Dear Prof. Renard,
thank you so much for your precious support.
Just a few extra question:
Dear
Domenico,
Le 05/03/2015 00:17, Domenico Notaro a écrit :
Dear GetFem Users,
I am completely new of GetFem++ I am trying to implement a mixed formulation for the 3D Darcy problem.
By simplifying
terms useless for this issue, the weak formulation of the problem is
Find ( u, p) in VxQ s.t.
(1) (1/k u, v) + (GRAD(p), v) = 0 in \Omega
(2) (u, GRAD(q)) - ((\alpha p, q)) + (f(p), q) = 0 in \Omega
where (.,.) and ((.,.)) indicate the L2 product on \Omega and \Gamma, respectively.
[I integrate by part the divergence term because of the Robin BC: u.n = \alpha p on \Gamma]
I have one question for each of the following step:
a) First of all, I tried to implement the
assembly procedure for (GRAD(p), v).
b) Then I tried to evaluate the satisfaction of the divergence constraint (||DIV(u)-f||), i.e. the strong equivalent of (2), for which I need to compute the
divergence of a vector field.
a) The implementation is the following, It seems to work properly - I have just a small doubt below - but I would like a double check from you expert users because this is my first implementation:
/// Build the mixed pressure term
/// $ G = \int GRAD(p).v dx $
template<typename MAT>
void
asm_darcy_G(MAT & G,
const mesh_im & mim,
const mesh_fem & mf_p,
const mesh_fem & mf_u,
const mesh_region & rg = mesh_region::all_convexes()
)
{
GMM_ASSERT1(mf_p.get_qdim() == 1, "invalid data mesh fem (Qdim=1 required)");
GMM_ASSERT1(mf_u.get_qdim() > 1, "invalid data mesh fem (Qdim=2,3 required)");
generic_assembly assem("M(#1,#2)+=comp(Grad(#1).vBase(#2))(:,i,:,i);");
assem.push_mi(mim);
assem.push_mf(mf_p);
assem.push_mf(mf_u);
assem.push_mat(G);
assem.assembly(rg);
} /* end of asm_darcy_G */
(?) The output of this asm procedure is G^T not G, isn't it? (because A(i,j)=a(\phi_j,\phi_i) is a(.,.) is a non symmetric bilinear form)
I mean, in this way I am assembling a mf_p.nb_dof() x mf_u.nb_dof() matrix while I need the transpose (to be multiplied then for the pressure vector P).
Yes, this is correct, and this is indeed a mf_p.nb_dof() x mf_u.nb_dof() matrix. You can also use the high level generic assembly to perform this with an assembly string of the type "Grad_Test_p.Test2_u".
OK, that's great!
b) I tried to address this issue in two ways.
b.1) By using the function
getfem::compute_gradient - that seems to be the only way to compute derivatives - I computed the gradient tensor of the vector velocity and then extracted the divergence:
// Compute GRAD(U)
getfem::mesh_fem mf_gradU(mesh);
bgeot::pgeometric_trans pgt_t = bgeot::geometric_trans_descriptor(MESH_TYPE);
size_type N = pgt_t->dim();
mf_gradU.set_qdim(bgeot::dim_type(N), bgeot::dim_type(N)); //3x3
//mf_gradUt.set_classical_finite_element(0);
mf_gradUt.set_classical_discontinuous_finite_element(0);
vector_type gradU(mf_gradU.nb_dof());
getfem::compute_gradient(mf_U, mf_gradU, U, gradU);
//mf_U is at this level 'FEM_PK(3,1)'
// Compute DIV(U)
getfem::mesh_fem mf_Ui(mesh);
mf_Ui.set_classical_discontinuous_finite_element(0);
size_type nb_dof_Ui = mf_Ui.nb_dof(); //= mf_gradU.nb_dof()/(N*N)! NOT
mf_U.nb_dof()/N
vector_type divU(nb_dof_Ui);
gmm::add(gmm::sub_vector(gradU, gmm::sub_interval(0*nb_dof_Ui, nb_dof_Ui)), divU);
gmm::add(gmm::sub_vector(gradU, gmm::sub_interval(4*nb_dof_Ui, nb_dof_Ui)), divU);
gmm::add(gmm::sub_vector(gradU, gmm::sub_interval(8*nb_dof_Ui, nb_dof_Ui)), divU);
This seems to be incorrect. the component of the gradient are consecutives in the vector gradU. So you should use a gmm::sub_slice instead.
OK, I will try to fix that. So, is the correct order
GRAD(U) = [DxUx,1 DyUx,1 DzUx,1 DxUx,2 DyUx,2 DzUx,2 ... ]
or
GRAD(U) = [DxUx,1 DyUx,1 DzUx,1 DxUy,1 DyUy,1 DzUy,1 ... ] ?
I mean, also the components are stored in this "sliced" way?
And, in general, which approach do you think is better between the "algebraic" and the "assembly" ones?
// Compute ||DIV(U)-F|| by using asm_L2_norm
(?) Here I assumed - but I am not sure at all - the function compute_gradient stores derivatives in the following order: GRAD(U) = [DxUx, DxUy, DxUz, DyUx, DyUy, ...]
b.2) In order to feel more confident about the previous implementation I tried also to compute ||DIV(u)-f|| with an
assembly approach:
/// Compute the L2 norm of the residual of the divergence constraint
/// $ ||DIV(u) - f|| = sqrt( \int (DIV(u) - f)^2 dx ) $
template<typename VEC>
scalar_type
asm_div_error_L2_norm( const VEC &U, const mesh_fem &mf_u,
const VEC &F, const mesh_fem &mf_f,
const mesh_im &mim,
const mesh_region &rg = mesh_region::all_convexes() )
{
GMM_ASSERT1(mf_u.get_qdim() > 1, "invalid data mesh fem (Qdim>1 required)");
GMM_ASSERT1(mf_f.get_qdim() == 1, "invalid data mesh fem (Qdim=1 required)");
GMM_ASSERT1(U.size() == mf_u.nb_dof(), "invalid vector data (size=mf.nb_dof() required)");
GMM_ASSERT1(F.size() == mf_f.nb_dof(), "invalid vector data (size=mf.nb_dof() required)");
GMM_ASSERT1(U.size() == (F.size()*mf_u.get_qdim()), "invalid vector data (U.size=Qdim*F.size required)" );
generic_assembly
assem("u=data$1(#1);"
"f=data$2(#2);"
"V()+=u(i).u(j).comp(vGrad(#1).vGrad(#1))(i,k,k,j,h,h)"
"-u(i).f(j).comp(vGrad(#1).Base(#2))(i,k,k,j)"
"-f(i).u(j).comp(Base(#2).vGrad(#1))(i,j,k,k)"
"+f(i).f(j).comp(Base(#2).Base(#2))(i,j);");
assem.push_mi(mim);
assem.push_mf(mf_u);
assem.push_mf(mf_f);
assem.push_data(U);
assem.push_data(F);
std::vector<scalar_type> v(1);
assem.push_vec(v);
assem.assembly(rg);
return sqrt(v[0]);
}
(?) The results are quite different from those of b.1 (also if the order is the same) so I don't know what to trust - if at least one is correct!
That's all! I am very sorry if it was too long and boring.
You can also use here the high level generic assembly instead of the low-level one, this should be simpler. You can use the assembly string "sqr(Trace(Grad_u) - f)"
OK, I will try that approach too!
Apart from that, do you think this implementation should work?
I am not sure about the index reduction:
"V()+=u(i).u(j).comp(vGrad(#1).vGrad(#1))(i,k,k,j,h,h)".
Yves.
Many thanks.
Best,
Domenico
user id: domenico_notaro
---
Domenico Notaro
mobile: (+1) 412 983 0940
address: 3765 Childs Street, Pittsburgh PA 15213
mailto:
address@hidden
Skype: domenico.not
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