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:: small_disturbance_surface_pressure_force(bool request_jacobian, DenseRealVector &f, DenseRealMatrix &jac, MAST::BoundaryCondition &p) { libmesh_assert(_elem.dim() < 3); // only applicable for lower dimensional elements. libmesh_assert(!follower_forces); // not implemented yet for follower forces libmesh_assert_equal_to(p.type(), MAST::SMALL_DISTURBANCE_MOTION); MAST::SmallDisturbanceMotion& sd_motion = dynamic_cast<MAST::SmallDisturbanceMotion&>(p); MAST::SurfaceMotionBase& surf_motion = sd_motion.get_deformation(); MAST::SmallDisturbanceSurfacePressure& surf_press = sd_motion.get_pressure(); FEMOperatorMatrix Bmat; // get the function from this boundary condition const std::vector<Real> &JxW = _fe->get_JxW(); // Physical location of the quadrature points const std::vector<libMesh::Point>& qpoint = _fe->get_xyz(); const std::vector<std::vector<Real> >& phi = _fe->get_phi(); const unsigned int n_phi = (unsigned int)phi.size(); const unsigned int n1=3, n2=6*n_phi; // normal for face integration libMesh::Point normal; // direction of pressure assumed to be normal (along local z-axis) // to the element face for 2D and along local y-axis for 1D element. normal(_elem.dim()) = -1.; Real press; ValType dpress; DenseRealVector phi_vec; libMesh::DenseVector<ValType> wtrans, utrans, dn_rot, force, local_f, vec_n2; wtrans.resize(3); utrans.resize(3); dn_rot.resize(3); phi_vec.resize(n_phi); force.resize(2*n1); local_f.resize(n2); vec_n2.resize(n2); libMesh::Point pt; for (unsigned int qp=0; qp<qpoint.size(); qp++) { this->global_coordinates(qpoint[qp], pt); // 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(2*n1, phi_vec); // get pressure and deformation information surf_press.surface_pressure<ValType>(_system.time, pt, press, dpress); surf_motion.surface_velocity(_system.time, pt, normal, wtrans, utrans, dn_rot); // libMesh::out << std::setw(15) << pt(0) // << std::setw(15) << std::real(press) // << std::setw(15) << std::imag(press) // << std::setw(15) << std::real(dpress) // << std::setw(15) << std::imag(dpress) // << std::setw(15) << std::real(utrans(1)) // << std::setw(15) << std::imag(utrans(1)) // << std::setw(15) << std::real(dn_rot(0)) // << std::setw(15) << std::imag(dn_rot(0)) << std::endl; // calculate force for (unsigned int i_dim=0; i_dim<n1; i_dim++) force(i_dim) = ( press * dn_rot(i_dim) + // steady pressure dpress * normal(i_dim)); // unsteady pressure Bmat.vector_mult_transpose(vec_n2, force); local_f.add(JxW[qp], vec_n2); } // now transform to the global system and add transform_to_global_system(local_f, vec_n2); MAST::add_to_assembled_vector(f, vec_n2); return (request_jacobian && follower_forces); }
bool MAST::StructuralElementBase::small_disturbance_surface_pressure_force(bool request_jacobian, DenseRealVector &f, DenseRealMatrix &jac, const unsigned int side, MAST::BoundaryCondition &p) { libmesh_assert(!follower_forces); // not implemented yet for follower forces libmesh_assert_equal_to(p.type(), MAST::SMALL_DISTURBANCE_MOTION); MAST::SmallDisturbanceMotion& sd_motion = dynamic_cast<MAST::SmallDisturbanceMotion&>(p); MAST::SurfaceMotionBase& surf_motion = sd_motion.get_deformation(); MAST::SmallDisturbanceSurfacePressure& surf_press = sd_motion.get_pressure(); FEMOperatorMatrix Bmat; // get the function from this boundary condition std::auto_ptr<libMesh::FEBase> fe; std::auto_ptr<libMesh::QBase> qrule; _get_side_fe_and_qrule(this->local_elem().local_elem(), side, fe, qrule); const std::vector<Real> &JxW = fe->get_JxW(); // Physical location of the quadrature points const std::vector<libMesh::Point>& qpoint = fe->get_xyz(); const std::vector<std::vector<Real> >& phi = fe->get_phi(); const unsigned int n_phi = (unsigned int)phi.size(); const unsigned int n1=3, n2=6*n_phi; // boundary normals const std::vector<libMesh::Point>& face_normals = fe->get_normals(); Real press; ValType dpress; DenseRealVector phi_vec; libMesh::DenseVector<ValType> wtrans, utrans, dn_rot, force, local_f, vec_n2; wtrans.resize(3); utrans.resize(3); dn_rot.resize(3); phi_vec.resize(n_phi); force.resize(2*n1); local_f.resize(n2); vec_n2.resize(n2); libMesh::Point pt; for (unsigned int qp=0; qp<qpoint.size(); qp++) { this->global_coordinates(qpoint[qp], pt); // 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(2*n1, phi_vec); // get pressure and deformation information surf_press.surface_pressure<ValType>(_system.time, pt, press, dpress); surf_motion.surface_velocity(_system.time, pt, face_normals[qp], wtrans, utrans, dn_rot); // press = 0.; // dpress = Complex(2./4.*std::real(dn_rot(0)), 2./4./.1*std::imag(utrans(1))); // libMesh::out << q_point[qp](0) // << " " << std::real(utrans(1)) // << " " << std::imag(utrans(1)) // << " " << std::real(dn_rot(0)) // << " " << std::imag(dn_rot(0)) // << " " << std::real(press) // << " " << std::imag(press) // << " " << std::real(dpress) // << " " << std::imag(dpress) << std::endl; // calculate force for (unsigned int i_dim=0; i_dim<n1; i_dim++) force(i_dim) = ( press * dn_rot(i_dim) + // steady pressure dpress * face_normals[qp](i_dim)); // unsteady pressure Bmat.vector_mult_transpose(vec_n2, force); local_f.add(JxW[qp], vec_n2); } // now transform to the global system and add if (_elem.dim() < 3) { transform_to_global_system(local_f, vec_n2); MAST::add_to_assembled_vector(f, vec_n2); } else MAST::add_to_assembled_vector(f, local_f); return (request_jacobian && follower_forces); }
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; }
bool MAST::StructuralElementBase::surface_pressure_force(bool request_jacobian, DenseRealVector &f, DenseRealMatrix &jac, MAST::BoundaryCondition &p) { libmesh_assert(_elem.dim() < 3); // only applicable for lower dimensional elements libmesh_assert(!follower_forces); // not implemented yet for follower forces FEMOperatorMatrix Bmat; // get the function from this boundary condition MAST::FieldFunction<Real>& func = dynamic_cast<MAST::FieldFunction<Real>&>(p.function()); const std::vector<Real> &JxW = _fe->get_JxW(); // Physical location of the quadrature points const std::vector<libMesh::Point>& qpoint = _fe->get_xyz(); const std::vector<std::vector<Real> >& phi = _fe->get_phi(); const unsigned int n_phi = (unsigned int)phi.size(); const unsigned int n1=3, n2=6*n_phi; // normal for face integration libMesh::Point normal; // direction of pressure assumed to be normal (along local z-axis) // to the element face for 2D and along local y-axis for 1D element. normal(_elem.dim()) = -1.; Real press; DenseRealVector phi_vec, force, local_f, vec_n2; phi_vec.resize(n_phi); force.resize(2*n1); local_f.resize(n2); vec_n2.resize(n2); libMesh::Point pt; for (unsigned int qp=0; qp<qpoint.size(); qp++) { this->global_coordinates(qpoint[qp], pt); // 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(2*n1, phi_vec); // get pressure value func(pt, _system.time, press); // calculate force for (unsigned int i_dim=0; i_dim<n1; i_dim++) force(i_dim) = press * normal(i_dim); Bmat.vector_mult_transpose(vec_n2, force); local_f.add(JxW[qp], vec_n2); } // now transform to the global system and add transform_to_global_system(local_f, vec_n2); f.add(1., vec_n2); return (request_jacobian && follower_forces); }
bool MAST::StructuralElementBase::surface_pressure_force(bool request_jacobian, DenseRealVector &f, DenseRealMatrix &jac, const unsigned int side, MAST::BoundaryCondition &p) { libmesh_assert(!follower_forces); // not implemented yet for follower forces FEMOperatorMatrix Bmat; // get the function from this boundary condition MAST::FieldFunction<Real>& func = dynamic_cast<MAST::FieldFunction<Real>&>(p.function()); std::auto_ptr<libMesh::FEBase> fe; std::auto_ptr<libMesh::QBase> qrule; _get_side_fe_and_qrule(this->get_elem_for_quadrature(), side, fe, qrule); const std::vector<Real> &JxW = fe->get_JxW(); // Physical location of the quadrature points const std::vector<libMesh::Point>& qpoint = fe->get_xyz(); const std::vector<std::vector<Real> >& phi = fe->get_phi(); const unsigned int n_phi = (unsigned int)phi.size(); const unsigned int n1=3, n2=6*n_phi; // boundary normals const std::vector<libMesh::Point>& face_normals = fe->get_normals(); Real press; DenseRealVector phi_vec, force, local_f, vec_n2; phi_vec.resize(n_phi); force.resize(2*n1); local_f.resize(n2); vec_n2.resize(n2); libMesh::Point pt; for (unsigned int qp=0; qp<qpoint.size(); qp++) { this->global_coordinates(qpoint[qp], pt); // 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(2*n1, phi_vec); // get pressure value func(pt, _system.time, press); // calculate force for (unsigned int i_dim=0; i_dim<n1; i_dim++) force(i_dim) = press * face_normals[qp](i_dim); Bmat.vector_mult_transpose(vec_n2, force); local_f.add(JxW[qp], vec_n2); } // now transform to the global system and add if (_elem.dim() < 3) { transform_to_global_system(local_f, vec_n2); f.add(1., vec_n2); } else f.add(1., local_f); return (request_jacobian && follower_forces); }
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:: _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; } } }