Non-incremental response evaluation in geometrically nonlinear structural dynamics using a space-time stiffness operator

摘要

This contribution presents a proper generalized decomposition-based nonlinear solver for an efficient solution of geometrically nonlinear dynamic problems. The solution is built as a sum of dyadic products of space and time modes, and this sum of so-called enrichments is truncated when the required accuracy is achieved. In the proposed algorithm, we apply a consistent linearization of the residual vectors around the currently known solution over the whole space-time domain. At first, the set of vectorized tangent stiffness matrices is separated in space and time using the singular value decomposition. Then, the left and right singular vectors are reshaped into matrices to separate the space-time stiffness operator. The latter can be incorporated into the alternating fixed-point algorithm to compute couples of space and time modes. Numerical examples of a two-dimensional geometrically exact beam model demonstrate the accuracy, efficiency, and limits of the method.