A method for predicting three-dimensional morphology of seabed grooves considering the dynamic bottoming effect of mooring lines

Abstract

The present invention discloses a method for predicting the three-dimensional morphology of a seabed groove taking into account the dynamic bottoming effect of a mooring line. The method comprises the following steps: step 1) simulating the forces of the mooring line in the soil middle section and the water middle section to obtain the soil range swept by the mooring line in one cycle, and obtaining a stable mooring line morphology and stress state after multiple cycles; step 2) based on the stable mooring line morphology, calculating the maximum point and the deepest point of the lateral displacement of the mooring line, and returning these two points to step 1) for recalculation, and obtaining the final stable groove profile through repeated iterations. The method of the present invention takes into account the lateral movement of the mooring line, and is used for describing the groove profile on site, and can also be used for the detailed description of the three-dimensional groove profile of the project, and can provide an important basis for the prevention and control of groove disasters on site.

Description

Technical Field

[0001] The invention belongs to the field of seabed disaster prediction, and in particular relates to a method for predicting the three-dimensional morphology of a seabed groove taking into account the dynamic bottoming effect of an anchor line. Background Art

[0002] Suction anchors are usually buried in the seabed soil at a certain depth, and anchor eyes are set below the seabed surface to increase the bearing capacity, so part of the anchor chain will be buried in the soil. In 2014, during a seabed survey in the Gulf of Guinea in West Africa, it was found that the movement of an FPSO caused the mooring line to repeatedly cut into the soil, resulting in a huge groove in the seabed in front of the suction anchor (Bhattacharjee et al., 2014). Considering the risk of suction anchor failure, the project reinstalled the mooring foundation and mooring line. The existence of seabed grooves brings the risk of suction anchor failure, which has attracted widespread attention from the engineering and academic circles.

[0003] Existing studies have shown that the presence of grooves has a serious adverse effect on the bearing capacity of suction anchors, so full attention must be paid. To prevent and control this engineering disaster, we should first understand its formation mechanism and evolution process, and then be able to predict and evaluate the groove profile in actual engineering. At present, the formation of seabed grooves is believed to be related to the interaction between the mooring line and the seabed (Sassi et al., 2017; Versteele et al., 2017), but because previous studies did not consider the dynamic interaction between the mooring line and the seabed, and did not analyze the three-dimensional morphology of the seabed groove, the calculated results have a large deviation from the actual engineering groove profile. Summary of the invention

[0004] In view of the defects in the above-mentioned background technology, the present invention provides a method for predicting the three-dimensional morphology of seabed grooves taking into account the dynamic bottoming effect of the mooring line. The method takes into account the lateral movement of the mooring line and can be used for describing the groove contour on site, and can also be used for the detailed description of the three-dimensional groove contour of the project. The method of the present invention can provide an important basis for the prevention and control of groove disasters on site.

[0005] The present invention adopts the following technical solutions:

[0006] A method for predicting the three-dimensional morphology of a seabed groove considering the dynamic bottoming effect of a mooring line, characterized in that it comprises the following steps:

[0007] 1): a. Calculate the initial mooring line shape and tension according to the mooring radius, water depth, anchor hole depth, mooring line parameters and soil properties; pre-tension the mooring line to achieve the initial required mooring line shape and tension;

[0008] b. Set the circular motion path of the fairlead hole to simulate the motion of the floating structure, so as to obtain the stress state of the mooring line in the soil and water sections and the soil area disturbed by the mooring line;

[0009] c. After several cycles of step b, the disturbed soil area caused by the mooring line becomes stable, thereby obtaining a stable mooring line shape and stress state;

[0010] 2): a. According to the stable mooring line shape obtained in step 1), calculate the deepest point of the groove and the point with the maximum lateral displacement;

[0011] b. Determine whether to end the prediction based on the calculation results in step a: If a stable groove profile is obtained after several cycles, directly output the calculation results of step a, where the groove profile is described by the deepest point of the groove and the point with the maximum lateral displacement; otherwise, return to step 1) to continue the calculation.

[0012] In the above technical solution, further, in the step 1), the mooring line shape and stress state are specifically:

[0013] Establish a rectangular coordinate system O XYZ and two local coordinate systems P xyz and P uvw ; In the rectangular coordinate system, the seabed plane is set as the OXY plane, and the origin O is located at the projection point of the anchor eye in the soil on the seabed surface; the Y axis is the intersection of the plane OYZ determined by the suction anchor axis and the anchor eye and the seabed surface OXY, the Z axis is perpendicular to the seabed surface OXY and points vertically upward, and the X axis is perpendicular to the OYZ plane to represent the lateral movement direction of the mooring line; the entire mooring line is divided to obtain each segment unit, and a local coordinate system P is established on each segment unit xyz , P xyz The origin is at point P, and the direction of its coordinate axis is the same as O XYZ Similarly, point P is located at the center point of the segment unit of the mooring line; uvw is a local coordinate system established on the mooring line segment unit. The coordinate system is divided into three components in the form of the mooring line: tangent, normal and binormal (u, v, w). The relationship between the two local coordinate systems can be expressed as:

[0014]

[0015] Where β = β(l) and θ = θ(l) are the elevation and azimuth of the mooring line, and l is the length of the mooring line unit;

[0016] Define u as the direction of the mooring line at point P, and the projection length of the mooring line unit in the x, y, and z directions can be expressed as:

[0017]

[0018] Under three-dimensional dynamic conditions, the governing equation of the force state of the mooring line unit is expressed as follows:

[0019] M(X,Y,Z,a)+C(X,Y,Z,v)+K(X,Y,Z)=F(X,Y,Z,v,t) (3)

[0020] Where M(X,Y,Z,a) is the inertial force, C(X,Y,Z,v) is the damping force, K(X,Y,Z) is the elastic force and F(X,Y,Z,v,t) is the external force, X, Y and Z represent the coordinates of the mooring line unit in the global coordinate system, a is the acceleration of the unit, v is the velocity of the unit, and t is the time;

[0021] External forces include fluid forces in three directions (F n ,F b ,F τ ), soil resistance in three directions (Q n ,Q b ,Q τ ) and the unit weight of the mooring line in water and seabed (W w ,W s );

[0022] Morrison equation is used to calculate the fluid force acting on the mooring line:

[0023]

[0024]

[0025]

[0026] Among them, ρ w is the density of water, d eff is the effective diameter of the anchor chain; dl is the unit length of the mooring section, V0 is the amount of water displaced by the mooring line, v n ,v b and v τ are the relative velocities of the mooring line and the seawater in the normal, subnormal and tangential directions, respectively, and C dn and C dτ are the normal and tangential resistance coefficients, respectively;

[0027] The soil resistance acting on the mooring line is decomposed into the normal component Q n , binormal component Q b and the axial component Q τ , expressed as:

[0028]

[0029]

[0030] Q τ =E τ d b fdl (5c)

[0031] Where, d b is the nominal diameter of the anchor chain, E n and E τ are the normal and axial equivalent width parameters respectively; q is the total normal resistance, which can be decomposed into normal and binormal components according to the direction, d w and d v are the displacements in the normal and binormal directions, respectively, and f is the axial resistance of the mooring line;

[0032] q and f are calculated by the following expressions:

[0033] q=N c s u (6a)

[0034] f=αs u (6b)

[0035] In the formula, s u is the undrained shear strength of the soil; N c is a function of soil depth z, expressed as

[0036]

[0037] Under dynamic conditions, the soil strain rate effect should be considered, that is, the shear strength of the soil increases with the increase of the shear strain rate. The influence of the strain rate is expressed by the parameter R f To characterize, R f is the relationship between the soil strength at a certain shear strain rate and the soil strength at a reference strain rate. The parameters R in the normal, binormal and axial directions are f It is expressed as:

[0038]

[0039]

[0040]

[0041] Where β is the strain rate parameter, which is between 0.034 and 0.140; the reference shear strain rate and is the shear strain rate in three directions, defined as:

[0042]

[0043]

[0044]

[0045] Where d is the characteristic diameter, d = d en =E n d s ; The real soil resistance considering the shear strain rate is:

[0046] S u =S ur R f (10)

[0047] Among them, S ur is the shear strength of remolded soil.

[0048] Furthermore, in step 2), the groove profile can be determined by the deepest point of the groove (0, Y dp ,Z dp ) and the maximum lateral displacement point (X tp ,Y tp ,0) and the description of the sidewall inclination angle α of the field trench statistics (α is a constant value, and its value is related to the seabed soil, such as α is 62.9° for clay seabed);

[0049] The deepest point of the groove (0,Y dp ,Z dp ) is: the first fixed point of the mooring line from top to bottom in a cycle, from which the position of the mooring line remains unchanged; for the position of the mooring line in a cycle, the deepest point (0, Y dp ,Z dp ) meets the following criteria:

[0050]

[0051] Among them, (X i ,Y i ,Z i ) is the position coordinate of the mooring line segment unit in a cycle;

[0052] Maximum lateral displacement point (X tp ,Y tp ,0) is determined as follows: Under dynamic conditions, a bottoming area is formed around the mooring line, and the point corresponding to the maximum X coordinate in the bottoming area is defined as the point with the maximum lateral displacement and recorded as (X tp ,Y tp ,0).

[0053] The present invention has the following advantages:

[0054] The present invention aims to solve the problem that in deepwater mooring systems, grooves are formed on the front side of the suction anchor due to the repeated cutting of the seabed by the cyclic movement of the mooring line, thereby reducing the bearing capacity of the anchor body. For the first time, a groove three-dimensional contour prediction method considering the dynamic interaction between the mooring line and the seabed is proposed. The three-dimensional morphology of the groove on site can be predicted, providing an important reference for the prevention and control of seabed groove disasters. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 Global and local coordinate systems in three-dimensional space

[0056] Figure 2 Force analysis of the mooring line unit in three-dimensional space, where (a) is the force of the water section and (b) is the force of the soil section;

[0057] Figure 3 The mooring line morphology at different times within a cycle, where (a) is the mooring line morphology and (b) is a local enlarged view;

[0058] Figure 4 Projection of the mooring line pattern in a cycle in the OZY plane

[0059] Figure 5 The projection of the mooring line shape in the OXY plane and the trajectory of the entry point, where (a) is the projection of the mooring line shape in the OXY plane, and (b) is the trajectory of the entry point;

[0060] Figure 6 A program for predicting the three-dimensional morphology of grooves. DETAILED DESCRIPTION

[0061] The technical solution of the present invention is further described below in conjunction with the embodiments and drawings:

[0062] The method of the present invention is implemented based on a groove three-dimensional morphology prediction model, which includes a three-dimensional mooring line dynamics module and a three-dimensional groove calculation module, and introduces how to use these two modules to calculate the groove three-dimensional morphology.

[0063] (1) 3D mooring line dynamics module

[0064] In the mooring line dynamics module, the anchor eye is fixed at a certain depth on the seabed. The mooring line is simulated using line elements, and a displacement load is applied to the upper end point (fairlead hole).

[0065] (1) Coordinate system

[0066] The present invention adopts a rectangular coordinate system O XYZ and two local coordinate systems P xyz and P uvw ,like Figure 1As shown. In the rectangular coordinate system, the seabed plane is set as the OXY plane, and the origin O is located at the projection point of the anchor eye in the soil on the seabed surface. The Y axis is the intersection of the plane OYZ determined by the suction anchor axis and the anchor eye and the seabed surface OXY. The Z axis is perpendicular to the seabed surface OXY and points vertically upward. The X axis is perpendicular to the OYZ plane and represents the lateral movement direction of the mooring line. The entire mooring line is divided into various segmented units, and a local coordinate system P is established on each segmented unit. xyz , P xyz The origin is at point P, and the direction of its coordinate axis is the same as O XYZ Similarly, point P is located at the center of the segment unit of the anchor line. In order to facilitate the description of the force direction, P uvw A local coordinate system based on the mooring line segment unit is defined. The coordinate system is divided into three components in the shape of the mooring line: tangent, normal and binormal (u, v, w). The relationship between the two local coordinate systems can be expressed as:

[0067]

[0068] Where β = β(l) and θ = θ(l) are the elevation and azimuth of the mooring line, and l refers to the unit length of the mooring line.

[0069] Define u as the direction of the mooring line at point P, and the projection length of the mooring line unit in the x, y, and z directions can be expressed as:

[0070]

[0071] (2) Force analysis

[0072] Under three-dimensional dynamic conditions, the stress state of the mooring line unit is as follows: Figure 2 , the control equation includes inertia force M(X,Y,Z,a), damping force C(X,Y,Z,v), elastic force K(X,Y,Z) and external force F(X,Y,Z,v,t), which can be expressed as:

[0073] M(X,Y,Z,a)+C(X,Y,Z,v)+K(X,Y,Z)=F(X,Y,Z,v,t) (3)

[0074] Where X, Y, and Z represent the coordinates of the mooring line unit in the global coordinate system, a is the acceleration of the unit, v is the velocity of the unit, and t is the time.

[0075] In the present invention, the external force includes fluid forces in three directions (F n ,F b ,F τ ), soil resistance in three directions (Q n ,Q b ,Q τ ) and the unit weight of the mooring line in water and seabed (Ww ,W s The above forces acting on the mooring line in water and soil are as follows: Figure 2 In the previous quasi-static analysis, the effects of inertia force, damping force and fluid force were ignored.

[0076] Normal, binormal and tangential fluid forces (F n ,F b ,F τ ) is calculated by the Morison equation. The Morrison equation is used to calculate the fluid forces on slender structures and can be used to calculate the fluid forces acting on the mooring line.

[0077]

[0078]

[0079]

[0080] Among them, ρ w is the density of water, d eff is the effective diameter of the anchor chain. eff =1.7d s (d s is the nominal diameter of the anchor chain). dl is the unit length of the mooring section, V0 is the amount of water displaced by the mooring line, v n ,v b and v τ are the relative velocities of the mooring line and the seawater in the normal, subnormal and tangential directions, respectively, and C dn and C dτ are normal and tangential resistance coefficients respectively. dn =1.2 and C dt / C dn =0.02. Additional mass coefficient C m =1.

[0081] like Figure 2 As shown in the figure, the soil resistance acting on the mooring line can be decomposed into the normal component Q n , binormal component Q b and the axial component Q τ , which can be expressed as:

[0082]

[0083]

[0084] Q τ =E τ d b fdl(5c)

[0085] Where, db is the nominal diameter of the anchor chain, E n and E τ are the normal and axial equivalent width parameters respectively. n =2.5 and E τ =8) is determined based on the anchor chain tensioning test in clay. q is the total normal resistance, which can be decomposed into normal and binormal components according to the direction, d w and d v are the displacements in the normal and binormal directions, respectively, and f is the axial resistance of the mooring line. q and f can be calculated by the following expressions:

[0086] q=N c s u (6a)

[0087] f=αs u (6b)

[0088] In the formula, s u is the undrained shear strength of the soil. The clay reduction factor α=1. N c It is a function of soil depth z and can be expressed as:

[0089]

[0090] Under dynamic conditions, the soil strain rate effect should be considered, that is, the phenomenon that the shear strength of the soil increases with the increase of the shear strain rate. This effect depends on the soil density, overconsolidation ratio, permeability, etc. In the present invention, the effect of strain rate is expressed by the parameter R f To characterize. f is the relationship between the soil strength at a certain shear strain rate and the soil strength at a reference strain rate. R f It can be expressed as:

[0091]

[0092]

[0093]

[0094] Wherein, β is the strain rate parameter, β is between 0.034-0.140, and β=0.07 is used in the present invention. The reference shear strain rate is and is the shear strain rate in three directions and can be defined as:

[0095]

[0096]

[0097]

[0098] Where d is the characteristic diameter, and the present invention selects d=d en =E n d s Therefore, the actual soil resistance considering the shear strain rate is:

[0099] S u =S ur R f (10)

[0100] Among them, S ur is the shear strength of remolded soil.

[0101] (2) 3D groove calculation module

[0102] Under dynamic conditions, the mooring line repeatedly cuts the soil, and the seabed soil softens and fluidizes, and the seabed soil is eroded and carried away by the water flow. As the size of the trench increases, the sidewalls of the trench will collapse and the size of the trench will increase.

[0103] Since the motion of mooring lines in three-dimensional space is very complex, it is difficult to describe the positions of all mooring lines. Figure 3 The position of the mooring line at different times within a cycle is shown. The study found that the mooring line in the water has obvious displacement with the movement of the fairlead. Due to the influence of the dynamic effect of the mooring line, the shape of the mooring line is complex and diverse. Unlike the two-dimensional calculation results, the disturbed soil area is in a three-dimensional space state.

[0104] As described above, the trench profile can be described by two points: the deepest point (0, Y dp ,Z dp ) and the maximum lateral displacement point (X tp ,Y tp ,0) and the sidewall inclination angle α of the field trench statistics. The deepest point (0,Y dp ,Z dp ) indicates the first fixed point of the mooring line from top to bottom in a cycle, that is, starting from this point, the position of the mooring line always remains unchanged. Figure 4 The mooring line pattern in the OZY plane in a cycle is shown. For the mooring line position in a cycle, the deepest point (0, Y dp ,Z dp ) meets the following criteria:

[0105]

[0106] (X i ,Y i ,Z i) is the position coordinate of any mooring line segment unit in a cycle.

[0107] In order to determine the maximum lateral displacement point (X tp ,Y tp ,0), Figure 5 The trajectory of the entry point in the OXY plane in a cycle is given. Figure 5 As shown in the figure, the mooring line near the anchor eye is embedded in the seabed soil. Under dynamic conditions, a bottoming area is formed around the mooring line, and the maximum X coordinate in the bottoming area is defined as the maximum point and recorded as (X tp ,Y tp ,0), as the feature points describing the groove profile. Figure 5 The typical touchdown point trajectory in a cycle is shown, and the coordinates of the maximum lateral displacement point are X tp = -1.58 and Y tp =127.

[0108] The method proposed in this invention is to predict the final three-dimensional shape of the trench. Due to the complexity of the trenching process, the specific trench formation process such as soil erosion and wall collapse is not considered; instead, the deepest point (0, Y dp ,Z dp ) and the maximum lateral displacement point (X tp ,Y tp ,0), briefly describe the three-dimensional morphology of the groove.

[0109] By combining the above two modules (mooring line dynamics module and 3D groove calculation module), the groove 3D morphology prediction program is as follows Figure 6 As shown. In the mooring line dynamics module, the necessary parameters (mooring radius, water depth, anchor eye depth, mooring line parameters, soil properties) are entered. In the first stage, the mooring line is pre-tensioned to achieve the initial required mooring line shape and tension. Then, the cyclic motion path of the fairlead hole is set to simulate the movement of the floating structure. After several cycles, the disturbed soil area caused by the mooring line becomes stable, and the stable mooring line shape is used as the input of the three-dimensional trench calculation module. In this process, the deepest point and the maximum point are determined by the mooring line shape in the YZ and XY planes, respectively, and the deepest point (0, Y dp ,Z dp ) and the maximum lateral displacement point (X tp ,Y tp ,0) and the slope angle α to update the groove profile. Based on the new groove profile, the mooring line dynamics module is executed again, and the above procedure is repeated until a stable groove profile is obtained.

Claims

1. A method for predicting the three-dimensional morphology of seabed grooves considering the dynamic bottoming effect of mooring lines, characterized in that: The following steps are involved: 1): a. Calculate the initial mooring line shape and tension according to the mooring radius, water depth, anchor hole depth, mooring line parameters and soil properties; Pre-tension the mooring line to achieve the initial required mooring line shape and tension; b. Set the circular motion path of the fairlead hole to simulate the motion of the floating structure, so as to obtain the stress state of the mooring line in the soil and water sections and the soil area disturbed by the mooring line; c. After several cycles of step b, the disturbed soil area caused by the mooring line becomes stable, thereby obtaining a stable mooring line shape and stress state; 2): a. According to the stable mooring line shape obtained in step 1), the deepest point of the groove and the point with the maximum lateral displacement are calculated to describe the groove profile; b. Determine whether to end the prediction based on the calculation result in step a: The calculation result of step a is returned to step 1 for further calculation; if a stable groove profile is obtained after several cycles, the calculation result of step a is directly output; otherwise, the calculation is returned to step 1) for further calculation.

2. The method for predicting the three-dimensional morphology of seabed grooves considering the dynamic bottoming effect of the mooring line according to claim 1 is characterized in that: In the step 1), the mooring line shape and stress state are specifically as follows: Establish a rectangular coordinate system O XYZ and two local coordinate systems P xyz and P uvw ; In the rectangular coordinate system, the seabed plane is set as the OXY plane, and the origin O is located at the projection point of the anchor eye in the soil on the seabed surface; the Y axis is the intersection of the plane OYZ determined by the suction anchor axis and the anchor eye and the seabed surface OXY, the Z axis is perpendicular to the seabed surface OXY and points vertically upward, and the X axis is perpendicular to the OYZ plane to represent the lateral movement direction of the mooring line; the entire mooring line is divided to obtain each segment unit, and a local coordinate system P is established on each segment unit xyz , P xyz The origin is at point P, and the direction of its coordinate axis is the same as O XYZ The same, point P is located at the center point of the mooring line segment unit; uvw is a local coordinate system established on the mooring line segment unit. The coordinate system is divided into three components in the form of the mooring line: tangent, normal and binormal (u, v, w). The relationship between the two local coordinate systems can be expressed as: Where β = β(l) and θ = θ(l) are the elevation and azimuth of the mooring line, and l is the length of the mooring line unit; Define u as the direction of the mooring line at point P, and the projection length of the mooring line unit in the x, y, and z directions can be expressed as: Under three-dimensional dynamic conditions, the governing equation of the force state of the mooring line unit is expressed as follows: M(X,Y,Z,a)+C(X,Y,Z,v)+K(X,Y,Z)=F(X,Y,Z,v,t) (3) Where M(X,Y,Z,a) is the inertial force, C(X,Y,Z,v) is the damping force, K(X,Y,Z) is the elastic force and F(X,Y,Z,v,t) is the external force, X, Y and Z represent the coordinates of the mooring line unit in the global coordinate system, a is the acceleration of the unit, v is the velocity of the unit, and t is the time; External forces include fluid forces in three directions (F n ,F b ,F τ ), soil resistance in three directions (Q n ,Q b ,Q τ ) and the unit weight of the mooring line in water and seabed (W w ,W s ); Morrison's equation is used to calculate the fluid force acting on the mooring line: Among them, ρ w is the density of water, d eff is the effective diameter of the anchor chain; dl is the unit length of the mooring section, V0 is the amount of water displaced by the mooring line, v n ,v b and v τ are the relative velocities of the mooring line and the seawater in the normal, subnormal and tangential directions, respectively, and C dn and C dτ are the normal and tangential resistance coefficients, respectively; The soil resistance acting on the mooring line is decomposed into the normal component Q n , binormal component Q b and the axial component Q τ , expressed as: Q τ =E τ d b fdl (5c) Where, d b is the nominal diameter of the anchor chain, E n and E τ are normal and axial equivalent width parameters respectively; q is the total normal resistance, which can be decomposed into normal and binormal components according to the direction, d w and d v are the displacements in the normal and binormal directions, respectively, and f is the axial resistance of the mooring line; q and f are calculated by the following expressions: q=N c s u (6a) f=αs u (6b) In the formula, s u is the undrained shear strength of the soil; N c is a function of soil depth z, expressed as Under dynamic conditions, the soil strain rate effect should be considered, that is, the shear strength of the soil increases with the increase of the shear strain rate. The influence of the strain rate is expressed by the parameter R f To characterize, R f is the relationship between the soil strength at a certain shear strain rate and the soil strength at a reference strain rate. The parameters R in the normal, binormal and axial directions are f It is expressed as: Where β is the strain rate parameter, which is between 0.034 and 0.140; the reference shear strain rate and is the shear strain rate in three directions, defined as: Where d is the characteristic diameter, d = d en =E n d s ; The real soil resistance considering the shear strain rate is: S u =S ur R f (10) Among them, S ur is the shear strength of remolded soil.

3. The method for predicting the three-dimensional morphology of seabed grooves considering the dynamic bottoming effect of the mooring line according to claim 2 is characterized in that: In step 2), the groove profile is defined by the deepest point of the groove (0, Y dp ,Z dp ) and the maximum lateral displacement point (X tp ,Y tp ,0) is calculated as follows: The deepest point of the groove (0,Y dp ,Z dp ) is expressed as: the first fixed point of the mooring line from top to bottom in a cycle, from which the position of the mooring line remains unchanged; for the position of the mooring line in a cycle, the deepest point (0, Y dp ,Z dp ) meets the following criteria: Among them, (X i ,Y i ,Z i ) is the position coordinate of any mooring line segment unit in a cycle; Maximum lateral displacement point (X tp ,Y tp ,0) is determined as follows: Under dynamic conditions, a bottoming area is formed around the mooring line, and the point corresponding to the maximum X coordinate in the bottoming area is defined as the point with the maximum lateral displacement and recorded as (X tp ,Y tp ,0).

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