Hydrostatics

Capytaine can compute some of the hydrostatic parameters of a given FloatingBody.

Note

Here the integration of a function over the immersed hull is approximated as the summation of the data function at face centers multiplied by the respective face areas.

(1)\[\iint_S f(x,y,z) dS \approx \sum_i^N f(x_i, y_i, z_i) \Delta S_i\]

where \(i\) is the face index, \((x_i, y_i, z_i)\) is \(i\) th face center, and \(S_i\) is \(i\) th face area.

Hydrostatic parameters

For each hydrostatic parameter a separate method is available in Capytaine. Some of them may require the definition of the center of mass of the body. It can be done by setting the attribute center_of_mass as in the example below.

Note

Before computing individual hydrostatic parameters, make sure to crop the body to only keep the immersed part.

import capytaine as cpt
import numpy as np

rigid_sphere = cpt.FloatingBody(
        mesh=cpt.mesh_sphere(radius=1.0, center=(0, 0, 0)),
        dofs=cpt.rigid_body_dofs(rotation_center=(0, 0, -0.3)),
        center_of_mass=(0, 0, -0.3)
        ).immersed_part()

print("Volume:", rigid_sphere.volume)
print("Center of buoyancy:", rigid_sphere.center_of_buoyancy)
print("Wet surface area:", rigid_sphere.wet_surface_area)
print("Displaced mass:", rigid_sphere.disp_mass(rho=1025))
print("Waterplane center:", rigid_sphere.waterplane_center)
print("Waterplane area:", rigid_sphere.waterplane_area)
print("Metacentric parameters:",
    rigid_sphere.transversal_metacentric_radius,
    rigid_sphere.longitudinal_metacentric_radius,
    rigid_sphere.transversal_metacentric_height,
    rigid_sphere.longitudinal_metacentric_height)

Hydrostatic stiffness

The equation to compute the hydrostatic stiffness of a floating body is

(2)\[C_{ij} = \iint_S (\hat{n} \cdot V_j) (w_i + z D_i) dS\]

where \(\hat{n}\) is the surface normal,

\(V_i = u_i \hat{n}_x + v_i \hat{n}_y + w_i \hat{n}_z\) is the dof vector and

\(D_i = \nabla \cdot V_i\) is the divergence of the DOF.

The method compute_hydrostatic_stiffness() computes the hydrostatic stiffness and returns a (DOF count x DOF count) 2D matrix as an xarray.DataArray.

rigid_sphere.hydrostatic_stiffness = rigid_sphere.compute_hydrostatic_stiffness()

Note

For rigid body dofs, the exact formula above can be evaluated. However, in general, the divergence \(D_i\) of an arbitrary dofs is not known. User can pass a value for D_i as an input to compute_hydrostatics. Otherwise the code assumes zero divergence \(D_{i} = 0\).

elastic_sphere = cpt.FloatingBody(
        mesh=cpt.mesh_sphere(radius=1.0, center=(0, 0, 0)),
        center_of_mass=(0, 0, -0.3)
        ).immersed_part()
elastic_sphere.dofs["elongate_in_z"] = np.array([(0, 0, z) for (x, y, z) in elastic_sphere.mesh.faces_centers])

dofs_divergence = {"elongate_in_z": np.ones(elastic_sphere.mesh.nb_faces)}

density = 1000.0
gravity = 9.81
elongate_in_z_hs = elastic_sphere.compute_hydrostatic_stiffness(divergence=dofs_divergence, rho=density, g=gravity)

analytical_hs = - density * gravity * (4 * elastic_sphere.volume * elastic_sphere.center_of_buoyancy[2])

print(np.isclose(elongate_in_z_hs.values[0, 0], analytical_hs))
# True

If the mass is not specified (as in the examples above), the body is assumed to be in buoyancy equilibrium. It’s mass is the mass of the displaced volume of water.

Non-neutrally buoyant bodies are partially implemented (only for a single rigid body). In this case, the mass of the body can be given by setting the mass attribute of the FloatingBody:

rigid_sphere.mass = 0.8 * rigid_sphere.disp_mass(rho=1000)
rigid_sphere.hydrostatic_stiffness = rigid_sphere.compute_hydrostatic_stiffness()

Inertia matrix

The method compute_rigid_body_inertia() is able to computes the 6 x 6 inertia matrix of a body with 6 rigid dofs. The inertia coefficient of other degrees of freedom are filled with NaN by default.

rigid_sphere.inertia_matrix = elastic_sphere.compute_rigid_body_inertia()

As for the hydrostatic stiffness, the mass is assumed to be the displaced mass of water, unless a mass attribute has been specified.

A custom matrix can be provided. For consistency with the data computed with Capytaine, it is recommended to wrap it in a xarray.DataArray with dof names as labels:

elastic_sphere.inertia_matrix = elastic_sphere.add_dofs_labels_to_matrix(np.array([[1000.0]]))

Compute all hydrostatics parameters

Instead of computing each hydrostatic parameters individually, compute_hydrostatics returns a dict containing all hydrostatic parameters.

Note

No need to apply immersed_part to use compute_hydrostatics.

hydrostatics = rigid_sphere.compute_hydrostatics()

print(hydrostatics.keys())
# dict_keys(['g', 'rho', 'center_of_mass', 'wet_surface_area', 'disp_volumes',
# 'disp_volume', 'disp_mass', 'center_of_buoyancy', 'waterplane_center',
# 'waterplane_area', 'transversal_metacentric_radius',
# 'longitudinal_metacentric_radius' , 'transversal_metacentric_height',
# 'longitudinal_metacentric_height', 'hydrostatic_stiffness',
# 'length_overall', 'breadt h_overall', 'depth', 'draught',
# 'length_at_waterline', 'breadth_at_waterline',
# 'length_overall_submerged', 'breadth_overall_submerged', 'inertia_matrix'])