Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton-Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

Sub-modelling for the ratchetting failure of insulated rail joints

Insulated rail joints are critical for train safety as they control electrical signalling systems; unfortunately they exhibit excessive ratchetting of the railhead near the endpost insulators. This paper reports a three-dimensional global model of these joints under wheel–rail contact pressure loading and a sub-model examining the ratchetting failures of the railhead. The sub-model employs a non-linear isotropic–kinematic elastic–plastic material model and predicts stress/strain levels in the localised railhead zone adjacent to the endpost which is placed in the air gap between the two rail ends at the insulated rail joint. The equivalent plastic strain plot is utilised to capture the progressive railhead damage adequately. Associated field and laboratory testing results of damage to the railhead material suggest that the simulation results are reasonable.