Classification and Comparison of Wave Impact Modes on Semi-Submersibles

Semi-submersibles for offshore oil exploration and exploitation often suffer from severe wave impacts in extreme ocean environments. Owing to the complex wave interactions among structural components of semi-submersibles, in-depth analyses on the characteristics of wave impact events are of significance for both industry and academia. An experimental study was carried out to investigate the local wave impact loads on a semi-submersible, with focus on understanding the wave impacts by identifying typical impact modes. Quantitative criteria are proposed to classify major wave impacts on the semi-submersible into six modes and two types. The results show that the classification is reasonable and provides valuable information for studying wave impacts on semi-submersibles. The incident wave characteristics at the fore column of the semi-submersible have important influence on the wave impact mode. The fore-column dominating wave impacts exert the most intense loads on the fore column and feature well-developed breaking waves or slightly breaking waves at the fore column. However, the aft-column dominating wave impacts exert the most intense loads on the aft column or the deck bottom and feature non-breaking waves at the fore column. Energy loss during the fore-column impact weakens the impact severity on the aft column in the fore-column dominating wave impacts. The shoaling effect of the submerged pontoon and different motion configurations of the platform result in higher occurrence rate of the aft-column dominating wave impacts. Different impact modes are also distinguished by different spatial distributions of wave impact loads.


Introduction
In-depth understanding of wave impact process and accurate determination of associated loads are crucial to the design of marine structures. The wave impact load is one of the most intense loads that marine structures could be subjected to during their lifetime. Recent accidents caused by wave impact loads have aroused renewed attentions on this issue, and have motivated the amendment of design regulations. In December 2015, the semi-submersible drilling rig COSL Innovator was subjected to an extreme wave which caused one fatality, four injuries and extensive damage to the living quarters (Viste-Ollestad et al., 2016).
The violent water impact problem has been studied for a long time. Von Karman (1929) and Wagner (1932) are the pioneering works about the water impact loading during water entry and lay the foundation for further theoretical stud-ies in related fields, such as the launching of life boats, bottom slamming of ships, wet-deck slamming, and wave impacts on various vertical structures. The water impact problem is involved in a large variety of marine applications. Researchers and engineers in coastal engineering have devoted decades of efforts into the violent impact of breaking waves on breakwaters and sea walls (Allsop et al., 1996;Oumeraci et al., 2001;Cuomo et al., 2010). For wave impacts on structures, the type of wave breaking or the breaker has significant influence on the magnitude of the impact pressure and its distribution. Different breaker types are determined by the distance between wave breaking point and the structure, and are characterized by different amounts of enclosed air. In their experiment, Chan and Melville (1988) changed the locations of a vertical wall relative to the wave breaking location and found that the elevation of the wave impact zone decreased with a downstream shift in wall location and the highest pressure occurred at the critical location where the wave crest hit approximately horizontally. Based on observations from model tests, Oumeraci et al. (1993) classified the breaker (i.e., breaking wave) into four main types: upward deflected breaker without air entrapped, plunging breaker with a small air cushion, well-developed plunging breaker with a large air cushion and turbulent bore with foamy front. They also attempted to build the correlation between the breaker type and the recorded loading characteristics. Wienke and Oumeraci (2005) investigated the breaking wave forces on a slender cylinder with a largescale model test. A theoretical description was developed based on the Wagner theory (Wagner, 1932) and compared with the experimental results. Also, the effect of the yaw angle and breaker type on the wave impact force was investigated. Bullock et al. (2007) demonstrated that the likelihood of high pressures was much the same for the breaker with small air pocket ("low-aeration") and the breaker with large air pocket ("high-aeration"), and that the demarcation between the two types relied on the subjectivity. The repeated compression and expansion of the air pocket trapped between the wave front and the structure surface would lead to the high frequency oscillations of the wave impact pressures (Kisacik et al., 2012;Vestbøstad et al., 2017;Mai et al., 2019). Sun et al. (2019) studied the air compressibility effects in breaking wave impacts numerically using a CIPbased model and reproduced the high frequency oscillation of the impact pressure. It was found that the oscillation frequency is related to the size of the air pocket.
Compared with those on solitary walls or cylinders, the characteristics of wave impacts on composite structures are less understood. Among the studies, Kisacik et al. (2012) investigated violent wave impacts on a combined structure consisting of both a vertical and a horizontal part, and classified breaking waves into four types according to impacts on the vertical part. In offshore engineering, the impact of steep or breaking waves is a key issue for the design of multi-column large-volume floaters like semi-submersibles. In the aftermath of the COSL Innovator accident, several model test campaigns were conducted to document the structural integrity for wave impact loads (Huang et al., 2017;Vestbøstad et al., 2017). Adversely to fixed breakwaters, these offshore structures are composed of more structural components, like columns, pontoons and decks. Various hydrodynamic interactions among different components, such as wave run-up along columns (Nielsen, 2003), wave trapping between the fore column and the aft column (Shan et al., 2012) and shoaling effect of the submerged pontoon (Iwata et al., 1997), have substantial influence on wave impacts on such structures. In addition, motions of floating structures complicate wave impacts on them. Abdussamie et al. (2017) carried out experimental investigations of wave-in-deck impact loads on a TLP model which was fixed rigidly and moored by four tendons, respectively. More recently, Guo et al. (2020) explored governing parameters for horizontal wave impact loads on a semi-submersible based on extensive experimental data, including both the wave characteristics and the platform's motion behaviors. As is done for breakwaters, classifying wave impact events based on the physics of the process into certain modes would be very helpful to deepen the understanding of wave impacts on semi-submersibles and to evaluate wave impact loads on such structures more systematically. However, to the authors' knowledge, very limited open literatures to date have been reported to focus on this aspect.
In this paper, a careful experimental investigation was carried out to study the wave impact loads on a semi-submersible. Based on the experimental results, wave impacts on the semi-submersible are classified into six modes and two types. To avoid subjectivity, quantitative criteria according to the affected components and the impact severities are adopted rather than visual observations of video records. The characteristics of the flow field, impact pressures on different components, and relative wave elevations at the fore column and the aft column during the evolution of typical wave impact events for each mode are analyzed. Furthermore, comparative studies are performed between different impact modes, focusing on motions of the platform at the occurrence of wave impacts, wave characteristics and spatial distribution of wave impact loads.

Experimental setup
The experiments were conducted in the Deep-water Offshore Basin at Shanghai Jiao Tong University in China at a scale ratio of 1:60. The basin is 50 m in length, 40 m in width and 10 m in maximum effective water depth, and is equipped with various advanced testing facilities and measuring instruments to ensure high-quality experimental campaign. Two sides of the basin are equipped with multi-flap wave makers to generate waves, and wave absorbing beaches were installed on the opposite sides to minimize reflected waves from the boundaries of the basin.
A semi-submersible model composed of two pontoons, four rectangular columns, four circular bracings and one deck box was employed for wave tests, as shown in Fig. 1. The platform model was fabricated with sufficient accuracy and robustness. Table 1 details the main particulars and mass properties of the semi-submersible under the survival loading condition, where the vertical position of the center of gravity is measured from the base line and the radius of gyration is calculated from the center of gravity of the semisubmersible. During the wave tests, the platform model was moored at the center of the basin with a horizontal mooring system, as shown in Fig. 2. The stiffness of each mooring line is 64.5 N/m (in model scale) which is considered to be soft enough to avoid the interference between motions in the horizontal plane (surge and sway) and motions in the vertic-al plane (heave, pitch and roll). Fig. 3 presents two examples under head seas and beam seas, respectively. It can be seen that the natural periods of horizontal-plane motions do not overlap with those of vertical-plane motions, indicating the selection of the stiffness of mooring springs does not affect the wave frequency motion responses. The wave impact loads are mainly determined by the waves and the motions of the platform. Therefore, the wave impact loads on the semi-submersible are also not affected by the selected mooring line stiffness.
Wave impact tests were carried out using 23 long crest irregular waves (belonging to six sea states), as listed in Table 2, where is the significant wave height, is the peak period, is the peakness factor and "Seed" is the number of irregular waves of corresponding sea state. The six sea states were chosen around the steepness limited flank of the − contour to include the governing sea state. The chosen irregular waves were simulated with typical JON-SWAP spectrum. Before placing the platform model in the wave basin, wave calibrations were performed to provide high quality input to wave tests. For all input irregular waves, the errors in terms of and do not exceed 5% and 3%, respectively. The duration of each irregular wave test is equivalent to 3 hours (prototype time), excluding a run-up time of 12 minutes. Although three wave headings were considered as shown in Fig. 2, only the wave test runs under the head sea conditions are analyzed in the paper. To measure the wave impact load, a total of 43 slam panels were installed on the fore column, the deck box, the aft column and the deck bottom of the semi-submersible model. The performance and reliability of the slam panel were validated by the wedge drop test (Guo et al., 2017). As shown in Figs. 1b−1d, only 15 out of them are used to study the successive wave impacts on the fore column, the aft column and the deck bottom. Fig. 4 depicts the detailed locations of the concerned slam panels. The overhanging part of the deck box is very small. Therefore, as shown in Fig. 4a, the deck box is regarded as the extension of the fore column unless otherwise stated. To measure wave-in-deck impact loads due to waves exceeding the available freeboard or wave run-up along the aft column, five slam panels were installed on the deck bottom along the length direction. A local coordinate system ( ) is adopted to measure the distance of slam panels on the deck bottom to the aft     (Fig. 4c). Furthermore, for convenience, the fore column, the aft column and the deck bottom are abbreviated as FC, AC, and DC, respectively. The slam panel consisted of a strain-gauge force transducer (KYOWA, LUX-B-100N-ID) and a sensing plate made of acrylic glass which was light enough to minimize the panel dynamic effects. The force transducer has the rated capacity of 100 N and the natural frequency of about 11 kHz. The dimension of the sensing plate was 2.0 cm× 2.0 cm in model scale (1.2 m×1.2 m in prototype) which was in compliance with the requirement of DNV•GL (2016) that panel dimensions should be smaller than 3.0 m×3.0 m (in prototype). Efforts were made to reduce hydro-elastic effects, such as pre-embedding steel base plates into the test model before installing slam panels, and to ensure leak proof by attaching watertight films. Similar applications of force transducers can be seen in a few published works (Huang et al., 2015;Kim et al., 2019). Hammer tests were conducted to qualify the slam panels after being installed on the semi-submersible model. The natural frequencies of the slam panels were found to be around 4 000 Hz and the noise effects were negligible. WG f WG a Two resistance-type wave gauges were installed to monitor the relative wave elevations at the FC ( ) and the AC ( ), as shown in Fig. 1c. The high-quality and reliability of the wave gauges used in the experiment have been proven in Zhao et al. (2017). The six-degree-of-freedom motions of the platform model at the center of gravity were measured with a non-contact optical capturing system. Ad-ditionally, two action cameras (GoPro HERO5 Black) were installed at a distance to the forward and port side of the semi model (see details in Fig. 2), to record the wave impact process at 120 frames per second.
During wave tests, all data channels were sampled and recorded with two data acquisition systems, which were synchronized with a shared signal. Ordinary signals such as relative wave elevations and motions, were sampled at the frequency of 200 Hz without hardware filtering. Owing to the short duration of the wave impact load, a high-speed system was employed to record wave impact loads with a sampling frequency of 20000 Hz and an anti-alias filter of 5000 Hz was used before sampling.

Threshold of major wave impact
For the issue of wave impacts, concerns are mainly given to severe wave impacts which could potentially endanger the structural integrity of the offshore unit. Therefore, on the basis of probability analysis, further processing with the peak-over-threshold (POT) method is applied to the maximum peak pressure data set to separate (the largest measured peak pressure among all slam panels during every single wave impact event) above the specified threshold from the rest of the data. The same strategy is also recommended in DNV•GL (2016) to obtain convergent Weibull probability distributions and to estimate the design load. Fig. 5a presents the exceedance probability distribution of the maximum peak pressure among all slam panels within each wave impact event under all sea states.

P(X > x)
The exceedance probability, is defined as follows: is the cumulative distribution function and represents the probability that the random variable takes on a value smaller than or equal to . The measured wave impact loads show a large statistical scatter. In this study, the threshold is set at the empirical 90th percentile of the maximum peak pressure, which are listed in Table 3 for different sea states. Instead of the empirical percentile, the threshold can also be extrapolated from the theoretical fit of the data samples, such as the extrapolation method used by Li et al. (2019) where the weight factor is used to focus on the more reliable sampled data. However, only empirical percentile is used in this study for simplicity due to the potential multi populations of wave impact pressures. The truncated data set is fitted to the Weibull distribution which is commonly used for the statistical analysis of wave-induced responses, as shown in Fig. 5b where the straight lines are the corresponding least square fittings. It is shown that the upper tail of the maximum peak pressure truncated by is well fitted by the Weibull distribution and the distribution parameters seem to be sea state independent.

Classification of major wave impact modes
The wave impact process on multi-column offshore structures like semi-submersibles is complex by the interactions among different structural components. Classifying wave impact events based on the physics of the process into certain modes would be very helpful to deepen the understanding of wave impacts on semi-submersibles and to evaluate wave impact loads on such structures more systematically.
Based on systematic examinations of the measurements, major wave impact events with the maximum peak pressure above are classified completely into two types and six modes according to the affected components and the impact severities, as shown in Fig. 6. Within the classification criteria, the ratio of the maximum peak pressure on the nondominating component to that on the dominating component is utilized to represent the energy evolution during the impact process. The wave impact load is localized in both time and space. Therefore, the ratio value should be selected reasonably. In DNV•GL (2016), the ratio of the wave impact pressure at upwell level (i.e., vertical position/up-well=1.0) to the largest impact pressure is about 0.2 (refer to Fig. 1 and Fig. 2 in DNV•GL (2016)). In the present study, the ratio value of 0.2 is also selected and is found to be crit-ical based on the observation of measured data set, and means substantial reduction of the impact intensity. Wave impacts of Type 1, including three modes, are the FC dominating wave impacts. The difference among the three modes is whether severe wave impacts occur (i.e. ratio ≥ 0.2) on the AC and the DC. Mode 4 and Mode 5 belong to Type 2, and are the AC dominating wave impacts, while Mode 6 are the DC dominating wave impacts. The DC dominating wave impacts are closely associated with the wave impacts on the AC. Therefore, wave impacts of Mode 6 are also classified into Type 2 and be treated as the AC dominating wave impacts.N wave Table 4 lists the number of major wave impacts of different modes and the average number of wave cycles (zero-up crossings of the wave elevation) during one test case under all the six sea states. Among all the wave impacts of Type 1, wave impacts of Mode 1 occur most frequently, which is ascribed to the wave energy dissipation. For wave impacts of Type 2, there are several times more wave impacts of Mode 4 than those of Modes 5−6. Moreover, the total number of Type 2 is larger than that of Type 1. This phenomenon could be explained by the assumption that some relatively small waves which do not impact the FC severely become larger and steeper under the shoaling effect of the submerged pontoon. Most of these relatively small waves mainly exert impact loads on the AC, Classification of major wave impacts modes ( , , and denote the maximum peak pressure on the FC, the AC, and the DC within each wave impact event, respectively). while a few waves run up and impact the DC.

FC dominating wave impacts
5.1 Mode 1 Fig. 7 presents the snapshots of a typical wave impact event of Mode 1 under sea state W6. From the snapshot in Fig. 7a, it can be seen that a fully-developed breaker is approaching before impacting the FC. The wave front has curled over, or even collapsed at somewhere along the wave crest line where whitewater appears. The breaker of Mode 1 is much similar to the "well-developed plunging breaker" or "plunging breaker" defined by Oumeraci et al. (1993). When the breaker arrives (Fig. 7b), it impacts the FC violently with an explosive "crack" sound, accompanied by rising water jets and splashes. After the FC impact (Fig. 7c), the breaker crest plunges back into the fluid. Finally, the broken wave impacts the AC moderately (Fig. 7d). It also can be seen that the platform is at a negative pitch angle for impact on the FC, but switches to a slightly positive pitch angle for impact on the AC. , where Z is the distance of the center of a slam panel to the baseline of the platform hull and D is the survival draft. It should be noted that all the time series involved in this study have been shifted to make the largest pressure of a wave impact event located at time zero. From Fig. 8a, it can be seen that the impact pressures on the FC reach up to 1.5 MPa at the second highest slam panel. The measured pressures incorporate the impulsive component and the pulsating component. The impulsive component is the so-called impact pressure which rises rapidly and is usually exerted by breaking waves, while the pulsating component is slowly varying and quasistatic, and is usually caused by the sloshing motion of nonbreaking waves. As stated in Cuomo et al. (2011), the distinction between impulsive and pulsating pressures is the duration of the loading process. The pulsating pressures sustain over a large time interval, while impulsive pressures act over a very short duration, usually tens of milliseconds. The contribution of the pulsating component increases with the decrease of the vertical elevation. Pressures measured by higher slam panels are almost totally impulsive impact pressures. The measured peak pressures on the AC are much smaller than those on the FC, which is mainly due to the energy loss after the FC impact. The energy loss comes from both the impact on the FC and the turbulence dissipation during wave breaking. In addition, the highly aerated turbulent bore causes the irregularity of pressure curves.

Z/D
The impact pressure curves in Fig. 8 do not show obvious evidence of the effect of air entrapment. However, air in the forms of large pockets or small bubbles is indeed found in many events of Mode 1 and influences the impact pressures substantially. As an example, Fig. 9 presents another wave impact event of Mode 1 where air trap matters a lot. Within this event, the impact pressure is first detected at the highest slam panel ( =1.75) and propagates downwards the fore column, indicating the curling of the wave tongue and containing of an air pocket. The propagation speed of the pressure wave is about 120 m/s (model scale) and is reasonable due to the air entrapment, compared with the sound speed of 1500 m/s in water (Chan and Melville, 1988;Bullock et al., 2007). Another common feature of such high-aeration impact is the damped oscillation in the pressure curves, which are caused by the alternate compression and expansion of the trapped air pocket. The oscillations are found in all slam panels on the FC and are in phase with H s T p Fig. 7. Event of Mode 1 (W6, =20.0 m, =15.9 s): snapshots (a) before impacting the FC; (b) impacting the FC; (c) before impacting the AC; (d) impacting the AC.  each other, indicating that the air is trapped over a large vertical extent. The oscillation frequency in this wave impact event is about 25 Hz, which is consistent with the experiments of similar structures in previous studies (Kisacik et al., 2012;Vestbøstad et al., 2017). Furthermore, negative pressures could be generated during the expanding process of the air pocket.

ω ω
The relative wave elevations around the wave impacts on the FC and the AC are shown in Fig. 10, in the forms of the time series (the magenta curve) and the wavelet transform-based time-frequency analysis (the contour plot). The wavelet transform provides a way to resolve the energy structure subtly in both time and frequency domain. The contour value at time t and frequency represents the wave energy distributed at frequency and time instant t. In Fig. 10, the vertical white dashed lines depict the time instant of the largest wave impact pressure, and the troughs of relative wave elevations are truncated due to the water exit of the pontoon. It can be seen that the relative wave crest height at the FC is larger and corresponding rise time is much smaller than those at the aft column, which reveals the energy loss of the wave cycle. From the time-frequency analysis, it can be seen that the energy magnitude is larger and the frequency band is wider at the occurrence of FC impact than those at the AC impact, indicating the wave energy loss from the FC to the AC. In addition, the larger energy at high frequencies at the FC impact means that, for wave impact Mode 1, the interaction between waves and the FC exhibits stronger nonlinearity.

Mode 2 and Mode 3
Wave impact Modes 2 and 3 resemble each other, except that more appreciable wave impact loads are measured at the DC in Mode 3 than those in Mode 2. Examples of Mode 2 and Mode 3 are presented in Figs. 11−13 and Figs. 14−16, respectively. From Fig. 11a and Fig. 14a, it can be seen that the incident wave is near breaking but has not developed as fully as in Mode 1. The wave has not developed a curling wave tongue that very little air is trapped between the wave and the front column surface. The breaker of Mode 2 and Mode 3 is more like the "upward deflected beaker" (Oumeraci et al., 1993) or the "slightly breaking waves" (Kisacik et al., 2012). The water level rises quickly as the wave crest approaches the front column, and a layer of vertical water jet emerges and rises to impact the helicopter deck right after the impact on the fore column ( Fig. 11b and Fig. 14b). After impacting the FC, the wave does not collapse over the pontoon between the FC and the AC, and the wave height is maintained and even get amplified due to shoaling effect of the pontoon (Fig. 11c and Fig. 14c). When the AC impact H s T p Fig. 11. Event of Mode 2 (W6, =20.0 m, =15.9 s): snapshots (a) before impacting the FC; (b) impacting the FC; (c) before impacting the AC; (d) impacting the AC.  (Fig. 11d and Fig. 14d), the impact water mass is rather vast.
The wave impact pressures are shown in Fig. 12 and Fig. 15. Compared with Mode 1, wave impact pressures of Mode 2 and Mode 3 on the FC are relatively smaller and have larger contributions from the pulsating component which lengthens the duration time of wave loads. The pressure curves also do not exhibit obvious air effects as in Mode 1. It is in accordance with the description of the flow field. The largest pressure in the event of Mode 2 is measured at the second highest slam panel on the FC, while it is measured at the highest slam panel in the event of Mode 3. For events of Mode 2 and Mode 3, wave energy is lost during the impact process on the FC, but extensive energy loss caused by the breaking dissipation is avoided because the wave does not break violently. Therefore, the AC also suffers from severe wave impact loads which reach up to 400 kPa in the two presented examples. In addition, large pressures are also measured at the DC in the event of Mode 3. The serious impact point on the AC in Mode 3 is higher than that in Mode 2. Fig. 13 and Fig. 16 illustrate the corresponding relative wave elevations for the presented events of Mode 2 and Mode 3. The measured time series support the observations of the flow field in respect of the breaker type and energy loss. The difference in the wave crest height and the rise time of relative wave elevations between the FC and the AC minishes, compared with Mode 1. The energy distribution at high frequencies is transferred well from upstream to downstream, strengthening the interaction between waves and the AC.

Aft-column dominating wave impacts
When waves approaching the FC are relatively small, they do not impact the FC severely and less energy is dissipated. But after further evolution, they could lead to the AC dominating wave impacts. As demonstrated in Section 4, there are numerous severe wave impacts which occur on the AC or the DC.

Mode 4
Among the AC dominating wave impacts, those of Mode 4 occur most frequently. An exemplary event of Mode 4 is shown in Figs. 17−19.
From Fig. 17a, it can be seen that the incident wave at the FC is small and totally non-breaking. The wave sloshes past the FC moderately (Fig. 17b). Then, the wave gets larger and steeper due to shoaling effect of the submerged pontoon (Fig. 17c). Finally, when the wave arrives at the AC, it has been near the breaking point and impacts the AC severely (Fig. 17d). Water jets follow the aft-column impact and hit the deck bottom vertically. The water jets are thin and mainly activate impact pressures on the slam panel which is closest to the AC. Fig. 18 represents the measured pressures of the event in Fig. 17. The pressures measured on the FC are only composed of the pulsating components and vary in time in ac-    Fig. 19. It is obvious that the wave elevations at the AC are larger than those at the FC. When the wave propagates from the FC to the AC, the wave gains more energy from the shoaling effect of the pontoon, especially at high frequencies.

Mode 5 and Mode 6
If the incident wave height of Mode 4 increases, more areas on the deck bottom will be exposed to severe wave impacts and the wave impacts change from Mode 4 to Mode 5 or Mode 6. The main difference between Mode 5 and Mode 6 is on which structural component the largest impact pressure is measured. Figs. 20−22 and Figs. 23−25 present typical events of Mode 5 and Mode 6, respectively.
From the snapshots before the wave impacts on the fore column ( Fig. 20a and Fig. 23a), it can be seen that the incident wave is not breaking, like that in Mode 4. Therefore, no severe wave impacts occur on the fore column. When the wave moves past the FC (Fig. 20b and Fig. 23b), water jets are produced due to the retardation of the column. Unlike those in Mode 2 and Mode 3, the water jets from FC impacts in Mode 4 and Mode 5 do not rise vertically, and their    velocities are small so that they drop into the wave basin quickly. The shape of the wave remains intact during the evolution between two columns ( Fig. 20c and Fig. 23c). The wave is very destructive when arriving at the AC, and impacts the corner between the column and the DC violently ( Fig. 20d and Fig. 23d). As shown in Fig. 21 and Fig. 24, wave impact pressures of Mode 5 and Mode 6 exhibit different characteristics from those of Mode 4. The impact pressures on the FC are asymmetric, especially in the event of Mode 6. The rise time is shorter than the fall time, which indicates the large steepness of the incident wave. The largest wave impact pressure is measured at the highest slam panel on the AC (Mode 5) or at the DC slam panel closest to the AC (Mode 6). The corresponding water jets in Mode 5 and Mode 6 are thicker than those in Mode 4. Therefore, more slam panels on the lower deck are activated. Fig. 22 and Fig. 25 show the corresponding relative wave elevations for the presented events of Mode 5 and Mode 6. As in Mode 4, steeper waves are found at the AC, instead of the FC. The energy loss due to the interaction between the wave and the FC is negligible, while the energy transfer to high frequencies is also detected in the relative wave elevations at the AC.

Comparative studies among different wave impact modes
From Table 4, it can be seen that major wave impact events of all the six modes occurred at sea state W6 during model tests, while some wave impact modes did not occur at other sea states. Therefore, without loss of generality, more discussions are given to the comparative studies among different modes based on wave impacts at sea state W6, including motions and waves at the occurrence of wave impacts and the spatial distribution of the wave impact pressures. Fig. 26 presents the heave and pitch motions of the semi-submersible with the time instants of wave impacts shifted to time zero.

Motions and waves
The pitch motions are the same at different parts of the semi-submersible, while heave motions are different at different positions. Therefore, the heave motions for the FC dominating wave impacts and the AC dominating wave impacts refer in particular to the local vertical motions of the FC and the AC, respectively, which can be calculated as follows: x c y c ξ(t) α(t) β (t) where ( , ) are the coordinates of the centers of the FC and AC; , , and are the measured heave, roll, and pitch motions, respectively. It can be seen that there are phase differences between motions of two wave impact types. In general, the motions under the AC dominating wave impacts precede those under the FC dominating wave impacts, whether it is for the heave motions or the pitch motions. Therefore, at the occurrence of the AC dominating   wave impacts, the semi-submersible rises up higher and the water above the submerged pontoon is shallower. Especially, the semi-submersible is almost at the peak negative pitch angle when wave impacts on the FC occur, while at a larger pitch angle when wave impacts on the AC occur. The same phenomenon is also revealed by the snapshots of different wave impact modes (Sections 5 and 6). The special motion configuration of the AC dominating wave impacts results in gradually decreasing water depth when the incident wave propagates from the FC to the AC, which explains why more wave impacts occur on the AC. Fig. 27 compares the relative wave elevations for the two types of wave impacts, where the upwell height and upwell velocity denote the crest height and rise velocity of the relative wave elevations, respectively. With respect to relative wave elevations, a clear boundary can be seen between the FC dominating wave impacts and the AC dominating wave impacts, with a few exceptions. Under the FC dominating wave impacts, the relative wave elevations at the FC exhibit higher crests and larger rise velocities than those at the AC. However, without extensive energy loss at the FC, waves become more and more energetic from the wavestructure interactions for the AC dominating wave impacts. Consequently, the difference between relative wave crest heights at the FC and those at the AC is mitigated, with the ratios between them scattering around 1. Furthermore, the rise velocities of the relative wave elevations at the AC even surpass those at the FC.    GUO Ying-hao et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 161-175 Based on the above discussions (Sections 5 to 7.1), the schematic diagrams of all the six wave impact modes are built and drawn in Fig. 28 where red solid lines and blue dashed lines are used to illustrate wave impacts on the FC and AC, respectively. The wave profile, degree of wave breaking and motions of the semi-submersible vary from one mode to another.

Spatial distribution
The spatial distributions of wave impact loads can be considered in the statistical sense by packing local peak pressures from numerous wave impact events as a whole. However, the largest impact pressure at each individual slam panel can be obtained from different events. Therefore, this kind of spatial distribution is useful but can be conservative. Moreover, the spatial distribution of a particular event is not provided, which is meaningful to build the prediction model of wave impact loads. Based on the horizontal impact loads on the column of a large-volume TLP, which are measured from extensive wave impact tests, Johannessen et al. (2017) proposed an empirical model (called DNV model in this study) of the dimensionless peak loads distribution for each individual wave impact events and applied it to develop the guidance provided in DNV•GL (2016). Within the empirical model, the local peak pres- sures of a particular event are normalized by their largest value and the vertical locations above the still water level ( ) of slam panels are normalized using the relative wave crest elevation and the absolute wave crest elevation . The normalizations can be written as follows: With the normalized parameters, the empirical model is given by the Beta distribution, of which the mean and the standard deviation are shown in Eqs. (5) and (6) where the corresponding coefficients are =0.2, =0.45, =0.25, =0.25, and = −0.08. The dimensionless local peak pressure profiles for individual FC and AC dominating wave impacts are presented in Fig. 29 and Fig. 30, respectively, and the benchmarking against the DNV model is performed. It can be seen that the largest wave impact load of each event occurs at slam pan-  (z − c rel )/c d < 0 μ ± σ els which are slightly lower than the relative crest elevations ( ), whether it is on the FC or the AC. For the FC dominating wave impacts, the local peak pressure profiles on the FC cover a large extent, and decay up and down the column from the worst-hit slam panel, while those on the AC are more irregular and show the influence of the DC (Figs. 29e and 29f). For the AC dominating wave impacts, the local peak pressures of each event on the FC conform to the variation of the pulsating component, increasing with the decrease of the vertical elevation, and the largest pressure is measured at the lowest slam panel, while the case for the AC is relatively more complex. In Mode 4, the local peak pressures have triangular distributions on the AC, similar with those on the FC in Modes 1−3. On the contrary, under the influence of the DC, the local peak pressures on the AC increase rapidly with the increase of the vertical elevation in Mode 5 and Mode 6. Compared with the DNV model, most wave impact data drop into the band of , but some differences are found. First, there are one-side distributions which cannot be fitted by the two-side DNV model, such as those on the FC within AC dominating impacts and those effected by the DC. Second, the worst-hit points determined by the experiment are somewhat lower than those predicted by the DNV model. The difference may be attributed to the fact that the DNV model is focused on the horizontal wave impact loads on the waveward side and is derived from the experiments of TLP. To be applied to different structural components of semi-submersibles more accurately, the DNV model needs to be adjusted accordingly.

Conclusions
To study wave impact loads on a semi-submersible, a series of scale model tests were conducted at various irregu-lar sea states. Quantitative criteria are proposed to classify major wave impacts on the semi-submersible into six modes and two types. Deterministic analyses of typical wave impact events for each mode and comparative studies among them are performed. The main characteristics of the scenarios represented by each impact mode are listed in Table 5.
The FC dominating wave impacts (i.e., Type 1) exert the most intense load on the FC and feature well-developed breaking waves (Mode 1) or slightly breaking waves (Mode 2 and Mode 3) at the FC. For impact Mode 1, the impact pressures on the FC are influenced by entrapped air, while those on the AC and the DC are relatively moderate due to serious energy loss after the FC impact. For impact Mode 2 and Mode 3, pressures on the FC have larger contributions from the pulsating component than those in Mode 1, and large impact pressures are also detected on the AC because the energy loss is relatively small.
The AC dominating wave impacts (i.e., Type 2) exert the most intense load on the AC or the DC, and occur more frequently than the FC dominating wave impacts. As been validated by comparative studies of motions and relative wave elevations, the shoaling effect of the submerged pontoon and different motion configurations of the platform between the FC impact and AC impact could account for the different occurrence rate. For impact Mode 4, even though the incident wave at the FC is small and totally non-breaking, it could still get larger and steeper, and finally impact the AC. For impact Mode 5 and Mode 6, waves at the FC are large but not breaking, and they can remain intact until impacting the AC and the DC severely.
From the perspective of individual events, the local peak pressure profiles on the FC in the FC dominating impacts are similar to the DNV model despite downward shift of serious impact points, while those in the AC dominating im-  pacts are clearly different and conform to one-side distributions. Therefore, the "impact mode" proposed in the study provides a novel approach to consider wave impact loads on different structural components of semi-submersibles and other floating offshore units. Accordingly, modifications can be made to the existing design guidelines after examining more model tests and different structures. It warrants further studies to propose the empirical model for more accurately predicting wave impact loads of semi-submersibles. − Well-developed plunging breakers − Waves collapse and plunge back into the fluid after the FC impact − FC impact: explosive impact with violent water jets − AC impact: weak impact exerted by highly aerated turbulent bore − Large energy loss from the FC impact and wave breaking dissipation Modes 2 and 3 − Near/slightly breaking waves − Waves do not collapse and wave heights are maintained between columns − FC impact: severe impact with quickly rising water level and violent waver jets − AC impact: moderate impact with rather vast water mass − Wave energy is lost from the FC impact, but that from breaking dissipation is negligible Mode 4 − Small and non-breaking waves − Waves are larger and steeper due to the shoaling effect − FC impact: waves slosh past the fore column moderately without impact − AC impact: severe impact with thin water jets hitting the DC − Waves gain more energy from the shoaling effect Modes 5 and 6 − Steep but non-breaking waves − Wave shapes remain intact between columns − FC impact: no severe wave impacts − AC impact: violent impact around the corner between the aft column and the DC − Waves gain more energy from the shoaling effect and the energy loss from the FC impact is negligible