Optimal simulations for a simplified skier in a cross-country track are considered. The skier is modeled considering air resistance drag, friction and normal forces when following a track of cubic splines in a vertical plane. The race is modeled as a time-evolution problem, where the acting forces give the movement. Based on an assumption on how the driving power variation along the track is limited by capacity measures, a mathematical optimization problem is formulated. This minimizes the race time under a constraint of maximum integrated cost of the mechanical work rate. The paper discusses the mathematical and numerical formulations of the problem, and shows some aspects of discretization and accuracy. It is obvious from the simulations, that significant reductions in race time can be reached by strategically using the available power resources, rather than using a uniform work rate. With the fatigue criterion used, the conclusion from simulations is that it is advantageous for the skier to input extra power in parts of the track where resistance is high, i.e., in up-slopes, in parts where friction is locally higher, and in parts where a head-wind is affecting the performance. Although the criterion used catches some aspects of how power production causes fatigue, it is concluded that better descriptions are needed for fatigue accumulation and reduction during a regime with different work rates.
Cross-country skiing; Mechanics; Power; Optimality; Pacing strategy.