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biharmonic_energy_intrinsic

biharmonic_energy_intrinsic(l_sq, F, n=None, bc='mixedfem_zero_neumann')

Constructs the biharmonic energy matrix Q such that for a per-vertex function u, the discrete biharmonic energy is u'Qu, using only intrinsic information from the mesh.

Parameters:

Name Type Description Default
l_sq (m,3) numpy array

squared halfedge lengths as computed by halfedge_lengths_squared

 required
F (m,3) numpy int array

face index list of a triangle mesh

 required
n int, optional (default None)

number of vertices in the mesh. If absent, will try to infer from F.

 None
bc string, optional (default 'mixedfem_zero_neumann')

Which type of discretization and boundary condition to use. Options are: {'mixedfem_zero_neumann', 'hessian', 'curved_hessian'} - 'mixedfem_zero_neumann' implements the mixed finite element discretization from Jacobson et al. 2010. "Mixed Finite Elements for Variational Surface Modeling". - 'curved_hessian' implements the Hessian energy from Stein et al. 2020 "A Smoothness Energy without Boundary Distortion for Curved Surfaces" via libigl C++ binding.

 'mixedfem_zero_neumann'

Returns:

Name Type Description
Q (n,n) scipy csr_matrix

biharmonic energy matrix

Examples:

# Mesh in V,F
import gpytoolbox as gpy
l_sq = gpy.halfedge_lengths_squared(V,F)
Q = biharmonic_energy_intrinsic(l_sq,F)
Source code in src/gpytoolbox/biharmonic_energy_intrinsic.py 
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def biharmonic_energy_intrinsic(l_sq,F,
    n=None,
    bc='mixedfem_zero_neumann'):
    """Constructs the biharmonic energy matrix Q such that for a per-vertex function u, the discrete biharmonic energy is u'Qu, using only intrinsic information from the mesh.

    Parameters
    ----------
    l_sq : (m,3) numpy array
        squared halfedge lengths as computed by halfedge_lengths_squared
    F : (m,3) numpy int array
        face index list of a triangle mesh
    n : int, optional (default None)
        number of vertices in the mesh.
        If absent, will try to infer from F.
    bc : string, optional (default 'mixedfem_zero_neumann')
         Which type of discretization and boundary condition to use.
         Options are: {'mixedfem_zero_neumann', 'hessian', 'curved_hessian'}
         - 'mixedfem_zero_neumann' implements the mixed finite element
         discretization from Jacobson et al. 2010.
         "Mixed Finite Elements for Variational Surface Modeling".
         - 'curved_hessian' implements the Hessian energy from Stein et al.
         2020 "A Smoothness Energy without Boundary Distortion for Curved Surfaces"
         via libigl C++ binding.

    Returns
    -------
    Q : (n,n) scipy csr_matrix
        biharmonic energy matrix

    Examples
    --------
    ```python
    # Mesh in V,F
    import gpytoolbox as gpy
    l_sq = gpy.halfedge_lengths_squared(V,F)
    Q = biharmonic_energy_intrinsic(l_sq,F)
    ```

    """

    assert F.shape[1] == 3, "Only works on triangle meshes."
    assert l_sq.shape == F.shape
    assert np.all(l_sq>=0)

    if n==None:
        n = np.max(F)+1

    if bc=='mixedfem_zero_neumann':
        Q = _mixedfem_neumann_laplacian_energy(l_sq, F, n)
    elif bc=='curved_hessian':
        Q = _curved_hessian_energy(l_sq, F, n)
    else:
        assert False, "Invalid bc"

    return Q