Efficient variable stiffness methods for cooling of hot-rolled steel sections

An industrial finite element package for the simulation of alternative cooling strategies for hot-rolled steel sections has been enhanced by the incorporation of a variable stiffness second-order time-integration scheme, based on a specially-developed family of extended-stability explicit Runge-Kutta methods, and an L-stable semi-implicit formula. The integration scheme uses local error estimation to vary step-size, and Runge-Kutta method selection is achieved by monitoring the numerical stiffness through the solution period, using a computationally inexpensive estimate for the spectral radius of the system Jacobian. Numerical tests on a range of section-cooling problems indicate that the variable stiffness RK code developed is considerably more efficient than standard first-order integration methods. For the levels of accuracy required for industrial simulations, the efficiency compares favorably with other high efficiency codes, such as the variable order backward differentiation formulae (VODE) and the ‘nearly stiff’ (RKC) explicit solver.