Hello all,
I am going to solve a generalized plane stain problem based on step 44. I
have three displacement components (3 dofs per node) on a 2D geometry (dim
=2) and mesh (compressible material hence no pressure). All displacements
are function of x and y coordinates only, i, e u = u(x,y), v=v(x,y), and
w = w(x,y). Hence solution_grads_u_total tensor and F tensor (deformation
gradient) are 3x3 tensors with the following forms (u, v, and w are
displacements, suffixes denote derivative; and semicolon denotes end of a
row):
solution_grads_u_total = [u_x, u_y, 0; v_x, v_y, 0; w_x, w_y, 0]
F = [F_11, F_12, 0; F_21, F_22, 0; F_31, F_32, 1]
I just wanted to confirm the following things. Please correct me if there
is a problem my approach:
Since my deformation grad, strain, stress tensors are 2nd rank and having
3x3 elements,
here I define the tensors (say F_inv) as (note that *dim = 2*)
Tensor<2, *dim+1*> F_inv;
Now in the function assemble_system_tangent_one_cell I need to compute
the grad_Nx and symm_grad_Nx
tensors which, as I understand, should also be of type Tensor<2, *dim+1*>
Thus my point is when we compute the gradient of the shape functions using
the piece of command in the below
(taken from function assemble_system_tangent_one_cell) we should expect to
get first two columns with non-zero
entries and last column with all zero entry as we not have derivatives of
shape functions w.r.t. x and y only (note that
the tensor has already been initialized with 0 in scratch as done in step
44). A part of the structure ScratchData_K is
also given in the below.
I shall be thankful if someone tells me if my approach about solving the
problem is correct and
if there is issue somewhere.
Thanks and regards,
Anup.
for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
{
const Tensor<2, *dim+1*> F_inv = lqph[q_point].get_F_inv();
for (unsigned int k = 0; k < dofs_per_cell; ++k)
{
scratch.grad_Nx[q_point][k] = scratch.fe_values_ref[u_fe].gradient(k,
q_point)
* F_inv;
scratch.symm_grad_Nx[q_point][k] =
symmetrize(scratch.grad_Nx[q_point][k]);
}
}
////////////////////////////////////// part of the structure Scratchdata_K
////////////////////////////////////////////////
template <int dim>
struct Solid<dim>::ScratchData_K
{
FEValues<dim> fe_values_ref;
std::vector<std::vector<double> > Nx;
std::vector<std::vector<Tensor<2, *dim+1*> > > grad_Nx;
std::vector<std::vector<SymmetricTensor<2, *dim+1*> > > symm_grad_Nx;
ScratchData_K(const FiniteElement<dim> &fe_cell,
const QGauss<dim> &qf_cell,
const UpdateFlags uf_cell)
:
fe_values_ref(fe_cell, qf_cell, uf_cell),
Nx(qf_cell.size(),
std::vector<double>(fe_cell.dofs_per_cell)),
grad_Nx(qf_cell.size(),
std::vector<Tensor<2, *dim+1*> >(fe_cell.dofs_per_cell)),
symm_grad_Nx(qf_cell.size(),
std::vector<SymmetricTensor<2, *dim+1*> >
(fe_cell.dofs_per_cell))
{}
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