 1 meter / square root of grid point number (dimension and square root resolution corrected)

is obtained.

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Figure 1: Periodic crystal of 20 grains (included in DAMASK as example) discretized at different resolutions. |

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Figure 2: Evolution of convergence criterion with iterations for the exemplary polycrystal at various resolutions (basic fix-point rev. 2232, first step of tensile test load case). |

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Figure 3: Polycrystal of 20 grains at different resolution periodic copies |

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Figure 4: Divergence over iterations for periodic copies of VE (basic fix-point rev. 2232, first step of tensile test load case). |

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Figure 5: Input Data |

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Figure 6: Evolution of divergence and plastic strain (basic PETSc rev. 2232) |

- [1]
- P. Eisenlohr, M. Diehl, R.A. Lebensohn, F. Roters

**A spectral method solution to crystal elasto-viscoplasticity at finite strains**

International Journal of Plasticity 46 (2013) 3753

Online version

- [2]
- M. Diehl

**A spectral method using fast Fourier transform to solve elastoviscoplastic mechanical boundary value problems**

Diploma Thesis, TU München (2010)

Download here

- [3]
- H. Moulinec, P. Suquet

**A numerical method for computing the overall response of nonlinear composites with complex microstructure**

Computer Methods in Applied Mechanics and Engineering 157 (1998) 6994

Online version

- [4]
- R.A. Lebensohn

**N-site modeling of a 3D viscoplastic polycrystal using Fast Fourier Transform**

Acta Materialia 49 (2001) 27232737

Online version

- [5]
- P. Shanthraj, P. Eisenlohr, M. Diehl, F. Roters

**Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials**

International Journal of Plasticity 66 (2015) 3145

Online version

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