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Fourier-Accelerated Nodal Solvers (FANS) for homogenization problems

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Abstract

Fourier-based homogenization schemes are useful to analyze heterogeneous microstructures represented by 2D or 3D image data. These iterative schemes involve discrete periodic convolutions with global ansatz functions (mostly fundamental solutions). The convolutions are efficiently computed using the fast Fourier transform. FANS operates on nodal variables on regular grids and converges to finite element solutions. Compared to established Fourier-based methods, the number of convolutions is reduced by FANS. Additionally, fast iterations are possible by assembling the stiffness matrix. Due to the related memory requirement, the method is best suited for medium-sized problems. A comparative study involving established Fourier-based homogenization schemes is conducted for a thermal benchmark problem with a closed-form solution. Detailed technical and algorithmic descriptions are given for all methods considered in the comparison. Furthermore, many numerical examples focusing on convergence properties for both thermal and mechanical problems, including also plasticity, are presented.

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Notes

  1. The first node according to the node numbering of the reference element Z, i.e., if \(\underline{\alpha }\) is the zero vector.

  2. Here, the fact is considered that \(n_1, \ldots , n_d\) are even (cf. Sect. 2.2).

  3. The index \( i^*\in \left\{ 1,\ldots ,n\right\} \) depends on the exact choice of the array representation in Sect. 2.6 and is uniquely determined by the requirement \( \mathscr {F}\left( \left[ \,{\mathcal {I}}\,\right] \right) \equiv 1 \) in view of (20). The DFT is generally defined such that \( {\varvec{x}}^{\langle i^*\rangle } \) corresponds to the first entry (with respect to all dimensions) of the input array. E.g., in the array representation of our implementation, this first entry corresponds to the first corner of the RVE \( (i^*=c_1) \).

  4. The confinement to isotropic (local) constitutive relations is made for simplicity but does not reflect a restriction of the FANS scheme.

  5. An assumption of this kind corresponds to a necessary Dirichlet condition that renders the boundary value problem well-defined.

  6. With the confinement to the local constitutive behavior specified by Fourier’s law (22), the effective constitutive relation is expressed by \( \bar{{\varvec{q}}} = - \bar{{\varvec{\kappa }}} \bar{{\varvec{g}}} \). The global conductivity \( \bar{{\varvec{\kappa }}}\in Sym({\mathbb {R}}^d) \) is a positive semidefinite second-order tensor, which is in general anisotropic despite the fact that the local conductivity is assumed to be isotropic at all positions \( {\varvec{x}}\in \varOmega \). Obviously, this type of global constitutive behavior is independent of the macroscopic temperature \( \bar{\theta } \), which can therefore assume arbitrary values in the present case. However, \(\bar{\theta }\) needs to be imposed to the microscopic problem when temperature-dependent conductivities are incorporated.

  7. In Eq. (28) it is exploited that \( {\underline{\underline{{\mathcal {P}}}}}^\mathsf{T} \underline{f} = \underline{0} \) holds, which is due to the assumed absence of heat sources.

  8. The gradient stencil defined in (46) is newly introduced for thermal problems here. Its Fourier space representation in (47) is inspired by results in [36] but is oriented rather to standard FE expressions than to formulations proposed in the literature on Fourier-based homogenization.

  9. The FANS fundamental solution is always computed using full integration.

  10. By exploiting the superposition principle, the number of DFTs can be reduced from \( 2^dd \) to \( 2^d \).

  11. Aside from the characteristic length \(l_\mathrm{c}\), the Fourier space expression of the error measure from Sect. 2.4 in [31] has been corrected by the factor \( 2 \pi \sqrt{n} \) in Eq. (52). Without this correction, a fixed error tolerance would be hit earlier when the discretization is refined for a given problem.

  12. The closed-form solution field is not only divergence-free, but also curl-free within both phases. Hence it is only applicable to heat conduction problems if both phases are isotropic. Note that the necessity to numerically compute the Jacobian elliptic delta function at each discretization point makes the closed-form solution computationally more demanding than FANS. To facilitate reproduction of our results, we mention a minor error in [32]: Both denominators in Eq. (3.5) must be replaced by their square roots.

  13. A violation of the HS bounds would of course not contradict the validity of the numerical results, but would indicate that the considered medium has anisotropy which is not negligible.

  14. Otherwise they loose their sparsity and, thus, efficiency.

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Acknowledgements

Funding by the German Research Foundation (DFG) via grant FR-2702/3 and within the Emmy Noether Programme of the DFG via grant FR-2702/6 is highly acknowledged. The stimulating discussions within the Cluster of Excellence Simulation Technology (funded by DFG in the scope of the excellence initiative under grant EXC 310) at the University of Stuttgart are highly appreciated.

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Authors and Affiliations

  1. Efficient Methods for Mechanical Analysis Group, Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Matthias Leuschner & Felix Fritzen

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  1. Matthias Leuschner
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  2. Felix Fritzen
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Correspondence to Felix Fritzen.

Appendices

Appendix A: Memory requirement

Taking into account symmetry, the number of nonzero entries that has to be stored for the periodized stiffness matrix \( \underline{\underline{{K^\mathrm{f}}}} \), defined for thermal problems in (28), is

$$\begin{aligned} \frac{3^d {\mathcal {D}}+ 1}{2}{\mathcal {D}}n, \end{aligned}$$
(A.1)

where n is the number of pixels/voxels and \( {\mathcal {D}} \) denotes the number of degrees of freedom per node, i.e., \( {\mathcal {D}}=1 \) for thermal problems and \( {\mathcal {D}}=d \) for general (non-) linear mechanical problems. When the storage demand is reduced by a matrix-free element-by-element (EBE) approach for thermal/mechanical linear problems with isotropic constituents, one/two stored scalar variables per element are sufficient. For more general problems, the EBE approach becomes less efficient in terms of memory requirement, and for nonlinear problems its profitability depends crucially on whether or not reduced integration can be used.

Appendix B: Construction of fundamental solutions for the SLS method

Algorithm 7 describes how the fields \( {\varvec{\psi }}^{(1)}, \ldots , {\varvec{\psi }}^{(d)} \), which constitute the fundamental solution for the original SLS, are constructed. For FA-SLS, these are re-organized in expanded fields \( {\varvec{\Psi }}^{(1)}, \ldots , {\varvec{\Psi }}^{(d)} \) that are defined on the full set of element centers \( {\mathcal {X}} \). To construct \( {\varvec{\Psi }}^{(j)} \), the field \( {\varvec{\psi }}^{(j)} \) is positioned in \( {\mathcal {X}} \) such that its central element is placed at the position \( {\varvec{x}}^{\left\{ i^*\right\} } \) characterizing the identity with respect to convolution (cf. Sect. 2.8), whereby periodic continuations across the boundaries are taken into account. Afterwards, the remainder of \( {\mathcal {X}} \) is padded with zeros. The procedure is illustrated for the two-dimensional case in Fig. 23.

figure g
Fig. 23
figure 23

Fundamental solution of the SLS method for two-dimensional heat conduction. Left: Components of the original field \( {\varvec{\psi }}^{(1)} \) on \( 15 \times 15 \) elements. Right: Components of the expanded field \( {\varvec{\Psi }}^{(1)} \) defined on the full domain of the RVE. Here, the position \( {\varvec{x}}^{\left\{ i^*\right\} } \) of the single nonzero entry of the identity with respect to convolution is located at the bottom left corner of the RVE

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Leuschner, M., Fritzen, F. Fourier-Accelerated Nodal Solvers (FANS) for homogenization problems. Comput Mech 62, 359–392 (2018). https://doi.org/10.1007/s00466-017-1501-5

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