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::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;
}
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::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;
}
}
}
