void Diffusion::flux_matrix(const FEMesh *mesh,
const Element *el,
const ElementFuncNodeIterator &j,
const Flux *flux,
const MasterPosition &pt,
double time,
SmallSystem *fluxdata)
const
{
// The atom flux matrix M_{ij} multiplies the vector of nodal
// concentrations to give the vector atom current J at point pt.
// M_{ij} = -D_{ik} grad_k N_j
// J_i = -M_{ij} T_j
// where N_j is the shapefunction at node j.
// printf("flux");
if (*flux != *atom_flux) {
throw ErrProgrammingError("Unexpected flux", __FILE__, __LINE__);
}
double sf = j.shapefunction( pt );
double dsf0 = j.dshapefunction( 0, pt );
double dsf1 = j.dshapefunction( 1, pt );
#if DIM==3
double dsf2 = j.dshapefunction( 2, pt );
#endif
const SymmMatrix3 cond( conductivitytensor( mesh, el, pt ) );
// Loop over flux components. Loop over all components, even if
// the flux is in-plane, because the out-of-plane components of
// the flux matrix are used to construct the constraint equation.
for(VectorFieldIterator i; !i.end(); ++i){
#if DIM==2
// in-plane concentration gradient contributions
fluxdata->stiffness_matrix_element( i, concentration, j ) -=
cond(i.integer(), 0) * dsf0 + cond(i.integer(), 1) * dsf1;
// out-of-plane concentration gradient contribution
if(!concentration->in_plane(mesh))
fluxdata->stiffness_matrix_element(i, concentration->out_of_plane(), j)
-= cond(i.integer(), 2) * sf;
#elif DIM==3
fluxdata->stiffness_matrix_element( i, concentration, j ) -=
cond( i.integer(), 0 ) * dsf0 +
cond( i.integer(), 1 ) * dsf1 +
cond( i.integer(), 2 ) * dsf2;
#endif
}
} // end of 'Diffusion::flux_matrix'
void CLargeStrainElasticity::flux_matrix(const FEMesh *mesh,
const Element *element,
const ElementFuncNodeIterator &node,
const Flux *flux,
const MasterPosition &pt,
double time,
SmallSystem *fluxmtx) const
{
int (*ij2voigt)(int,int) = &SymTensorIndex::ij2voigt; // shorter func name
SmallMatrix dU(3); // gradient of displacement
double Fval; // the value of the shape function (for node)
DoubleVec dF(3); // and its derivative at the given pt
bool inplane = false; // needed both in 2D & 3D versions regardless,
// passed to contract_C_dU_dF
#if DIM==2
// in 2D, check if it is an in-plane eqn or a plane-flux eqn.
static CompoundField *displacement =
dynamic_cast<CompoundField*>(Field::getField("Displacement"));
inplane = displacement->in_plane( mesh );
#endif
// check for unexpected flux, flux should be a stress flux
if (*flux != *stress_flux) {
throw ErrProgrammingError("Unexpected flux", __FILE__, __LINE__);
}
// evaluate the shape function and its gradient (of node) at the given pt
Fval = node.shapefunction( pt ); // value of the shape function
dF[0] = node.dshapefunction( 0, pt ); // x-deriv of the shape function
dF[1] = node.dshapefunction( 1, pt ); // y-deriv of the shape function
#if DIM==3
dF[2] = node.dshapefunction( 2, pt ); // z-deriv of the shape function
#endif
computeDisplacementGradient( mesh, element, pt, dU );
const Cijkl CC = cijkl( mesh, element, pt ); // elasticity modulus
// add the flux contributions to stiffness matrix element
// k_indx is needed for fluxmtx->stifness_matrix_element function,
// which does not take int k as argument
VectorFieldIndex k_indx;
for (SymTensorIterator ij_iter; !ij_iter.end(); ++ij_iter) {
int k0, k1, k2, ij = ij_iter.integer();
double nonlinear_part; // to store the sum from the nonlinear terms
// TODO: Use tensor iterators for k0, k1, k2.
#if DIM==2
// sum CC(i,j,k,l)*dF(l), k=0 over l=0,1, then add to stiffness_mtx
k_indx.set( 0 );
k0 = ij2voigt( 0,0 );
k1 = ij2voigt( 0,1 );
nonlinear_part = contract_C_dU_dF(CC, dU, dF, ij, 0, inplane ); // at ij, k=0
fluxmtx->stiffness_matrix_element( ij_iter, displacement, k_indx, node )
+= CC( ij,k0 ) * dF[0] + CC( ij,k1 ) * dF[1]
+ nonlinear_part;
// sum CC(i,j,k,l)*dF(l), k=1 over l=0,1, then add to stiffness_mtx
k_indx.set( 1 );
k0 = ij2voigt( 1,0 );
k1 = ij2voigt( 1,1 );
nonlinear_part = contract_C_dU_dF( CC, dU, dF, ij, 1, inplane ); // at ij, k=1
fluxmtx->stiffness_matrix_element( ij_iter, displacement, k_indx, node )
+= CC( ij,k0 ) * dF[0] + CC( ij,k1 ) * dF[1]
+ nonlinear_part;
#elif DIM==3
// sum CC(i,j,k,l)*dF(l), k=0 over l=0,1,2 then add to stiffness_mtx
k_indx.set( 0 );
k0 = ij2voigt( 0,0 );
k1 = ij2voigt( 0,1 );
k2 = ij2voigt( 0,2 );
nonlinear_part = contract_C_dU_dF( CC, dU, dF, ij, 0, inplane ); // at ij, k=0
fluxmtx->stiffness_matrix_element( ij_iter, displacement, k_indx, node )
+= CC( ij,k0 ) * dF[0] + CC( ij,k1 ) * dF[1] + CC( ij,k2 ) * dF[2]
+ nonlinear_part;
// sum CC(i,j,k,l)*dF(l), k=1 over l=0,1,2 then add to stiffness_mtx
k_indx.set( 1 );
k0 = ij2voigt( 1,0 );
k1 = ij2voigt( 1,1 );
k2 = ij2voigt( 1,2 );
nonlinear_part = contract_C_dU_dF( CC, dU, dF, ij, 1, inplane ); // at ij, k=1
fluxmtx->stiffness_matrix_element( ij_iter, displacement, k_indx, node )
+= CC( ij,k0 ) * dF[0] + CC( ij,k1 ) * dF[1] + CC( ij,k2 ) * dF[2]
+ nonlinear_part;
// sum CC(i,j,k,l)*dF(l), k=2 over l=0,1,2 then add to stiffness_mtx
k_indx.set( 2 );
k0 = ij2voigt( 2,0 );
k1 = ij2voigt( 2,1 );
k2 = ij2voigt( 2,2 );
nonlinear_part = contract_C_dU_dF( CC, dU, dF, ij, 2, inplane ); // at ij, k=2
fluxmtx->stiffness_matrix_element( ij_iter, displacement, k_indx, node )
+= CC( ij,k0 ) * dF[0] + CC( ij,k1 ) * dF[1] + CC( ij,k2 ) * dF[2]
+ nonlinear_part;
#endif
#if DIM==2
if ( !inplane ) // now contributions from z-deriv of displacement field
{
Field *disp_z_deriv = displacement->out_of_plane();
for(IteratorP k_iter = disp_z_deriv->iterator( ALL_INDICES ); !k_iter.end(); ++k_iter)
{
double diag_factor = ( k_iter.integer()==2 ? 1.0 : 0.5 );
k2 = ij2voigt( 2, k_iter.integer() );
fluxmtx->stiffness_matrix_element( ij_iter, disp_z_deriv, k_iter, node )
+= diag_factor * Fval * CC( ij,k2 );
}
} // end of 'if (!inplane)'
#endif
} // end of loop over ij
} // end of 'CLargeStrainElasticity::flux_matrix'