Unconditionally Stable Convergence with Power Principle-based Time-Integration Schemes

This manuscript introduces a novel sufficient condition for the unconditionally stable convergence of the general class of trapezoidal integrators. Contrary to standard energy-based approaches, this convergence criterion is derived from the power principles, in terms of both balance and dissipation. This result improves the well-known condition of stable convergence based on the energy method, extending its applicative spectrum to a variety of physical problems, whose constitutive prescriptions may be more appropriately characterized by means of evolving field equations. Our treatment, tailored for generalized trapezoidal integrators, addresses both linear and nonlinear problems, extending its applicability to contexts where standard energy-based schemes present loss of stable convergence, as well as an uncontrolled increase in terms of energy. To appreciate such novel result, the Newmark-based numerical solution scheme is applied to a simplified nonlinear problem, with hardening plasticity and finite deformations within a one-dimensional description. The proposed test shows how the power-based method attains a stable convergence and overcomes the requirement for additional conservative invariants, such as energy and angular momentum.