void AssembleWillmoreFlow_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
TransientNonlinearImplicitSystem* mlPdeSys = &ml_prob.get_system<TransientNonlinearImplicitSystem> ("Willmore"); // 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 iproc = msh->processor_id(); // get the process_id (for parallel computation)
//solution variable
unsigned solRIndex[3];
solRIndex[0] = mlSol->GetIndex("X"); // get the position of "X" in the ml_sol object
solRIndex[1] = mlSol->GetIndex("Y"); // get the position of "Y" in the ml_sol object
solRIndex[2] = mlSol->GetIndex("Z"); // get the position of "Z" in the ml_sol object
unsigned solRType[3];
solRType[0]= mlSol->GetSolutionType(solRIndex[0]); // get the finite element type for "R"
solRType[1]= mlSol->GetSolutionType(solRIndex[1]); // get the finite element type for "R"
solRType[2]= mlSol->GetSolutionType(solRIndex[2]); // get the finite element type for "R"
unsigned solRPdeIndex[3];
solRPdeIndex[0] = mlPdeSys->GetSolPdeIndex("X"); // get the position of "X" in the pdeSys object
solRPdeIndex[1] = mlPdeSys->GetSolPdeIndex("Y"); // get the position of "Y" in the pdeSys object
solRPdeIndex[2] = mlPdeSys->GetSolPdeIndex("Z"); // get the position of "Z" in the pdeSys object
vector < adept::adouble > solR[3]; // local solution
vector < double > solR_old[3];
unsigned solHIndex;
solHIndex = mlSol->GetIndex("H"); // get the position of "H" in the ml_sol object
unsigned solHType = mlSol->GetSolutionType(solHIndex); // get the finite element type for "H"
unsigned solHPdeIndex;
solHPdeIndex = mlPdeSys->GetSolPdeIndex("H"); // get the position of "H" in the pdeSys object
vector < adept::adouble > solH; // local solution
vector < vector < double > > x(dim); // local coordinates
unsigned xType = 2; // get the finite element type for "x", it is always 2 (LAGRANGE QUADRATIC)
vector< int > sysDof; // local to global pdeSys dofs
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
vector< double > Res; // local redidual vector
vector< adept::adouble > aResR[3]; // local redidual vector
vector< adept::adouble > aResH; // local redidual vector
// reserve memory for the local standar vectors
const unsigned maxSize = static_cast< unsigned >(ceil(pow(3, dim))); // conservative: based on line3, quad9, hex27
for(int i=0;i<3;i++){
solR[i].reserve(maxSize);
solR_old[i].reserve(maxSize);
}
solH.reserve(maxSize);
for (unsigned i = 0; i < dim; i++)
x[i].reserve(maxSize);
sysDof.reserve(4 * maxSize);
phi.reserve(maxSize);
phi_x.reserve(maxSize * dim);
unsigned dim2 = (3 * (dim - 1) + !(dim - 1)); // dim2 is the number of second order partial derivatives (1,3,6 depending on the dimension)
phi_xx.reserve(maxSize * dim2);
Res.reserve(4 * maxSize);
for(int i=0;i<3;i++){
aResR[i].reserve(maxSize);
}
aResH.reserve(maxSize);
vector < double > Jac; // local Jacobian matrix (ordered by column, adept)
Jac.reserve(4 * maxSize * 4 * 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->_elementOffset[iproc]; iel < msh->_elementOffset[iproc + 1]; iel++) {
short unsigned ielGeom = el->GetElementType(iel); // element geometry type
unsigned nDofs = el->GetElementDofNumber(iel, solHType); // number of solution element dofs
unsigned nDofs2 = el->GetElementDofNumber(iel, xType); // number of coordinate element dofs
// resize local arrays
sysDof.resize(4 * nDofs);
for(int i = 0; i < 3; i++){
solR[i].resize(nDofs);
solR_old[i].resize(nDofs);
}
solH.resize(nDofs);
for (int i = 0; i < dim; i++) {
x[i].resize(nDofs2);
}
Res.resize(4 * nDofs); //resize
for(int i = 0; i < 3; i++){
aResR[i].resize(nDofs); //resize
}
aResH.resize(nDofs); //resize
for(int i = 0; i < 3; i++){
std::fill(aResR[i].begin(), aResR[i].end(), 0); //set aRes to zero
}
std::fill(aResH.begin(), aResH.end(), 0); //set aRes to zero
// local storage of global mapping and solution
for (unsigned i = 0; i < nDofs; i++) {
unsigned solDof = msh->GetSolutionDof(i, iel, solHType); // global to global mapping between solution node and solution dof
for(int k = 0; k < 3; k++){
solR[k][i] = (*sol->_Sol[solRIndex[k]])(solDof); // global extraction and local storage for the solution
solR_old[k][i] = (*sol->_SolOld[solRIndex[k]])(solDof); // global extraction and local storage for the solution
}
solH[i] = (*sol->_Sol[solHIndex])(solDof); // global extraction and local storage for the solution
for(int k = 0; k < 3; k++){
sysDof[k*nDofs + i] = pdeSys->GetSystemDof(solRIndex[k], solRPdeIndex[k], i, iel); // global to global mapping between solution node and pdeSys dof
}
sysDof[3*nDofs + i] = pdeSys->GetSystemDof(solHIndex, solHPdeIndex, i, iel); // global to global mapping between solution node and pdeSys dof
}
// local storage of coordinates
for (unsigned i = 0; i < nDofs2; i++) {
unsigned xDof = msh->GetSolutionDof(i, iel, xType); // global to global mapping between coordinates node and coordinate dof
for (unsigned idim = 0; idim < dim; idim++) {
x[idim][i] = (*msh->_topology->_Sol[idim])(xDof); // global extraction and local storage for the element coordinates
}
}
if (level == levelMax || !el->GetRefinedElementIndex(iel)) { // do not care about this if now (it is used for the AMR)
// 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][solHType]->GetGaussPointNumber(); ig++) {
// *** get gauss point weight, test function and test function partial derivatives ***
msh->_finiteElement[ielGeom][solHType]->Jacobian(x, ig, weight, phi, phi_x, phi_xx);
// evaluate the solution, the solution derivatives and the coordinates in the gauss point
adept::adouble solRGauss[3];
double solRGaussOld[3];
adept::adouble solRGauss_x[3][2];
adept::adouble solRGauss_xx[3][2][2];
adept::adouble sol_x[2];
sol_x[0]=sol_x[1]=0.;
for(int k=0; k<3; k++){
solRGauss[k]=0.;
solRGaussOld[k]=0.;
for(int i=0; i<dim; i++){
solRGauss_x[k][i]=0.;
for(int j=0; j<dim; j++){
solRGauss_xx[k][i][j]=0.;
}
}
}
adept::adouble solHGauss = 0;
adept::adouble solHGauss_x[2]={0.,0.};
for (unsigned i = 0; i < nDofs; i++) {
for(int k = 0; k < 3; k++){
solRGauss[k] += phi[i] * solR[k][i];
solRGaussOld[k] += phi[i] * solR_old[k][i];
}
solHGauss += phi[i] * solH[i];
for (unsigned u = 0; u < dim; u++) {
sol_x[u] += phi[i] * x[u][i];
}
for (unsigned u = 0; u < dim; u++) { // gradient
for(int k=0; k < 3; k++){
solRGauss_x[k][u] += phi_x[i * dim + u] * solR[k][i];
}
solHGauss_x[u] += phi_x[i * dim + u] * solH[i];
}
for( unsigned u = 0; u < dim; u++ ) { // hessian
for( unsigned v = 0; v < dim; v++ ) {
unsigned uvindex = 0; //_uu
if( u != v ) uvindex = 2; //_uv or _vu
else if( u == 1 ) uvindex = 1; //_vv
for(int k = 0; k < 3; k++){
solRGauss_xx[k][u][v] += phi_xx[i * dim2 + uvindex] * solR[k][i];
}
}
}
}
adept::adouble g[2][2];
g[0][0] = g[0][1] = g[1][0] = g[1][1] = 0.;
for(int k = 0; k < 3; k++){
for(int u = 0; u < dim; u++){
for(int v = 0; v < dim; v++){
g[u][v] += solRGauss_x[k][u] * solRGauss_x[k][v];
}
}
}
adept::adouble detg = g[0][0]*g[1][1]-g[0][1]*g[1][0];
adept::adouble A = sqrt(detg);
adept::adouble gI[2][2];
gI[0][0] = g[1][1]/detg;
gI[0][1] = -g[0][1]/detg;
gI[1][0] = -g[1][0]/detg;
gI[1][1] = g[0][0]/detg;
adept::adouble N[3];
N[0] = ( solRGauss_x[1][0] * solRGauss_x[2][1] - solRGauss_x[1][1] * solRGauss_x[2][0] ) / A;
N[1] = ( solRGauss_x[2][0] * solRGauss_x[0][1] - solRGauss_x[2][1] * solRGauss_x[0][0] ) / A;
N[2] = ( solRGauss_x[0][0] * solRGauss_x[1][1] - solRGauss_x[0][1] * solRGauss_x[1][0] ) / A;
adept::adouble h[2][2];
h[0][0]=h[0][1]=h[1][0]=h[1][1]=0.;
for(int k=0; k<3; k++){
for(int u=0; u<dim; u++){
for(int v=0; v<dim; v++){
h[u][v] += solRGauss_xx[k][u][v] * N[k];
}
}
}
//adept::adouble K = cos(sol_x[0])/(a+cos(sol_x[0]));//(h[0][0]*h[1][1]-h[0][1]*h[1][0])/detg;
adept::adouble K = (h[0][0]*h[1][1]-h[0][1]*h[1][0])/detg;
adept::adouble H_exact = 0.5*(1. + cos(sol_x[0])/(a+cos(sol_x[0])));
// *** phi_i loop ***
for (unsigned i = 0; i < nDofs; i++) {
for(int k=0; k<3; k++){
for(int u=0; u<dim; u++){
adept::adouble AgIgradRgradPhi=0;
for(int v=0; v<dim; v++){
AgIgradRgradPhi += A * gI[u][v].value() * solRGauss_x[k][v];
}
aResR[k][i] += AgIgradRgradPhi * phi_x[i * dim + u] * weight;
}
aResR[k][i] += 2.* A * solHGauss.value() * N[k] * phi[i] * weight;
}
for(int u=0; u<dim; u++){
adept::adouble AgIgradHgradPhi=0;
for(int v=0; v<dim; v++){
AgIgradHgradPhi += A * gI[u][v].value() * solHGauss_x[v];
}
aResH[i] -= AgIgradHgradPhi * phi_x[i * dim + u] * weight;
}
aResH[i] += A * ( -0*(solRGauss[0]-solRGaussOld[0])*N[0].value()
-0*(solRGauss[1]-solRGaussOld[1])*N[1].value()
-0*(solRGauss[2]-solRGaussOld[2])*N[2].value()
+ 2. * solHGauss * ( solHGauss * solHGauss - K.value() ) )* phi[i] * weight;
} // 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
for (int i = 0; i < nDofs; i++) {
for( int k=0;k<3;k++){
Res[ k * nDofs + i] = -aResR[k][i].value();
}
Res[ 3 * nDofs + i] = -aResH[i].value();
}
RES->add_vector_blocked(Res, sysDof);
if (assembleMatrix) {
Jac.resize((4 * nDofs) *(4 * nDofs));
// define the dependent variables
for( int k=0;k<3;k++){
s.dependent(&aResR[k][0], nDofs);
}
s.dependent(&aResH[0], nDofs);
// define the independent variables
for( int k=0;k<3;k++){
s.independent(&solR[k][0], nDofs);
}
s.independent(&solH[0], nDofs);
// get the jacobian matrix (ordered by row)
s.jacobian(&Jac[0], true);
KK->add_matrix_blocked(Jac, sysDof, sysDof);
s.clear_independents();
s.clear_dependents();
}
} //end element loop for each process
RES->close();
if (assembleMatrix) KK->close();
// ***************** END ASSEMBLY *******************
}
