void
MAST::MindlinBendingOperator::
_transverse_shear_operations(const std::vector<std::vector<Real> >& phi,
const std::vector<std::vector<libMesh::RealVectorValue> >& dphi,
const std::vector<Real>& JxW,
const unsigned int qp,
const RealMatrixX& material,
FEMOperatorMatrix& Bmat,
RealVectorX& phi_vec,
RealVectorX& vec_n2,
RealVectorX& vec_2,
RealMatrixX& mat_n2n2,
RealMatrixX& mat_2n2,
bool request_jacobian,
RealVectorX& local_f,
RealMatrixX& local_jac) {
// initialize the strain operator
for ( unsigned int i_nd=0; i_nd<phi.size(); i_nd++ )
phi_vec(i_nd) = dphi[i_nd][qp](0); // dphi/dx
Bmat.set_shape_function(0, 2, phi_vec); // gamma-xz: w
for ( unsigned int i_nd=0; i_nd<phi.size(); i_nd++ )
phi_vec(i_nd) = dphi[i_nd][qp](1); // dphi/dy
Bmat.set_shape_function(1, 2, phi_vec); // gamma-yz : w
for ( unsigned int i_nd=0; i_nd<phi.size(); i_nd++ )
phi_vec(i_nd) = phi[i_nd][qp]; // phi
Bmat.set_shape_function(0, 4, phi_vec); // gamma-xz: thetay
phi_vec *= -1.0;
Bmat.set_shape_function(1, 3, phi_vec); // gamma-yz : thetax
// now add the transverse shear component
Bmat.vector_mult(vec_2, _structural_elem.local_solution());
vec_2 = material * vec_2;
Bmat.vector_mult_transpose(vec_n2, vec_2);
local_f += JxW[qp] * vec_n2;
if (request_jacobian) {
// now add the transverse shear component
Bmat.left_multiply(mat_2n2, material);
Bmat.right_multiply_transpose(mat_n2n2, mat_2n2);
local_jac += JxW[qp] * mat_n2n2;
}
}
bool
MAST::StructuralElementBase::inertial_force (bool request_jacobian,
DenseRealVector& f,
DenseRealMatrix& jac)
{
FEMOperatorMatrix Bmat;
const std::vector<Real>& JxW = _fe->get_JxW();
const std::vector<libMesh::Point>& xyz = _fe->get_xyz();
const std::vector<std::vector<Real> >& phi = _fe->get_phi();
const unsigned int n_phi = (unsigned int)phi.size(), n1=6, n2=6*n_phi;
DenseRealMatrix material_mat, mat1_n1n2, mat2_n2n2, local_jac;
DenseRealVector phi_vec, vec1_n1, vec2_n2, local_f;
mat1_n1n2.resize(n1, n2); mat2_n2n2.resize(n2, n2); local_jac.resize(n2, n2);
phi_vec.resize(n_phi); vec1_n1.resize(n1); vec2_n2.resize(n2);
local_f.resize(n2);
std::auto_ptr<MAST::FieldFunction<DenseRealMatrix > > mat_inertia
(_property.get_property(MAST::SECTION_INTEGRATED_MATERIAL_INERTIA_MATRIX,
*this).release());
if (_property.if_diagonal_mass_matrix()) {
// as an approximation, get matrix at the first quadrature point
(*mat_inertia)(xyz[0], _system.time, material_mat);
Real vol = 0.;
const unsigned int nshp = _fe->n_shape_functions();
for (unsigned int i=0; i<JxW.size(); i++)
vol += JxW[i];
vol /= (1.* nshp);
for (unsigned int i_var=0; i_var<6; i_var++)
for (unsigned int i=0; i<nshp; i++)
local_jac(i_var*nshp+i, i_var*nshp+i) =
vol*material_mat(i_var, i_var);
local_jac.vector_mult(local_f, local_acceleration);
}
else {
libMesh::Point p;
for (unsigned int qp=0; qp<JxW.size(); qp++) {
this->global_coordinates(xyz[qp], p);
(*mat_inertia)(p, _system.time, material_mat);
// now set the shape function values
for ( unsigned int i_nd=0; i_nd<n_phi; i_nd++ )
phi_vec(i_nd) = phi[i_nd][qp];
Bmat.reinit(_system.n_vars(), phi_vec);
Bmat.left_multiply(mat1_n1n2, material_mat);
mat1_n1n2.vector_mult(vec1_n1, local_acceleration);
Bmat.vector_mult_transpose(vec2_n2, vec1_n1);
local_f.add(JxW[qp], vec2_n2);
if (request_jacobian) {
Bmat.right_multiply_transpose(mat2_n2n2,
mat1_n1n2);
local_jac.add(JxW[qp], mat2_n2n2);
}
}
}
// now transform to the global coorodinate system
if (_elem.dim() < 3) {
transform_to_global_system(local_f, vec2_n2);
f.add(1., vec2_n2);
if (request_jacobian) {
transform_to_global_system(local_jac, mat2_n2n2);
jac.add(1., mat2_n2n2);
}
}
else {
f.add(1., local_f);
if (request_jacobian)
jac.add(1., local_jac);
}
return request_jacobian;
}
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;
}
bool
MAST::StructuralElementBase::inertial_force_sensitivity(bool request_jacobian,
DenseRealVector& f,
DenseRealMatrix& jac)
{
// this should be true if the function is called
libmesh_assert(this->sensitivity_param);
libmesh_assert(!this->sensitivity_param->is_shape_parameter()); // this is not implemented for now
// check if the material property or the provided exterior
// values, like temperature, are functions of the sensitivity parameter
bool calculate = false;
calculate = calculate || _property.depends_on(*(this->sensitivity_param));
// nothing to be calculated if the element does not depend on the
// sensitivity parameter.
if (!calculate)
return false;
FEMOperatorMatrix Bmat;
const std::vector<Real>& JxW = _fe->get_JxW();
const std::vector<libMesh::Point>& xyz = _fe->get_xyz();
const std::vector<std::vector<Real> >& phi = _fe->get_phi();
const unsigned int n_phi = (unsigned int)phi.size(), n1=6, n2=6*n_phi;
DenseRealMatrix material_mat, mat1_n1n2, mat2_n2n2, local_jac;
DenseRealVector phi_vec, vec1_n1, vec2_n2, local_f;
mat1_n1n2.resize(n1, n2); mat2_n2n2.resize(n2, n2); local_jac.resize(n2, n2);
phi_vec.resize(n_phi); vec1_n1.resize(n1); vec2_n2.resize(n2);
local_f.resize(n2);
std::auto_ptr<MAST::FieldFunction<DenseRealMatrix > > mat_inertia
(_property.get_property(MAST::SECTION_INTEGRATED_MATERIAL_INERTIA_MATRIX,
*this).release());
if (_property.if_diagonal_mass_matrix()) {
mat_inertia->total(*this->sensitivity_param,
xyz[0], _system.time, material_mat);
Real vol = 0.;
const unsigned int nshp = _fe->n_shape_functions();
for (unsigned int i=0; i<JxW.size(); i++)
vol += JxW[i];
vol /= (1.* nshp);
for (unsigned int i_var=0; i_var<6; i_var++)
for (unsigned int i=0; i<nshp; i++)
local_jac(i_var*nshp+i, i_var*nshp+i) =
vol*material_mat(i_var, i_var);
local_jac.vector_mult(local_f, local_acceleration);
}
else {
libMesh::Point p;
for (unsigned int qp=0; qp<JxW.size(); qp++) {
this->global_coordinates(xyz[qp], p);
mat_inertia->total(*this->sensitivity_param,
p, _system.time, material_mat);
// now set the shape function values
for ( unsigned int i_nd=0; i_nd<n_phi; i_nd++ )
phi_vec(i_nd) = phi[i_nd][qp];
Bmat.reinit(_system.n_vars(), phi_vec);
Bmat.left_multiply(mat1_n1n2, material_mat);
mat1_n1n2.vector_mult(vec1_n1, local_acceleration);
Bmat.vector_mult_transpose(vec2_n2, vec1_n1);
local_f.add(JxW[qp], vec2_n2);
if (request_jacobian) {
Bmat.right_multiply_transpose(mat2_n2n2,
mat1_n1n2);
local_jac.add(JxW[qp], mat2_n2n2);
}
}
}
// now transform to the global coorodinate system
if (_elem.dim() < 3) {
transform_to_global_system(local_f, vec2_n2);
f.add(1., vec2_n2);
if (request_jacobian) {
transform_to_global_system(local_jac, mat2_n2n2);
jac.add(1., mat2_n2n2);
}
}
else {
f.add(1., local_f);
if (request_jacobian)
jac.add(1., local_jac);
}
return request_jacobian;
}
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;
}
}
}
