Application to Military Vessels
Accidents caused by the failure modes considered in the second generation intact stability criteria may be fatal (see the report of the accident occurred to the Chicago Express off Hong-Kong in 2008, ) or may cause significant financial losses (APL China in October 1998 , Maersk Carolina in January 2003 ), but they are fortunately rare. Since the number of naval ships in service is significantly smaller than the number of merchant vessels and their time at sea is smaller too, it is not surprising that none of the serious accidents causing the development of the new criteria has occurred on a naval ship. However it cannot be excluded in principle that naval ships be vulnerable to such stability failures. Although the new regulations are not intended for naval ships, it seems interesting to assess the outcome of their applications. In fact the hull geometry and the high speed of naval ship typology are in principle a remarkable combination worthy of attention. Hence, a set of three military vessels, chosen for their variety of typology and size, has been included in this study: a 12,000-ton helicopter carrier, a 9,000-ton destroyer and a 1,500-ton Offshore Patrol Vessel.
All figures showing the KGmax curves in paragraphs 3.2.1 (pure loss of stability) and 3.3.1 (parametric roll) have the same graphic design:
• The KGmax curve associated with the current IMO regulation  is drawn as a grey dashed line.
• The KGmax curve associated with the French military regulation (IG 6018A DGA ), computed only for naval vessels, is drawn as a grey dotted line.
• The light blue plain line indicates the height of the transverse metacenter above the baseline (KMT) which allows to determine the minimum GM required by all criteria.
• The vertical grey lines indicate the full-load displacement and, when it is known, the light displacement. Otherwise, a black dot corresponds to the standard loading condition of the ship.
• The KGmax curve associated with the first method of level one (parallel waterplane) criteria of both pure loss of stability and parametric roll failure modes is drawn in blue with square markers.
• The KGmax curve associated with the second method of level-one criteria (ship balanced in trim and sinkage on a wave with the same length) is drawn in red with round markers.
• The KGmax curve associated with the level-two criteria (first check only for parametric roll) is drawn as a green solid line with diamond markers.
• The KGmax curve associated with the second check of level-two parametric roll criterion is drawn as a green dashed line with diamond markers.
Since neither the 319 m container vessel nor the tanker fulfill the condition on Froude number (Fn>0.24), their KGmax curves associated to level 1 and 2 criteria are not presented. Both vessels are assessed as non-vulnerable to the pure loss of stability by the new regulation. Parts of the content of this section have already been presented in  (influence of the watertight deck height),  (results on naval vessels) and [2, 6] (results on some civilian vessels and one naval vessel).
Focus on Second Check
The content of this section has already been presented in .
For any ship at any draft, the KGmax associated with the second check of level-two criterion (C2) is defined as the highest value of KG for which the value of C2 is lower than RPR0=0.06. Thus, it is interesting to check the curve of C2 versus KG. These curves are shown in Figure 54 to Figure 58 for five of the vessels studied in this thesis. For the non-vulnerable ships (Ro-Ro, tanker, DTMB-5415), they are computed at full-load draft for an interval containing KGmax with a step of 2 centimeters. For both container vessels, assessed as vulnerable, they are computed for a larger interval of KG with a step of 1 centimeter, at a draft equal to 10 m (C11 container vessel, Figure 54) and 12 m (319 m container vessel, Figure 55). Figure 57 shows the curve of the tanker. C2 is equal to 0 for all values of KG lower than KGmax (13.70 m) and to 1 for all higher values. This shows that the parametric roll never occurs on this ship. The value of C2 is forced to 1 when the average value of GM in waves becomes negative (see Section 1.2.4 page 33).
Figure 58 shows the curve for the DTMB-5415. We observe a small interval of KG (centered approximatively at 8.70 m) in which C2 is non-zero. This shows that the parametric roll occurs for some lightly-weighted waves. For higher values of KG, C2 tends to zero and then rapidly increases to 1. Parametric roll occurs in these conditions of KG but the average value of GM is near zero: the ship becomes statically unstable on waves.
Figure 56 shows the curve C2 versus KG for the Ro-Ro vessel. We observe that the increasing part of the curve is longer than those of the tanker and the naval ship. We also observe that two values of KG larger than KGmax (12.57 m) give values of C2 lower than RPR0 (KG = 12.60 and 12.62 m, marked with * in Figure 56).
Figure 54 and Figure 55 show the same curves respectively for the C11 and the 319 m container vessels. On both, we observe many peaks and relatively large intervals of KG larger than KGmax for which the value of C2 is lower than RPR0, thus for which the associated criterion is fulfilled. These intervals are colored in grey in the corresponding figures. This non-monotonically-increasing configuration of the C2 curve makes the starting value of KG (KGstart in this thesis, 15 m for both container vessels) very important in the process of finding KGmax. The value of the increment used in this process is also very important. Both parameters must be chosen to avoid overlooking a small zone of KG for which C2 is larger than RPR0.
We observe that the more the ship is vulnerable to parametric roll, the more the curve C2 versus KG has peaks and the longer the interval where C2 increase from 0 to 1 is.
Influence of Computation Parameters
During the 12th International Conference on the Stability of Ships and Ocean Vehicles, held in Glasgow (UK) in June 2015, Peters et al.  formulated some recommendations to solve the parametric roll differential equation (23) and calculate the associated maximum roll angle required in the second check of level-two criterion (coefficient C2, seen as a separate criterion here). Their proposals have been included in the explanatory notes of the new regulation (SDC 3/WP.5, Annex 4, Appendix 3 ).
Among other recommendations, Peters et al. propose to solve the differential equation with a simulation time equal to 15 natural roll periods of the ship and an initial roll angle equal to 5 degrees. They also recommended to consider a non-linear GZ.
In this section, we propose to study the influence of each of these proposals on the KGmax curves associated with the second check of level-two criterion for four selected ships: both container vessels (assessed as vulnerable to parametric roll by the new criteria) the Ro-Ro vessel (assessed as slightly vulnerable, although neither the test in the towing tank nor direct assessment computation have proven this yet) and the tanker (clearly non-vulnerable).
Note: in this section, the service speed of the C11 container vessel is set to 20 knots. The content of this section has been presented at the 15th International Ship Stability Workshop held in Stockholm (Sweden) in June 2016 .
Since parametric roll is a resonance phenomenon due to the repetition of the encounter of waves, attaining the steady state roll amplitude is essential to determine the vulnerability to this failure mode. Thus, the duration of the simulation is important. The KGmax curves associated with the second check of level-two criterion are computed for 6 different si ulatio du atio s, gi e as a u e of the ship’s atu al oll pe iod. The follo i g durations are tested: 3, 4, 6, 10, 15 and 20 natural roll periods. Peters et al.  and SDC 3/WP.5  recommend a simulation duration equal to 15 roll periods.
Figure 63 and Figure 64 show the results for both container ships. We observe that the KGmax significantly varies with the duration of the simulation, but the curves associated with 10, 15 and 20 roll periods are fully coincident for both ships. This proves that the steady state roll amplitude has been attained between 6 and 10 roll periods. Figure 65 shows the results for the Ro-Ro vessel. We observe that all curves are close together. The KGmax is slightly affected by the simulation duration. As above, the curves associated with 10, 15 and 20 periods are fully coincident.
Figure 66 shows the results for the tanker. We observe that all curves are coincident and correspond to zero-GM. This proves again that the tanker is not vulnerable to parametric roll: parametric roll never occurs, regardless of the wave and speed (the C2 coefficient is set to 1 when the ship becomes statically unstable in waves, see Section 1.2.4 page 33). The simulation duration has no effect on KGmax curves.
Table of contents :
Chapter 1. Second Generation Intact Stability Criteria
1.1. Pure Loss of Stability Failure Mode
1.1.1. Physical Background and General Information
1.1.2. Level One
1.1.3. Level Two
1.2. Parametric Roll Failure Mode
1.2.1. Historical and Physical Background
1.2.2. Level One
1.2.3. Level Two
1.2.4. Maximum Roll Angle and KGmax computation
Chapter 2. Hydrostatic Computation
2.1. Generation of Volume Mesh
2.1.1. First Step
2.1.2. Second Step
2.2. Cutting a Volume Mesh by a Plane
2.2.1. Decomposition of Elementary Volumes
2.2.2. Cutting a Tetrahedron by a Plane
2.3. Finding the Balance Position
2.3.1. Definition of the Balance Position
2.3.2. Inclined-Ship Planes
2.3.3. Hydrostatic Computation in Calm Water
2.3.4. Hydrostatic Computation in Waves
2.3.6. Calculation of Transverse Metacentric Height
Chapter 3. Results
3.1. Preliminary Information
3.1.1. General Information
3.1.2. Application to Military Vessels
3.1.3. Graphic Design
3.2. Pure Loss of Stability
3.2.1. General Results
3.2.2. Influence of the Watertight Deck Height
3.2.3. Influence of Speed
3.3. Parametric Roll
3.3.1. General Results
3.3.2. Focus on Second Check
3.3.3. Influence of Speed
3.3.4. Influence of Computation Parameters
3.3.5. Comparison with 6-Degrees-of-Freedom Simulation
Chapter 4. Energy Analysis of Parametric Roll
4.2. Parametric Roll in Resonance Condition
4.2.1. Equation of Parametric Roll
4.2.3. Distribution of Energy
4.2.4. Direct Calculation of the Maximum Roll Angle in Resonance Condition
4.3. Parametric Roll in Other Conditions
4.3.1. Non-Synchronized Parametric Roll
4.3.2. Lock-in Field
4.3.3. Second and Third Modes of Parametric Roll
4.3.4. Shift Angle in the Lock-in Field
4.3.5. Width of the Lock-in Field
4.4. Method Providing Steady-State Parametric Roll Amplitude at Any Speed
4.4.1. Energy Method
4.4.2. Improvement of the Energy Method
4.4.3. Application to Second Generation Intact Stability Criteria
Annex 1. Calcoque Software
Annex 2. Presentation of Ships
Annex 3. Mathematical Proofs
Annex 4. Ship-Fixed Coordinate System