/**
* This function assemble the stiffnes matrix KK and the residual vector Res
* Using automatic differentiation for Newton iterative scheme
* J(u0) w = - F(u0) ,
* with u = u0 + w
* - F = f(x) - J u = Res
* J = \grad_u F
*
* thus
* J w = f(x) - J u0
**/
void AssemblePoissonProblem_AD(MultiLevelProblem& ml_prob) {
// ml_prob is the global object from/to where get/set all the data
// level is the level of the PDE system to be assembled
// levelMax is the Maximum level of the MultiLevelProblem
// assembleMatrix is a flag that tells if only the residual or also the matrix should be assembled
// call the adept stack object
adept::Stack& s = FemusInit::_adeptStack;
// extract pointers to the several objects that we are going to use
LinearImplicitSystem* mlPdeSys = &ml_prob.get_system<LinearImplicitSystem> ("Poisson"); // pointer to the linear implicit system named "Poisson"
const unsigned level = mlPdeSys->GetLevelToAssemble();
Mesh* msh = ml_prob._ml_msh->GetLevel(level); // pointer to the mesh (level) object
elem* el = msh->el; // pointer to the elem object in msh (level)
MultiLevelSolution* mlSol = ml_prob._ml_sol; // pointer to the multilevel solution object
Solution* sol = ml_prob._ml_sol->GetSolutionLevel(level); // pointer to the solution (level) object
LinearEquationSolver* pdeSys = mlPdeSys->_LinSolver[level]; // pointer to the equation (level) object
SparseMatrix* KK = pdeSys->_KK; // pointer to the global stifness matrix object in pdeSys (level)
NumericVector* RES = pdeSys->_RES; // pointer to the global residual vector object in pdeSys (level)
const unsigned dim = msh->GetDimension(); // get the domain dimension of the problem
unsigned dim2 = (3 * (dim - 1) + !(dim - 1)); // dim2 is the number of second order partial derivatives (1,3,6 depending on the dimension)
const unsigned maxSize = static_cast< unsigned >(ceil(pow(3, dim))); // conservative: based on line3, quad9, hex27
unsigned iproc = msh->processor_id(); // get the process_id (for parallel computation)
//solution variable
unsigned soluIndex;
soluIndex = mlSol->GetIndex("u"); // get the position of "u" in the ml_sol object
unsigned soluType = mlSol->GetSolutionType(soluIndex); // get the finite element type for "u"
unsigned soluPdeIndex;
soluPdeIndex = mlPdeSys->GetSolPdeIndex("u"); // get the position of "u" in the pdeSys object
vector < adept::adouble > solu; // local solution
solu.reserve(maxSize);
vector < vector < double > > x(dim); // local coordinates
unsigned xType = 2; // get the finite element type for "x", it is always 2 (LAGRANGE QUADRATIC)
for (unsigned i = 0; i < dim; i++) {
x[i].reserve(maxSize);
}
vector <double> phi; // local test function
vector <double> phi_x; // local test function first order partial derivatives
vector <double> phi_xx; // local test function second order partial derivatives
double weight; // gauss point weight
phi.reserve(maxSize);
phi_x.reserve(maxSize * dim);
phi_xx.reserve(maxSize * dim2);
vector< adept::adouble > aRes; // local redidual vector
aRes.reserve(maxSize);
vector< int > l2GMap; // local to global mapping
l2GMap.reserve(maxSize);
vector< double > Res; // local redidual vector
Res.reserve(maxSize);
vector < double > Jac;
Jac.reserve(maxSize * maxSize);
KK->zero(); // Set to zero all the entries of the Global Matrix
// element loop: each process loops only on the elements that owns
for (int iel = msh->_elementOffset[iproc]; iel < msh->_elementOffset[iproc + 1]; iel++) {
short unsigned ielGeom = msh->GetElementType(iel);
unsigned nDofu = msh->GetElementDofNumber(iel, soluType); // number of solution element dofs
unsigned nDofx = msh->GetElementDofNumber(iel, xType); // number of coordinate element dofs
// resize local arrays
l2GMap.resize(nDofu);
solu.resize(nDofu);
for (int i = 0; i < dim; i++) {
x[i].resize(nDofx);
}
aRes.resize(nDofu); //resize
std::fill(aRes.begin(), aRes.end(), 0); //set aRes to zero
// local storage of global mapping and solution
for (unsigned i = 0; i < nDofu; i++) {
unsigned solDof = msh->GetSolutionDof(i, iel, soluType); // global to global mapping between solution node and solution dof
solu[i] = (*sol->_Sol[soluIndex])(solDof); // global extraction and local storage for the solution
l2GMap[i] = pdeSys->GetSystemDof(soluIndex, soluPdeIndex, i, iel); // global to global mapping between solution node and pdeSys dof
}
// local storage of coordinates
for (unsigned i = 0; i < nDofx; i++) {
unsigned xDof = msh->GetSolutionDof(i, iel, xType); // global to global mapping between coordinates node and coordinate dof
for (unsigned jdim = 0; jdim < dim; jdim++) {
x[jdim][i] = (*msh->_topology->_Sol[jdim])(xDof); // global extraction and local storage for the element coordinates
}
}
// start a new recording of all the operations involving adept::adouble variables
s.new_recording();
// *** Gauss point loop ***
for (unsigned ig = 0; ig < msh->_finiteElement[ielGeom][soluType]->GetGaussPointNumber(); ig++) {
// *** get gauss point weight, test function and test function partial derivatives ***
msh->_finiteElement[ielGeom][soluType]->Jacobian(x, ig, weight, phi, phi_x, phi_xx);
// evaluate the solution, the solution derivatives and the coordinates in the gauss point
adept::adouble solu_gss = 0;
vector < adept::adouble > gradSolu_gss(dim, 0.);
vector < double > x_gss(dim, 0.);
for (unsigned i = 0; i < nDofu; i++) {
solu_gss += phi[i] * solu[i];
for (unsigned jdim = 0; jdim < dim; jdim++) {
gradSolu_gss[jdim] += phi_x[i * dim + jdim] * solu[i];
x_gss[jdim] += x[jdim][i] * phi[i];
}
}
// *** phi_i loop ***
for (unsigned i = 0; i < nDofu; i++) {
adept::adouble laplace = 0.;
for (unsigned jdim = 0; jdim < dim; jdim++) {
laplace += phi_x[i * dim + jdim] * gradSolu_gss[jdim];
}
double srcTerm = - GetExactSolutionLaplace(x_gss);
aRes[i] += (srcTerm * phi[i] - laplace) * weight;
} // end phi_i loop
} // end gauss point loop
//--------------------------------------------------------------------------------------------------------
// Add the local Matrix/Vector into the global Matrix/Vector
//copy the value of the adept::adoube aRes in double Res and store
Res.resize(nDofu); //resize
for (int i = 0; i < nDofu; i++) {
Res[i] = - aRes[i].value();
}
RES->add_vector_blocked(Res, l2GMap);
// define the dependent variables
s.dependent(&aRes[0], nDofu);
// define the independent variables
s.independent(&solu[0], nDofu);
// get the jacobian matrix (ordered by row major )
Jac.resize(nDofu * nDofu); //resize
s.jacobian(&Jac[0], true);
//store K in the global matrix KK
KK->add_matrix_blocked(Jac, l2GMap, l2GMap);
s.clear_independents();
s.clear_dependents();
} //end element loop for each process
RES->close();
KK->close();
// ***************** END ASSEMBLY *******************
}
void GetL2Norm ( MultiLevelProblem& ml_prob, MultiLevelProblem& ml_prob2 )
{
LinearImplicitSystem* mlPdeSys = &ml_prob.get_system<LinearImplicitSystem> ( "NonLocal" );
const unsigned level = mlPdeSys->GetLevelToAssemble();
Mesh* msh = ml_prob._ml_msh->GetLevel ( level );
elem* el = msh->el;
MultiLevelSolution* mlSol = ml_prob._ml_sol;
Solution* sol = ml_prob._ml_sol->GetSolutionLevel ( level );
LinearImplicitSystem* mlPdeSys2 = &ml_prob2.get_system<LinearImplicitSystem> ( "Local" );
MultiLevelSolution* mlSol2 = ml_prob2._ml_sol;
Solution* sol2 = ml_prob2._ml_sol->GetSolutionLevel ( level );
const unsigned dim = msh->GetDimension();
unsigned xType = 2; // get the finite element type for "x", it is always 2 (LAGRANGE QUADRATIC)
double error_solExact_norm2 = 0.;
double error_solExact_local_norm2 = 0.;
double error_solLocal_norm2 = 0.;
double solNonlocal_norm2 = 0.;
double solLocal_norm2 = 0.;
double sol_exact_norm2 = 0.;
unsigned soluIndex;
soluIndex = mlSol->GetIndex ( "u" );
unsigned soluType = mlSol->GetSolutionType ( soluIndex );
unsigned soluIndexLocal;
soluIndexLocal = mlSol2->GetIndex ( "u_local" );
unsigned iproc = msh->processor_id();
unsigned nprocs = msh->n_processors();
for ( int iel = msh->_elementOffset[iproc]; iel < msh->_elementOffset[iproc + 1]; iel++ ) {
short unsigned ielGeom = msh->GetElementType ( iel );
unsigned nDofu = msh->GetElementDofNumber ( iel, soluType );
unsigned nDofx = msh->GetElementDofNumber ( iel, xType );
vector < vector < double > > x1 ( dim );
for ( int i = 0; i < dim; i++ ) {
x1[i].resize ( nDofx );
}
vector < double > soluNonLoc ( nDofu );
vector < double > soluLoc ( nDofu );
for ( unsigned i = 0; i < nDofu; i++ ) {
unsigned solDof = msh->GetSolutionDof ( i, iel, soluType );
soluNonLoc[i] = ( *sol->_Sol[soluIndex] ) ( solDof );
soluLoc[i] = ( *sol2->_Sol[soluIndexLocal] ) ( solDof );
}
for ( unsigned i = 0; i < nDofx; i++ ) {
unsigned xDof = msh->GetSolutionDof ( i, iel, xType );
for ( unsigned jdim = 0; jdim < dim; jdim++ ) {
x1[jdim][i] = ( *msh->_topology->_Sol[jdim] ) ( xDof );
}
}
vector <double> phi; // local test function
vector <double> phi_x; // local test function first order partial derivatives
double weight; // gauss point weight
// *** Gauss point loop ***
for ( unsigned ig = 0; ig < msh->_finiteElement[ielGeom][soluType]->GetGaussPointNumber(); ig++ ) {
// *** get gauss point weight, test function and test function partial derivatives ***
msh->_finiteElement[ielGeom][soluType]->Jacobian ( x1, ig, weight, phi, phi_x );
double soluNonLoc_gss = 0.;
double soluLoc_gss = 0.;
double exactSol_gss_x = 0.;
double exactSol_gss_y = 0.;
for ( unsigned i = 0; i < nDofu; i++ ) {
soluNonLoc_gss += phi[i] * soluNonLoc[i];
soluLoc_gss += phi[i] * soluLoc[i];
exactSol_gss_x += phi[i] * x1[0][i]; // this is x at the Gauss point
// exactSol_gss_y += phi[i] * x1[1][i]; // this is y at the Gauss point
}
exactSol_gss_x = exactSol_gss_x * exactSol_gss_x * exactSol_gss_x * exactSol_gss_x + 0.1 * exactSol_gss_x * exactSol_gss_x; // this is x^4 + delta * x^2
// exactSol_gss_x = exactSol_gss_x * exactSol_gss_x; // this is x^2
// exactSol_gss_x = exactSol_gss_x * exactSol_gss_x * exactSol_gss_x; // this is x^3
// exactSol_gss_y = exactSol_gss_y * exactSol_gss_y * exactSol_gss_y; // this is y^3
// exactSol_gss_x = exactSol_gss_x * exactSol_gss_x * exactSol_gss_x * exactSol_gss_x; // this is x^4
// exactSol_gss_x = 2 * exactSol_gss_x + exactSol_gss_x * exactSol_gss_x * exactSol_gss_x * exactSol_gss_x * exactSol_gss_x ; // this is 2x + x^5
error_solExact_norm2 += ( soluNonLoc_gss - exactSol_gss_x ) * ( soluNonLoc_gss - exactSol_gss_x ) * weight;
// error_solExact_norm2 += (soluNonLoc_gss - (exactSol_gss_x + exactSol_gss_y)) * (soluNonLoc_gss - (exactSol_gss_x + exactSol_gss_y)) * weight; //error L2 norm of x^3 + y^3
error_solExact_local_norm2 += ( soluLoc_gss - exactSol_gss_x ) * ( soluLoc_gss - exactSol_gss_x ) * weight;
error_solLocal_norm2 += ( soluNonLoc_gss - soluLoc_gss ) * ( soluNonLoc_gss - soluLoc_gss ) * weight;
solNonlocal_norm2 += soluNonLoc_gss * soluNonLoc_gss * weight;
solLocal_norm2 += soluLoc_gss * soluLoc_gss * weight;
// sol_exact_norm2 += (exactSol_gss_x + exactSol_gss_y) * (exactSol_gss_x + exactSol_gss_y) * weight; //L2 norm of x^3 + y^3
sol_exact_norm2 += exactSol_gss_x * exactSol_gss_x * weight;
}
}
double norm2 = 0.;
MPI_Allreduce ( &error_solExact_norm2, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
double norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "L2 norm of ERROR: Nonlocal - exact = " << norm << std::endl;
norm2 = 0.;
MPI_Allreduce ( &error_solExact_local_norm2, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "L2 norm of ERROR: Local - exact = " << norm << std::endl;
norm2 = 0.;
MPI_Allreduce ( &error_solLocal_norm2, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "L2 norm of ERROR: Nonlocal - local = " << norm << std::endl;
norm2 = 0.;
MPI_Allreduce ( &solNonlocal_norm2, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "L2 norm of NONLOCAL soln = " << norm << std::endl;
norm2 = 0.;
MPI_Allreduce ( &solLocal_norm2, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "L2 norm of LOCAL soln = " << norm << std::endl;
norm2 = 0.;
MPI_Allreduce ( &sol_exact_norm2, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "L2 norm of EXACT soln = " << norm << std::endl;
double littleL2norm = 0.;
std::vector<double> littleLInfinitynorm ( nprocs, 0. );
for ( unsigned i = msh->_dofOffset[soluType][iproc]; i < msh->_dofOffset[soluType][iproc + 1]; i++ ) {
double nonLocalNodalValue = ( *sol->_Sol[soluIndex] ) ( i );
double LocalNodalValue = ( *sol2->_Sol[soluIndexLocal] ) ( i );
double difference = fabs ( nonLocalNodalValue - LocalNodalValue );
if ( difference > littleLInfinitynorm[iproc] ) littleLInfinitynorm[iproc] = difference;
littleL2norm += difference * difference;
}
norm2 = 0.;
MPI_Allreduce ( &littleL2norm, &norm2, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD );
norm = sqrt ( norm2 );
std::cout.precision ( 14 );
std::cout << "l2 norm of ERROR: Nonlocal - local = " << norm << std::endl;
for ( int kproc = 0; kproc < nprocs; kproc++ ) {
MPI_Bcast ( &littleLInfinitynorm[iproc], 1, MPI_DOUBLE, kproc, MPI_COMM_WORLD );
}
double littleLInfinityNorm = littleLInfinitynorm[0];
for ( unsigned kproc = 0; kproc < nprocs; kproc++ ) {
if ( littleLInfinitynorm[kproc] > littleLInfinityNorm ) littleLInfinityNorm = littleLInfinitynorm[kproc];
}
std::cout.precision ( 14 );
std::cout << "linfinity norm of ERROR: Nonlocal - local = " << littleLInfinityNorm << std::endl;
}
/**
* This function assemble the stiffnes matrix Jac and the residual vector Res
* such that
* Jac w = RES = F - Jac u0,
* and consequently
* u = u0 + w satisfies Jac u = F
**/
void AssemblePoissonProblem(MultiLevelProblem& ml_prob) {
// ml_prob is the global object from/to where get/set all the data
// level is the level of the PDE system to be assembled
// levelMax is the Maximum level of the MultiLevelProblem
// assembleMatrix is a flag that tells if only the residual or also the matrix should be assembled
// extract pointers to the several objects that we are going to use
LinearImplicitSystem* mlPdeSys = &ml_prob.get_system<LinearImplicitSystem> ("Poisson"); // pointer to the linear implicit system named "Poisson"
const unsigned level = mlPdeSys->GetLevelToAssemble();
const unsigned levelMax = mlPdeSys->GetLevelMax();
const bool assembleMatrix = mlPdeSys->GetAssembleMatrix();
Mesh* msh = ml_prob._ml_msh->GetLevel(level); // pointer to the mesh (level) object
elem* el = msh->el; // pointer to the elem object in msh (level)
MultiLevelSolution* mlSol = ml_prob._ml_sol; // pointer to the multilevel solution object
Solution* sol = ml_prob._ml_sol->GetSolutionLevel(level); // pointer to the solution (level) object
LinearEquationSolver* pdeSys = mlPdeSys->_LinSolver[level]; // pointer to the equation (level) object
SparseMatrix* KK = pdeSys->_KK; // pointer to the global stifness matrix object in pdeSys (level)
NumericVector* RES = pdeSys->_RES; // pointer to the global residual vector object in pdeSys (level)
const unsigned dim = msh->GetDimension(); // get the domain dimension of the problem
unsigned dim2 = (3 * (dim - 1) + !(dim - 1)); // dim2 is the number of second order partial derivatives (1,3,6 depending on the dimension)
const unsigned maxSize = static_cast< unsigned >(ceil(pow(3, dim))); // conservative: based on line3, quad9, hex27
unsigned iproc = msh->processor_id(); // get the process_id (for parallel computation)
//solution variable
unsigned soluIndex;
soluIndex = mlSol->GetIndex("u"); // get the position of "u" in the ml_sol object
unsigned soluType = mlSol->GetSolutionType(soluIndex); // get the finite element type for "u"
unsigned soluPdeIndex;
soluPdeIndex = mlPdeSys->GetSolPdeIndex("u"); // get the position of "u" in the pdeSys object
vector < double > solu; // local solution
solu.reserve(maxSize);
vector < vector < double > > x(dim); // local coordinates
unsigned xType = 2; // get the finite element type for "x", it is always 2 (LAGRANGE QUADRATIC)
for (unsigned i = 0; i < dim; i++) {
x[i].reserve(maxSize);
}
vector <double> phi; // local test function
vector <double> phi_x; // local test function first order partial derivatives
vector <double> phi_xx; // local test function second order partial derivatives
double weight; // gauss point weight
phi.reserve(maxSize);
phi_x.reserve(maxSize * dim);
phi_xx.reserve(maxSize * dim2);
vector< double > Res; // local redidual vector
Res.reserve(maxSize);
vector< int > l2GMap; // local to global mapping
l2GMap.reserve(maxSize);
vector < double > Jac;
Jac.reserve(maxSize * maxSize);
if (assembleMatrix)
KK->zero(); // Set to zero all the entries of the Global Matrix
// element loop: each process loops only on the elements that owns
for (int iel = msh->IS_Mts2Gmt_elem_offset[iproc]; iel < msh->IS_Mts2Gmt_elem_offset[iproc + 1]; iel++) {
unsigned kel = msh->IS_Mts2Gmt_elem[iel]; // mapping between paralell dof and mesh dof
short unsigned kelGeom = el->GetElementType(kel); // element geometry type
unsigned nDofu = el->GetElementDofNumber(kel, soluType); // number of solution element dofs
unsigned nDofx = el->GetElementDofNumber(kel, xType); // number of coordinate element dofs
// resize local arrays
l2GMap.resize(nDofu);
solu.resize(nDofu);
for (int i = 0; i < dim; i++) {
x[i].resize(nDofx);
}
Res.resize(nDofu); //resize
std::fill(Res.begin(), Res.end(), 0); //set Res to zero
Jac.resize(nDofu * nDofu); //resize
std::fill(Jac.begin(), Jac.end(), 0); //set Jac to zero
// local storage of global mapping and solution
for (unsigned i = 0; i < nDofu; i++) {
unsigned iNode = el->GetMeshDof(kel, i, soluType); // local to global solution node
unsigned solDof = msh->GetMetisDof(iNode, soluType); // global to global mapping between solution node and solution dof
solu[i] = (*sol->_Sol[soluIndex])(solDof); // global extraction and local storage for the solution
l2GMap[i] = pdeSys->GetKKDof(soluIndex, soluPdeIndex, iNode); // global to global mapping between solution node and pdeSys dof
}
// local storage of coordinates
for (unsigned i = 0; i < nDofx; i++) {
unsigned iNode = el->GetMeshDof(kel, i, xType); // local to global coordinates node
unsigned xDof = msh->GetMetisDof(iNode, xType); // global to global mapping between coordinates node and coordinate dof
for (unsigned jdim = 0; jdim < dim; jdim++) {
x[jdim][i] = (*msh->_coordinate->_Sol[jdim])(xDof); // global extraction and local storage for the element coordinates
}
}
if (level == levelMax || !el->GetRefinedElementIndex(kel)) { // do not care about this if now (it is used for the AMR)
// *** Gauss point loop ***
for (unsigned ig = 0; ig < msh->_finiteElement[kelGeom][soluType]->GetGaussPointNumber(); ig++) {
// *** get gauss point weight, test function and test function partial derivatives ***
msh->_finiteElement[kelGeom][soluType]->Jacobian(x, ig, weight, phi, phi_x, phi_xx);
// evaluate the solution, the solution derivatives and the coordinates in the gauss point
double solu_gss = 0;
vector < double > gradSolu_gss(dim, 0.);
vector < double > x_gss(dim, 0.);
for (unsigned i = 0; i < nDofu; i++) {
solu_gss += phi[i] * solu[i];
for (unsigned jdim = 0; jdim < dim; jdim++) {
gradSolu_gss[jdim] += phi_x[i * dim + jdim] * solu[i];
x_gss[jdim] += x[jdim][i] * phi[i];
}
}
// *** phi_i loop ***
for (unsigned i = 0; i < nDofu; i++) {
double laplace = 0.;
for (unsigned jdim = 0; jdim < dim; jdim++) {
laplace += phi_x[i * dim + jdim] * gradSolu_gss[jdim];
}
double srcTerm = - GetExactSolutionLaplace(x_gss);
Res[i] += (srcTerm * phi[i] - laplace) * weight;
if (assembleMatrix) {
// *** phi_j loop ***
for (unsigned j = 0; j < nDofu; j++) {
laplace = 0.;
for (unsigned kdim = 0; kdim < dim; kdim++) {
laplace += (phi_x[i * dim + kdim] * phi_x[j * dim + kdim]) * weight;
}
Jac[i * nDofu + j] += laplace;
} // end phi_j loop
} // endif assemble_matrix
} // end phi_i loop
} // end gauss point loop
} // endif single element not refined or fine grid loop
//--------------------------------------------------------------------------------------------------------
// Add the local Matrix/Vector into the global Matrix/Vector
//copy the value of the adept::adoube aRes in double Res and store
RES->add_vector_blocked(Res, l2GMap);
if (assembleMatrix) {
//store K in the global matrix KK
KK->add_matrix_blocked(Jac, l2GMap, l2GMap);
}
} //end element loop for each process
RES->close();
if (assembleMatrix) KK->close();
// ***************** END ASSEMBLY *******************
}
