Real
AuxChemElastic::computeDifferential(const Real & coupled_conserved, const Real & coupled_nonconserved)
{
// partial derivative of f_chem with respect to conserved variable
Real dfchem_dcons(0.0);
// partial derivative of f_chem with respect to nonconserved variable
Real dfchem_dnoncons(0.0);
// partial derivatives of self-elastic energies
Real dself_dcons(0.0);
Real dself_dnoncons(0.0);
// partial derivative of interaction elastic energy
Real dint_dcons(0.0);
Real dint_dnoncons(0.0);
Real first_term(0.0);
Real second_term(0.0);
dfchem_dcons = computeDfchemDcons(coupled_conserved, coupled_nonconserved);
dself_dcons = computeDselfDcons();
dint_dcons = computeDintDcons();
dfchem_dnoncons = computeDfchemDnoncons(coupled_conserved, coupled_nonconserved);
dself_dnoncons = computeDselfDnoncons();
dint_dnoncons = computeDintDnoncons();
first_term = (dfchem_dcons + dself_dcons + dint_dcons)*(_precip_conserved - coupled_conserved);
second_term = (dfchem_dnoncons + dself_dnoncons + dint_dnoncons)*(_precip_nonconserved - coupled_nonconserved);
return first_term + second_term;
}
Real
AuxChem::computeDifferential(const Real & coupled_conserved, const Real & coupled_nonconserved)
{
// partial derivative of f_chem with respect to conserved variable
Real dfchem_dcons(0.0);
// partial derivative of f_chem with respect to nonconserved variable
Real dfchem_dnoncons(0.0);
Real first_term(0.0);
Real second_term(0.0);
dfchem_dcons = computeDfchemDcons(coupled_conserved, coupled_nonconserved);
dfchem_dnoncons = computeDfchemDnoncons(coupled_conserved, coupled_nonconserved);
first_term = (dfchem_dcons)*(_precip_conserved - coupled_conserved);
second_term = (dfchem_dnoncons)*(_precip_nonconserved - coupled_nonconserved);
return first_term + second_term;
}
/// @brief
/// Main evaluation routine. Computes the inverse of the
/// matrix representation of the mimetic inner product in a
/// single cell with kown permeability @f$K@f$. Adds a
/// regularization term in order to guarantee a positive
/// definite matrix.
///
/// @tparam RockInterface
/// Type representing rock properties. Assumed to
/// expose a method @code permeability(i) @endcode which
/// retrieves the static permeability tensor of cell @code
/// i @endcode. The permeability tensor, @$K@$, is in
/// turn, assumed to expose a method @code operator()(int
/// i, int j) @endcode such that the call @code K(i,j)
/// @endcode retrieves the @f$ij@f$'th component of the
/// cell permeability @f$K@f$.
///
/// @param [in] c
/// Cell for which to evaluate the inverse of the mimetic
/// inner product.
///
/// @param [in] r
/// Specific reservoir properties. Only the permeability
/// is used in method @code buildMatrix() @endcode.
///
/// @param [in] nf
/// Number of faces (i.e., number of neighbours) of cell
/// @code *c @endcode.
void buildStaticContrib(const CellIter& c,
const RockInterface& r,
const typename CellIter::Vector& grav,
const int nf)
{
// Binv = (N*lambda*K*N' + t*diag(A)*(I - Q*Q')*diag(A))/vol
// ^ ^^^^^^^^^^^^^^^^^^^^^^^^^^
// precompute: n_ precompute: second_term_
// t = 6/dim * trace(lambda*K)
typedef typename CellIter::FaceIterator FI;
typedef typename CellIter::Vector CV;
typedef typename FI ::Vector FV;
// Now we need to remember the rocks, since we will need
// the permeability for dynamic assembly.
prock_ = &r;
const int ci = c->index();
static_assert (FV::dimension == int(dim), "");
assert (int(t1_.size()) >= nf * dim);
assert (int(t2_.size()) >= nf * dim);
assert (int(fa_.size()) >= nf * nf);
SharedFortranMatrix T2 (nf, dim, &t2_ [0]);
SharedFortranMatrix fa (nf, nf , &fa_ [0]);
SharedFortranMatrix second_term(nf, nf, &second_term_[ci][0]);
SharedFortranMatrix n(nf, dim, &n_[ci][0]);
// Clear matrices of any residual data.
zero(second_term); zero(n); zero(T2); zero(fa);
// Setup: second_term <- I, n <- N, T2 <- C
const CV cc = c->centroid();
int i = 0;
for (FI f = c->facebegin(); f != c->faceend(); ++f, ++i) {
second_term(i,i) = Scalar(1.0);
fa(i,i) = f->area();
FV fc = f->centroid(); fc -= cc; fc *= fa(i,i);
FV fn = f->normal (); fn *= fa(i,i);
for (int j = 0; j < dim; ++j) {
n (i,j) = fn[j];
T2(i,j) = fc[j];
}
}
assert (i == nf);
// T2 <- orth(T2)
if (orthogonalizeColumns(T2) != 0) {
assert (false);
}
// second_term <- second_term - T2*T2' == I - Q*Q'
symmetricUpdate(Scalar(-1.0), T2, Scalar(1.0), second_term);
// second_term <- diag(A) * second_term * diag(A)
symmetricUpdate(fa, second_term);
// Gravity term: Kg_ = K * grav
vecMulAdd_N(Scalar(1.0), r.permeability(ci), &grav[0],
Scalar(0.0), &Kg_[ci][0]);
}
