Computing Bounds on the Eigenvalues of the Preconditioned X‐FFT System in 3D Linear Elastic Homogenization

ABSTRACT

The previously introduced X‐FFT solver permits solving three‐dimensional linear elastic homogenization problems with high accuracy and efficiency. Due to the extended finite element discretization (X‐FEM), matrix‐inclusion problems may be approximated with the error convergence of interface‐conforming finite elements. By leveraging FFT‐based computational homogenization methods, efficient numerical computations are achieved. A crucial ingredient of the X‐FFT solver is its preconditioning strategy, which prevents ill‐conditioning by bounding the eigenvalues of the preconditioned system with some constant from above and below independently of the mesh spacing. This work presents an element‐based approach for computing explicit bounds on these constants and deduces a step size for the basic scheme using these bounds. Computational experiments analyze the sharpness of the derived step size, and we discuss its influence on other solution schemes.