In the liquefied natural gas (LNG) shipping industry, the phenomenon of sloshing can lead to the occurrence of very high pressures in the tanks of the vessel. The issue of modelling or estimating the probability of the simultaneous occurrence of such extremal pressures is now crucial from the risk assessment point of view. In this paper, heavy-tail modelling, widely used as a conservative approach to risk assessment and corresponding to a worst-case risk analysis, is applied to the study of sloshing. Multivariate heavy-tailed distributions are considered, with Sloshing pressures investigated by means of small-scale replica tanks instrumented with d >1 sensors. When attempting to fit such nonparametric statistical models, one naturally faces computational issues inherent in the phenomenon of dimensionality. The primary purpose of this article is to overcome this barrier by introducing a novel methodology. For d-dimensional heavy-tailed distributions, the structure of extremal dependence is entirely characterised by the angular measure, a positive measure on the intersection of a sphere with the positive orthant in Rd. As d increases, the mutual extremal dependence between variables becomes difficult to assess. Based on a spectral clustering approach, we show here how a low dimensional approximation to the angular measure may be found. The nonparametric method proposed for model sloshing has been successfully applied to pressure data. The parsimonious representation thus obtained proves to be very convenient for the simulation of multivariate heavy-tailed distributions, allowing for the implementation of Monte-Carlo simulation schemes in estimating the probability of failure. Besides confirming its performance on artificial data, the methodology has been implemented on a real data set specifically collected for risk assessment of sloshing in the LNG shipping industry.