void
MAST::NoniterativeUGFlutterSolver::calculate_sensitivity(MAST::FlutterRootBase& root,
const libMesh::ParameterVector& params,
const unsigned int i) {
// make sure that the aero_structural_model is a valid pointer
libmesh_assert(aero_structural_model);
libMesh::out
<< " ====================================================" << std::endl
<< "UG Sensitivity Solution" << std::endl
<< " k_red = " << std::setw(10) << root.k_red << std::endl
<< " V_ref = " << std::setw(10) << root.V << std::endl;
Complex eig = root.root, sens = 0., k_sens = 0., vref_sens = 0., den = 0.;
// matrix to store the sensitivity of constraint wrt kref and vref
// first row is constraint f1(k,v) = g = im(lambda)/re(lambda) = 0
// second row is constraint f2(k,v) = vref*sqrt(re(lambda))-1. = 0
RealMatrixX jac;
// residual vector for the sensitivity problem, and total derivative vec
// first row is for constraint f1(k,v)
// first row is for constraint f2(k,v)
RealVectorX res, dsens;
jac.setZero(2,2);
res.setZero(2);
// get the sensitivity of the matrices
ComplexMatrixX mat_A, mat_B, mat_A_sens, mat_B_sens;
ComplexVectorX v;
// initialize the baseline matrices
initialize_matrices(root.k_red_ref, root.V_ref, mat_A, mat_B);
// calculate the eigenproblem sensitivity
initialize_matrix_sensitivity_for_param(params, i,
root.k_red_ref,
root.V_ref,
mat_A_sens,
mat_B_sens);
// the eigenproblem is A x - lambda B x = 0
// therefore, the denominator is obtained from the inner product of
// x^T B x
// sensitivity is
// -dlambda/dp x^T B x = - x^T (dA/dp - lambda dB/dp)
// or
// dlambda/dp = [x^T (dA/dp - lambda dB/dp)]/(x^T B x)
// now calculate the quotient for sensitivity
// numerator = ( dA/dp - lambda dB/dp)
mat_B_sens *= -eig;
mat_B_sens += mat_A_sens;
v = mat_B_sens*root.eig_vec_right;
den = root.eig_vec_left.dot(mat_B*root.eig_vec_right);
sens = root.eig_vec_left.dot(v)/den;
// sensitivity of f1(k,v) = im(lambda)/re(lambda) = 0
res(0) =
sens.imag()/eig.real() - eig.imag()/pow(eig.real(),2) * sens.real();
// sensitivity of f2(k,v) = vref*sqrt(real(lambda)) - 1 = 0
res(1) = root.V_ref*0.5*pow(eig.real(),-0.5)*sens.real();
// next we need the sensitivity of k_red before we can calculate
// the sensitivity of flutter eigenvalue
initialize_matrix_sensitivity_for_reduced_freq(root.k_red_ref,
root.V_ref,
mat_A_sens,
mat_B_sens);
// now calculate the quotient for sensitivity wrt k_red
// calculate numerator
mat_B_sens *= -eig;
mat_B_sens += mat_A_sens;
v = mat_B_sens*root.eig_vec_right;
k_sens = root.eig_vec_left.dot(v) / den;
// use this to calculate the partial derivative of f1 wrt k_red
jac(0,0) =
k_sens.imag()/eig.real() - eig.imag()/pow(eig.real(),2)*k_sens.real();
jac(1,0) = root.V_ref*0.5*pow(eig.real(),-0.5)*k_sens.real();
// next we need the sensitivity of Vref constraint before we can calculate
// the sensitivity of flutter eigenvalue
initialize_matrix_sensitivity_for_V_ref(root.k_red_ref,
root.V_ref,
mat_A_sens,
mat_B_sens);
// now calculate the quotient for sensitivity wrt k_red
// calculate numerator
mat_B_sens *= -eig;
mat_B_sens += mat_A_sens;
v = mat_B_sens*root.eig_vec_right;
vref_sens = root.eig_vec_left.dot(v) / den;
// use this to calculate the partial derivative of f2 wrt vref
jac(0,1) =
vref_sens.imag()/eig.real() - eig.imag()/pow(eig.real(),2)*vref_sens.real();
jac(1,1) =
sqrt(eig.real()) + root.V_ref*0.5*pow(eig.real(),-0.5)*vref_sens.real();
// now invert the Jacobian to calculate the sensitivity
dsens = -jac.inverse()*res;
// finally add the correction to the flutter sensitivity
sens += k_sens * dsens(0) + vref_sens * dsens(1);
// set value in the return root
root.has_sensitivity_data = true;
root.root_sens = sens;
root.V_sens = -.5*sens.real()/pow(eig.real(), 1.5);
libMesh::out
<< "Finished UG Sensitivity Solution" << std::endl
<< " ====================================================" << std::endl;
}
std::pair<bool, MAST::FlutterSolutionBase*>
MAST::NoniterativeUGFlutterSolver::newton_search(const MAST::FlutterSolutionBase& init_sol,
const unsigned int root_num,
const Real tol,
const unsigned int max_iters) {
std::pair<bool, MAST::FlutterSolutionBase*> rval(false, NULL);
// assumes that the upper k_val has +ve g val and lower k_val has -ve
// k_val
Real k_red, v_ref;
unsigned int n_iters = 0;
std::auto_ptr<MAST::FlutterSolutionBase> new_sol;
RealVectorX res, sol, dsol;
RealMatrixX jac;
ComplexMatrixX mat_A, mat_B, mat_A_sens, mat_B_sens;
ComplexVectorX v;
res.resize(2); sol.resize(2); dsol.resize(2);
jac.resize(2,2);
// initialize the solution to the values of init_sol
k_red = init_sol.get_root(root_num).k_red_ref;
v_ref = init_sol.get_root(root_num).V_ref;
sol(0) = k_red;
sol(1) = v_ref;
const MAST::FlutterSolutionBase* prev_sol = &init_sol;
bool if_continue = true;
while (if_continue) {
// evaluate the residual and Jacobians
std::auto_ptr<MAST::FlutterSolutionBase> ug_sol =
this->analyze(k_red, v_ref, prev_sol);
ug_sol->print(_output);
// add the solution to this solver
_insert_new_solution(v_ref, ug_sol.release());
// now get a pointer to the previous solution
// get the solution from the database for this reduced frequency
std::map<Real, MAST::FlutterSolutionBase*>::iterator it =
_flutter_solutions.find(k_red);
libmesh_assert(it != _flutter_solutions.end());
rval.second = it->second;
prev_sol = it->second;
// solve the Newton update problem
const MAST::FlutterRootBase& root = prev_sol->get_root(root_num);
Complex eig = root.root, eig_k_red_sens = 0., den = 0., eig_V_ref_sens = 0.;
// initialize the baseline matrices
initialize_matrices(k_red, v_ref, mat_A, mat_B);
// solve the sensitivity problem
// first with respect to k_red
// next we need the sensitivity of k_red before we can calculate
// the sensitivity of flutter eigenvalue
initialize_matrix_sensitivity_for_reduced_freq(k_red,
v_ref,
mat_A_sens,
mat_B_sens);
// now calculate the quotient for sensitivity wrt k_red
// calculate numerator
mat_B_sens *= -eig;
mat_B_sens += mat_A_sens;
v = mat_B_sens*root.eig_vec_right;
den = root.eig_vec_left.dot(mat_B*root.eig_vec_right);
eig_k_red_sens = root.eig_vec_left.dot(v) / den;
// next, sensitivity wrt V_ref
initialize_matrix_sensitivity_for_V_ref(k_red,
v_ref,
mat_A_sens,
mat_B_sens);
// now calculate the quotient for sensitivity wrt V_ref
// calculate numerator
mat_B_sens *= -eig;
mat_B_sens += mat_A_sens;
v = mat_B_sens*root.eig_vec_right;
den = root.eig_vec_left.dot(mat_B*root.eig_vec_right);
eig_V_ref_sens = root.eig_vec_left.dot(v) / den;
// residual
res(0) = eig.imag()/eig.real();
res(1) = v_ref*sqrt(eig.real()) - 1.;
// Jacobian
jac(0,0) = eig_k_red_sens.imag()/eig.real() -
eig.imag()*pow(eig.real(),-2)*eig_k_red_sens.real();
jac(0,1) = eig_V_ref_sens.imag()/eig.real() -
eig.imag()*pow(eig.real(),-2)*eig_V_ref_sens.real();
jac(1,0) = 0.5*v_ref*pow(eig.real(),-0.5)*eig_k_red_sens.real();
jac(1,1) = sqrt(eig.real()) + 0.5*v_ref*pow(eig.real(),-0.5)*eig_V_ref_sens.real();
/*std::auto_ptr<MAST::FlutterSolutionBase> ug_dsol =
this->analyze(k_red+.001, v_ref, prev_sol);
Complex deig = ug_dsol->get_root(root_num).root;
deig -= eig;
deig /= .001;
std::cout << "fd deig/dk: " << deig << std::endl;
std::cout << "analytical: " << eig_k_red_sens << std::endl;
ug_dsol.reset(this->analyze(k_red, v_ref+.001, prev_sol).release());
deig = ug_dsol->get_root(root_num).root;
deig -= eig;
deig /= .001;
std::cout << "fd deig/dV: " << deig << std::endl;
std::cout << "analytical: " << eig_V_ref_sens << std::endl;*/
// now calculate the updates
// r0 + J *dx = 0
// => dx = - inv(J) * r0
// => x1 = x0 + dx
dsol = -jac.inverse()*res;
sol += dsol;
// get the updated parameter values
k_red = sol(0);
v_ref = sol(1);
// increment the iteration counter
n_iters++;
// set the flag
if (fabs(root.g) < tol) {
rval.first = true;
return rval;
}
if (n_iters >= max_iters)
if_continue = false;
}
// return false, along with the latest sol
rval.first = false;
return rval;
}