by Riccardo Broglia, Matteo Diez (CNR-INM) and Lorenzo Tamellini (CNR-IMATI)
The design of efficient seagoing vessels is key to a sustainable blue growth. Computer simulations are routinely used to explore different designs, but a reliable analysis must take into account the unavoidable uncertainties that are intrinsic to the maritime environment. We investigated two ways of performing this analysis in an effective, CPU-time parsimonious way.
The sustainable growth of the marine and maritime sectors involves ship designers, shipyards and operators in the quest for energy-efficient ships and safe operations at sea. Computational models, such as computational fluid and structural dynamics, can provide accurate predictions of complex physical phenomena and, combined with design/operational space exploration methods, drive the decision-making process.
However, even assuming perfect models and error-free solvers, multiple sources of uncertainty (such as variability of sea and operational conditions) impact the performance analyses. Ship design and performance prediction must take into account this uncertainty to provide meaningful information to decision-makers. For instance, the goal might be to design a ship that displays optimal resistance and seakeeping on average for a wide range of sea conditions, or alternatively to optimise worst-case scenario performance.
Such results can only be obtained by first testing several scenarios, and then computing statistical indicators such as expected value, variance, quantiles, that summarise the variability of the quantities of interest (i.e. the outputs of the analysis, such as ship resistance, motion, safety-related quantities). In other words, an uncertainty quantification analysis must be performed and possibly repeated for multiple ship designs until a satisfactory design is found.
Even with today’s computers, uncertainty quantification analyses of complex problems are extremely computationally intensive. Performing them in an effective, CPU-time parsimonious way is key to a successful design process – a challenge that we are addressing in a collaboration between CNR institutes: CNR-INM [L1] and CNR-IMATI [L2]. CNR-INM has a long tradition in computational methods for ship performance assessment (as well as in experimental validation, thanks to large experimental facilities), while CNR-IMATI is an applied mathematics institute. A first set of results is reported in , where we consider the problem of computing expected value, variance and probability density function of the ship resistance of a roll-on/roll-off passenger ferry, subject to two uncertain operational parameters: ship speed and draught, which are representative of operating conditions including payload. The design of the ferry, shown in panel (a) of Figure 1, is being made available in the framework of the H2020 EU Project Holiship [L3]. More complex problems with a larger number of uncertainties are subject to ongoing investigation.
Two ingredients required for a successful uncertainty quantification analysis are an effective strategy to decide which scenarios should be tested (i.e. which values of ship speed and draught), and an efficient solver to evaluate each scenario. For the latter, we employ X-navis, a multi-grid RANS code developed at CNR-INM (panel (a) of Figure 1 shows the free surface elevation computed by X-navis). The multi-grid nature of the solver is also crucial for the former point, i.e., for the strategy that decides the scenarios to be tested. Indeed, we employ a multi-fidelity strategy: we first solve a substantial number of scenarios on a coarse RANS grid (hence with a limited computational cost) to get a rough estimate of the variability of the ship’s resistance over the range of considered values of ship speed and draught, and then iteratively refine the RANS grid and further solve a few additional scenarios to get a more precise estimate for some critical combinations of ship speed and draught. In  we employed four grids, ranging from 5.5 million grid cells for the finest mesh down to 11,000 for the coarsest. In addition to the multi-fidelity strategy, we also employed an adaptive paradigm, where the scenarios to be solved for each RANS grid are not determined a-priori, but decided on-the-go as the computation proceeds, based on suitable criteria.
Figure 1: From left to right: (a) the ferry used for the numerical tests, and free surface waves computed by X-navis; (b) and (c) the sampling of the ship speed/draught space provided by MISC and SRBF strategies respectively (i.e. the scenarios solved by the methods); (d) the resulting probability density function of the ship-resistance (we report the results obtained by SRBF only, as those obtained by MISC are comparable). Source: 
Two different non-intrusive approaches to uncertainty quantification, resulting in two batches of simulations, have been considered in : the Multi-Index Stochastic Collocation (MISC) method  and an adaptive Stochastic Radial Basis Function (SRBF) method . The former samples the ship speed/draught space in a rather structured way (panel (b) of Figure 1), while the latter results in a more scattered sampling (panel (c) of Figure 1). Different colours denote the different RANS grids used at each ship speed/draught combination. Despite the differences in the sampling strategies, both methods give similar results: the average (expected value) ship resistance (at model scale) is around 52 Newton, while the standard deviation is around 22 Newton. The full probability density function, shown in panel (d) of Figure 1, features a heavy tail towards large resistances. Our experience suggests that MISC seems to reach a good accuracy for slightly smaller computational costs than SRBF, while SRBF seems to be more robust to the numerical noise that affects the RANS simulations, especially on the coarser grids. All in all, however, both methods seem to be viable candidates to tackle more complex problems with a larger number of uncertain parameters.
In summary, the design of efficient seagoing vessels is vital for sustainable blue growth. Ship performance analyses become more meaningful if they report the uncertainty quantification analysis, i.e. the variability of the performance due to the unavoidable uncertainties in the operating regime, which means that several scenarios should be tested, and results condensed in statistical terms (average, standard deviation, etc). Recently developed multi-fidelity, adaptive algorithms can provide these figures in a time-effective way, and further research in this area is mandatory to make these methods even more reliable, effective and ready for the general public.
 C. Piazzola et al.: “Uncertainty Quantification of Ship Resistance via Multi-Index Stochastic Collocation and Radial Basis Function Surrogates: A Comparison”, Proc. of the AIAA Aviation Forum 2020. Also available as arXiv e-prints, 2005.07405.
 J. Beck et al.: “IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains”, Computer Methods in Applied Mechanics and Engineering, 2019.
 J. Wackers et al.: “Adaptive N-Fidelity Metamodels for Noisy CFD Data”, Proc. of the AIAA Aviation Forum 2020.