Floating body

As described in the Tutorial, a floating body is defined as a mesh with degrees of freedom and optionally other properties, such as a mass and a center of mass.

Initialization

The floating body is set up by initializing the FloatingBody class:

body = cpt.FloatingBody(mesh=mesh, dofs={})

The above example creates a new body without degrees of freedom.

Mesh

The mesh can be a Mesh object, or any of the variants defined in Capytaine, such as a ReflectionSymmetricMesh. Meshes from meshio can also be given directly to the FloatingBody constructor without calling load_from_meshio().

Lid mesh

For irregular frequencies removal, a second mesh can be provided when defining the body. It is meant to be a mesh of a lid of the part of the free surface that is inside the body. The lid can either on the free surface or slightly below. This mesh is ignored for many computations (such as hydrostatics) and is only used when solving a BEM problem.

It is set as in the following example:

body = cpt.FloatingBody(mesh=mesh, lid_mesh=lid_mesh)

Once a lid mesh has been defined, it is automatically used for irregular frequencies removal without any other action from the user.

For irregular frequencies removal, the lid is expected to have normals going down (towards the fluid, through the body). If the lid appears to be horizontal with normal going up, Capytaine will switch the direction of its normal vectors.

Currently, meshes with a symmetry are not supported, in the sense that the computation will be done without using the symmetries when a lid is added. This should be improved to support at least vertical symmetry plane in a future version.

Dofs

The degrees of freedom are defined as a Python dictionary associating the name of each dof to a Numpy array of shape (nb_faces, 3). This array stores the displacement vector at the center of each face of the mesh:

body = cpt.FloatingBody(
        mesh=mesh,
        dofs={
            "heave": np.array([(0, 0, 1) for x, y, z in mesh.faces_centers]),
            "x-shear": np.array([(np.cos(np.pi*z/2), 0, 0) for x, y, z in mesh.faces_centers])
            },
        )

cpt.rigid_body_dofs() can be used to automatically give a body the six degrees of freedom of a rigid body:

body = cpt.FloatingBody(mesh=mesh, dofs=cpt.rigid_body_dofs(rotation_center=(0, 0, -1)))
print(body.dofs.keys())
# dict_keys(['Surge', 'Sway', 'Heave', 'Roll', 'Pitch', 'Yaw'])

Other parameters

Besides the mandatory mesh and dofs, a floating body can also be defined with a mass, given a floating point number and a center_of_mass, given a three coordinates. These two arguments are required for Hydrostatics but not for first-order wave-structure interaction. Their only role then is for the definition of the rotation degrees of freedom. When defining a rotation dof, the code looks for attributes called rotation_center, center_of_mass or geometric_center (in that order), and use them to define the rotation axis. If none of them are define, the rotation is defined around the origin of the domain \((0, 0, 0)\).

Finally, as the mesh objects, the floating body can be assigned a name.

Display and animation

The methods show() and show_matplotlib() of meshes can also be used on FloatingBody.

Once a FloatingBody with dofs has been defineds, the animate() method can be used to visualize a given motion of the body:

anim = body.animate(motion={"Heave": 0.1, "Surge": 0.1j}, loop_duration=1.0)
anim.run()

The above example will present an interactive animation of the linear combination of heave and surge.

Jupyter notebooks can also include a (non-interactive) video of the animation:

anim.embed_in_notebook(camera_position=(-1.0, -1.0, 1.0), resolution=(400, 300))

Geometric transformations

All the geometric transformation defined on meshes in Meshes can also be applied to FloatingBody. Beside updating the mesh, they also update the definition of the degrees of freedom and the center of mass (if relevant).

Multiple bodies

Multiple bodies problems can be defined by combining several bodies with the join_bodies method:

all_bodies = cpt.FloatingBody.join_bodies(body_1, body_2, body_3, body_4)

For two-body problems, the + operator can also be used:

two_bodies = body_1 + body_2

But it is not recommended to use it for large number of bodies as it is not strictly associative (that is body_1 + (body_2 + body_3) has some internal differences with (body_1 + body_2) + body_3).

When two floating bodies with dofs are merged, the resulting body inherits from the dofs of the individual bodies with the new name body_name__dof_name:

print(two_bodies.nb_dofs)
# 12
print(two_bodies.dofs.keys())
# dict_keys(['body_1__Surge', 'body_1__Sway', 'body_1__Heave', 'body_1__Roll', 'body_1__Pitch', 'body_1__Yaw', 'body_2__Surge', 'body_2__Sway', 'body_2__Heave', 'body_2__Roll', 'body_2__Pitch', 'body_2__Yaw'])

Capytaine also include helper functions to create arrays of identical bodies:

array = body.assemble_regular_array(distance=1.0, nb_bodies=(4, 5))

places copies of the body on a regular grid of \(4 \times 5\) with distance between bodies of 1 meter, and:

locations = np.array([[0.0, 0.0], [1.0, 2.0], [3.0, 4.5], [3.0, -0.5]])
array = body.assemble_arbitrary_array(locations)

places copies of the body at the list of locations specified.

Warning

As currently implemented in Capytaine, the multiple bodies are stored as a single body with a non-connex mesh and generalized degrees of freedom. Hence some information about the individual bodies is lost. It includes the center of mass and the center of rotation of the individual bodies (although the latter could be recovered indirectly by studying the definition of the rotation dof). Although it does not affect first order wave-structure interaction, it hinders the computation of hydrostatics for multiple rigid bodies and will need to be fixed in the future.