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; } }
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; } } }