# An Inter-Model Comparison for Wave Interactions with Sea Dikes on Shallow Foreshores

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## Abstract

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^{®}(OF), (ii) the weakly compressible smoothed particle hydrodynamics model DualSPHysics (DSPH) and (iii) the non-hydrostatic nonlinear shallow water equations model SWASH. An inter-model comparison is performed to determine the (standalone) applicability of the three models for this specific case, which requires the simulation of many processes simultaneously, including wave transformations over the foreshore and wave-structure interactions with the dike, promenade and vertical wall. A qualitative comparison is done based on the time series of the measured quantities along the wave flume, and snapshots of bore interactions on the promenade and impacts on the vertical wall. In addition, model performance and pattern statistics are employed to quantify the model differences. The results show that overall, OF provides the highest model skill, but has the highest computational cost. DSPH is shown to have a reduced model performance, but still comparable to OF and for a lower computational cost. Even though SWASH is a much more simplified model than both OF and DSPH, it is shown to provide very similar results: SWASH exhibits an equal capability to estimate the maximum quasi-static horizontal impact force with the highest computational efficiency, but does have an important model performance decrease compared to OF and DSPH for the force impulse.

## 1. Introduction

^{®}) can provide very similar results to large-scale experiments of overtopped wave impacts on coastal dikes with a very shallow foreshore (i.e., from the WAve LOads on WAlls (WALOWA) project [4]). Yet, such Eulerian numerical methods require often expensive mesh generation and need to implement conservative multi-phase schemes to capture the nonlinearities and rapidly changing geometries. Conversely, meshless schemes can efficiently handle problems characterised by large deformations at interfaces, including moving boundaries and do not require special tracking to detect the free surface. Methods such as smoothed particle hydrodynamics (SPH) [5] and the particle finite element method (PFEM) [6] are examples, of which SPH is the most commonly applied in coastal engineering applications [7]. In SPH, the continuum is replaced by particles, which are calculation nodal points free to move in space according to the governing dynamics laws. Although, differently from Eulerian grid-based methods, multiphase air–water SPH models are still quite scarce and have a high computational cost [8,9]. Several studies on coastal engineering applications based on single-phase SPH have been published during the last decades, for example, wave propagation over a beach [10], solitary waves [11], modelling of surf zone hydrodynamics [12], wave run-up on dikes [13], tsunamis forces [14] and wave forces on vertical walls and storm walls [15,16]. Still, single-phase SPH is also inherently expensive computationally, therefore high-performance computing is required. In particular, graphics processing units (GPUs) are employed to accelerate the computations, as, for example, in GPU-SPH [17] and DualSPHysics [18].

^{®}[35,36,37]). They investigated non-breaking, impulsive breaking, and broken monochromatic wave interactions with elevated coastal structures, and found that the numerical accuracy of wave shoaling and breaking processes played a key role for the accuracy of the forces and pressures on the structure. Both models provided similarly good results, but validation was again mostly limited to a qualitative visual comparison of time series. One exception was the model performance in terms of force and pressure, which was quantified by calculating a normalised residual impulse of force/pressure. González-Cao et al. [38] both validated and inter-compared DualSPHysics and IHFOAM to experiments of breaking monochromatic waves impacting a vertical sea wall with a hanging horizontal cantilever slab, placed on a steep foreshore. They applied model performance and pattern statistics and showed that both models provide comparable results, with IHFOAM narrowly obtaining higher skill scores for low and medium resolutions, whereas for high spatial resolutions both models provided a similar level of accuracy. Finally, Lashley et al. [39] applied a broad range of wave models, including both SWASH and OpenFOAM

^{®}, to irregular wave overtopping on dikes with shallow mildly sloping foreshores (similar to the case considered in this paper). They found that accurate modelling of the LWs was essential to obtain accurate results for the mean overtopping discharge q and that the most computationally expensive model is not always necessary to obtain an accurate result. However, the analysis was strictly limited to the bulk parameters of wave transformation until the dike toe and q, and did not consider time series nor individual wave related events.

^{®}), (ii) a weakly compressible SPH model (i.e., DualSPHysics) and (iii) a non-hydrostatic NLSW equations model (i.e., SWASH). We chose to investigate the performance of each model as standalone for the present work in order to provide a detailed overview of model capabilities and limitations applied to wave–structure interaction phenomena in very shallow water conditions. The RANS model is the same one as presented by Gruwez et al. [3], which was validated with large-scale experiments of overtopped wave impacts on coastal dikes with a very shallow foreshore from the WALOWA project [4]. In this paper, the same experiment and RANS model are used as a basis for the inter-model comparison with the (until now untested for this case) DualSPHysics and SWASH models. The analysis is done both (i) qualitatively, based on a comparison of the time series of the main measured parameters and snapshots of bore interactions and impacts on the dike, and (ii) quantitatively, based on model performance and pattern statistics, to critically assesses the performance of all three models to reproduce the large-scale experiment. The computational cost of each numerical model is also evaluated in terms of computational and model setup time. Finally, the results are discussed by comparing to results of the numerical models for the individual processes in other available literature, and the applicability of each numerical model for a design case is investigated.

## 2. Methods

#### 2.1. Large-Scale Laboratory Experiments

_{x}with an electromagnetic current meter (ECM) positioned on the promenade [40], and the horizontal wave impact force F

_{x}and pressures p by load cells (LC) and pressure sensors (PS), respectively. Both bichromatic and irregular wave tests were conducted with active reflection compensation (ARC), of which the repeated bichromatic wave test Bi_02_6 (Table 1) was chosen for the inter-model comparison. The test included mostly plunging breakers on the 1:10 transition slope and spilling breakers on the 1:35 foreshore slope in front of the dike. All other relevant details of the tests and the processing of the experimental data used in this paper for the inter-model comparison are provided by Gruwez et al. [3]. For further information on the experimental model setup, the reader is referred to the work in [4]. The WALOWA experimental dataset is available open access [41].

#### 2.2. Numerical Models

#### 2.2.1. OpenFOAM

^{®}model and model setup as described by Gruwez et al. [3] is used. To summarise, and citing the work in [3], the solver interFoam of OpenFOAM

^{®}v6 [42] is applied, “where the advection and sharpness of the water–air interface are handled by the algebraic volume of fluid (VOF) method [43] based on the multidimensional universal limiter with explicit solution (MULES)” [44,45,46]. The boundary conditions for wave generation and absorption are managed by olaFlow [47], while “the turbulence is modelled by the k-ω SST turbulence closure model” that was “stabilised in nearly potential flow regions by Larsen and Fuhrman [48], with their default parameter values [49]”. Hereafter, OF refers to the OpenFOAM

^{®}numerical model as presented by the authors of [3].

#### 2.2.2. DualSPHysics

_{SPH}; and with artificial viscosity [52], tuned with parameter α

_{av}, which represents the fluid viscosity, prevents particles from interpenetrating, and provides numerical stability for free surface flows [53]. Moreover, employing the artificial viscosity scheme has been shown to exhibit interesting features related to the turbulence field under breaking waves [12]. The weakly compressible SPH method requires that the speed of sound is usually maintained at least 10 times higher than the maximum velocity in the system (managed by the empirical coefficient coeff

_{sound}). One consequence is that numerical pressure noise tends to develop [54]. To combat this, a density diffusion term (DDT) was introduced in the continuity equation [54]. This so-called δ-SPH approach increases the smoothness and the accuracy of pressure profiles. The δ-SPH method is applied in this study, by using the improved DDT of Fourtakas et al. [55] where the dynamic density is substituted with the total one. The modified Dynamic Boundary Conditions (mDBC) are employed for the fluid–boundary interactions [56]. Waves are generated by means of moving boundaries that mimic the movement of a laboratory wavemaker. The model also has its own embedded wave generation and absorption system capable for generation of random sea states, monochromatic waves and multiple solitary waves [11,57]. Hereafter, the DualSPHysics numerical model as presented here is simply referred to as DSPH.

_{av}= 0.01, which is most commonly used for sea wave modelling [16], and h

_{SPH}/dp = 2.12, where the smoothing length is calculated in DSPH according to the initial interparticle distance as h

_{SPH}= coefh √2dp in 2DV. In the present calculations, coefh = 1.8 was assumed (usually in the range 1.2 to 1.8 [59]). The recommended and default density diffusion parameter value of 0.1 was chosen. The results of a sensitivity analysis of these parameters showed negligible influence (not shown). The so-calculated kernel size is equal to 0.051 m, which can be considered as the effective model resolution since, citing Lowe et al. (2019), “the kernel size effectively reduces the model resolution by smoothing the results over the length-scale h

_{SPH}”. It is therefore twice the finest resolution used on the promenade in the OF model (i.e., dx = dz = 0.0225 m).

#### 2.2.3. SWASH

_{o}(where k is the wavenumber) [61]. For both primary wave components, kh

_{o}is below 1.0 and a one-layer approach (or depth-averaged, K = 1) is acceptable with respect to frequency dispersion. Although a one-layer approach also appeared to be sufficient in terms of accuracy of water surface elevation for the wave–structure interactions with the dike and vertical wall, a second SWASH simulation was done as well using eight layers (K = 8) to resolve more the flow on top of the dike and in front of the wall. This allows a comparison of the velocity field in the snapshot comparisons with the other two numerical models (Section 3.5) and an evaluation of the model performance of a multi-layer model. Discretisation of the vertical pressure gradient is done by the implicit Keller-box scheme for SW1L, while the explicit central differences layout was applied for SW8L to ensure robustness.

#### 2.3. Data Sampling and Processing

_{x}and p time series of both the experiment and numerical model results. This removed the high frequency oscillations caused by stochastic processes during dynamic or impulsive impacts, so that the experimental signal can be reproduced by the deterministic numerical models [3,64].

_{x}was calculated by integration of p along the height of the LC (by using the OpenFOAM

^{®}library “libforces.so”). In DSPH, p is calculated by interpolating the fluid particle pressure at a distance from the wall equal to h

_{SPH}and forces are calculated as the summation of the acceleration values (solving the momentum equation) multiplied by the mass of each boundary particle belonging to the vertical wall. In SWASH, the total pressure, including both the hydrostatic and non-hydrostatic pressures, exhibited strong oscillations in the grid cells closest to the vertical wall (not shown). Contrary to OF and DSPH, the (numerical noise) oscillations were not removed completely by the applied filtering, with significant residual—and in some cases even exacerbated—spurious oscillations. No immediate explanation was found to their root cause. In any case, it was found that these oscillations are attributable to the non-hydrostatic part of the pressure. Therefore, they disappeared entirely when only considering the hydrostatic pressure. The SW1L/SW8L p and F

_{x}time series are therefore limited to the hydrostatic part in further analyses. For SW1L, the hydrostatic pressures at the pressure sensor locations were then calculated by ρg(η-z

_{PS}), where ρ is the water density (1000 kg/m

^{3}), g the gravitational acceleration (9.81 m/s

^{2}), η is taken from the grid cell closest to the vertical wall (which represents most closely the bore run-up height against the vertical wall) and z

_{PS}is the z-coordinate of the considered pressure sensor. For SW8L, the hydrostatic pressure was interpolated between the 8 vertical grid cell values closest to the PS locations. The horizontal impact force F

_{x}was obtained by integration of the hydrostatic pressure along the vertical wall.

_{SW}and η

_{LW}by applying a 3rd order Butterworth high- and low-pass filter, respectively. A separation frequency of 0.09 Hz was employed, which is in between the bound long wave frequency (f

_{1}− f

_{2}= 0.035 Hz) and the lowest frequency of the primary wave components (f

_{2}= 0.155 Hz).”

#### 2.4. Inter-Model Comparison Method

_{r}(Equation (A5)).

_{r}-value between predicted and observed maximum horizontal force per impact event F

_{x,max}(i.e., d

_{r,Fx,max}). The duration of the wave impact can be evaluated by the impulse of the total horizontal force I:

_{N}is the total duration of the test. To evaluate the model performance, a normalised predicted impulse is considered:

_{p}and I

_{o}are the predicted and observed force impulses calculated by Equation (4), respectively. The observed total horizontal force impulse is overestimated, equal to or underestimated by the prediction when I

^{*}> 1, I

^{*}= 1 or I

^{*}< 0, respectively. Note that I

^{*}is evaluated for the complete F

_{x}time series, so that phase differences are disregarded by this parameter. Therefore, I

^{*}purely evaluates the correspondence of the total impulse on the vertical wall during the complete test.

## 3. Results

#### 3.1. Time Series Comparison

_{LW}time series are shown separately in Figure 5. The time series of U

_{x}are compared in Figure 6 at the ECM location on the promenade. For the numerical models, actually the depth-averaged horizontal velocity ${\overline{U}}_{x}$ is shown instead, as it was shown to deliver a better correspondence to EXP than U

_{x}for OF [3]. The same was found to be the case for DSPH (not shown), and SW1L only provides ${\overline{U}}_{x}$, since it is a depth-averaged model. In Figure 7, the time series of F

_{x}are compared to the LC measurements, and in Figure 8 the time series of p are compared at the PS locations selected at approximately equidistant positions along the array (i.e., 0.28 m from PS01 up to PS09 and 0.24 m up to PS13).

_{x}appears to be more challenging than η for all numerical models. Most of the positive U

_{x}peak values (i.e., flow towards the vertical wall) are reproduced, while some of the return flow durations (i.e., t for U

_{x}< 0) are modelled longer by OF than SW1L, with DSPH being in between (Figure 6). Unfortunately, return flow velocities were often not captured by the ECM measurements in EXP, mostly by too thin flow layers (no data).

_{x}(Figure 7) and p time series (Figure 8), differences become more distinctive. DSPH shows (small) negative or sub-atmospheric p peaks, not observed in EXP, that occur before some of the dynamic impact peaks and mostly for the lowest PSs (Figure 8a). Both OF and DSPH appear to underestimate most peak values for both F

_{x}and p, while phase differences with EXP are most apparent for DSPH and SW1L.

#### 3.2. Spatial Distribution of Wave Characteristics

_{rms}, the SW and LW components (i.e., H

_{rms,sw}and H

_{rms,lw}), and the mean surface elevation $\overline{\eta}$ (wave setup) over the wave flume up to the toe of the dike are compared in Figure 9. All models agree on the general evolution of the H

_{rms}curves along the flume. The wave height slightly decreases from the wave paddle up to the toe of the foreshore, more so in case of OF than DSPH (with DSPH closest to EXP): from the wave paddle location to the foreshore toe, H

_{rms}decreases about 10% more for OF than DSPH. SW1L shows a similar behaviour from its offshore boundary (i.e., at WG02) until the foreshore toe and corresponds most with DSPH. On the other hand, SW shoaling is overestimated by both DSPH and SW1L (H

_{rms,sw}at WG07 in Figure 9). In the surf zone, DSPH reproduces the energy loss due to SW breaking best of all three numerical models (i.e., closest result to (R)EXP for H

_{rms,sw}at WG13 and WG14 in Figure 9), while SW1L overestimates and OF underestimates H

_{rms,sw}there.

_{rms,lw}evolution along the flume in the experiment is also reproduced by all three numerical models. An unexpected peak appears in the DSPH result near x = 126 m, which is not found in the results by OF and SW1L. Moreover DSPH significantly overestimates H

_{rms,lw}at WG13, while OF underestimates H

_{rms,lw}at the dike toe (WG14).

_{rms}followed by SW1L, while OF clearly underestimates it for all measured locations. In terms of the wave setup $\overline{\eta}$, however, SW1L shows the best correspondence with (R)EXP, while OF and DSPH, respectively, over- and underestimate it until at least WG07. In the inner surf zone, OF corresponds better with (R)EXP for $\overline{\eta}$ together with SW1L, while DSPH significantly overestimates it ($\overline{\eta}$ at WG13–14 in Figure 9).

#### 3.3. Model Performance and Pattern Statistics

_{r}provides a single dimensionless measure of average error, it is suitable to provide insight into the spatial variation of model error in the flume. In addition to the d

_{r}of each numerical model, the d

_{r}of the repeated experiment (REXP) is also included in this analysis. A numerical model d

_{r}higher than the d

_{r}of REXP means that the numerical error cannot be reduced further compared to the experimental repeatability error and a near “perfect” model performance would be achieved with regard to the experiment [3]. Therefore, a relative refined index of agreement d’

_{r}(Equation (A8)) and a corresponding rating (Table A1) was defined by Gruwez et al. [3] which provides the performance of the numerical model relative to the experimental model uncertainty. Table 2 and Table 3 provide the d

_{r}-values and the pattern statistics at key locations (dike toe and on the promenade, respectively). It is noted that the statistics for η reported in Table 3 were averaged over the four measured locations (WLDM01—WLDM04), for the sake of brevity and because it better represents the statistics for the processes on the promenade. The evolution of d

_{r}and R at the WG locations along the wave flume up to the toe of the dike is shown, respectively, in Figure 10 and Figure 11, for η

_{SW}(d

_{r,sw}and R

_{sw}), for η

_{LW}(d

_{r,lw}and R

_{lw}) and for η (d

_{r,tot}and R), and of d

_{r}for η and U

_{x}on the dike in Figure 12.

_{r,tot}over the surf zone (WG07-13) and increases back up to the dike toe (WG13-14). Over the surf zone, DSPH gradually becomes the least performing numerical model (WG13-14, rated Good) followed by SW1L (WG13, rated Good). The relative model performance of SW1L increases back to Very Good at the dike toe (WG14). The performance of OF is not as good as DSPH in the offshore area (WG02-04), but becomes the highest of all three numerical models in the surf zone (WG13-14, rated Very Good) and continues to perform the best on the dike as well (Figure 12), where the d

_{r}of η remains more or less constant for all models, with exception of DSPH which increases slightly back to a rating of Very Good.

_{r,sw}mostly follows the same trend as d

_{r,tot}, with the exception that DSPH performs better than SW1L at the dike toe for d

_{r,sw}(Figure 10). On the other hand, d

_{r,lw}clearly has a different behaviour: OF does not reproduce the incident LWs as good as DSPH and SW1L, but its LW performance steadily increases towards the dike toe (Figure 10), where the LW energy increases (Figure 9). SW1L shows the overall best LW performance as it shows similar d

_{r,lw}values offshore to DSPH and similar values to OF in the surf zone. It is also revealed that the increase in SW1L error at WG07 is mostly caused by a decrease in η

_{sw}performance. Even though SW1L shows the least performance in modelling η

_{sw}over the foreshore, that does not seem to affect its capability of reproducing the LW shoaling and energy transfer from the SW to LW components, with a similar accuracy to OF for modelling η

_{lw}in the surf zone. Increased accuracy of SWASH of the SW modelling can be obtained however, with increased vertical resolution: the SW8L model exhibits much better performance in the shoaling zone (i.e., SW1L: d

_{r,sw,WG07}= 0.73, SW8L: d

_{r,sw,WG07}= 0.86, not shown), and attains the same model performance for η at the toe of the dike as DSPH (d

_{r,sw,WG14}= 0.53). However, because of the smaller wave height of the SW components at the dike toe compared to the LW components (Figure 9), this improvement only slightly increases the overall model performance at the dike toe (Table 2, SW1L: d

_{r,sw,WG14}= 0.85, SW8L: d

_{r,sw,WG14}= 0.86).

_{rms}graphs in Figure 9. Both were already discussed in Section 3.2. The result is that at the toe of the dike, DSPH has the best result of the three numerical models in terms of reproduction of the wave height (Table 2, ${\sigma}^{*}$ = 1.02), but the worst result in terms of the wave setup (Table 2, B* = 0.21). OF has the worst result for the wave height (Table 2, ${\sigma}^{*}$ = 0.89), while delivering a close second-best result with SW1L for the wave setup (Table 2, B* = 0.05). SW1L provides the lowest wave setup error (Table 2, B* = 0.04).

_{LW}result of DSPH as well, which is not observed in the other numerical model results. Consequently, at the toe of the dike the phase error is largest for DSPH (i.e., lowest R value in Table 2). However, at the dike toe the difference with SW1L is small, while OF provides the best phase correspondence with EXP.

_{r}of U

_{x}is provided in Figure 12 and Table 3, and indicates a lower model performance for all three numerical models than obtained for η. However, for the relative model performance d’

_{r}this difference significantly reduces, so that the same rating is obtained for U

_{x}as for η in case of DSPH and OF (Table 3, rated Very Good). SW1L (and SW8L) has the lowest d’

_{r}for U

_{x}and rating (Table 3, rated Good). Although the wave setup at the dike toe is overestimated by each numerical model (Table 2, B* > 0), η on the promenade is generally underestimated and U

_{x}as well (Table 3, B* < 0). The bore wave height is best represented by OF (indicated by ${\sigma}^{*}$), closely followed by DSPH and SW1L. Phase differences are observed for all numerical models (i.e., R < 1.00 in Table 3), but are lowest for OF, followed by DSPH and SW1L.

_{r}-values of the pressures at the vertical wall are compared in Figure 13 and the statistics in Table 4. Again, all models show a lower model performance than REXP, and OF obtained the highest value followed by SW1L and DSPH, both of which have very similar model performance along the PS array. Model performances of all models tend to decrease and converge to each other towards higher sensor locations on the vertical wall.

_{r}value of F

_{x}for each model is shown in an overview table (Table 4) together with the other pressure and force related statistics. Even though OF has the highest model performance in terms of the F

_{x}time series (Table 4, d

_{r,Fx}= 0.76), the force peaks are better estimated by SW1L (Table 4, d

_{r,Fx,max,OF}= 0.85 and d

_{r,Fx,max,SW1L}= 0.88), while it has the largest errors in the F

_{x}time series (Table 4, d

_{r,Fx}= 0.64). Moreover, SW1L underestimates the total impulse much more than OF does (Table 4, I

^{*}

_{OF}= 0.85 and I

^{*}

_{SW1L}= 0.62). DSPH has a similar model performance as OF for the force peaks F

_{x,max}, while I* is in between OF and SW1L and its overall model performance for F

_{x}is similar to SW1L. Consequently, the relative model performance for F

_{x}is rated Very Good for OF and Good for DSPH and SW1L. While the model performance slightly increases for SW8L compared to SW1L at the dike toe (Table 2) and on the promenade (Table 3), this does not translate into a better model performance for the wave impact on the vertical wall; in fact, almost every F

_{x}statistic is lower for SW8L than for SW1L (Table 4).

_{x}are slightly worse for the multi-layer model SW8L compared to the single layer model SW1L, except for ${\sigma}^{*}$.

#### 3.4. Skill Target Diagrams

_{x}and F

_{x}only one representative observed location is available. They are visualised as pentagrams together with the model data points for η and p, respectively.

_{x}pentagrams suggest a better U

_{x}model skill of DSPH than SW1L. For p, OF remains the numerical model with the highest model skill, followed by DSPH and SW1L. The F

_{x}pentagrams have the same ranking.

_{x}skill of all numerical models is one of a clear underestimation (pentagrams in lower left quadrant), of both the mean value (B* < 0) and standard deviation (${\sigma}_{d}$ < 0), and increased phase difference (low R-values). The same is valid for p and F

_{x}, where all numerical model skill points are also positioned in the lower left quadrant.

#### 3.5. Snapshot Inter-Model Comparison

_{x}and U

_{x}time series were selected during this series of impacts and are listed chronologically in Table 5. These time instants were selected from each model result independent of time, because due to phase errors these key time instants have occurred at (slightly) different times between the models.

_{x}impact peak with EXP (see F

_{x}graph insets in Figure 15c). The timing of the smaller bore impact corresponds with EXP in case of OF and SW8L, but is late for DSPH (time instant 1, Figure 15a). For all numerical models, the large incoming bore arrived later than was observed in EXP. This means that the collision between the reflected small bore and incoming large bore was timed differently for each model, with repercussions for the subsequent impact of the overturned wave on the vertical wall.

_{x}peak during the dynamic impact (time instant 4, Figure 15c), but the pressure distribution remains hydrostatic and therefore no local pressure peaks were captured at all.

_{x}peak (time instant 6, Figure 16a). In both the OF and DSPH results, a clockwise vortex formed near the bottom of the wall during the run-up process. However, in case of DSPH, this vortex was much stronger and lasted during the quasi-static F

_{x}peak as well. The p distribution of EXP was mostly hydrostatic, except for a small local peak at PS06 (time instant 5), which seems to have been captured qualitatively by DSPH. On the other hand, the strong vortex modelled by DSPH also caused a non-hydrostatic p distribution during time instant 6, while it was mostly hydrostatic in EXP. In this time instant, again OF was closest to the EXP observation. SW8L was not successful in correctly predicting the wave run-up against the vertical wall during reflection of the large impacting bore (time instant 5) and consequently underestimated p and F

_{x}more than the other numerical models for time instants 5 and 6. During return flow (time instant 8, Figure 16b) the pressure distribution was mostly hydrostatic, and all numerical models were able to predict the p distribution well. Only DSPH shows a pressure decrease near the bottom of the wall (PS01), possibly caused by the persistent vortex modelled there, but now further removed from the wall compared to the previous time instants 5 and 6.

_{x}event (time instant 2, Figure 15b) followed by a maximum return flow U

_{x}event (time instant 7, Figure 17b), it can be seen that all numerical models underestimate U

_{x}of EXP close to the bottom for the incident bore (time instant 2), while OF predicts it very well, DSPH slightly underestimates it, and SW8L overestimates it during the return flow (time instant 7).

#### 3.6. Computational Cost

_{m,t}/260 (with H the wave height and L

_{m,t}the mean wave length of the SW components at the dike toe), for DSPH by dp = H/50 = L

_{m,t}/585 and for SWASH by Δx = L

_{m,o}/170 = L

_{m,t}/60 (with L

_{m,o}the mean wave length of the SW components offshore).

## 4. Discussion

#### 4.1. Inter-Model Comparison of Wave Transformation Processes

#### 4.1.1. Wave Transformation Processes Until the Dike Toe

_{rms}decreased slightly (4%) between the offshore region (Figure 9, WG02-WG04) and the shoaling zone (WG07), indicating an energy loss greater than the H

_{rms}gain due to shoaling, possibly due to bottom friction and/or model effects (e.g., lateral wall friction, sediment transport on the sandy slope). However, too few EXP measurement locations were available in this area to confirm this. In any case, all three applied numerical models agreed that H

_{rms}first decreased towards the foreshore toe, and then increased due to SW shoaling (Figure 9). In case of both OF and SW1L, the H

_{rms}decrease was more pronounced than DSPH and could be explained by wave energy losses due to bottom friction to a certain degree (i.e., in OF by the no-slip boundary and in SW1L by parameterised bottom friction). However, these energy losses cannot be explained by the bottom friction alone due to the relatively short propagation distance from the offshore model boundary to the foreshore toe of less than three wave lengths. Viscous and/or diffusive numerical schemes could have contributed as well, especially in the case of OF [45], which mostly used second order discretization schemes but also included a first order scheme (i.e., Euler time discretisation for the volume fraction of the VOF method) for numerical stability reasons [3]. Such numerical diffusion was limited as much as possible in each model by a careful selection of parameters and schemes, which is a balancing exercise between model accuracy and efficiency. DSPH is presently unable to model bottom friction and the numerical diffusion was least noticeable of all three models, causing the H

_{rms}overestimation at WG07 by DSPH. SW1L/SW8L tended to overestimate the shoaling (mostly for the LWs) so that an overestimation similar to DSPH at WG07 was obtained. Clearly OF suffered most from (numerical) energy losses, as it started with approximately the same wave energy at the wave paddle as DSPH, but underestimated EXP at WG07. DSPH and OF did agree on the location of the mean breaking point (x

_{b}= ~120 m, Figure 9) and simulated both spilling and plunging breakers, also observed in EXP [3]. Indeed, both models previously have been shown to predict the breaking point (and hydrodynamics) similarly well for both spilling and plunging breakers (i.e., OF [69] and DSPH [12]). SW1L/SW8L, on the other hand, predicted the breaking point to be located more offshore (by about 10 m). Contrary to OF and DSPH, SWASH does not explicitly model the turbulent wave front during the breaking process, but treats it at the sub-grid level instead by assuming similarity to a hydraulic jump [25]. SW1L/SW8L therefore did not reproduce the overturning of the wave front. Moreover, the vertical resolution (or K) for both SW1L and SW8L was too low to be able to model wave breaking without the use of the HFA, which has been shown before to cause the breaking point to be predicted too much offshore in case of plunging breakers (see their Figure 5 in [25]). Based on these observations, the breakpoint was most likely better predicted by OF and DSPH than by SW1L/SW8L. Although, it should be noted that SWASH has the potential to match the breakpoint location in H

_{rms}by increasing K to 20 layers [25] (not tested). In the surf zone, DSPH and SW1L/SW8L predicted very similar wave heights until the dike toe and both models ended up slightly overestimating H

_{rms,EXP}at the dike toe, SW1L more than DSPH (σ* in Table 2). The evolution of H

_{rms}was also very similar for OF, but the values were lower because H

_{rms}at the breakpoint was lower as well, with OF the only model that underestimated H

_{rms,EXP}at the dike toe.

_{sw}result in the surf zone observed for SW1L in Figure 11.

_{r}in Table 2). Although DSPH achieved the best wave amplitude result, the errors in both wave setup and wave phase caused it to have the lowest model performance at the dike toe location. OF also achieved the best overall model skill in terms of η over the foreshore (and promenade), with the mean skill circle having the smallest radius in the target diagram (Figure 14, left), again, respectively, followed by SW1L and DSPH.

_{r}value fell just below the limit for a Very Good rating (i.e., d’

_{r,WG14,DSPH}= 0.79 < 0.80, Table 2 and Table A1). This means that all three numerical models are shown to be able to represent frequency dispersion, and the nonlinear wave transformation processes: SW shoaling (Figure 4b), breaking (Figure 4c,d) and energy transfer to the subharmonic bound LW (Figure 4b–d). This is a confirmation of what has been proven before by Torres-Freyermuth et al. [70,71] for RANS modelling, by Lowe et al. [12] for DualSPHysics and Rijnsdorp et al. [26] for SWASH.

#### 4.1.2. Bore Interactions on the Promenade and Impacts on the Vertical Wall

_{LW}at the dike toe was lower than SW1L (Figure 10), while it was higher than SW1L for η

_{SW}at the same location, and the model performance for the total η increased on the promenade for DSPH relative to SW1L (Figure 12). This was mostly because of a better wave amplitude and phase accuracy by DSPH than SW1L/SW8L on the promenade (Table 3, σ* and R, respectively). Moreover, in terms of U

_{x}for the bore interactions on the promenade, DSPH had a higher model performance than SW1L and was comparable to OF. In any case, the highest model performance for both η and U

_{x}on the promenade was achieved by OF. This is mostly indicated by a higher phase accuracy than both DSPH and SW1L/SW8L and a higher amplitude accuracy than SW1L/SW8L (Table 3, R and σ*, respectively). OF obtained the highest phase accuracy on the promenade as a result of the highest wave phase accuracy achieved at the dike toe (Figure 11, WG14), especially because of the SW phase accuracy (i.e., R

_{sw}) at the dike toe which was notably higher for OF than both DSPH and SW1L/SW8L. At the dike toe, DSPH had the highest error in $\overline{\eta}$ by overestimating it (Table 2, B* = 0.21) more than OF and SW1L/SW8L. Nevertheless, on the promenade DSPH underestimated $\overline{\eta}$ with a very similar result as OF and SW1L (Table 3, ${\overline{B}}_{WLDM01-WLDM04}^{*}$ = −0.04). A possible cause for this change in behaviour is that thin layered flows with a water depth on the promenade were not captured by DSPH because the water depth was smaller than a couple of particles high. The fact that SW1L/SW8L was also able to achieve a Very Good model performance for the overtopped flow layer thickness on the promenade (Table 3) indicates that a very good wave overtopping accuracy has been demonstrated as well, which is a confirmation of previous validation works on wave overtopping over a dike with a shallow foreshore modelled with SWASH [27,28].

_{x}on the promenade (Table 3), and consequently also for F

_{x}and p at the vertical wall (Table 4, Figure 13 and Figure 14b). Conversely, DSPH had the best wave amplitude, but the worst wave setup and phase prediction at the dike toe and showed a lower performance in terms of F

_{x}at the vertical wall. In addition, the snapshot comparison of the first series of impacts at the vertical wall (Section 3.5) confirmed that the wave phase accuracy was critical to obtain the correct bore interaction pattern on top of the promenade, which was—in its turn—shown to be vital to the accuracy of the impact itself. Likewise, SW1L showed similar SW phase errors at the dike toe (Figure 11, R

_{sw}), which consequently caused phase errors in the overtopped bores (R for SW1L/SW8L is mostly lower than OF and DSPH for both η and U

_{x}in Table 3) and their impacts on the vertical wall (R for SW1L/SW8L is mostly lower than OF and DSPH for both p and F

_{x}in Table 4). In Section 3.5, this manifested itself in the SW8L result by a clear phase difference in the U

_{x}time series (Figure 17) and in the F

_{x}time series of the main impact (Figure 15c), which were both a result of a phase error in η for the largest bore of the first wave group arriving at the dike toe (see t ~55 s in Figure 4d).

_{x,max}and the total horizontal force impulse I is of particular interest. The model performance for F

_{x,max}was found to be similar for all three applied numerical models (Table 4, d

_{r,Fx,max}). However, for I* important differences were noted, with the best total impulse prediction obtained by OF, followed by DSPH and SW1L/SW8L, respectively. For SWASH, the best result was obtained by the depth-averaged model SW1L. Adding vertical resolution (i.e., SW8L) generally improved the model performance for η along the wave flume (not shown, except at the dike toe in Table 2) and η and U

_{x}along the promenade (Table 3). However, SW8L unexpectedly decreased the model performance for F

_{x}compared to SW1L, with slightly worse performance for F

_{x,max}and a higher error in I* (Table 4). The cause for this is unclear, but this might indicate that SW1L’s good estimation of F

_{x}could be—in part—caused by chance due to numerical errors (e.g., overestimation of the wave height at the toe of the dike, Table 2, σ* > 1.00). Nevertheless, the SW1L/SW8L model results show that SWASH can provide similar or only slightly worse model performance for F

_{x}compared to OF and DSPH, including the best estimation of F

_{x,max}per impact event, albeit with an important underestimation of I.

_{x}, is able to provide a similarly accurate prediction of F

_{x}(especially F

_{x,max}) compared to more complex and computationally expensive RANS or SPH models. Xie and Chu [19] already showed that with the hydrostatic pressure assumption, a tsunami bore impact force on a vertical wall can be obtained similar to OF (which includes non-hydrostatic effects as well) and experimental data. This is confirmed and extended here for overtopped bore impacts on dike-mounted vertical walls, determined from the SW1L/SW8L results based on the hydrostatic pressure only. Still, SW1L/SW8L had a lower model performance than OF and DSPH for F

_{x}, most probably due to the hydrostatic pressure assumption. Indeed, while Whittaker et al. [23] found that the perturbed hydrostatic pressure gives an accurate approximation to the pulsating horizontal force on a gently sloped seawall, they expected the hydrodynamic contributions in F

_{x}to increase in importance for steeper slopes (including vertical walls), particularly in the case of breaking waves. In addition, it is important to note that a low-pass filter was applied to both the experimental and numerical p and F

_{x}time series that mostly removed high frequency oscillations in the dynamic impact peak of the double peaked signal caused by a bore impact (Section 2.3). Because these oscillations are stochastic by nature, not even REXP was able to reproduce them exactly and they cannot be reproduced by deterministic numerical models (which is why they were filtered out). However, in both the OF and DSPH results such high-frequency oscillations were present in the unfiltered signals (not shown). Although it is unclear whether they are caused by numerical errors or actual physical processes (or a combination thereof), it still might suggest that they are able to represent this phenomenon in some capacity. Conversely, in the case of SW1L/SW8L, even in the unfiltered F

_{x}signal no such high-frequency oscillations were observed, because of the hydrostatic assumption.

^{®}[74] (however, presently not available open source).

#### 4.2. Application Feasibility of the Numerical Models for a Design Case

_{max}and corresponding impulse I for given design conditions has to be predicted. Generally, two methods can be used to achieve this:

- Modelling of an irregular wave train of sufficiently long duration (i.e., 1000 waves or more) to obtain statistically relevant results for F
_{max}.

_{max}(more on that later), the computational time necessary to achieve a result becomes the determinative factor in the choice between the three considered models. In the first method, a single wave group is modelled to obtain a certain focus location relative to the structure. Such a test duration has a similar length to the test considered here. For this design approach a standalone application of all three considered numerical models is therefore possible. However, many combinations of the focus location and phase at focus might be necessary to obtain the “true” F

_{max}for a given offshore design wave state [23], which would be more challenging (in terms of runtime) for standalone application of especially OF, but also DSPH.

_{x}itself. In terms of I, SWASH clearly underperformed compared to both DSPH and OF. Therefore, when I is important, OF is the recommended model to apply (i.e., highest value for I*, Table 4). However, the inter-model comparison provided in this paper has shown that all three models have a similar model performance in terms of estimation of F

_{x,max}for individual impact events (Table 4, d

_{r,Fx,max}). This means that for the estimation of F

_{x,max}, SWASH is actually the most recommended model to apply, because of the lowest computational cost. Although, it is important to note that this conclusion is only valid for a relatively straightforward geometry of a dike slope, promenade and vertical wall. For more complex geometries of dikes (e.g., presence of roughness elements, small storm walls with or without parapets, etc.), all numerical models considered in this paper remain untested and especially the simplified model SWASH is expected to be insufficiently accurate, because the importance of vertical flows would increase. However, a meshless approach could potentially be the most capable to capture nonlinearities of fluid–structure interactions derived from extremely complex dike geometries.

## 5. Conclusions

^{®}(OF), (ii) the weakly compressible SPH model DualSPHysics (DSPH) and (iii) the non-hydrostatic NLSW equations model SWASH (depth-averaged (K = 1): SW1L, and multi-layered (K = 8): SW8L). The inter-model comparison of those three numerical models to the experiment (EXP) demonstrated that they are all capable of modelling the dominant wave transformation (i.e., propagation, shoaling, wave breaking, energy transfer from the SW components to the bound LW via nonlinear wave–wave interactions) and the wave–structure interaction (i.e., individual wave overtopping, bore interactions, and reflection processes) processes involved leading up to the impacts on the vertical wall, albeit with a varying degree of accuracy. Based on a time series comparison, all three applied numerical models initially appeared to have a good correspondence of η, U

_{x}, p and F

_{x}to EXP. However, consistent differences between the models were hard to distinguish in this purely qualitative way. The accuracy was subsequently quantified more objectively by employing model performance statistics and the nature of the errors was exposed by pattern statistics. These statistics were plotted over the wave flume to provide spatial insight into the model performance, and the pattern statistics were plotted in a skill diagram, which visualised both the model performance and pattern statistics in a summarised way. In all statistics, the original EXP was used as the comparison reference, so that the repeated experiment (REXP) statistics could be used as reference for an ideal numerical model performance. While none of the numerical models managed to achieve such an ideal model performance, a rating of Good to Very Good was achieved by all three of them for most parameters and measured locations. The best overall model performance was achieved by OF, but required the highest computational cost. Although DPSH managed the best reproduction of the wave height until the dike toe, accumulation of errors in the wave setup and wave phase in the surf zone and near the dike toe caused a lower model performance than OF at the dike toe and for the processes on the dike. From this, it followed that accurate modelling of the wave setup and wave phases at the dike toe seem to be most important for accurate modelling of the bore interactions on the promenade. An analysis and comparison of snapshots of the numerical results on the dike revealed that these bore interactions are determinative for the impacts on the vertical wall. Even though SWASH is a much more simplified model than both OF and DSPH, it is shown to provide very similar results for the wave transformations until the dike toe and even for the processes on the dike and impacts on the vertical wall. When the impulse of the force on the structure is of lesser importance, SWASH is even most recommended for this application, because it is able to predict F

_{x,max}relatively accurate for each individual impact, with a significantly reduced computational cost, compared to OF and DSPH. However, SWASH is limited to hydrostatic pressure profiles for the impacts on the vertical wall, which is not always valid during more dynamic impact events. In addition, when the force impulse is of importance and more accurate and detailed wave–dike interactions are needed, OF is most recommended for this application. For future work, it is suggested to investigate whether the same conclusion is valid (particularly regarding applicability of SWASH) in the case of more complex dike geometries.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Numerical Convergence Analyses

#### Appendix A.1. Model Convergence Statistics

- Freeboard normalised bias (NB):

_{c}is the freeboard, and B is the bias between the considered and reference time series.

- Residual error of the normalised standard deviation (RNSD):

- Residual error of the correlation coefficient (RCC):

- Normalised mean absolute error (NMAE) given by

_{max}and O

_{min}are the maximum and minimum value of the reference time series.

#### Appendix A.2. Convergence Analyses

#### Appendix A.2.1. DSPH

_{SPH}, the artificial viscosity parameter α

_{av}and the initial particle distance dp. However, the current case also includes wave transformations over a beach with decreasing water depth towards the dike, for which dp is found to be the most important parameter. The convergence analysis for DSPH is therefore focused on dp (Figure A1).

**Figure A1.**DSPH model inter-particle distance dp convergence analysis of the η time series at the WG locations along the flume up to the dike toe (WG14), based on (

**a**) the normalised bias, (

**b**) the residual normalised standard deviation, (

**c**) the residual correlation coefficient, and (

**d**) the normalised mean-absolute-error. The reference is the highest resolution simulation with dp = 0.02 m.

#### Appendix A.2.2. SWASH

## Appendix B. Numerical Model Performance and Pattern Statistics

_{r}[81]:

_{r}is bounded by [−1.0, 1.0], c is a scaling factor and is taken equal to 2, MAE is the mean absolute error defined by

_{r}was defined by [3] which provides the performance of the numerical model relative to the experimental model uncertainty (in case a repetition of the experiment is available):

_{num}− MAE

_{rexp}is negative (i.e., <0), the numerical error compared to the experiment is smaller than the experimental uncertainty, which means that the numerical model performance cannot be improved. In that case MAE

_{num}− MAE

_{rexp}= 0 is forced, so that d’

_{r}= 1. A classification of d’

_{r}and corresponding rating terminology as proposed by the authors of [3] is provided in Table A1.

**Table A1.**Proposed classification of the relative refined index of agreement d’

_{r}and corresponding rating. Reproduced from [3], with permission from the authors, 2020.

d’_{r} Classification [-] | Rating |
---|---|

0.90–1.00 | Excellent |

0.80–0.90 | Very Good |

0.70–0.80 | Good |

0.50–0.70 | Reasonable/Fair |

0.30–0.50 | Poor |

(−1.00)–0.30 | Bad |

- The normalised standard deviation σ*:

_{p}and σ

_{o}are the standard deviations of the predicted and observed time series, respectively.

- The normalised bias B*:

- The correlation coefficient R:

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**Figure 1.**(

**a**) Overview of the geometrical parameters of the wave flume and WALOWA model setup, with indicated WG locations. Reproduced from the work in [4], with permission from the authors, 2020. (

**b**) Front view of the vertical wall on the promenade with indication of the LCs and PS array.

**Figure 2.**Definition of the DSPH 2DV computational domain, with coloured indication of the model fixed and movable boundaries. The still water level (SWL) is indicated in blue (z = 4.14 m). Note: The axes are in a distorted scale.

**Figure 3.**Definition of the SWASH computational domain, with coloured indication of the model boundaries. The wavemaker and weakly reflective boundary is positioned at the most offshore wave gauge WG02 location (x = 43.5 m). The SWL is indicated in blue (z = 4.14 m). Note: The axes are in a distorted scale.

**Figure 4.**Comparison of the η time series at selected sensor locations (

**a**–

**e**). The zero-reference is the SWL for panels (

**a**–

**d**) and the promenade bottom for panel (

**e**). Note: η

_{LW}is shown as well, but only for EXP (bold dotted lines in panels (

**a**–

**d**)). Adapted from the work in [3], with permission from the authors, 2020.

**Figure 5.**Comparison of the η

_{LW}time series at selected sensor locations (

**a**–

**d**). The zero-reference is the SWL. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 6.**Comparison of the U

_{x}time series at the ECM location. The zero-reference is the promenade bottom. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 7.**Comparison of the F

_{x}time series at the vertical wall. The experiment is the load cell force measurement. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 8.**Comparison of the p time series at selected sensor locations (

**a**–

**e**), PS01 being the bottom PS (

**a**) and PS13 the top-most PS (

**e**). Adapted from the work in [3], with permission from the authors, 2020.

**Figure 9.**Comparison of the root mean square wave height H

_{rms}between each numerical model and the (repeated) experiment up to the toe of the dike. From top to bottom: H

_{rms,sw}for the short wave components, H

_{rms,lw}for the long wave components, H

_{rms}for the total surface elevation, mean surface elevation or wave setup $\overline{\eta}$ and an overview of the sensor locations, SWL and bottom profile. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 10.**Comparison of d

_{r}, evaluated for REXP, OF, DSPH and SW1L with reference to EXP, up to the toe of the dike. From top to bottom: d

_{r,sw}for η

_{SW}, d

_{r,lw}for η

_{LW}, d

_{r,tot}for η, and finally an overview of the sensor locations, SWL, and bottom profile. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 11.**Comparison of R evaluated for η (of REXP, OF, DSPH and SW1L with reference to EXP) up to the dike toe. From top to bottom: R

_{sw}for η

_{SW}, R

_{lw}for η

_{LW}, R for η, and finally an overview of the sensor locations, SWL, and bottom profile. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 12.**Comparison of d

_{r}, evaluated for η and U

_{x}(of REXP, OF, DSPH and SW1L with reference to EXP) from the toe of the dike up to the vertical wall. From top to bottom: d

_{r}for η and U

_{x}, and finally an overview of the sensor locations, SWL, and bottom profile. Adapted from the work in [3], with permission from the authors, 2020.

**Figure 13.**Comparison of d

_{r}evaluated for p (of REXP, OF, DSPH and SW1L with reference to EXP) at the vertical wall (horizontal axis). Adapted from the work in [3], with permission from the authors, 2020.

**Figure 14.**Numerical model skill target diagrams (target: EXP) for selected sensor locations along the flume. All markers are colour-filled according to the d

_{r}colour scale. The circles represent the mean value of all markers for a specific model. The data points of REXP are not plotted for clarity, but only the mean (black circle). (

**a**) Target diagram with pattern statistics for η at locations WG04, 07, 13, 14 and averaged pattern statistics over WLDM01-WLDM04 for the promenade (OF: circles, DSPH: squares; and SW1L: triangles) and U

_{x}at the ECM location (pentagrams). The magenta arrows indicate the markers representing the model performance of η on the promenade (WLDM02). (

**b**) p at approx. equidistant locations PS01, 03, 05, 07, 09, 10–13 (OF: circles, DSPH: squares; and SW1L: triangles) and F

_{x}(pentagrams). Note: the target diagrams have different axis ranges.

**Figure 15.**Numerical comparative snapshots of the water flow on the dike. Colours are the velocity magnitude |U| according to the colour scale shown at the top of each figure. The red arrows are the velocity vectors, which are scaled for a clear visualisation. Each model snapshot has two inset graphs: at the top is a time series plot of F

_{x}in which a marker indicates the time of the snapshot, and along the vertical wall p is plotted at each pressure sensor location (vertical axis is z [m]). Adapted from the work in [3], with permission from the authors, 2020. (

**a**) Time instant 1; (

**b**) time instant 3; (

**c**) time instant 4; (

**d**) time instant 5.

**Figure 17.**Numerical comparative snapshots of the water flow on the dike. Colours are the velocity magnitude |U| according to the colour scale shown at the top of each figure. The red arrows are the velocity vectors, which are scaled for a clear visualisation. Each model snapshot has two inset graphs of U

_{x}at the ECM location on the promenade: at the top is a time series plot of U

_{x}in which a marker indicates the time of the snapshot, and next to the vertical wall the numerical U

_{x}-profile is plotted over the water column at the ECM location (vertical axis is z [m]) together with the single point U

_{x}measurement by the ECM (+marker). Adapted from the work in [3], with permission from the authors, 2020. (

**a**) Time instant 2; (

**b**) time instant 7.

**Table 1.**Hydraulic parameters for the WALOWA bichromatic wave test (EXP) and its repetition (REXP): h

_{o}is the offshore water depth, h

_{t}the water depth at the dike toe, H

_{m0,o}the incident offshore significant wave height, R

_{c}the dike crest freeboard, f

_{i}the SW component frequency, a

_{i}the SW component amplitude and δ (=a

_{2}/a

_{1}) the modulation factor. Reproduced from the work in [3], with permission from the authors, 2020.

TestID [-] | Duration [s] | h_{o} [m] | h_{t} [m] | h_{t}/H_{m0,o} [-] | R_{c} [m] | f_{1} [Hz] | a_{1} [m] | f_{2} [Hz] | a_{2} [m] | δ [-] |
---|---|---|---|---|---|---|---|---|---|---|

Bi_02_6 (EXP) & Bi_02_6_R (REXP) | 209 | 4.14 | 0.43 | 0.33 | 0.117 | 0.19 | 0.45 | 0.155 | 0.428 | 0.951 |

**Table 2.**Model performance and pattern statistics evaluated for η of REXP, OF, DSPH, SW1L and SW8L at measured location WG14 (dike toe location). Adapted from the work in [3], with permission from the authors, 2020.

Model [-] | B* [-] | σ* [-] | R [-] | d_{r} [-] | d′_{r} [-] | Rating [-] |
---|---|---|---|---|---|---|

REXP | 0.00 | 1.00 | 0.98 | 0.92 | 1.00 | Excellent |

OF | 0.05 | 0.89 | 0.91 | 0.82 | 0.90 | Very Good |

DSPH | 0.21 | 1.02 | 0.85 | 0.71 | 0.79 | Good |

SW1L | 0.04 | 1.09 | 0.87 | 0.77 | 0.85 | Very Good |

SW8L | 0.08 | 1.00 | 0.88 | 0.78 | 0.86 | Very Good |

**Table 3.**Model performance and pattern statistics evaluated for η of REXP, OF, DSPH, SW1L and SW8L averaged over all measured locations on the promenade (WLDM01—WLDM04) and for U

_{x}at the measured location ECM. Adapted from the work in [3], with permission from the authors, 2020.

Model [-] | Parameter [-] | B* [-] | σ* [-] | R [-] | d_{r} [-] | d′_{r} [-] | Rating [-] |
---|---|---|---|---|---|---|---|

REXP | η | −0.01 | 0.99 | 0.99 | 0.92 | 1.00 | Excellent |

U_{x} | −0.02 | 1.05 | 0.87 | 0.81 | 1.00 | Excellent | |

OF | η | −0.04 | 1.00 | 0.89 | 0.81 | 0.89 | Very Good |

U_{x} | −0.25 | 0.94 | 0.73 | 0.63 | 0.82 | Very Good | |

DSPH | η | −0.04 | 1.01 | 0.81 | 0.72 | 0.80 | Very Good |

U_{x} | −0.26 | 0.92 | 0.68 | 0.62 | 0.81 | Very Good | |

SW1L | η | −0.03 | 0.96 | 0.78 | 0.74 | 0.82 | Very Good |

U_{x} | −0.22 | 0.84 | 0.51 | 0.55 | 0.74 | Good | |

SW8L | η | −0.14 | 0.91 | 0.82 | 0.75 | 0.83 | Very Good |

U_{x} | −0.09 | 0.86 | 0.62 | 0.59 | 0.78 | Good |

**Table 4.**Model performance and pattern statistics evaluated for p and F

_{x}of REXP, OF, DSPH, SW1L and SW8L at the respective measured locations PS05 and LC. Adapted from the work in [3], with permission from the authors, 2020.

Model [-] | Variable [-] | B* [-] | σ* [-] | R [-] | I* [-] | d_{r,Fx,max} [-] | d_{r} [-] | d′_{r} [-] | Rating [-] |
---|---|---|---|---|---|---|---|---|---|

REXP | p | 0.01 | 1.00 | 0.96 | - | - | 0.91 | 1.00 | Excellent |

F_{x} | 0.00 | 0.97 | 0.90 | 0.99 | 0.92 | 0.90 | 1.00 | Excellent | |

OF | p | −0.11 | 0.75 | 0.61 | - | - | 0.69 | 0.78 | Good |

F_{x} | −0.12 | 0.74 | 0.73 | 0.85 | 0.85 | 0.76 | 0.86 | Very Good | |

DSPH | p | −0.21 | 0.93 | 0.30 | - | - | 0.51 | 0.60 | Reasonable/Fair |

F_{x} | −0.21 | 0.78 | 0.55 | 0.73 | 0.84 | 0.66 | 0.76 | Good | |

SW1L | p | −0.43 | 0.74 | 0.43 | - | - | 0.53 | 0.62 | Reasonable/Fair |

F_{x} | −0.31 | 0.66 | 0.55 | 0.62 | 0.88 | 0.64 | 0.74 | Good | |

SW8L | p | −0.56 | 0.81 | 0.25 | - | - | 0.46 | 0.55 | Reasonable/Fair |

F_{x} | −0.37 | 0.71 | 0.40 | 0.54 | 0.87 | 0.60 | 0.70 | Reasonable/Fair |

Time Instant | Figure | Description |
---|---|---|

1 | Figure 15a | Dynamic F_{x} peak of the small wave impact preceding the main bore impact. |

2 | Figure 17a | Local positive U_{x} peak at the ECM location preceding the dynamic force peak of the main bore impact. |

3 | Figure 15b | Local F_{x} peak preceding the dynamic force peak of the main bore impact. |

4 | Figure 15c | Dynamic F_{x} peak of the main bore impact. |

5 | Figure 15d | Local F_{x} minimum between the dynamic and quasi-static force peaks of the main bore impact. |

6 | Figure 16a | Quasi-static F_{x} peak of the main bore impact. |

7 | Figure 17b | Local negative U_{x} peak at the ECM location after the quasi-static force peak of the main bore impact. |

8 | Figure 16b | Local F_{x} minimum after the quasi-static force peak of the main bore impact. |

**Table 6.**Grid resolution, number of cells and approximate computational times of each applied numerical model.

Parameter | OF | DSPH | SW1L | SW8L |
---|---|---|---|---|

Δx [m] | 0.0225–0.18 | n/a | 0.20 | 0.20 |

Δz [m] | 0.0225–0.18 | n/a | - | - |

dp [m] | - | 0.02 | - | - |

K [-] | - | - | 1 | 8 |

# computational points [-] | 318,381 | 1,309,056 | 1032 | 8256 |

hardware [-] | 24 cores (CPU) | 2880 cores (GPU) | 1 core (CPU) | 1 core (CPU) |

computational time [DD HH:MM:SS] | 03 12:00:00 | 00 15:30:00 | 00 00:00:53 | 00 00:06:31 |

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**MDPI and ACS Style**

Gruwez, V.; Altomare, C.; Suzuki, T.; Streicher, M.; Cappietti, L.; Kortenhaus, A.; Troch, P.
An Inter-Model Comparison for Wave Interactions with Sea Dikes on Shallow Foreshores. *J. Mar. Sci. Eng.* **2020**, *8*, 985.
https://doi.org/10.3390/jmse8120985

**AMA Style**

Gruwez V, Altomare C, Suzuki T, Streicher M, Cappietti L, Kortenhaus A, Troch P.
An Inter-Model Comparison for Wave Interactions with Sea Dikes on Shallow Foreshores. *Journal of Marine Science and Engineering*. 2020; 8(12):985.
https://doi.org/10.3390/jmse8120985

**Chicago/Turabian Style**

Gruwez, Vincent, Corrado Altomare, Tomohiro Suzuki, Maximilian Streicher, Lorenzo Cappietti, Andreas Kortenhaus, and Peter Troch.
2020. "An Inter-Model Comparison for Wave Interactions with Sea Dikes on Shallow Foreshores" *Journal of Marine Science and Engineering* 8, no. 12: 985.
https://doi.org/10.3390/jmse8120985