This study explores the statistical dependence between wind speed and surge height along the Dutch coast using a large synthetic dataset. Storms were clustered based on wind direction, tidal offset, wind rotation, tidal peak, surge and wind exceedance duration, resulting in 16 clusters per wind direction and per location. Apart from wind direction, comparing clusters revealed a limited impact of clustering based on these storm characteristics on the choice of the best-fitting copula model, suggesting sub-clustering may not be necessary for accurately representing the statistical dependence between extreme wind speeds and surge heights. The BB8 copula generally provided the best fit to the data. However, the observed upper tail dependence did not decrease to zero, particularly for western to northern wind directions, indicating non-negligible dependence in joint extremes of wind speed and surge height. Therefore, applying the BB8 copula (or any other copula model without upper tail dependence) may lead to underestimation of the flood risk, when applied in probabilistic analyses. The findings from this study provide valuable insights for refining hydraulic load models for reliability assessments and design of flood defenses.

Current wind design codes incorporate turbulence through gust factors and rely on historical wind data, including tropical cyclones. While generally conservative, standard code wind profiles and spectra do not fully reproduce the vertical distribution and dynamic characteristics of hurricane winds, particularly in the supergradient region near the eyewall, and can sometimes underestimate tail risks, low-probability, high-impact events, as observed during Hurricane Otis in Acapulco (2023). This study probabilistically evaluates wind-induced vibrations in high-rise buildings with different lateral resisting systems equipped with fluid viscous dampers (FVDs), under non-tropical storm and tropical cyclone conditions. Along-wind loads were modeled in the time domain as stationary, multidimensional stochastic processes and analyzed using one million Monte Carlo simulations and Incremental Dynamic Analysis on the DelftBlue supercomputer. Statistical distributions of responses, bivariate dependence via copulas, and fragility curves were obtained. Results show that wind type, structural deformation mode, and damper properties significantly affect response distributions, correlation structures, and failure probabilities. FVDs effectively reduce structural dynamic response, improving serviceability, while increased shear stiffness further reduces fragility. Modeling hurricane winds as non-tropical storms can overestimate damper effectiveness. These findings provide insights for refining wind codes and designing high-rise buildings that remain safe and functional under extreme events.

Extreme storms over the North Sea drive coastal flood risk in the Netherlands, causing high waves and extreme sea levels. Designing flood defenses requires accurate statistical extrapolation of hydraulic load conditions with return periods of 1,000 years or more. This is a challenging task given limited observational data. This study uses a large, simulated dataset (~9,000 years) to explore the statistical dependence between extreme wind speed u and surge height s. Storms were clustered using several techniques. Self-organizing maps (SOM) effectively captured physical relationships, such as the influence of wind direction and tidal offset on storm dynamics, however variability in statistical dependence between u and s for different clusters was better represented using manual clusters. Copula models were fitted to the cluster data, with the BB8 copula outperforming others. This study illustrates the potential of machine learning to identify patterns in large datasets while emphasizing the relevance of manual clustering approaches for revealing nuanced statistical dependencies critical to flood risk assessment.

Field data of armour damage in mound breakwaters is scarce, and experimental testing methods usually neglect the influence of pre-existing damage on subsequent damage increments. This study proposes a probabilistic framework, based on a dataset of 44 cumulative damage experiments, to estimate long-term damage progression in a full-scale, non-overtopped, cube-armoured mound breakwater located in the depth-induced wave breaking zone. A Gaussian copula-based Bayesian network is constructed to model the multivariate relationships between existing armour damage (S_e), the increment of armour damage (ΔS_e), the dimensionless water depth at the toe of the structure (h_s/H_m0), wave steepness (H_m0/L_0p), and the stability number (N_s). Each variable is modelled with a univariate parametric distribution, enabling inference of probabilities for values not directly observed in the experimental dataset. Model validation includes testing the Gaussian copula assumption, which is deemed a reasonable model, and assessing the defined graph with satisfactory results. The model is conditionalized using a historical wave dataset to generate synthetic damage curves, which are subsequently used to quantify a gamma process to model the survivability of the structure along its design life. A case study of a hypothetical breakwater with D_n=2 m close to the port of Tarragona (Spain) illustrates the methodology. According to the results of the model, the probability of observing a dimensionless damage S_e>5 corresponding to Initiation of Destruction after 10 years is 0.29. Overall, the obtained results are deemed conservative; this could be caused by the use of data from 2D experiments which do not take into account oblique wave attack. However, the approach is adaptable to other datasets with additional variables, breakwaters in different conditions or with other configurations, and can also be used in combination with simulations of synthetic wave data, making it relevant under changing climate conditions.

Extreme sea level events pose significant risks to coastal regions, with non-tidal residuals (NTRs) being a primary driver in low-lying areas like the Netherlands, where shallow seas amplify their impact. This study investigates the spatial patterns of NTRs along the Dutch coast using time series clustering on historical NTR hydrographs. The design of hydraulic boundary conditions divides the Netherlands into three coastal regions. To evaluate whether this division sufficiently captures regional variability, three clustering scenarios (k = 3, k = 4, and k = 5) were explored. The analysis identified k = 5 as the optimal configuration based on the Davies-Bouldin index. This result emphasized the importance of fine-scale approaches to understanding regional spatial variations in NTR dynamics. Regional bathymetry and tide-surge interactions were explored as drivers of these spatial patterns. Southern stations near river systems and deeper waters displayed characteristics distinct from northern stations in the Wadden Sea, which are influenced by shallow tidal flats. Analysis of the M2 tidal constituent and the timing of NTR maxima relative to high tides underscored the role of tidal dynamics in shaping spatial clusters. Future research will focus on integrating spatio-temporal patterns and environmental drivers into clustering methodologies, providing deeper insights for coastal risk management and adaptation strategies.

Offshore floating structures are experiencing harsh environmental conditions risking their safety. Therefore, mooring lines are crucial for ensuring structures’ stability. Sudden increases in tensions after temporarily slack of the mooring line are called snap loads and are the most critical load states. These snap loads and their dependence to various factors are investigated in the present study. 12 study locations in the south-eastern North Sea are selected. For each location, wave and current variables are extracted from a three-dimensional large-scale numerical model covering the European Shelf. Mooring tensions at different rope positions are calculated via a Finite Element model for flexible mooring lines for different hydrodynamic conditions and used subsequently to obtain tension rates as indicator for snap loads. The dependence among 13 variables per study location is modelled via Gaussian copula-based Bayesian Networks (GCBN). This allows for spatial analysis of the relationships between hydrodynamic variables and tension rates, but also to determine the influence of hydrodynamic variables on expected tension rates. Furthermore, distributions of tension rates are obtained under specific constant hydrodynamic conditions. The results indicate that conditionalising on certain hydrodynamic variables can reduce the expected tension rates, as their marginal distributions are characterised by heavy tails. Still, mooring systems should be designed conservatively. However, once specific hydrodynamic information is available, uncertainties can be minimised, enhancing safety and reliability. Thus, accounting for the dependence among hydrodynamic variables and tension rates is crucial for improving the safety of structures under varying environmental conditions.

Manufacturing imperfections are an inherent aspect of the production process, affecting the reliability and performance of all engineered components, including actuators. This work investigates the effects of manufacturing uncertainty in coil assembly. Given the manufacturing tolerances of geometrical and material properties of wires and coils, the coil’s electrical response is studied from a probabilistic perspective. While many researchers acknowledge these uncertainties, they often do not quantify their effects on the system’s response. We propose the use of copulas to model nonlinear relationships and quantify the probabilistic dependence between design variables (i.e., wire and coil’s properties) and their effects on electrical responses. Additionally, tolerances of the design variables are determined through unstructured expert elicitation. The results show the impact of the thickness of the wire’s insulation in the performance of the coil. Furthermore, we establish a design space that is studied probabilistically, allowing for probabilistic design optimization.

Effective resource planning in higher education requires anticipating student demand for courses, especially when dealing with elective programs. Monitoring student preference is a recurring topic in the literature; however, to the authors’ knowledge, no simple methods for estimating student preferences when choosing courses in higher education have been proposed. This study develops and explores the use of a simple questionnaire to capture patterns in student course preferences within a university context. The research is developed in the context of the nine Cross-Over modules offered as part of the curriculum of the master’s programs (MSc) of the Faculty of Civil Engineering and Geosciences of Delft University of Technology (The Netherlands). No prior registration is required far in advance for these courses, making an accurate estimation of student numbers critical for the planning and allocation of educational resources. The developed questionnaire is applied three times in two different academic years to the students’ choice of Cross-Over modules. The questionnaire was shared in 2021, with 225 responses out of 339 students, in 2022, with 159 responses out of 365 students, and in 2024, with 94 responses out of 272 students. Student enrollment in the academic year 2023/2024 is used to assess the performance of the questionnaire. The questionnaire is able to capture general preferences of the students, providing fair estimates of the number of students per course; larger differences are observed in courses with a lower number of students. In addition, some patterns were identified in student preferences: there is a relationship between the first and second choices, and students usually choose modules closer to their own disciplines. The developed questionnaire provides with a reasonable first estimation of the expected number of students in courses, allowing for better planning and allocation of educational resources beforehand.

Laser powder bed fusion (LPBF), a prominent metal-based additive manufacturing (AM) technique, enables the production of complex, neat-net-shape components with minimal material waste and reduced lead times. However, achieving high final product quality is challenging due to numerous process variables and intricate, nonlinear interactions introduced by LPBF’s thermal-mechanical mechanisms. This study employs a copula-based analysis using multi-physics numerical simulations to probabilistically map relationships among process and part quality variables. Dependence and tail-dependence analyses are performed to provide deeper insights into variable interactions, enabling the identification of preferred operational windows with a balanced trade-off between product quality and productivity. The developed methodology advances the understanding of uncertainty propagation in LPBF, contributing toward improved process optimization, repeatability, and reliability.

Base isolation of high-rise buildings has growing popularity to limit peak floor accelerations under seismic loads; however, it may increase susceptibility to wind-induced vibrations due to the increase in fundamental vibration period. This study presents an equivalent coupled-two-beam (CTB) model incorporating base isolation (BI) and a tuned mass damper inerter (TMDI) to evaluate passive vibration control under lateral wind loads for various lateral resisting systems. A 144-meter-tall building was analyzed under along-wind and across-wind loads simulated as Gaussian processes, considering six isolator-damper configurations: (1) fixed-base (FB), (2) FB with a top TMDI (FB-TTMDI), (3) BI, (4) BI with a top TMDI (BI-TTMDI), (5) BI with a bottom TMDI (BI-BTMDI), and (6) BI with double TMDI (BI-DTMDI). TMDIs were compared to traditional tuned mass dampers (TMDs) to assess mass amplification under varying base isolator damping. Optimization strategies were explored to enhance vibration control: for FB-TTMDI, the TMDI placement minimized RMS accelerations, while for BI-TTMDI, it was optimized to reduce peak displacement. Finally, design guidelines are provided for ultimate and serviceability limit states. Results indicate hybrid control systems are most effective when lateral deformation resembles pure bending, making them suited for shear wall-frame and tubular systems.

In absence of sufficient data, structured expert judgment is a suitable method to estimate uncertain quantities. While such methods are well established for individual variables, eliciting their dependence in a structured manner is a less explored field of research. We tested the performance of experts in constructing and quantifying a nonparametric Bayesian network, describing the correlation between river tributary discharges. Specialized software was provided to assist the experts. Expert performance was investigated using the dependence calibration score (a correlation matrix distance metric) and the likelihood of the joint distribution. Desirable properties of the dependence calibration score were investigated theoretically. Individual expert judgments were combined based on performance into a group opinion aka decision maker. All experts were able to create and quantify a correlation matrix between 10 variables that resembled the correlations between observed discharges well. The decision makers performed similarly to the best expert. Based on the metrics investigated, it mattered little which expert opinions and with what weight were combined in a decision maker. This is partly because all experts performed well. Adding a bad performing expert increased the positive effect of performance-based weighting, underscoring the importance of developing scoring rules for dependence elicitation. The overall results are promising: Aided by specialized graphical software, the experts in this study were able to quickly create and quantify dependence structures.

This study evaluates five scoring rules, or measures of statistical accuracy, for assessing uncertainty estimates from expert judgment studies and model forecasts. These rules — the Continuously Ranked Probability Score ((Formula presented.)), Kolmogorov-Smirnov ((Formula presented.)), Cramer-von Mises ((Formula presented.)), Anderson Darling ((Formula presented.)), and chi-square test — were applied to 6864 expert uncertainty estimates from 49 Classical Model (CM) studies. We compared their sensitivity to various biases and their ability to serve as performance-based weight for expert estimates. Additionally, the piecewise uniform and Metalog distribution were evaluated for their representation of expert estimates because four of the five rules require interpolating the experts' estimates. Simulating biased estimates reveals varying sensitivity of the considered test statistics to these biases. Expert weights derived using one measure of statistical accuracy were evaluated with other measures to assess their performance. The main conclusions are (1) (Formula presented.) overlooks important biases, while chi-square and (Formula presented.) behave similarly, as do (Formula presented.) and (Formula presented.). (2) All measures except (Formula presented.) agree that performance weighting is superior to equal weighting with respect to statistical accuracy. (3) Neither distributions can effectively predict the position of a removed quantile estimate. These insights show the behavior of different scoring rules for combining uncertainty estimates from expert or models, and extent the knowledge for best-practices.

Due to the inherent uncertainties in manufacturing properties and intrinsic variability of materials, the assumption of homogeneous input variables is generally not justified. As a result, stochastic forward problems have emerged as a tool to incorporate these uncertainties into numerical simulations, improving model prediction capability in structural analyses. Although most of the existing methods focus on the consideration of stochastic loading, the recently developed statFEM employs a Bayesian paradigm to incorporate data and propagate uncertainties from random physical properties in finite element models. This tool, however, is not developed for cases when Gaussian assumptions are inadequate. The present work provides a copula-based approach embedded into the statFEM methodology to propagate uncertainty from arbitrarily distributed physical properties. The random variables are defined in terms of a Gaussian Copula Process, where samples are drawn from a latent variable governed by a Gaussian Process and then brought respectively to copula and marginal spaces, producing random variables with the desired distribution while retaining the usually desired smooth Gaussian dependence in the spatial domain. The quality of the approximated results is then assessed in a simplified 1D Poisson problem by comparing with Monte Carlo sampling results for different random diffusion coefficients, demonstrating that the method is capable of providing good responses for non-Gaussian physical parameters.

Immersed tunnels are positive buoyant structures during installation and negative buoyant after installation. A tunnel is composed of sequential immersed elements that are coupled to each other in joints. Tunnel elements consist of segments which are compressed to each other by longitudinal post-tensioning. After immersion the tunnel is supported by the seabed and the longitudinal post-tension is cut at the joints between segments. Therefore, the structure is a segmented lining which is sensitive for settlements due to non uniform circumstances over the length of the tunnel. An uneven response of the bedding underneath the tunnel introduce shear forces in joints of an immersed tunnel. Because immersed tunnels need to be buoyant during installation, they have limitations on weight and geometry, the size and therefore the capacity of these shear keys is limited because the height of the tunnel, as shear keys are applied in the walls of the tunnel. The foundation response is influenced by many factors related to subsoil but also to construction and dredging tolerances. The shear forces were derived as a function of different covariance lengths for subsoil stiffness and dredging tolerances for different tunnel layouts. In reliability analyses, using two different probabilistic methods, exceedance probabilities of maximum shear forces are derived for one lay out using Non Parametric Bayesian Networks and Vine Copulas. The analyses give more insight in to the magnitude of the shear forces in joints both in conditioned and unconditioned situations and this can be used for the design of immersed tunnels.

The magnitude of flood impacts is regulated not only by hydrometeorological hazard and exposure, but also flood protection levels (primarily from structural flood defenses) and vulnerability (relative loss at given intensity of hazard). Here, we infer the variation of protection levels and vulnerability from data on historical riverine, coastal, and compound floods and associated impacts obtained from the HANZE database, in 42 European countries over the period 1950–2020. We contrast actual damaging floods, which imply flood protection was locally inadequate, with modelled potential floods, i.e. events that were hydrologically extreme but did not lead to significant impacts, which imply that flood protection was sufficient to prevent losses. Further, we compare the reported magnitude of impacts (fatalities, population affected, and economic losses) with potential impacts computed with depth-damage functions. We finally derive the spatial and temporal drivers of both flood protection and vulnerability through a multivariate statistical analysis. We apply vine-copulas to derive the best predictors out of a set of candidate variables, including hydrological parameters of floods, exposure to floods, socioeconomic development, and governance indicators. Our results show that riverine flood protection levels are much lower than assumed in previous pan-European studies. North-western Europe is shown to have better riverine protection than the south and east, while the divide is not so clear for coastal protection. By contrast, many parts of western Europe have relatively high vulnerability, with lowest value observed in central and northern Europe. Still, a strong decline in flood vulnerability over time is also observed for all three indicators of relative losses, suggesting improved flood adaptation. Flood protection levels have also improved since 1950, particularly for coastal floods.

Aquaculture at sea is gaining increasing importance, not only as a (local) food source but also due to its potential of being combined with other offshore activities such as wind parks. Nevertheless, experience of offshore aquaculture is limited. This study aims to provide a framework to evaluate offshore aquaculture suitability accounting for the probabilistic dependence between relevant variables. This framework is applied to obtain suitability maps of aquaculture for the North Sea for the blue mussel Mytilus edulis and the sugar kelp Saccharina latissima. For each of these species, three ecological variables are selected and the optimal growth and critical survival limits are defined. Here, suitability is defined as the probability of meeting these conditions. Data on the selected variables is extracted from a large-scale 3D hydrodynamic and ecological model of the northwest European Shelf, of which daily extremes are sampled. The probabilistic model is developed using bivariate copula models, which are fitted to each variable pair to describe their joint distribution function at each studied location. Empirical distribution functions are used to describe the univariate distribution function of each variable and location. Using Monte-Carlo simulations, the probability of meeting the optimal and critical limits is estimated and suitability maps accounting for the probabilistic dependence between the variables are generated. In addition, suitability maps disregarding the dependence are generated and compared to those accounting for the probabilistic dependence. It was found that considering the dependence between variables significantly improves the accuracy of the results for optimal and critical growth conditions for both species. The presented method allows to identify potential areas where blue mussel and sugar kelp cultivation is the most suitable. For instance, in this study, a north-south elongated area west of the German and Danish coast appears to be most suitable for blue mussels, while estuaries and rivers are found the most suitable for the sugar kelp.

This article describes the development of a GUI that addresses the challenge of eliciting dependencies between uncertain quantities elicited by experts. While software for eliciting univariate uncertainties is widely available, the mathematical complexity of multivariate dependence models makes direct elicitation difficult. To overcome this, we developed Matlatzinca,1 a GUI built on top of the Python module PyBANSHEE. The GUI facilitates the elicitation process and allows experts to model dependencies using a non-parametric Bayesian network without the need for ad hoc programming. A recent practical application shows that the developed GUI is a useful tool for performing dependence elicitations, highlighting the significance of the program for dependence assessment with expert judgment.

The severity of damages to riverine structures, such as groynes or revetments, across German estuaries has increased in the past years due to the increase in the ship-induced loads. However, few studies can be found in the literature focused on the damage of rock slopes under ship wave attack. In this study, the field data of a rock-armored groyne (lateral slope 1/4 and rocks with nominal diameter Dn50»12.6cm and high density ρs=3.7t/m3) tracked for a year by Melling et al. (2020) is analyzed; the field campaign began after the structure was rebuilt and finished when the structure already presented severe damage. During this field campaign, the incident ship-induced primary waves and water levels were recorded, and laser scans of the groyne armor were taken. Using those field laser scans, damage curves along the life of the structure were derived. Also, ship data from the AIS was retrieved. Then, each increment of the damage (increment of the dimensionless eroded area, ΔSe) was related to a ship-wave event and a passing ship. The most significant variable to describe ΔSe was found to be the primary wave height Hp, while the best explanatory variables for Hp were the partial blockage factor, the ship length and width and the relative velocity of the ship. The shape of the dependence between these variables is also analyzed by pairs using copula space. A clear tail dependence is observed between several pairs. For instance, the pair ΔSe and Hp presents upper tail dependence, meaning that the high values of ΔSe and Hp are more correlated than the smaller ones. This implies that models more complex than Gaussian copula, which is commonly used in Coastal Engineering applications, might be needed to model the probabilistic dependence between the variables.

The Netherlands has traditionally focused on managing flood risk. However, the frequent occurrence of droughts in recent years has brought attention to managing both extremes. Transitions between these opposite extremes pose additional challenges to water management, requiring a trade-off between water storage during dry periods and flood control during wet periods. In this study, we develop a framework to define wet and dry meteorological events and study their transitions using timeseries of meteorological data namely, precipitation, temperature and potential evapotranspiration. The magnitudes of event characteristics are retained, which presents a different approach to the normalized climate indices (like the Standardized Precipitation Index) commonly used in literature. We apply this framework to the Dutch part of the Meuse River basin in northwestern Europe using climate observations between 1951 and 2022. Our analysis shows a statistically significant increase in the amount of water lost from potential evapotranspiration compared to water gained from precipitation between April and September of the water year and an increase in the length of this drying period over the past decades. Such trends in the drying period are related to variability in potential evapotranspiration caused by rising temperatures in the region, indicating the potential for increased water shortage in Spring and Summer due to future temperature increases. We also identify abrupt transitions between opposite extreme events where there is a lack of water at the end of the second event as meteorological situations that challenge water management due to overlapping impacts like flash flooding, less time for water storage, and reduced water availability. We see such conditions occur in 6% of the wet-dry transitions and 20% of the dry-wet transitions, highlighting meteorological scenarios to which the hydrological response of the catchment can be simulated to increase our understanding of the combined risk of floods and droughts.

The rapid changes in the shipping fleet during the last decades has increased the ship-induced loads and, thus, their impact on infrastructures, margin protections and ecosystems. Primary waves have been pointed out as the cause of those impacts, with heights that can exceed 2 m and periods around 2 minutes. Consequently, extensive literature can be found on their estimation mainly from a deterministic perspective with methods based on datasets limited to one location, making difficult their generalization. These studies propose either computationally expensive numerical models or empirical equations which often underestimate the extreme primary waves, hindering their use for design purposes. Moreover, a framework to allow the design of infrastructure under ship-wave attack based on probabilistic concepts such as return periods is still missing. In this study, a probabilistic model based on bivariate copulas is proposed to model the joint distribution of the primary wave height, the peak of the total energy flux, the ship length, the ship width, the relative velocity of the ship and the blockage factor. This model, a vine-copula, is developed and validated for four different deployments along the Savannah river (USA), with different locations and times. To do so, the model is quantified using part of the data in one deployment and validated using the rest of the data from this deployment and data of the other three. The vine-copula is validated from both a predictive performance point of view and with respect to the statistical properties. We prove that the probabilistic dependence of the data is preserved spatially and temporally in the Savannah river.