Superaccurate effective elastic moduli via postprocessing in computational homogenization

With the complexity of modern microstructured materials, computational homogenization methods have been shown to provide accurate estimates of their effective mechanical properties, reducing the involved experimental effort considerably. After solving the balance of linear momentum on the microscale, the effective stress is traditionally computed through volume averaging the microscopic stress field. In the work at hand, we exploit the idea that averaging the elastic energy may lead to much more accurate effective elastic properties than through stress averaging. We show that the accuracy is roughly doubled when using energy equivalence instead of strain equivalence for compatible iterates of iterative schemes. Thus, to achieve a prescribed accuracy, the necessary effort is roughly reduced by a factor of two. In addition to the theory, we provide a handbook for utilizing these ideas for modern solvers prominent in FFT-based micromechanics. We demonstrate the superiority of energy averaging through computational examples, discuss the peculiarities of polarization methods with their non-compatible iterates and expose a superaccuracy phenomenon occurring for the linear conjugate gradient method.