Jacobian-Free Newton Krylov based iterative strategies in frequency-domain analyses

This study examines the theory and performance of Jacobian-Free Newton Krylov (JFNK) methods for the efficient, iterative solution of steady state dynamics of linear vibrating systems in the frequency domain. Currently, most commercial FEM algorithms employ use direct factorization of large system matrices to achieve steady state solution. Such approaches are usually quite demanding on memory and CPU requirements. Some implementations exploit iterative solutions, but still use large, assembled system matrices, requiring significant memory. The methods investigated in this work avoid the formation of a matrix completely, minimizing memory requirements and enabling much larger problems to be performed on desktop computers. Here, the Conjugate-Gradient (CG) and Transpose Free Quasi Minimal Residual (TFQMR) algorithms are studied as possibilities. These methods are of particular interest for adaptation to finite element software which uses explicit transient dynamics, because such software's optimal architecture prevents the formation and solution of a stiffness (or a tangent stiffness) matrix. In order to demonstrate the advantages of these algorithms, we choose examples that use the JFNK-CG and JFNK-TFQMR technique to show computational advantages over regular matrix based solutions, with significant decrease in memory requirements.