Linear Elasticity
=================

Pressed Hollow Rectangle 
------------------------

.. code-block:: python 

    from torch_fem import Mesh, ElementAssembler

    class KAssembler(ElementAssembler):
        def precompute(self):
            E = 1
            nu = 0.3

            D = E/(1 - nu*nu) * torch.tensor([
                [1, nu, 0],
                [nu, 1, 0],
                [0, 0, 0.5*(1-nu)]
            ])
            self.D = D


        def forward(self, gradu, gradv):
            """
                Parameters:
                -----------
                    gradu: torch.Tensor[n_basis, n_dim]
                    gradv: torch.Tensor[n_basis, n_dim]
                Returns:
                --------
                    K: torch.Tensor[n_basis, n_basis, n_dim, n_dim]
            """

            D = self.D.type(gradu.dtype).to(gradu.device)

            zeros = torch.zeros_like(gradu[:, 0]) # [n_basis]
            B = matrix([
                [gradu[:, 0], zeros],
                [zeros, gradu[:, 1]],
                [gradu[:, 1], gradu[:, 0]]
            ]) # [n_basis, 3, 2]
        
            C  = matmul(D, B) # [3, 3] @ [n_basis, 3, 2] = [n_basis, 3, 2]

            K  = dot(C, B, reduce_dim=-2) # [n_basis, n_basis, 2, 2]
        
            return K

    mesh = Mesh.gen_hollow_rectangle(chara_length=0.02, element_type="quad").double()

    K_asm = KAssembler.from_mesh(mesh, quadrature_order=3)

    K = K_asm(mesh.points)
    F = torch.zeros_like(mesh.points)
    F[mesh.point_data['is_outer_top_boundary'], 1] = 0.05
    F = F.flatten()

    dirichlet_mask = torch.zeros_like(mesh.points, dtype=torch.bool)
    dirichlet_mask[mesh.point_data['is_outer_bottom_boundary'], :] = True

    condenser = Condenser(dirichlet_mask.flatten())

    K_, f_ = condenser(K, F)

    u_ = K_.solve(f_)

    u = condenser.recover(u_)

    mesh.plot(
        {"ux":u[0::2],
         "uy":u[1::2]},
         backend="matplotlib",
        save_path="press_hollow_rectangle.png",
        show_mesh=True
    )


.. image:: ../_static/press_hollow_rectangle.png
    :width: 600
    :align: center