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Efficient fixed point and Newton–Krylov solvers for FFT-based homogenization of elasticity at large deformations

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In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton–Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton’s method using linear fixed point solvers and roughly \(100\) times less iterations than the nonlinear fixed point method. However, the Newton–Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant–Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

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  3. It is possible to work with three deformation gradients but only at the expense of two matrix operations per iteration.

  4. One should not confuse the position variable \(X\) with the solution vector \(X\) from (38).




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The authors benefited from many fruitful discussions with Heiko Andrä and Andreas Günnel.

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Correspondence to Matthias Kabel.

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This work was conducted while MK enjoyed a sabbatical leave at the Karlruhe Institute of Technology (KIT). MS gratefully acknowledges financial support by the German Research Foundation (DFG), Federal Cluster of Excellence EXC \(1075\) “MERGE Technologies for Multifunctional Lightweight Structures”.

Green’s operator for large deformations in Fourier space

Green’s operator for large deformations in Fourier space

An efficient implementation of the basic scheme (Algorithm 1) needs two main ingredients. A fast implementation of the discrete Fourier transformations as provided for example by the FFTWFootnote 6 library [20] and an explicit formula for the Fourier coefficients \(\hat{\varGamma }^0\) of Green’s operator.

The operators \(\nabla ,\) \(\mathrm{Div}\) and \(G^0\) are understood in Fourier space via

$$\begin{aligned}&\left[ \widehat{\nabla d}(\xi )\right] _{kl} = i \xi _l \hat{d}_k(\xi ),\end{aligned}$$
$$\begin{aligned}&\left[ \widehat{\mathrm{Div}D}(\xi )\right] _{k} = i \xi _l \hat{D}_{kl}(\xi ),\end{aligned}$$
$$\begin{aligned}&\left[ \widehat{G^0 d}(\xi )\right] _{k} = \hat{G}^0(\xi )_{kl}\hat{d}_{l}(\xi ), \end{aligned}$$

for \(k,l = 1,\ldots ,d\), where \(\xi = 2\pi z/L\), \(z \in {\mathbb {Z}}^d\), denotes a wave vector with the short notation \(z/L=(z_1/L_1,\ldots z_d/L_d)\). Therefore, in Fourier space the constitutive Eq. (4) and the equilibrium condition (1) have the form

$$\begin{aligned}&\hat{P}_{kL}(\xi ) = i {\mathbb {C}}^0_{kLmN} \xi _N \hat{u}_m(\xi ) + \hat{\tau }_{kL}(\xi ), \end{aligned}$$
$$\begin{aligned}&i \hat{P}_{kL}(\xi ) \xi _L = 0, \end{aligned}$$

Eliminating \(\hat{P}_{kL}\) yields

$$\begin{aligned} {\mathbb {C}}^0_{kLmN} \xi _L \xi _N \hat{u}_m(\xi ) = i \hat{\tau }_{kL}(\xi ) \xi _L. \end{aligned}$$

Due to the definition of the solution operator \(G^0\) this implies

$$\begin{aligned}&(\hat{G}^0)^{-1}_{km}(\xi ) = {\mathbb {C}}^0_{kLmN} \xi _L \xi _N, \end{aligned}$$
$$\begin{aligned}&\hat{u}_m(\xi ) = \hat{G}^0_{mk}(\xi ) i \hat{\tau }_{kL}\xi _L. \end{aligned}$$

Using (9) yields additionally

$$\begin{aligned} \hat{\varGamma }^0_{kLmN}(\xi ) = \xi _L \xi _N \hat{G}^0_{km}(\xi ). \end{aligned}$$

In the case of an isotropic reference material \({\mathbb {C}}^0\) with Lamé moduli \(\lambda _0\) and \(\mu _0\), i.e. \({\mathbb {C}}^0=\lambda _0 I \otimes I + 2 \mu _0 {\mathbb {I}}^S\) or \({\mathbb {C}}^0_{kLmN} = \lambda _0 \delta _{kL}\delta _{mN} + \mu _0 (\delta _{km}\delta _{LN} + \delta _{kN}\delta _{Lm})\) for \(k,L,m,N = 1,\ldots ,d\), the Fourier coefficients \(\hat{G}^0\) of the solution operator read (cf. [33])

$$\begin{aligned}&(\hat{G}^0)^{-1}_{km}(\xi ) \!=\! \lambda _0 \delta _{kL}\delta _{mN} \xi _L \xi _N \!+\! \mu _0 (\delta _{km}\delta _{LN} \!+\! \delta _{kN}\delta _{Lm})\xi _L \xi _N, \end{aligned}$$
$$\begin{aligned}&(\hat{G}^0)^{-1}(\xi ) = (\lambda _0 + \mu _0) \xi \otimes \xi + \mu _0 |\xi |^2 I,\end{aligned}$$
$$\begin{aligned}&\hat{G}^0(\xi ) = \frac{(\lambda _0 + 2 \mu _0) |\xi |^2 I - (\lambda _0 + \mu _0) \xi \otimes \xi }{\mu _0(\lambda _0+2\mu _0)|\xi |^4}. \end{aligned}$$

By applying (62) we arrive at an explicit formula for the Fourier coefficients of the Green’s operator

$$\begin{aligned}&\hat{\varGamma }^0(\xi )\hat{\tau } = \frac{\hat{\tau }\xi \otimes \xi }{\mu _0 |\xi |^2} -\frac{\lambda _0+\mu _0}{\mu _0\left( \lambda _0+2\mu _0\right) } \frac{\hat{\tau }\xi \cdot \xi }{|\xi |^4} \xi \otimes \xi , \end{aligned}$$
$$\begin{aligned}&\hat{\varGamma }^0_{kLmN}(\xi ) = \frac{\delta _{km}\xi _L\xi _N}{\mu _0 |\xi |^2} - \frac{\lambda _0+\mu _0}{\mu _0\left( \lambda _0+2\mu _0\right) } \frac{\xi _k \xi _L \xi _m \xi _N}{|\xi |^4}. \end{aligned}$$

Symmetrizing \(\hat{\varGamma }^0\) gives the Fourier coefficients of Green’s operator for linear elasticity.

If the isotropic reference material \({\mathbb {C}}^0\) with Lamé moduli \(\lambda _0\) and \(\mu _0\) is not symmetrized (cf. Sects. 3.1 and 3.2.5), i.e. \({\mathbb {C}}^0=\lambda _0 I \otimes I + 2 \mu _0 {\mathbb {I}}\) or \({\mathbb {C}}^0_{kLmN} = \lambda _0 \delta _{kL}\delta _{mN} + 2\mu _0 \delta _{km}\delta _{LN}\) for \(k,L,m,N = 1,\ldots ,d\), the Fourier coefficients \(\hat{G}^0\) of the solution operator read

$$\begin{aligned}&(\hat{G}^0)^{-1}_{km}(\xi ) = \lambda _0 \delta _{kL}\delta _{mN} \xi _L \xi _N + 2 \mu _0 \delta _{km}\delta _{LN}\xi _L \xi _N, \end{aligned}$$
$$\begin{aligned}&(\hat{G}^0)^{-1}(\xi ) = \lambda _0 \xi \otimes \xi + 2\mu _0 |\xi |^2 I,\end{aligned}$$
$$\begin{aligned}&\hat{G}^0(\xi ) = \frac{(\lambda _0 + 2 \mu _0) |\xi |^2 I - \lambda _0 \xi \otimes \xi }{2\mu _0(\lambda _0+2\mu _0)|\xi |^4}. \end{aligned}$$

By applying (62) we arrive at an explicit formula for the Fourier coefficients of the Green’s operator

$$\begin{aligned}&\hat{\varGamma }^0(\xi )\hat{\tau } = \frac{\hat{\tau }\xi \otimes \xi }{2\mu _0 |\xi |^2} - \frac{\lambda _0}{2\mu _0\left( \lambda _0+2\mu _0\right) } \frac{\hat{\tau }\xi \cdot \xi }{|\xi |^4} \xi \otimes \xi , \end{aligned}$$
$$\begin{aligned}&\hat{\varGamma }^0_{kLmN}(\xi ) = \frac{\delta _{km}\xi _L\xi _N}{2\mu _0 |\xi |^2} - \frac{\lambda _0}{2\mu _0\left( \lambda _0+2\mu _0\right) } \frac{\xi _k \xi _L \xi _m \xi _N}{|\xi |^4}. \end{aligned}$$

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Kabel, M., Böhlke, T. & Schneider, M. Efficient fixed point and Newton–Krylov solvers for FFT-based homogenization of elasticity at large deformations. Comput Mech 54, 1497–1514 (2014).

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