Building upon the equivalence of the basic scheme in the work of Moulinec and Suquet with gradient descent methods, we investigate the effect of using the celebrated Barzilai-Borwein step size selection technique in this context. We provide an overview of recent convergence theory and present efficient implementations in the context of computational micromechanics, with and without globalization. In contrast to polarization schemes and fast gradient methods, no lower bound on the eigenvalues of the material tangent is necessary for the Barzilai-Borwein scheme. We demonstrate the power of the proposed method for linear elastic and inelastic large scale problems with finite and infinite material contrast.
KEYWORDSBarzilai-Borwein method, FFT-based solver, inelastic material behavior, micromechanics, porous material
INTRODUCTIONThe fast Fourier transform (FFT)-based computational homogenization method 1,2 was introduced to efficiently solve the cell problem of mathematical homogenization for microstructures given on a regular (voxel) grid, for instance, as typical results of microcomputed tomography. Its success rests on three pillars. Firstly, it heavily relies upon the FFT, for which fast implementations are available. Secondly, the basic (solution) scheme in the works of Moulinec and Suquet 1,2 can be implemented, keeping a single strain field stored in memory, which is critical for large digital volume images. Last but not the least, incorporating nonlinear and inelastic material behavior has been a focus from the very beginning.Since its inception, the areas of application of the FFT-based techniques have been extended significantly, incorporating finite strain mechanics, 3 piezoelectic materials, 4 crystal viscoplasticity, 5 thermo-magneto-electro-elasticity, 6 trabecular bones, 7 explosive materials, 8 and brittle fracture. [9][10][11] The required iteration count of the basic scheme in the work of Moulinec and Suquet depends strongly on the material contrast, ie, the ratio of the largest and the smallest eigenvalue of the material tangents (evaluated for all voxels). For large or infinite contrast, iteration counts can become excessive. Thus, accelerated solution techniques were developed. Actually, these solution methods come in two species. For the first species, each iterate satisfies the mechanical compatibility condition, whereas, for the second species, compatibility is only ensured upon convergence. The first species includes the conjugate gradient method, 12 the Newton-scheme (coupled to different linear solvers), 3,13,14 and the fast 482