7) The rigid-body equations of motion are solved by numerical integration;
8) Iterations are performed at each time step to find a balance between the indentation into ice and the contact forces.
In the following section, the construction of present numerical model is described based on the above items.
2.2 Construction of the numerical model
As shown in Fig. 2.1, the ice forces encountered by a ship transiting a level sheet of ice depend primarily on the processes, by which its hull breaks and displaces ice. First, when the ice sheet contacts the hull, crushing happens. The crushing force will keep growing with an increasing contact area until its vertical component is large enough to cause a bending failure of ice. After the ice floes have been broken from the ice sheet, the advance of ship forces them to turn on edge until parallel with the hull. Then, the floes will become submerged and slide along the hull until they can not maintain contact with the hull. In some hull zones, typically at the very bow and at the shoulders which have large slope angles (almost vertical), crushing may be the only failure mode (Lindqvist, 1989). This can lead to a considerable ice resistance.
Fig. 2.1 The overall process of ice–hull interaction in level ice (Riska, 2010b)
Fig. 2.2 Idealized time histories of ice forces and the definition of ice resistance (Riska, 2010a)
7) The rigid-body equations of motion are solved by numerical integration;
8) Iterations are performed at each time step to find a balance between the indentation into ice and the contact forces.
In the following section, the construction of present numerical model is described based on the above items.
2.2 Construction of the numerical model
As shown in Fig. 2.1, the ice forces encountered by a ship transiting a level sheet of ice depend primarily on the processes, by which its hull breaks and displaces ice. First, when the ice sheet contacts the hull, crushing happens. The crushing force will keep growing with an increasing contact area until its vertical component is large enough to cause a bending failure of ice. After the ice floes have been broken from the ice sheet, the advance of ship forces them to turn on edge until parallel with the hull. Then, the floes will become submerged and slide along the hull until they can not maintain contact with the hull. In some hull zones, typically at the very bow and at the shoulders which have large slope angles (almost vertical), crushing may be the only failure mode (Lindqvist, 1989). This can lead to a considerable ice resistance.
Fig. 2.1 The overall process of ice–hull interaction in level ice (Riska, 2010b)
Fig. 2.2 Idealized time histories of ice forces and the definition of ice resistance (Riska, 2010a)
16 Chapter 2
7) The rigid-body equations of motion are solved by numerical integration;
8) Iterations are performed at each time step to find a balance between the indentation into ice and the contact forces.
In the following section, the construction of present numerical model is described based on the above items.
2.2 Construction of the numerical model
As shown in Fig. 2.1, the ice forces encountered by a ship transiting a level sheet of ice depend primarily on the processes, by which its hull breaks and displaces ice. First, when the ice sheet contacts the hull, crushing happens. The crushing force will keep growing with an increasing contact area until its vertical component is large enough to cause a bending failure of ice. After the ice floes have been broken from the ice sheet, the advance of ship forces them to turn on edge until parallel with the hull. Then, the floes will become submerged and slide along the hull until they can not maintain contact with the hull. In some hull zones, typically at the very bow and at the shoulders which have large slope angles (almost vertical), crushing may be the only failure mode (Lindqvist, 1989). This can lead to a considerable ice resistance.
Fig. 2.1 The overall process of ice–hull interaction in level ice (Riska, 2010b)
16 Chapter 2
7) The rigid-body equations of motion are solved by numerical integration;
8) Iterations are performed at each time step to find a balance between the indentation into ice and the contact forces.
In the following section, the construction of present numerical model is described based on the above items.
2.2 Construction of the numerical model
As shown in Fig. 2.1, the ice forces encountered by a ship transiting a level sheet of ice depend primarily on the processes, by which its hull breaks and displaces ice. First, when the ice sheet contacts the hull, crushing happens. The crushing force will keep growing with an increasing contact area until its vertical component is large enough to cause a bending failure of ice. After the ice floes have been broken from the ice sheet, the advance of ship forces them to turn on edge until parallel with the hull. Then, the floes will become submerged and slide along the hull until they can not maintain contact with the hull. In some hull zones, typically at the very bow and at the shoulders which have large slope angles (almost vertical), crushing may be the only failure mode (Lindqvist, 1989). This can lead to a considerable ice resistance.
Fig. 2.1 The overall process of ice–hull interaction in level ice (Riska, 2010b)
Early research on level ice resistance was usually carried out based on this break–
displace process (see e.g. Fig. 2.2). Although it may be questionable (Enkvist et al., 1979), most of the ice resistance formulas were established on this assumption (e.g.
Enkvist, 1972, Lewis, 1982, and Lindqvist, 1989). In the present numerical model, the icebreaking forces are numerically detected, while the ice forces induced after the ice wedges are broken from the ice edge are taken into account by the Lindqvist’s ice resistance formula (the submersion and friction components given in Lindqvist (1989)).
2.2.1 Geometric model for ice–hull interaction
The basic geometric model for ice–hull interaction includes the full-size waterline of the ship and the edge of the ice. As shown in Fig. 2.3, the waterline of the ship is discretized into a closed polygon and the edge of the ice is discretized into a polyline in the established simulation program. At each time step, the simulation program is set to detect the ice nodes which are inside the hull polygon. Then, each contact zone can be found. To check whether the ice node is inside the hull polygon, the specific geometric tools for computer graphics are adopted. The detailed algorithm can be found in Schneider et al. (2002).
Fig. 2.3 Geometrical idealization of ice–hull interaction
At each contact zone shown in Fig. 2.3, it is assumed that the contact surface between ice and hull is flat, and the contact area, Ac, is simply determined by the contact length, Lh, and the indentation depth, Ld. Herein, L h is calculated from the distance between adjacent hull nodes, and L d is calculated from the perpendicular distance from the cusp of ice nodes to the contact surface (see e.g. in Fig. 2.4). As shown in Fig. 2.5, two cases must be considered in the calculation of contact area:
1 Case 1: tan( ) zones (i.e. different locations of hull nodes).
Early research on level ice resistance was usually carried out based on this break–
displace process (see e.g. Fig. 2.2). Although it may be questionable (Enkvist et al., 1979), most of the ice resistance formulas were established on this assumption (e.g.
Enkvist, 1972, Lewis, 1982, and Lindqvist, 1989). In the present numerical model, the icebreaking forces are numerically detected, while the ice forces induced after the ice wedges are broken from the ice edge are taken into account by the Lindqvist’s ice resistance formula (the submersion and friction components given in Lindqvist (1989)).
2.2.1 Geometric model for ice–hull interaction
The basic geometric model for ice–hull interaction includes the full-size waterline of the ship and the edge of the ice. As shown in Fig. 2.3, the waterline of the ship is discretized into a closed polygon and the edge of the ice is discretized into a polyline in the established simulation program. At each time step, the simulation program is set to detect the ice nodes which are inside the hull polygon. Then, each contact zone can be found. To check whether the ice node is inside the hull polygon, the specific geometric tools for computer graphics are adopted. The detailed algorithm can be found in Schneider et al. (2002).
Fig. 2.3 Geometrical idealization of ice–hull interaction
At each contact zone shown in Fig. 2.3, it is assumed that the contact surface between ice and hull is flat, and the contact area, Ac, is simply determined by the contact length, Lh, and the indentation depth, Ld. Herein, L h is calculated from the distance between adjacent hull nodes, and L d is calculated from the perpendicular distance from the cusp of ice nodes to the contact surface (see e.g. in Fig. 2.4). As shown in Fig. 2.5, two cases must be considered in the calculation of contact area:
1 Case 1: tan( ) zones (i.e. different locations of hull nodes).
Description of the Numerical Model 17
Early research on level ice resistance was usually carried out based on this break–
displace process (see e.g. Fig. 2.2). Although it may be questionable (Enkvist et al., 1979), most of the ice resistance formulas were established on this assumption (e.g.
Enkvist, 1972, Lewis, 1982, and Lindqvist, 1989). In the present numerical model, the icebreaking forces are numerically detected, while the ice forces induced after the ice wedges are broken from the ice edge are taken into account by the Lindqvist’s ice resistance formula (the submersion and friction components given in Lindqvist (1989)).
2.2.1 Geometric model for ice–hull interaction
The basic geometric model for ice–hull interaction includes the full-size waterline of the ship and the edge of the ice. As shown in Fig. 2.3, the waterline of the ship is discretized into a closed polygon and the edge of the ice is discretized into a polyline in the established simulation program. At each time step, the simulation program is set to detect the ice nodes which are inside the hull polygon. Then, each contact zone can be found. To check whether the ice node is inside the hull polygon, the specific geometric tools for computer graphics are adopted. The detailed algorithm can be found in Schneider et al. (2002).
Fig. 2.3 Geometrical idealization of ice–hull interaction
At each contact zone shown in Fig. 2.3, it is assumed that the contact surface between ice and hull is flat, and the contact area, Ac, is simply determined by the contact length, Lh, and the indentation depth, Ld. Herein, L h is calculated from the distance between adjacent hull nodes, and L d is calculated from the perpendicular distance from the cusp of ice nodes to the contact surface (see e.g. in Fig. 2.4). As shown in Fig. 2.5, two cases must be considered in the calculation of contact area:
1 Case 1: tan( ) zones (i.e. different locations of hull nodes).
Description of the Numerical Model 17
Early research on level ice resistance was usually carried out based on this break–
displace process (see e.g. Fig. 2.2). Although it may be questionable (Enkvist et al., 1979), most of the ice resistance formulas were established on this assumption (e.g.
Enkvist, 1972, Lewis, 1982, and Lindqvist, 1989). In the present numerical model, the icebreaking forces are numerically detected, while the ice forces induced after the ice wedges are broken from the ice edge are taken into account by the Lindqvist’s ice resistance formula (the submersion and friction components given in Lindqvist (1989)).
2.2.1 Geometric model for ice–hull interaction
The basic geometric model for ice–hull interaction includes the full-size waterline of the ship and the edge of the ice. As shown in Fig. 2.3, the waterline of the ship is discretized into a closed polygon and the edge of the ice is discretized into a polyline in the established simulation program. At each time step, the simulation program is set to detect the ice nodes which are inside the hull polygon. Then, each contact zone can be found. To check whether the ice node is inside the hull polygon, the specific geometric tools for computer graphics are adopted. The detailed algorithm can be found in Schneider et al. (2002).
Fig. 2.3 Geometrical idealization of ice–hull interaction
At each contact zone shown in Fig. 2.3, it is assumed that the contact surface between ice and hull is flat, and the contact area, Ac, is simply determined by the contact length, Lh, and the indentation depth, Ld. Herein, L h is calculated from the distance between adjacent hull nodes, and L d is calculated from the perpendicular distance from the cusp of ice nodes to the contact surface (see e.g. in Fig. 2.4). As shown in Fig. 2.5, two cases must be considered in the calculation of contact area:
1 Case 1: tan( ) zones (i.e. different locations of hull nodes).
Fig. 2.4 Illustration of the contact length (Lh) and indentation depth (Ld) at each contact zone
ij
ij Ld
Ld
Lh
Lh
(1)
(2)
hi
hi
Contact Area Ac
Fig. 2.5 Two cases for the calculation of contact area
2.2.2 Ice crushing force
As shown in Fig. 2.6, it is assumed that the ice is uniformly crushed on the contact surface. The crushing force, Fcr, is normal to the contact surface and calculated as the product of the effective ice crushing strength, Vc, and the contact area, Ac:
cr c c
F V A (2.2)
where the effective ice crushing strength, Vc, is derived from the measured ice crushing pressure on ship hull (Kujala, 1994).
Herein, the frictional force is also taken into account, which is divided into two components, fH and fV, according to the relative motion between ice and hull:
2 ,1
2
rel rel rel
H i cr t t n
f PF v v v (2.3)
Fig. 2.4 Illustration of the contact length (Lh) and indentation depth (Ld) at each contact zone
ij
ij Ld
Ld
Lh
Lh
(1)
(2)
hi
hi
Contact Area Ac
Fig. 2.5 Two cases for the calculation of contact area
2.2.2 Ice crushing force
As shown in Fig. 2.6, it is assumed that the ice is uniformly crushed on the contact surface. The crushing force, Fcr, is normal to the contact surface and calculated as the product of the effective ice crushing strength, Vc, and the contact area, Ac:
cr c c
F V A (2.2)
where the effective ice crushing strength, Vc, is derived from the measured ice crushing pressure on ship hull (Kujala, 1994).
Herein, the frictional force is also taken into account, which is divided into two components, fH and fV, according to the relative motion between ice and hull:
2 ,1
2
rel rel rel
H i cr t t n
f P F v v v (2.3)
18 Chapter 2
Fig. 2.4 Illustration of the contact length (Lh) and indentation depth (Ld) at each contact zone
ij
ij Ld
Ld
Lh
Lh
(1)
(2)
hi
hi
Contact Area Ac
Fig. 2.5 Two cases for the calculation of contact area
2.2.2 Ice crushing force
As shown in Fig. 2.6, it is assumed that the ice is uniformly crushed on the contact surface. The crushing force, Fcr, is normal to the contact surface and calculated as the product of the effective ice crushing strength, Vc, and the contact area, Ac:
cr c c
F V A (2.2)
where the effective ice crushing strength, Vc, is derived from the measured ice crushing pressure on ship hull (Kujala, 1994).
Herein, the frictional force is also taken into account, which is divided into two components, fH and fV, according to the relative motion between ice and hull:
2
2
rel rel rel
f PF v v v (2.3)
18 Chapter 2
Fig. 2.4 Illustration of the contact length (Lh) and indentation depth (Ld) at each contact zone
ij
ij Ld
Ld
Lh
Lh
(1)
(2)
hi
hi
Contact Area Ac
Fig. 2.5 Two cases for the calculation of contact area
2.2.2 Ice crushing force
As shown in Fig. 2.6, it is assumed that the ice is uniformly crushed on the contact surface. The crushing force, Fcr, is normal to the contact surface and calculated as the product of the effective ice crushing strength, Vc, and the contact area, Ac:
cr c c
F V A (2.2)
where the effective ice crushing strength, Vc, is derived from the measured ice crushing pressure on ship hull (Kujala, 1994).
Herein, the frictional force is also taken into account, which is divided into two components, fH and fV, according to the relative motion between ice and hull:
2
2
rel rel rel
f P F v v v (2.3)
2
2
,1 ,1
rel rel rel
V i cr n t n
f P F v v v (2.4)
where Pi is the frictional coefficient, vrel is the relative velocity between ice and hull, and the decomposed force and velocity components are illustrated in Fig. 2.7.
The horizontal and vertical components, FH and FV, of the total contact force are then calculated as:
sin( ) cos( )
H cr V
F F M f M (2.5)
cos( ) sin( )
V cr V
F F M f M (2.6)
Fig. 2.6 Calculation of the ice crushing force
Fig. 2.7 Decomposed force and velocity components
2.2.3 Ice bending failure
If the vertical component of the contact force between ice and hull (FV, shown in Fig.
2.7) exceeds the bending failure load of ice cover, Pf , given in Equation (2.7), the ice wedge (as shown in Fig. 2.3) will be broken from the edge of the ice:
2 2
f f f i
P C T V h
S
§ ·¨ ¸
© ¹ (2.7)
where T is the opening angle of the idealized ice wedge shown in Fig. 2.8, Vf is the flexural strength of the ice, hi is the thickness of the ice, and Cf is an empirical
2
2
,1 ,1
rel rel rel
V i cr n t n
f P F v v v (2.4)
where Pi is the frictional coefficient, vrel is the relative velocity between ice and hull, and the decomposed force and velocity components are illustrated in Fig. 2.7.
The horizontal and vertical components, FH and FV, of the total contact force are then calculated as:
sin( ) cos( )
H cr V
F F M f M (2.5)
cos( ) sin( )
V cr V
F F M f M (2.6)
Fig. 2.6 Calculation of the ice crushing force
Fig. 2.7 Decomposed force and velocity components
2.2.3 Ice bending failure
If the vertical component of the contact force between ice and hull (FV, shown in Fig.
2.7) exceeds the bending failure load of ice cover, Pf, given in Equation (2.7), the ice wedge (as shown in Fig. 2.3) will be broken from the edge of the ice:
2 2
f f f i
P C T V h S
§ ·¨ ¸
© ¹ (2.7)
where T is the opening angle of the idealized ice wedge shown in Fig. 2.8, Vf is the flexural strength of the ice, hi is the thickness of the ice, and Cf is an empirical
Description of the Numerical Model 19
2
2
,1 ,1
rel rel rel
V i cr n t n
f P F v v v (2.4)
where Pi is the frictional coefficient, vrel is the relative velocity between ice and hull, and the decomposed force and velocity components are illustrated in Fig. 2.7.
The horizontal and vertical components, FH and FV, of the total contact force are then calculated as:
sin( ) cos( )
H cr V
F F M f M (2.5)
cos( ) sin( )
V cr V
F F M f M (2.6)
Fig. 2.6 Calculation of the ice crushing force
Fig. 2.7 Decomposed force and velocity components
2.2.3 Ice bending failure
If the vertical component of the contact force between ice and hull (FV, shown in Fig.
2.7) exceeds the bending failure load of ice cover, Pf , given in Equation (2.7), the ice wedge (as shown in Fig. 2.3) will be broken from the edge of the ice:
2 2
f f f i
P C T V h
S
§ ·¨ ¸
© ¹ (2.7)
where T is the opening angle of the idealized ice wedge shown in Fig. 2.8, Vf is the flexural strength of the ice, hi is the thickness of the ice, and Cf is an empirical
Description of the Numerical Model 19
2
2
,1 ,1
rel rel rel
V i cr n t n
f P F v v v (2.4)
where Pi is the frictional coefficient, vrel is the relative velocity between ice and hull, and the decomposed force and velocity components are illustrated in Fig. 2.7.
The horizontal and vertical components, FH and FV, of the total contact force are then calculated as:
sin( ) cos( )
H cr V
F F M f M (2.5)
cos( ) sin( )
V cr V
F F M f M (2.6)
Fig. 2.6 Calculation of the ice crushing force
Fig. 2.7 Decomposed force and velocity components
2.2.3 Ice bending failure
If the vertical component of the contact force between ice and hull (FV, shown in Fig.
2.7) exceeds the bending failure load of ice cover, Pf, given in Equation (2.7), the ice wedge (as shown in Fig. 2.3) will be broken from the edge of the ice:
2 2
f f f i
P C T V h S
§ ·¨ ¸
© ¹ (2.7)
where T is the opening angle of the idealized ice wedge shown in Fig. 2.8, Vf is the flexural strength of the ice, hi is the thickness of the ice, and Cf is an empirical
parameter. Equation (2.7) accounts for the opening angle, and it is an empirical equation (introduced by Kashtelyan (Kerr, 1975), applied in Wang (2001), Liu et al. (2006), and Nguyen et al. (2009)). Thus, the constant Cf must be obtained from measurements.
The geometrical idealization of the ice wedge in contact with the hull is illustrated in Fig. 2.8, where the bending crack is determined by the interpolation of the icebreaking radius at the first and last contact node (i.e., Rf and Rl). The icebreaking radius R is
The geometrical idealization of the ice wedge in contact with the hull is illustrated in Fig. 2.8, where the bending crack is determined by the interpolation of the icebreaking radius at the first and last contact node (i.e., Rf and Rl). The icebreaking radius R is