linear_elasticity

linear_elasticity(V, F, U0, dt=0.1, bb=None, bc=None, Ud0=None, fext=None, K=1.75, mu=0.0115, volumes=None, mass=None)

Linear elastic deformation

Compute the deformation of a 2D solid object according to the usual linear elasticity model.

Parameters:

Name	Type	Description	Default
V	numpy double array	Matrix of vertex coordinates	required
F	numpy int array	Matrix of triangle indices	required
U0	numpy double array	Matrix of previous displacements	required
dt	double (optional, default 0.1)	Timestep	0.1
bb	numpy int array (optional, default None)	Fixed vertex indices into V	None
bc	numpy double array (optional, default None)	Fixed vertex displacements	None
fext	numpy double array (optional, default None)	Matrix of external forces (for example, gravity or a load)	None
Ud0	numpy double array (optional, default None)	Matrix of previous velocity	None
K	double (optional, default 1.75)	Bulk modulus	1.75
mu	double (optional, default 0.0115)	Material shear modulus	0.0115
volumes	numpy double array (optional, default None)	Vector with the volumes (in 3D) or areas (in 2D) of each mesh element (if None, will be computed)	None
mass	scipy sparse_csr (optional, default None)	The mesh's sparse mass matrix (if None, will be computed)	None

Returns:

Name	Type	Description
U	numpy double array	Matrix of new displacements
sigma_v	numpy double array	Vector of per-element Von Mises stresses

See Also

linear_elasticity_stiffness.

Notes

This implementation only works for 2D triangle meshes. Tetrahedral meshes will be supported soon.

Examples:

from gpytoolbox import regular_square_mesh, linear_elasticity
# Build very small mesh
V, F = regular_square_mesh(3)
V = (V + 1.)/2.
# Initial conditions
fext = 0*V
Ud0 = 0*V
U0 = V.copy()
U0[:,1] = 0.0
U0[:,0] = U0[:,0] - 0.5
# print(np.reshape(U0,(-1,1),order='F'))
dt = 0.2
U, sigma_v = linear_elasticity(V,F,U0,fext=fext,dt=dt,Ud0=Ud0)

Source code in src/gpytoolbox/linear_elasticity.py

def linear_elasticity(V,F,U0,dt=0.1,bb=None,bc = None
    ,Ud0=None,fext=None,K=1.75,mu=0.0115,volumes=None,mass=None):
    """Linear elastic deformation

    Compute the deformation of a 2D solid object according to the usual linear elasticity model.

    Parameters
    ----------
    V : numpy double array
        Matrix of vertex coordinates
    F : numpy int array
        Matrix of triangle indices
    U0 : numpy double array 
        Matrix of previous displacements
    dt : double (optional, default 0.1) 
        Timestep
    bb : numpy int array (optional, default None)
        Fixed vertex indices into V
    bc : numpy double array (optional, default None)
        Fixed vertex *displacements*
    fext : numpy double array (optional, default None)
        Matrix of external forces (for example, gravity or a load)
    Ud0 : numpy double array (optional, default None)
        Matrix of previous velocity
    K : double (optional, default 1.75)
        Bulk modulus
    mu : double (optional, default 0.0115)
        Material shear modulus
    volumes : numpy double array (optional, default None)
        Vector with the volumes (in 3D) or areas (in 2D) of each mesh element (if None, will be computed)
    mass : scipy sparse_csr (optional, default None)
        The mesh's sparse mass matrix (if None, will be computed)

    Returns
    -------
    U : numpy double array
        Matrix of new displacements
    sigma_v : numpy double array
        Vector of per-element Von Mises stresses

    See Also
    --------
    linear_elasticity_stiffness.

    Notes
    -----
    This implementation only works for 2D triangle meshes. Tetrahedral meshes will be supported soon.

    Examples
    --------
    ```python
    from gpytoolbox import regular_square_mesh, linear_elasticity
    # Build very small mesh
    V, F = regular_square_mesh(3)
    V = (V + 1.)/2.
    # Initial conditions
    fext = 0*V
    Ud0 = 0*V
    U0 = V.copy()
    U0[:,1] = 0.0
    U0[:,0] = U0[:,0] - 0.5
    # print(np.reshape(U0,(-1,1),order='F'))
    dt = 0.2
    U, sigma_v = linear_elasticity(V,F,U0,fext=fext,dt=dt,Ud0=Ud0)
    ```
    """

    if (Ud0 is None):
        Ud0 = 0*V
    if (fext is None):
        fext = 0*V
    if (bb is not None):
        # This is the bit that assumes 2D
        bb = np.concatenate((bb,bb+V.shape[0]))
        bc = np.concatenate((bc[:,0],bc[:,1]))

    K, C, strain, A, M = linear_elasticity_stiffness(V,F,K=K,volumes=volumes,mass=mass,mu=mu)


    A = M + (dt**2)*K
    B = M*((dt**2)*np.reshape(fext,(-1, 1),order='F') + np.reshape(U0,(-1, 1),order='F') + dt*np.reshape(Ud0,(-1, 1),order='F'))

    U = min_quad_with_fixed(A,c=-1.0*np.squeeze(B),k=bb,y=bc)
    #print(U)
    # https://en.m.wikipedia.org/wiki/Von_Mises_yield_criterion
    face_stress_vec = C*strain*U
    if V.shape[1]==2:
        sigma_11 = face_stress_vec[0:F.shape[0]]
        sigma_22 = face_stress_vec[F.shape[0]:(2*F.shape[0])]
        sigma_12 = face_stress_vec[(2*F.shape[0]):(3*F.shape[0])]
        sigma_v = np.sqrt(0.5*(sigma_11*sigma_11 - sigma_11*sigma_22 + sigma_22*sigma_22 + 3*sigma_12*sigma_12))
    elif V.shape[1]==3:
        print("TET MESHES UNSUPPORTED")
    return U, sigma_v