void
MAST::StructuralElement2D::
initialize_von_karman_strain_operator_sensitivity(const unsigned int qp,
RealMatrixX &vk_dwdxi_mat_sens) {
const std::vector<std::vector<libMesh::RealVectorValue> >& dphi = _fe->get_dphi();
const unsigned int n_phi = (unsigned int)dphi.size();
libmesh_assert_equal_to(vk_dwdxi_mat_sens.rows(), 3);
libmesh_assert_equal_to(vk_dwdxi_mat_sens.cols(), 2);
Real dw=0.;
vk_dwdxi_mat_sens.setConstant(0.);
RealVectorX phi_vec = RealVectorX::Zero(n_phi);
dw = 0.;
for ( unsigned int i_nd=0; i_nd<n_phi; i_nd++ ) {
phi_vec(i_nd) = dphi[i_nd][qp](0); // dphi/dx
dw += phi_vec(i_nd)*_local_sol_sens(2*n_phi+i_nd); // dw/dx
}
vk_dwdxi_mat_sens(0, 0) = dw; // epsilon-xx : dw/dx
vk_dwdxi_mat_sens(2, 1) = dw; // gamma-xy : dw/dx
dw = 0.;
for ( unsigned int i_nd=0; i_nd<n_phi; i_nd++ ) {
phi_vec(i_nd) = dphi[i_nd][qp](1); // dphi/dy
dw += phi_vec(i_nd)*_local_sol_sens(2*n_phi+i_nd); // dw/dy
}
vk_dwdxi_mat_sens(1, 1) = dw; // epsilon-yy : dw/dy
vk_dwdxi_mat_sens(2, 0) = dw; // gamma-xy : dw/dy
}
void
MAST::OrthotropicProperty3D::
StiffnessMatrix::operator() (const libMesh::Point& p,
const Real t,
RealMatrixX& m) const {
RealMatrixX
A = RealMatrixX::Zero(3, 3),
Tinv = RealMatrixX::Zero(6, 6);
_material_stiffness (p, t, m);
_orient (p, t, A);
_orient.stress_strain_transformation_matrix(A.transpose(), Tinv);
// vk' = vj ej.ek' = vj Ajk = A^T v
// v = A vk'
// sij' = skl ek.ei' el.ej' = skl Aki Alj = A^T s A
// s' = T s
// s = Tinv s'
// s' = C' e'
// T s = C' Rinv T R s
// s = Tinv C Rinv T R e
// C = Tinv C Rinv T R
// T R scales last three columns by 1/2
// Rinv T scales last three rows by 2
// therefore, Rinv T R scales top right 3x3 block by 1/2,
// and bottom left 3x3 block by 2.
// Also, Rinv T R = T^{-T}
m = Tinv * m * Tinv.transpose();
}
void MAST::TimeDomainFlutterSolver::initialize_matrices(Real ref_val,
RealMatrixX& a, // LHS
RealMatrixX& b) // RHS
{
bool has_matrix = false;
RealMatrixX mat1, mat2;
Real q0 = flight_condition->q0();
// mass matrix forms the LHS of the eigenvalue problem
has_matrix = aero_structural_model->get_structural_mass_matrix(mat1);
const unsigned int n = (int)mat1.rows();
a.block(n,n,n,n) = mat1;
libmesh_assert(has_matrix);
// structural and aerodynamic stiffness matrices
has_matrix = aero_structural_model->get_structural_stiffness_matrix(mat1);
libmesh_assert(has_matrix);
has_matrix = aero_structural_model->get_aero_stiffness_matrix(mat2);
libmesh_assert(has_matrix);
b.block(n, 0, n, n) = (q0*mat2-mat1);
// structural and aerodynamic stiffness matrices
has_matrix = aero_structural_model->get_structural_damping_matrix(mat1);
libmesh_assert(has_matrix);
has_matrix = aero_structural_model->get_aero_damping_matrix(mat2);
libmesh_assert(has_matrix);
b.block(n, 0, n, n) = (q0*mat2-mat1);
// finally set the identity blocks in the matrix
for (unsigned int i=0; i<n; i++) {
a(i, i) = 1.;
b(i,i+n) = 1.;
}
}
void
MAST::StructuralElement2D::_convert_prestress_B_mat_to_vector(const RealMatrixX& mat,
RealVectorX& vec) const {
libmesh_assert_equal_to(mat.rows(), 2);
libmesh_assert_equal_to(mat.cols(), 2);
vec = RealVectorX::Zero(3);
vec(0) = mat(0,0); // sigma x
vec(1) = mat(1,1); // sigma y
vec(2) = mat(0,1); // tau xy
}
void
MAST::OrthotropicProperty3D::ThermalConductanceMatrix::
operator() (const libMesh::Point& p,
const Real t,
RealMatrixX& m) const {
RealMatrixX
A = RealMatrixX::Zero(3, 3);
_mat_cond (p, t, m);
_orient (p, t, A);
m = A.transpose() * m * A;
}
virtual void operator() (const libMesh::Point& p,
const Real t,
RealMatrixX& m) const {
// add the values of each matrix to get the integrated value
RealMatrixX mi;
for (unsigned int i=0; i<_layer_mats.size(); i++) {
(*_layer_mats[i])(p, t, mi);
// use the size of the layer matrix to resize the output
// all other layers should return the same sized matrices
if (i==0)
m = RealMatrixX::Zero(mi.rows(), mi.cols());
m += mi;
}
}
virtual void derivative ( const MAST::FunctionBase& f,
const libMesh::Point& p,
const Real t,
RealMatrixX& m) const {
// add the values of each matrix to get the integrated value
RealMatrixX mi;
m = RealMatrixX::Zero(2,2);
for (unsigned int i=0; i<_layer_mats.size(); i++) {
_layer_mats[i]->derivative( f, p, t, mi);
// use the size of the layer matrix to resize the output
// all other layers should return the same sized matrices
if (i==0)
m = RealMatrixX::Zero(mi.rows(), mi.cols());
m += mi;
}
}
void
MAST::StructuralElement2D::
initialize_von_karman_strain_operator(const unsigned int qp,
RealVectorX& vk_strain,
RealMatrixX& vk_dwdxi_mat,
MAST::FEMOperatorMatrix& Bmat_vk) {
const std::vector<std::vector<libMesh::RealVectorValue> >& dphi = _fe->get_dphi();
const unsigned int n_phi = (unsigned int)dphi.size();
libmesh_assert_equal_to(vk_strain.size(), 3);
libmesh_assert_equal_to(vk_dwdxi_mat.rows(), 3);
libmesh_assert_equal_to(vk_dwdxi_mat.cols(), 2);
libmesh_assert_equal_to(Bmat_vk.m(), 2);
libmesh_assert_equal_to(Bmat_vk.n(), 6*n_phi);
Real dw=0.;
vk_strain.setConstant(0.);
vk_dwdxi_mat.setConstant(0.);
RealVectorX phi_vec = RealVectorX::Zero(n_phi);
dw = 0.;
for ( unsigned int i_nd=0; i_nd<n_phi; i_nd++ ) {
phi_vec(i_nd) = dphi[i_nd][qp](0); // dphi/dx
dw += phi_vec(i_nd)*_local_sol(2*n_phi+i_nd); // dw/dx
}
Bmat_vk.set_shape_function(0, 2, phi_vec); // dw/dx
vk_dwdxi_mat(0, 0) = dw; // epsilon-xx : dw/dx
vk_dwdxi_mat(2, 1) = dw; // gamma-xy : dw/dx
vk_strain(0) = 0.5*dw*dw; // 1/2 * (dw/dx)^2
vk_strain(2) = dw; // (dw/dx)*(dw/dy) only dw/dx is provided here
dw = 0.;
for ( unsigned int i_nd=0; i_nd<n_phi; i_nd++ ) {
phi_vec(i_nd) = dphi[i_nd][qp](1); // dphi/dy
dw += phi_vec(i_nd)*_local_sol(2*n_phi+i_nd); // dw/dy
}
Bmat_vk.set_shape_function(1, 2, phi_vec); // dw/dy
vk_dwdxi_mat(1, 1) = dw; // epsilon-yy : dw/dy
vk_dwdxi_mat(2, 0) = dw; // gamma-xy : dw/dy
vk_strain(1) = 0.5*dw*dw; // 1/2 * (dw/dy)^2
vk_strain(2) *= dw; // (dw/dx)*(dw/dy)
}
void
MAST::TimeDomainFlutterRoot::init(const Real ref_val, const Real b_ref,
const Complex num,
const Complex den,
const RealMatrixX& Bmat,
const ComplexVectorX& evec_right,
const ComplexVectorX& evec_left)
{
V = ref_val;
if (std::abs(den) > 0.)
{
root = num/den;
if (std::real(root) > 0.)
{
V = sqrt(1./std::real(root));
g = std::imag(root)/std::real(root);
omega = k_red*V/b_ref;
if_nonphysical_root = false;
}
else
{
V = 0.;
g = 0.;
omega = 0.;
if_nonphysical_root = true;
}
}
// calculate the modal participation vector
const unsigned int nvals = (int)Bmat.rows();
eig_vec_right = evec_right;
eig_vec_left = evec_left;
ComplexVectorX k_q;
k_q = Bmat * evec_right;
modal_participation.resize(nvals, 1);
for (unsigned int i=0; i<nvals; i++)
modal_participation(i) = std::abs(std::conj(evec_right(i)) * k_q(i));
modal_participation *= (1./modal_participation.sum());
}
void
MAST::LevelSetReinitializationTransientAssembly::
residual_and_jacobian (const libMesh::NumericVector<Real>& X,
libMesh::NumericVector<Real>* R,
libMesh::SparseMatrix<Real>* J,
libMesh::NonlinearImplicitSystem& S) {
libmesh_assert(_system);
libmesh_assert(_discipline);
libmesh_assert(_elem_ops);
MAST::TransientSolverBase
&solver = dynamic_cast<MAST::TransientSolverBase&>(*_elem_ops);
MAST::NonlinearSystem
&transient_sys = _system->system();
MAST::LevelSetTransientAssemblyElemOperations
&level_set_elem_ops = dynamic_cast<MAST::LevelSetTransientAssemblyElemOperations&>
(solver.get_elem_operation_object());
// make sure that the system for which this object was created,
// and the system passed through the function call are the same
libmesh_assert_equal_to(&S, &transient_sys);
if (R) R->zero();
if (J) J->zero();
// iterate over each element, initialize it and get the relevant
// analysis quantities
RealVectorX vec, ref_sol;
RealMatrixX mat;
std::vector<libMesh::dof_id_type> dof_indices;
const libMesh::DofMap& dof_map = transient_sys.get_dof_map();
// stores the localized solution, velocity, acceleration, etc. vectors.
// These pointers will have to be deleted
std::vector<libMesh::NumericVector<Real>*>
local_qtys;
std::unique_ptr<libMesh::NumericVector<Real> > localized_solution;
localized_solution.reset(build_localized_vector(transient_sys,
*_ref_sol).release());
// if a solution function is attached, initialize it
if (_sol_function)
_sol_function->init( X);
// ask the solver to localize the relevant solutions
solver.build_local_quantities(X, local_qtys);
libMesh::MeshBase::const_element_iterator el =
transient_sys.get_mesh().active_local_elements_begin();
const libMesh::MeshBase::const_element_iterator end_el =
transient_sys.get_mesh().active_local_elements_end();
for ( ; el != end_el; ++el) {
const libMesh::Elem* elem = *el;
dof_map.dof_indices (elem, dof_indices);
solver.init(*elem);
// get the solution
unsigned int ndofs = (unsigned int)dof_indices.size();
vec.setZero(ndofs);
ref_sol.setZero(ndofs);
mat.setZero(ndofs, ndofs);
for (unsigned int i=0; i<dof_indices.size(); i++)
ref_sol(i) = (*localized_solution)(dof_indices[i]);
solver.set_element_data(dof_indices, local_qtys);
level_set_elem_ops.set_elem_reference_solution(ref_sol);
// if (_sol_function)
// physics_elem->attach_active_solution_function(*_sol_function);
// perform the element level calculations
solver.elem_calculations(J!=nullptr?true:false, vec, mat);
solver.clear_elem();
// copy to the libMesh matrix for further processing
DenseRealVector v;
DenseRealMatrix m;
if (R)
MAST::copy(v, vec);
if (J)
MAST::copy(m, mat);
// constrain the quantities to account for hanging dofs,
// Dirichlet constraints, etc.
if (R && J)
dof_map.constrain_element_matrix_and_vector(m, v, dof_indices);
else if (R)
dof_map.constrain_element_vector(v, dof_indices);
else
dof_map.constrain_element_matrix(m, dof_indices);
// add to the global matrices
if (R) R->add_vector(v, dof_indices);
if (J) J->add_matrix(m, dof_indices);
}
// delete pointers to the local solutions
for (unsigned int i=0; i<local_qtys.size(); i++)
delete local_qtys[i];
// if a solution function is attached, clear it
if (_sol_function)
_sol_function->clear();
if (R) R->close();
if (J) J->close();
}
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;
}
void
MAST::StructuralElement2D::
_linearized_geometric_stiffness_sensitivity_with_static_solution
(const unsigned int n2,
const unsigned int qp,
const std::vector<Real>& JxW,
RealMatrixX& local_jac,
FEMOperatorMatrix& Bmat_mem,
FEMOperatorMatrix& Bmat_bend,
FEMOperatorMatrix& Bmat_vk,
RealMatrixX& stress_l,
RealMatrixX& vk_dwdxi_mat,
RealMatrixX& material_A_mat,
RealMatrixX& material_B_mat,
RealVectorX& vec1_n1,
RealVectorX& vec2_n1,
RealMatrixX& mat1_n1n2,
RealMatrixX& mat2_n2n2,
RealMatrixX& mat3)
{
this->initialize_direct_strain_operator(qp, Bmat_mem);
_bending_operator->initialize_bending_strain_operator(qp, Bmat_bend);
// first handle constant throught the thickness stresses: membrane and vonKarman
Bmat_mem.vector_mult(vec1_n1, _local_sol_sens);
vec2_n1 = material_A_mat * vec1_n1; // linear direct stress
// copy the stress values to a matrix
stress_l(0,0) = vec2_n1(0); // sigma_xx
stress_l(0,1) = vec2_n1(2); // sigma_xy
stress_l(1,0) = vec2_n1(2); // sigma_yx
stress_l(1,1) = vec2_n1(1); // sigma_yy
// get the von Karman operator matrix
this->initialize_von_karman_strain_operator(qp,
vec2_n1, // epsilon_vk
vk_dwdxi_mat,
Bmat_vk);
// sensitivity of the vk_dwdxi matrix due to solution sensitivity
this->initialize_von_karman_strain_operator_sensitivity(qp,
vk_dwdxi_mat);
// membrane - vk
mat3 = RealMatrixX::Zero(vk_dwdxi_mat.rows(), n2);
Bmat_vk.left_multiply(mat3, vk_dwdxi_mat);
mat3 = material_A_mat * mat3;
Bmat_mem.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// vk - membrane
Bmat_mem.left_multiply(mat1_n1n2, material_A_mat);
mat3 = vk_dwdxi_mat.transpose() * mat1_n1n2;
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// vk - vk: first order term
mat3 = RealMatrixX::Zero(2, n2);
Bmat_vk.left_multiply(mat3, stress_l);
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// bending - vk
mat3 = RealMatrixX::Zero(vk_dwdxi_mat.rows(), n2);
Bmat_vk.left_multiply(mat3, vk_dwdxi_mat);
mat3 = material_B_mat.transpose() * mat3;
Bmat_bend.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// vk - bending
Bmat_bend.left_multiply(mat1_n1n2, material_B_mat);
mat3 = vk_dwdxi_mat.transpose() * mat1_n1n2;
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
}
void
MAST::LAPACK_DGGEV::compute(const RealMatrixX &A,
const RealMatrixX &B,
bool computeEigenvectors) {
libmesh_assert(A.cols() == A.rows() &&
B.cols() == A.rows() &&
B.cols() == B.rows());
_A = A;
_B = B;
RealMatrixX
Amat = A,
Bmat = B;
int n = (int)A.cols();
char L='N',R='N';
if (computeEigenvectors) {
L = 'V'; R = 'V';
VL.setZero(n, n);
VR.setZero(n, n);
}
int
lwork=16*n;
info_val=-1;
alpha.setZero(n);
beta.setZero(n);
RealVectorX
work,
aval_r,
aval_i,
bval;
RealMatrixX
vecl,
vecr;
work.setZero(lwork);
aval_r.setZero(n);
aval_i.setZero(n);
bval.setZero(n);
vecl.setZero(n,n);
vecr.setZero(n,n);
Real
*a_vals = Amat.data(),
*b_vals = Bmat.data(),
*alpha_r_v = aval_r.data(),
*alpha_i_v = aval_i.data(),
*beta_v = bval.data(),
*vecl_v = vecl.data(),
*vecr_v = vecr.data(),
*work_v = work.data();
dggev_(&L, &R, &n,
&(a_vals[0]), &n,
&(b_vals[0]), &n,
&(alpha_r_v[0]), &(alpha_i_v[0]), &(beta_v[0]),
&(vecl_v[0]), &n, &(vecr_v[0]), &n,
&(work_v[0]), &lwork,
&info_val);
// now sort the eigenvalues for complex conjugates
unsigned int n_located = 0;
while (n_located < n) {
// if the imaginary part of the eigenvalue is non-zero, it is a
// complex conjugate
if (aval_i(n_located) != 0.) { // complex conjugate
alpha( n_located) = std::complex<double>(aval_r(n_located), aval_i(n_located));
alpha(1+n_located) = std::complex<double>(aval_r(n_located), -aval_i(n_located));
beta ( n_located) = bval(n_located);
beta (1+n_located) = bval(n_located);
// copy the eigenvectors if they were requested
if (computeEigenvectors) {
std::complex<double> iota = std::complex<double>(0, 1.);
VL.col( n_located) = (vecl.col( n_located).cast<Complex>() +
vecl.col(1+n_located).cast<Complex>() * iota);
VL.col(1+n_located) = (vecl.col( n_located).cast<Complex>() -
vecl.col(1+n_located).cast<Complex>() * iota);
VR.col( n_located) = (vecr.col( n_located).cast<Complex>() +
vecr.col(1+n_located).cast<Complex>() * iota);
VR.col(1+n_located) = (vecr.col( n_located).cast<Complex>() -
vecr.col(1+n_located).cast<Complex>() * iota);
}
// two complex conjugate roots were found
n_located +=2;
}
else {
alpha( n_located) = std::complex<double>(aval_r(n_located), 0.);
beta ( n_located) = bval(n_located);
// copy the eigenvectors if they were requested
if (computeEigenvectors) {
VL.col(n_located) = vecl.col(n_located).cast<Complex>();
VR.col(n_located) = vecr.col(n_located).cast<Complex>();
}
// only one real root was found
n_located++;
}
}
if (info_val != 0)
libMesh::out
<< "Warning!! DGGEV returned with nonzero info = "
<< info_val << std::endl;
}
void
MAST::NonlinearImplicitAssembly::
residual_and_jacobian (const libMesh::NumericVector<Real>& X,
libMesh::NumericVector<Real>* R,
libMesh::SparseMatrix<Real>* J,
libMesh::NonlinearImplicitSystem& S) {
libMesh::NonlinearImplicitSystem& nonlin_sys =
dynamic_cast<libMesh::NonlinearImplicitSystem&>(_system->system());
// make sure that the system for which this object was created,
// and the system passed through the function call are the same
libmesh_assert_equal_to(&S, &(nonlin_sys));
if (R) R->zero();
if (J) J->zero();
// iterate over each element, initialize it and get the relevant
// analysis quantities
RealVectorX vec, sol;
RealMatrixX mat;
std::vector<libMesh::dof_id_type> dof_indices;
const libMesh::DofMap& dof_map = _system->system().get_dof_map();
std::auto_ptr<MAST::ElementBase> physics_elem;
std::auto_ptr<libMesh::NumericVector<Real> > localized_solution;
localized_solution.reset(_build_localized_vector(nonlin_sys,
X).release());
// if a solution function is attached, initialize it
if (_sol_function)
_sol_function->init_for_system_and_solution(*_system, X);
libMesh::MeshBase::const_element_iterator el =
nonlin_sys.get_mesh().active_local_elements_begin();
const libMesh::MeshBase::const_element_iterator end_el =
nonlin_sys.get_mesh().active_local_elements_end();
for ( ; el != end_el; ++el) {
const libMesh::Elem* elem = *el;
dof_map.dof_indices (elem, dof_indices);
physics_elem.reset(_build_elem(*elem).release());
// get the solution
unsigned int ndofs = (unsigned int)dof_indices.size();
sol.setZero(ndofs);
vec.setZero(ndofs);
mat.setZero(ndofs, ndofs);
for (unsigned int i=0; i<dof_indices.size(); i++)
sol(i) = (*localized_solution)(dof_indices[i]);
physics_elem->set_solution(sol);
if (_sol_function)
physics_elem->attach_active_solution_function(*_sol_function);
//_check_element_numerical_jacobian(*physics_elem, sol);
// perform the element level calculations
_elem_calculations(*physics_elem,
J!=NULL?true:false,
vec, mat);
physics_elem->detach_active_solution_function();
// copy to the libMesh matrix for further processing
DenseRealVector v;
DenseRealMatrix m;
if (R)
MAST::copy(v, vec);
if (J)
MAST::copy(m, mat);
// constrain the quantities to account for hanging dofs,
// Dirichlet constraints, etc.
if (R && J)
nonlin_sys.get_dof_map().constrain_element_matrix_and_vector(m, v, dof_indices);
else if (R)
nonlin_sys.get_dof_map().constrain_element_vector(v, dof_indices);
else
nonlin_sys.get_dof_map().constrain_element_matrix(m, dof_indices);
// add to the global matrices
if (R) R->add_vector(v, dof_indices);
if (J) J->add_matrix(m, dof_indices);
}
// if a solution function is attached, clear it
if (_sol_function)
_sol_function->clear();
if (R) R->close();
if (J) J->close();
}
bool
MAST::NonlinearImplicitAssembly::
sensitivity_assemble (const libMesh::ParameterVector& parameters,
const unsigned int i,
libMesh::NumericVector<Real>& sensitivity_rhs) {
libMesh::NonlinearImplicitSystem& nonlin_sys =
dynamic_cast<libMesh::NonlinearImplicitSystem&>(_system->system());
sensitivity_rhs.zero();
// iterate over each element, initialize it and get the relevant
// analysis quantities
RealVectorX vec, sol;
RealMatrixX mat;
std::vector<libMesh::dof_id_type> dof_indices;
const libMesh::DofMap& dof_map = nonlin_sys.get_dof_map();
std::auto_ptr<MAST::ElementBase> physics_elem;
std::auto_ptr<libMesh::NumericVector<Real> > localized_solution;
localized_solution.reset(_build_localized_vector(nonlin_sys,
*nonlin_sys.solution).release());
// if a solution function is attached, initialize it
if (_sol_function)
_sol_function->init_for_system_and_solution(*_system, *nonlin_sys.solution);
libMesh::MeshBase::const_element_iterator el =
nonlin_sys.get_mesh().active_local_elements_begin();
const libMesh::MeshBase::const_element_iterator end_el =
nonlin_sys.get_mesh().active_local_elements_end();
for ( ; el != end_el; ++el) {
const libMesh::Elem* elem = *el;
dof_map.dof_indices (elem, dof_indices);
physics_elem.reset(_build_elem(*elem).release());
// get the solution
unsigned int ndofs = (unsigned int)dof_indices.size();
sol.setZero(ndofs);
vec.setZero(ndofs);
mat.setZero(ndofs, ndofs);
for (unsigned int i=0; i<dof_indices.size(); i++)
sol(i) = (*localized_solution)(dof_indices[i]);
physics_elem->sensitivity_param = _discipline->get_parameter(parameters[i]);
physics_elem->set_solution(sol);
if (_sol_function)
physics_elem->attach_active_solution_function(*_sol_function);
// perform the element level calculations
_elem_sensitivity_calculations(*physics_elem, vec);
physics_elem->detach_active_solution_function();
// copy to the libMesh matrix for further processing
DenseRealVector v;
MAST::copy(v, vec);
// constrain the quantities to account for hanging dofs,
// Dirichlet constraints, etc.
nonlin_sys.get_dof_map().constrain_element_vector(v, dof_indices);
// add to the global matrices
sensitivity_rhs.add_vector(v, dof_indices);
}
// if a solution function is attached, initialize it
if (_sol_function)
_sol_function->clear();
sensitivity_rhs.close();
return true;
}
void
MAST::StructuralElement2D::_internal_residual_operation
(bool if_bending,
bool if_vk,
const unsigned int n2,
const unsigned int qp,
const std::vector<Real>& JxW,
bool request_jacobian,
bool if_ignore_ho_jac,
RealVectorX& local_f,
RealMatrixX& local_jac,
FEMOperatorMatrix& Bmat_mem,
FEMOperatorMatrix& Bmat_bend,
FEMOperatorMatrix& Bmat_vk,
RealMatrixX& stress,
RealMatrixX& stress_l,
RealMatrixX& vk_dwdxi_mat,
RealMatrixX& material_A_mat,
RealMatrixX& material_B_mat,
RealMatrixX& material_D_mat,
RealVectorX& vec1_n1,
RealVectorX& vec2_n1,
RealVectorX& vec3_n2,
RealVectorX& vec4_2,
RealVectorX& vec5_2,
RealMatrixX& mat1_n1n2,
RealMatrixX& mat2_n2n2,
RealMatrixX& mat3,
RealMatrixX& mat4_2n2)
{
this->initialize_direct_strain_operator(qp, Bmat_mem);
// first handle constant throught the thickness stresses: membrane and vonKarman
Bmat_mem.vector_mult(vec1_n1, _local_sol);
vec2_n1 = material_A_mat * vec1_n1; // linear direct stress
// copy the stress values to a matrix
stress_l(0,0) = vec2_n1(0); // sigma_xx
stress_l(0,1) = vec2_n1(2); // sigma_xy
stress_l(1,0) = vec2_n1(2); // sigma_yx
stress_l(1,1) = vec2_n1(1); // sigma_yy
stress = stress_l;
// get the bending strain operator
vec2_n1.setConstant(0.); // used to store vk strain, if applicable
if (if_bending) {
_bending_operator->initialize_bending_strain_operator(qp, Bmat_bend);
Bmat_bend.vector_mult(vec2_n1, _local_sol);
vec1_n1 = material_B_mat * vec2_n1;
stress_l(0,0) += vec2_n1(0); // sigma_xx
stress_l(0,1) += vec2_n1(2); // sigma_xy
stress_l(1,0) += vec2_n1(2); // sigma_yx
stress_l(1,1) += vec2_n1(1); // sigma_yy
stress(0,0) += vec2_n1(0); // sigma_xx
stress(0,1) += vec2_n1(2); // sigma_xy
stress(1,0) += vec2_n1(2); // sigma_yx
stress(1,1) += vec2_n1(1); // sigma_yy
if (if_vk) { // get the vonKarman strain operator if needed
this->initialize_von_karman_strain_operator(qp,
vec2_n1, // epsilon_vk
vk_dwdxi_mat,
Bmat_vk);
vec1_n1 = material_A_mat * vec2_n1; // stress
stress(0,0) += vec1_n1(0); // sigma_xx
stress(0,1) += vec1_n1(2); // sigma_xy
stress(1,0) += vec1_n1(2); // sigma_yx
stress(1,1) += vec1_n1(1); // sigma_yy
}
}
// add the linear and nonlinear direct strains
Bmat_mem.vector_mult(vec1_n1, _local_sol);
vec2_n1 += vec1_n1; // epsilon_mem + epsilon_vk
// copy the total integrated stress to the vector
vec1_n1(0) = stress(0,0);
vec1_n1(1) = stress(1,1);
vec1_n1(2) = stress(0,1);
// now the internal force vector
// this includes the membrane strain operator with all A and B material operators
Bmat_mem.vector_mult_transpose(vec3_n2, vec1_n1);
local_f += JxW[qp] * vec3_n2;
if (if_bending) {
if (if_vk) {
// von Karman strain
vec4_2 = vk_dwdxi_mat.transpose() * vec1_n1;
Bmat_vk.vector_mult_transpose(vec3_n2, vec4_2);
local_f += JxW[qp] * vec3_n2;
}
// now coupling with the bending strain
// B_bend^T [B] B_mem
vec1_n1 = material_B_mat * vec2_n1;
Bmat_bend.vector_mult_transpose(vec3_n2, vec1_n1);
local_f += JxW[qp] * vec3_n2;
// now bending stress
Bmat_bend.vector_mult(vec2_n1, _local_sol);
vec1_n1 = material_D_mat * vec2_n1;
Bmat_bend.vector_mult_transpose(vec3_n2, vec1_n1);
local_f += JxW[qp] * vec3_n2;
}
if (request_jacobian) {
// membrane - membrane
Bmat_mem.left_multiply(mat1_n1n2, material_A_mat);
Bmat_mem.right_multiply_transpose(mat2_n2n2, mat1_n1n2);
local_jac += JxW[qp] * mat2_n2n2;
if (if_bending) {
if (if_vk) {
// membrane - vk
mat3 = RealMatrixX::Zero(vk_dwdxi_mat.rows(), n2);
Bmat_vk.left_multiply(mat3, vk_dwdxi_mat);
mat3 = material_A_mat * mat3;
Bmat_mem.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// vk - membrane
Bmat_mem.left_multiply(mat1_n1n2, material_A_mat);
mat3 = vk_dwdxi_mat.transpose() * mat1_n1n2;
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// if only the first order term of the Jacobian is needed, for
// example for linearized buckling analysis, then the linear
// stress combined with the variation of the von Karman strain
// is included. Otherwise, all terms are included
if (if_ignore_ho_jac) {
// vk - vk: first order term
mat3 = RealMatrixX::Zero(2, n2);
Bmat_vk.left_multiply(mat3, stress_l);
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
}
else {
// vk - vk
mat3 = RealMatrixX::Zero(2, n2);
Bmat_vk.left_multiply(mat3, stress);
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
mat3 = RealMatrixX::Zero(vk_dwdxi_mat.rows(), n2);
Bmat_vk.left_multiply(mat3, vk_dwdxi_mat);
mat3 = vk_dwdxi_mat.transpose() * material_A_mat * mat3;
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
}
// bending - vk
mat3 = RealMatrixX::Zero(vk_dwdxi_mat.rows(), n2);
Bmat_vk.left_multiply(mat3, vk_dwdxi_mat);
mat3 = material_B_mat.transpose() * mat3;
Bmat_bend.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
// vk - bending
Bmat_bend.left_multiply(mat1_n1n2, material_B_mat);
mat3 = vk_dwdxi_mat.transpose() * mat1_n1n2;
Bmat_vk.right_multiply_transpose(mat2_n2n2, mat3);
local_jac += JxW[qp] * mat2_n2n2;
}
// bending - membrane
Bmat_mem.left_multiply(mat1_n1n2, material_B_mat);
Bmat_bend.right_multiply_transpose(mat2_n2n2, mat1_n1n2);
local_jac += JxW[qp] * mat2_n2n2;
// membrane - bending
Bmat_bend.left_multiply(mat1_n1n2, material_B_mat);
Bmat_mem.right_multiply_transpose(mat2_n2n2, mat1_n1n2);
local_jac += JxW[qp] * mat2_n2n2;
// bending - bending
Bmat_bend.left_multiply(mat1_n1n2, material_D_mat);
Bmat_bend.right_multiply_transpose(mat2_n2n2, mat1_n1n2);
local_jac += JxW[qp] * mat2_n2n2;
}
}
}