# Coupled SPH–FEM Modeling of Tsunami-Borne Large Debris Flow and Impact on Coastal Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. SPH Governing Equations

#### 2.1.1. Kernel Approximation

- The smoothing function is normalized:

- 2.
- There is a compact support for the smoothing function:

- 3.
- $W\left(x-{x}^{\prime},h\right)$ is non-negative for any ${x}^{\prime}$ within the support domain. This is necessary to achieve physically meaningful results in hydrodynamic computations.

- 4.
- The smoothing length increases as particles separate and reduces as the concentration increases.
- 5.
- With the smoothing length approaching zero, the kernel approaches the Dirac delta function:

- 6.
- The smoothing function should be an even function.

#### 2.1.2. Particle Approximation

#### 2.2. SPH for Viscous Fluid

#### 2.3. Sorting

#### 2.4. Equation of State (EOS)

#### 2.5. Time Integration

#### 2.6. Contact Definitions

_{i}is given in terms of the bulk modulus K

_{i}, the volume V

_{i}, and the face area A

_{i}of the element that contains s

_{i}as:

## 3. Experimental Work

_{erf.}

## 4. Coupled SPH–FEM Modeling

#### 4.1. Numerical Settings

#### 4.2. Accuracy of Numerical Modeling

#### 4.2.1. Free Surface and Fluid Velocities

_{1}= 2.496 m and h

_{2}= 0.13 m, and T

_{erf}= 30 s from [65], where h

_{1}and h

_{2}corresponds to the initial water depth offshore (at wavemaker location) and on the coast close to the debris, respectively. It can be observed that both the free-surface and fluid velocity histories computed by the numerical model are in good agreement with the experimental data, both in terms of the peak values and temporal evolution. There are some underpredictions of the maximum wave height and some overpredictions of the velocity but those do not exceed 6% for the selected wave. In addition to this good agreement, the encouraging thing is that the numerical model can predict the relative increase between wg2 and wg1 of the maximum free surface, indicating that the SPH–FEM coupled approach can capture the interaction of the fluid with the sloped part of the flume and result in a similar non-linear transformation of the wave during the shoaling process.

_{erf}= 30 s, 40 s, 45 s) for h

_{1}= 2.496 m and h

_{2}= 0.13 m. Promising agreement with the experimental was achieved, with the maximum deviation from the average value of the measured data at wg1 being 6.3%, 11.7%, and 13.7% for T

_{erf}= 30, 40 and 45 s, respectively. At wg2, the maximum differences are 4%, 15.9%, and 15.2%. As expected, both the numerical and experimental data show that the shorter T

_{erf,}i.e., T

_{erf}= 30 s gives greater inundation depths at the two locations. This also seems to be true for the peak velocities, especially those measured at adv2, as shown in Figure 5. The latter figure presents the experimentally and numerically recorded maximum velocities at (a) adv 1 and (b) adv2 for the same hydrodynamic conditions as in Figure 4 (h

_{1}= 2.496 m). Similarly to the free surfaces, the SPH–FEM approach can predict reasonably the maximum fluid velocities with maximum deviations of 3%, 4.7%, and 2% at adv1 for the three different tsunami bores (T

_{erf}= 30, 40, and 45 s), respectively. At adv2, the maximum differences are 6%, 14%, and 22%, respectively. It is noteworthy that (i) the experimentally recorded peak fluid velocities in Ko and Cox [65] had significantly larger variability than the measurements of the free surface, and (ii) the most sensitive experimental results corresponded to the measured velocities of the slower flows (e.g., for T

_{erf}= 45 s), for which the numerical modeling yielded the largest deviations (underprediction of approximately 25%).

#### 4.2.2. Debris Motion

_{2}= 0.13 m and T

_{erf}= 30 s. The debris velocity was generated based on the publicly available videos on DesignSafe (i.e., Ko and Cox [65]), using a color-based tracking algorithm. However, it must be noted that the estimated velocities (from the videos of the experiments) could potentially entail some errors due to the fact that the ceiling cameras could not be perpendicular to the top surface of the debris for the whole propagation process (from its initial position up to the coastal structure). A more accurate estimate would require a perspective correction, as in Ko [37], which was not done herein due to the lack of adequate information.

#### 4.2.3. Debris Impact Force

_{erf}= 30 s and T

_{erf}= 45 s for the same initial water depth h

_{2}= 0.13 m. Subfigures (a) and (b) correspond to T

_{erf}= 30 s for the numerical and experimental data respectively, and subfigures (c) and (d) show the same results but for T

_{erf}= 45 s.The agreement between the computations and the different trials of the physical tests is reasonable, both in terms of the peak impact force and in the overall trends. For instance, for the case of T

_{erf}= 30 s, the SPH–FEM models predict a relatively higher impact force than the experiments by approximately 15%. However, this can be explained by the fact that as shown in Figure 6 the numerically predicted impact velocity was approximately 20% higher than the experimental one. Given the fact that the majority of the available simplified equations for debris impact loads, such as those presented in FEMA P646 [3] and ASCE [79], are a linear function of the impact velocity, it is reasonable to obtain larger impact forces from the numerical simulations since they predict larger velocities. Apart from the similarities in the results, there are also two noticeable differences:

- In the numerical simulations, the impact force on the column is applied earlier for T
_{erf}= 30 s and later for T_{erf}= 45 s compared to the physical tests. However, these differences in the instants could be justified by the differences in the debris velocities, which were most likely overpredicted and underpredicted, respectively, as indicated by the trends in the maximum values. In other words, it is reasonable for the debris impact to occur earlier when the numerical models overpredict the magnitude of the impact, because the reason is the larger debris velocity. - Immediately after the primary impact force, the column in the physical tests experienced a second short-duration impact force, which is relatively small compared to the main impact. However, this trend was not observed in the numerical results. In contrast, the simulations show a long duration load after the initial impact, which seems to have a nearly constant magnitude. This difference can be attributed again to the 2D simplification made in the numerical models, which are unable to allow the fluid to escape from the sides of the debris after the initial impact on the column and relieve the pressures applied on its offshore face. This leads consequently to the stagnation of the flow in front of the offshore face, resulting in a nearly steady-state horizontal damming load.

_{erf}= 30, 40, and 45 s) with h

_{1}= 2.496 m and h

_{2}= 0.13 m. In fact, this figure reveals that the maximum deviation from the measured values is 14%, 25%, and 19% for the three flows, respectively, which is promising given the differences in the debris velocities. Moreover, another observation that reinforces the confidence in the SPH–FEM modeling approach is that it presents similar trends with the physical tests (as a function of T

_{erf,}) since both of them give larger debris velocities and impact forces for the smallest T

_{erf}= 30 s representing the more transient flow with the largest fluid velocities.

## 5. Role of Debris Restraints

_{erf}= 30 s for h

_{2}= 0.13 m as the free debris propagates towards the column location, from the instant that the bore starts moving the debris up to the instant of the second impact. Among these snapshots, ‘e’, ‘f’, ‘g’ correspond to following instants: (i) slightly before the first impact on the column, (ii) after the 1st impact, and (iii) at the instant of the 2nd impact on the column, respectively. As shown, the debris starts pitching in the clockwise direction up to the point that the onshore bottom corner tends to touch the bottom of the flume, after which it immediately changes the direction of pitching. It is also revealed that the large counter-clockwise pitching continues until the debris reaches the column location, which results in a non-normal impact angle and a consequently non-uniform contact of the onshore vertical face of the debris with the column, affecting the contact area and consequently the maximum impact forces. After the initial impact, the debris bounces back and re-impacts the column with a clockwise pitching angle, which generates a smaller magnitude impulse.

## 6. Effect of Hydraulic Conditions on Debris Motion and Impact Forces

#### 6.1. Tsunami Flow Characteristics

_{erf}= 30 s, T

_{erf}= 40 s, and T

_{erf}= 45 s for h

_{1}= 2.496 m and h

_{2}= 0.13 m tested in [37]. In this section, the debris will be considered free in the 2D plane since it is considered more realistic, despite the fact that the restrained model captured better the experimental data. Comparison of the free surface and fluid velocity at a location close to the container, i.e., x = 62 m, is depicted in Figure 12. As discussed in [37], the error function (T

_{erf}) affects the wave characteristics, and particularly the wave height and fluid velocity. The new figure shows that although there are similarities in the free-surface histories for all three waves, the smallest T

_{erf}(which corresponds to the faster movement of the wavemaker) exhibits the largest peak values for the free surface and the fluid velocity, with the most noticeable differences observed in the fluid velocities. Interestingly, as the T

_{erf}increases the flow height and fluid velocity is reducing, while the duration of the inundation is elongated, which indicates that the flow is switching from a highly transient bore to a more steady-state flow.

_{erf}= 30 s, and T

_{erf}= 45 s, respectively. While both waves present similar trends in the debris motion, which comprises of a clockwise rotation followed by a counter-clockwise one, the faster moving bore results in larger particle velocities around the debris that in turn cause larger pitching of the container. This would indicate that the pitching of the debris is mainly caused by the larger velocities, and not that much by the differences in the free surface, which are much smaller. Figure 14 plots the vertical movement of the lower-right corner of the container and its rotation throughout the propagation inland. This figure illustrates that the debris flow is indeed highly dependent on the tsunami characteristics and that the fastest bore (T

_{erf}= 30 s) can cause pitching angles that are approximately 85% larger than those of the slowest wave (T

_{erf}= 45 s). This larger rotation leads to an increase in the upward vertical movement of the offshore face of the container, which enables it to impact structural locations at higher elevations.

_{erf}= 30 s, as shown below:

- T1: This time instant represents the initiation of the debris rotation. It occurs slightly after the tsunami has started pushing the debris inland. After that initial contact with the bore, the debris starts accelerating and as the flow below the debris increases and the bore front surpasses the debris, the latter one starts rotating clockwise (see also snapshots b. and c. in Figure 10).
- T2: This instant corresponds to the largest clockwise rotation, at which point the lower right (onshore) corner has displaced downward so much that it impacts the floor of the flume. When this impact takes place, a restoring force is applied to the debris causing it to start rotating in the opposite direction (counter-clockwise).
- T3: After the primary debris impact on the flume floor and the initiation of the counter-clockwise rotation, the debris continues rotating in this direction until it reaches the maximum pitch angle, which tends to occur slightly before the primary debris impact on the column. At this instant, it is possible for the lower left (offshore) corner of the debris to touch the floor of the flume before it impacts the column. However, this will depend on the initial relative distance between the debris and the coastal structure, as well as, the hydrodynamic conditions. This means that instant T3 represents the maximum clockwise pitching angle, which might be close to the impact angle, but not necessarily the same. Future studies should investigate different debris–structure relative distances, and a larger range of hydrodynamic conditions in order to determine the dependence of the maximum pitching angle and the impact angle on these parameters. Ideally, such studies should employ three-dimensional models, which are expected to be more accurate than two-dimensional models.

_{2}= 0.13 m. As expected, larger tsunami waves have more energy and higher particle velocities, which lead to higher values of the debris impact velocities and impulsive forces on the column. This is why the wave with T

_{erf}= 30 s exerts the largest impact force on the column. Interestingly, the largest/fastest tsunami bores also result in higher damming loads, with T

_{erf}= 30 s giving almost 3-fold larger values than T

_{erf}= 45 s. However, this observation must be taken with caution, since, as explained earlier, the 2D nature of the numerical models can lead to over-prediction of the damming loads. Last but not least, the right subplot of Figure 15 reveals that contrary to the existing simplified equations of debris loads (e.g., FEMA P626 [3]), the impact forces might not necessarily be a linear function of the impact velocity, at least for the specific hydrodynamic conditions. In fact, when the debris velocity increases above a certain limit (e.g., 1.1 m/s for this water depth) the rate of the increase in the impact force with the velocity decreases, resulting in a non-linear increase in the force. This behavior can be explained by the trends observed in the previous figures, according to which, the largest impact velocity corresponds to the fastest bore (T

_{erf}= 30 s) that cause significant pitching of the debris and non-normal impact on the column. Ultimately, the results presented herein indicate that the debris impact forces might be a function of both the debris velocity and pitching angle at the instant of the impact on the coastal structure. However, this indication must be further verified with validated three-dimensional models. Ideally, such future models should simulate the debris and the coastal structure as flexible bodies, since that will enable a more realistic prediction of the impact duration and determine its dependence on the debris velocity and pitch angle.

#### 6.2. Initial Water Depth

_{1}= 2.496 and 2.664 m and a new depth with h

_{1}= 2.8 m were considered for a range of tsunami flows. These offshore water depths translated into local initial water depths of 0.13, 0.30 and 0.43 m, respectively, at the debris location. Figure 16 illustrates (a) the variation of the free surface, (b) thefluid velocity, (c) the debris vertical displacement, (d) the debris rotation, (e) the debris velocity and (f) the impact force. The free-surface histories are plotted close to the offshore side of the debris, at x = 62 m, and are calculated relative to the initial water level, while the fluid velocities are plotted at the same x coordinate at the level of the initial free surface (i.e., 9.1 cm above the bottom of the debris). This means that the absolute elevation of the locations at which the fluid velocities are plotted are different for each water depth, but the relative distance from the bottom of the debris is the same. The figure reveals small differences in the maximum bore heights and nearly negligible differences in the fluid velocity histories close to the debris location, for three water depths. There are some differences in the free-surface history of the shallower water level relative to the two larger depths, with the most obvious difference in the bore front and the instant of the arrival at x = 62 m. In the case of the deeper water, the tsunami waves are arriving slightly faster, which is attributable to the increase in the wave celerity offshore caused by the increase in the water depth. Nonetheless, the tsunami bores at the debris location are similar enough for the three water depths, enabling the proper investigation of the effect of this parameter.

_{1}= 2.496 m relative to the other depths, this case exhibits the largest vertical displacement of the debris (at its onshore corner) and the largest pitching. In fact, the maximum pitching angle seems to consistently decrease with the increase in the water depth, with the shallowest case inducing an approximately 5-fold larger maximum pitching angle relative to the deepest case of 2.8 m. This demonstrates that the rotation of the debris is highly dependent on the initial water depth. Moreover, in addition to the differences in the magnitudes, it can be observed that the previously identified pattern in the debris motion, which involved an initial clockwise debris rotation followed by a counter-clockwise one before the impact on the column, is not consistent for all the water depths. For larger initial depths it is possible to notice an opposite sequence of debris rotations (i.e., for h

_{1}= 2.66 m) or just a clockwise rotation before the initial debris contact with the column (i.e., for h

_{1}= 2.8 m). Last but not least, in contrast to the differences in the vertical displacement and rotation of the debris, the debris horizontal velocities present more similarities. The major difference is observed in the fact that the deeper cases exhibit a gradual increase in the debris velocity, which becomes nearly constant as it approaches the coastal structure, while in the shallowest case more abrupt increases in the debris velocity are observed that result in a larger impact velocity on the column. Despite the larger debris impact velocity for h

_{1}= 2.496 m (for the specific tsunami bore), this case gives similar impact forces with the large water depths, which can be justified by the higher level of pitching in the former case.

_{erf}= 40 s, as the debris moves inland and impacts the coastal structure. This visual representation of the phenomenon verifies the previously observed trends, with the two larger water depths being associated with nearly a consistent debris orientation (small rotation) contrary to the shallow water that is dominated by debris pitching effects. Moreover, despite the similar bore velocities at x = 62 m (a few meters from the debris) observed in Figure 16, the fringe plots of the fluid particle velocities reveal that there are significant differences in the flow around the debris, since in the shallow case the flow seems to accelerate more below the debris. This is probably due to the fact that in the latter case the bore is more restricted and does not have as much space to propagate below the debris as in deeper waters, resulting in faster flows horizontally that tend to uplift on one side of the debris and consequently rotate it. Another major difference lies in the fact that when the initial water level is low, the pitching of the debris can move one of its corners downwards so much that it impacts the bottom of the flume. This contact between the debris and the flume complicates further the debris–fluid interaction and is a distinguishable feature of the small water depths only. Last, but not least the snapshots reveal major differences in the fluid flow below the debris after the initial impact on the column, which affects the number of secondary impacts and their magnitude, as well as, the damming loads. For example, smaller damming loads can be noticed in the case of larger water depths, because less of the bore gets reflected on the structure and more of it propagates onshore by moving below the debris. However, these differences might be exaggerated by the 2D assumption made in the development of the numerical models.

_{1}= 2.66 and 2.8 m) the relationship between the maximum debris impact force and impact velocity is nearly linear, which agrees with existing predictive equations (e.g., FEMA P646 [3] and ASCE [79]), for small water depth the trend seems to be non-linear. In fact, in the latter case, after the exceedance of the debris velocities above a certain limit, the impact force increases less than what a linear force–velocity assumption would suggest, indicating that predictive equations that utilize such an assumption might yield conservative results. To investigate whether such an indication is true, additional parametric investigations are required, followed by direct comparisons with available simplified equations, which is beyond to scope of this manuscript. However, the results presented herein raise the question on whether future predictive equations of debris loads should: (i) be a function of the water depth, so that for large initial water depths a linear impact force–velocity relationship is used and for shallow depths a non-linear one that will account for the possibility of debris pitching and non-normal impact angle on the structure, especially if the structure is located close to the location of the debris entrainment, and (ii) limit the applicability of the linear force–velocity equations to a certain velocity limit, above which, the rate of the force increase with the velocity will drop significantly.

## 7. Summary and Conclusions

- The free surface and fluid velocities had good agreement with the experimentally recorded results, both offshore and during the wave propagation along the slope. The deviation of the maximum wave height from the average value of the experimental tests ranged between 4% and 15.2%, while the deviation of the maximum fluid velocities was between 2% and 22%, depending on the location along the flume and the tsunami flow. These results showed that the numerical model can predict the relative increase in the free surface and fluid velocities as the wave propagates over the sloped part and undergoes a non-linear transformation, indicating that the fluid–flume contact worked properly.
- The SPH–FEM models estimate similar debris velocities with the experiments, especially when the bore reaches the container and starts transporting it. However, as the debris propagation inland continues, the numerical model tends to accelerate more and reach an impact velocity that is approximately 20% larger than in the experiments, leading consequently to some differences in the arrival time at the column location. One possible explanation for these differences lies in the 2D formulation of the current numerical model, which implies that the pressures applied from the bore on the offshore face of the debris is uniform across the debris width, which is not necessarily the case in real 3D environments.
- The deviation of the numerically predicted maximum debris impact forces on the column from the experimental data was in the range of 14–25% for the investigated hydrodynamic flows. However, these differences are consistent with the observed differences in the debris impact velocities. Overall, the numerical results presented similar trends with the physical tests since both gave larger impact forces for the more transient and faster tsunami flows.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{erf}= 30 s. This appendix (Figure A1, Figure A2 and Figure A3) shows the numerically predicted free surface, debris velocity and impact force for particle sizes equal to 1, 2 and 3 cm.

**Figure A1.**Variation of the free surface at different locations along the flume. Experimental [65] and numerical results for three particle sizes, for h

_{1}= 2.496 m and T

_{erf}= 30 s.

**Figure A2.**Debris velocity histories. Estimated based on the experimental tests of [65] and numerical results for three particle sizes, for h

_{1}= 2.496 m and T

_{erf}= 30 s.

**Figure A3.**Debris impact forces. Experimental [65] and numerical results for three particle sizes, for h

_{1}= 2.496 m and T

_{erf}= 30 s.

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**Figure 1.**Cross-section of the Large Wave Flume (LWF) depicting the bathymetry, column location and flume instrumentation of the experiments of Ko and Cox [65] used for the validation study.

**Figure 2.**Numerical models with large debris: side-views of the debris, the column and the wave maker.

**Figure 3.**Variation of the free surface and fluid velocity at different locations along the flume. Experimental [65] and numerical results for h

_{1}= 2.496 m and T

_{erf}= 30 s.

**Figure 4.**Maximum free-surface values of experiments [65] and numerical simulations for h

_{1}= 2.496 m, and three wave cases with T

_{erf}= 30, 40 and 45 s.

**Figure 5.**Maximum fluid velocities of experimental tests [65] and numerical simulations for h

_{1}= 2.496 m, and three wave cases with T

_{erf}= 30, 40, and 45 s.

**Figure 6.**Debris velocity histories: Estimated based on the experimental tests of [65] and results from the numerical simulations for h

_{1}= 2.496 m, T

_{erf}= 30 s.

**Figure 7.**Comparison of the debris impact forces: experimental tests [65] and numerical simulations for h

_{1}= 2.496 m, and two wave cases with T

_{erf}= 30 s and 45 s.

**Figure 8.**Maximum values of debris impact force on column: experimental tests [65] and numerical simulation for h

_{1}= 2.496 m, and three wave cases with T

_{erf}= 30, 40, and 45 s.

**Figure 9.**Selected instants of the debris−wave interaction and impact on the column of the free debris. Numerical results for the hydrodynamic case with h

_{1}= 2.496 m, h

_{2}= 0.13 m and T

_{erf}= 30 s.

**Figure 10.**Debris trajectory (

**left**) and rotation (

**right**) for the restrained debris and free debris. Numerical results for the hydrodynamic case with h

_{1}= 2.496 m, and h

_{2}= 0.13 m and T

_{erf}= 30 s.

**Figure 11.**Debris velocity (

**left**) and forces on the column (

**right**). Experimental and numerical results (of restrained and free debris) for the hydrodynamic case h

_{1}= 2.496 m and T

_{erf}= 30 s.

**Figure 12.**Variation of free surface (

**left**) and fluid velocity (

**right**) at x = 62 m: Numerical results for h

_{1}= 2.496 m and three wave cases with T

_{erf}= 30, 40, and 45 s.

**Figure 13.**Snapshots of debris−tsunami interaction and impact on the column. Numerical results for h

_{1}= 2.496 m and two wave cases with T

_{erf}= 30 s (

**top**) and 45 s (

**bottom**).

**Figure 14.**Numerical results of the free debris: vertical displacement (

**left**) and rotation (

**right**), for h

_{1}= 2.496 m and three wave cases with T

_{erf}= 30, 40, and 45 s.

**Figure 15.**Free−debris histories (

**left**) and maximum values of debris impact on the column vs. the impact velocity (

**right**). Numerical results for h

_{1}= 2.496 m h

_{2}= 0.13 m, and T

_{erf}= 30, 40, and 45 s.

**Figure 16.**Time histories of the free surface, fluid velocity, motion of the debris (y displacement and rotation), debris velocity and debris impact force on the column. Numerical results of free debris for T

_{erf}= 40 s and three water depths with h

_{1}= 2.496, 2.664 and 2.8 m.

**Figure 17.**Snapshots of debris-tsunami interaction and impact on the column. Numerical results of free debris for T

_{erf}= 40 s and three water depths with h

_{1}= 2.496 (

**top**), 2.664 (

**center**) and 2.8 m (

**bottom**).

**Figure 18.**Maximum values of the debris impact force as a function of the impact velocity for three water depths and nine tsunami bores.

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**MDPI and ACS Style**

Hasanpour, A.; Istrati, D.; Buckle, I.
Coupled SPH–FEM Modeling of Tsunami-Borne Large Debris Flow and Impact on Coastal Structures. *J. Mar. Sci. Eng.* **2021**, *9*, 1068.
https://doi.org/10.3390/jmse9101068

**AMA Style**

Hasanpour A, Istrati D, Buckle I.
Coupled SPH–FEM Modeling of Tsunami-Borne Large Debris Flow and Impact on Coastal Structures. *Journal of Marine Science and Engineering*. 2021; 9(10):1068.
https://doi.org/10.3390/jmse9101068

**Chicago/Turabian Style**

Hasanpour, Anis, Denis Istrati, and Ian Buckle.
2021. "Coupled SPH–FEM Modeling of Tsunami-Borne Large Debris Flow and Impact on Coastal Structures" *Journal of Marine Science and Engineering* 9, no. 10: 1068.
https://doi.org/10.3390/jmse9101068