Multiscale modeling of polycrystalline materials with Jacobian-free multiscale method (JFMM)

The non-linear constitutive response of polycrystalline aggregate of single crystal grains is obtained using a micro-macro transition scheme, which is realized within a Jacobian-free multiscale method (JFMM). The Jacobian-free approach circumvents computation of the tangent matrix at the macroscale using a Newton–Krylov process. This has a major advantage in terms of storage requirements and computational cost over existing approaches based on homogenized material coefficients, which may require considerable effort to form the exact Jacobian at every Newton step. The rate-independent constitutive response of polycrystalline copper and rate-dependent stress–strain response of \(\upalpha \)-RDX (cyclotrimethylene trinitramine) polycrystal is verified against the computational homogenization based two-level finite element \((\hbox {FE}^{2})\) multiscale method. Numerical examples demonstrate that while the rate of convergence of Newton iterations for the JFMM and \(\hbox {FE}^{2}\) method is comparable, the computational cost of JFMM is nearly constant as opposed to exponential increase for the latter with increasing number of degrees of freedom \((n)\) at the macroscale. The storage requirement for the JFMM increases linearly with increasing \(n\), whereas, it increases as approximately \(O(n^{7/4})\) for the \(\hbox {FE}^{2}\) method.