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From: Roy Stogner <roystgnr@ic...>  20120907 22:21:50

On Fri, 7 Sep 2012, Andrew E Slaughter wrote: > I am working on solving the level set equation using DG methods, form the > looks of example Miscellaneous Example 5 this should be possible. It certainly should be. > Currently, I am simply trying to formulate the mass matrix, which is simply > the integral of N^T N over the domain, where N is the shape function > vector. I would like this matrix to be diagonal so it can be inverted > locally allowing me to solve the system with an explicit scheme. Does > libMesh have a set of basis functions that could offer me his behavior? Not yet. We'd love a patch adding them. ;) I'm not sure how you'd want to go about it, though. Discontinuous orthogonal polynomials are easy to derive on rectangles, but they'd no longer be orthogonal once they went through a nonaffine transformation onto physical elements. I suppose you could just work with triangles/tets; if the inner product is a plain L2 integration then orthogonality ought to be preserved by affine maps. Start with the MONOMIAL bases and then do GramSchmidt to get an orthonormal basis? > The book I am following for this problem recommends "orthogonal > hierarchical shape functions". So, I tried to using L2_HIERARCHIC > via the add_system(...) command, but the mass matrix was not > diagonal. Could it be a problem with how I am assembling the matrix > via quadrature points, my assemble() function is shown at the end. I > am new to FEM and mostly selftaught so I am naive in many aspects. No  the L2_HIERARCHIC shapes are discontinuous and hierarchic, but they aren't orthogonal in L2  that name was just supposed to distinguish them from the "default" continuous (H1conforming) hierarchics that share the same local numbering and shapes. I suppose you could do a numerical stabilized GramSchmidt process on *any* set of discontinuous bases (maybe start from XYZ?), doing projections with the same quadrature you use for integration. That would be expensive enough that you'd want to do it once and cache the results, but you might want to cache shape function evaluations anyway if you're trying an explicit solve.  Roy 
From: Andrew E Slaughter <andrew.e.slaughter@gm...>  20120907 21:07:29

I am working on solving the level set equation using DG methods, form the looks of example Miscellaneous Example 5 this should be possible. Question 1: Currently, I am simply trying to formulate the mass matrix, which is simply the integral of N^T N over the domain, where N is the shape function vector. I would like this matrix to be diagonal so it can be inverted locally allowing me to solve the system with an explicit scheme. Does libMesh have a set of basis functions that could offer me his behavior? The book I am following for this problem recommends "orthogonal hierarchical shape functions". So, I tried to using L2_HIERARCHIC via the add_system(...) command, but the mass matrix was not diagonal. Could it be a problem with how I am assembling the matrix via quadrature points, my assemble() function is shown at the end. I am new to FEM and mostly selftaught so I am naive in many aspects. Question 2: Is there any professional help available for developing libMesh code. I have a research grant and could offer payment for assistance, which I need and will need as I continue on this project. There is also the potential for collaboration and publications if anyone is interested. Thanks and I appreciate any help I can get. Andrew  ASSEMBLE FUNCTION  void LevelSetSystem :: assemble(){ // Get a constant reference to the mesh object. const MeshBase& mesh = get_mesh(); // The dimension that we are running const unsigned int dim = mesh.mesh_dimension(); // Get a constant reference to the Finite Element type // for the first (and only) variable in the system. FEType fe_type = variable_type(0); // Build a Finite Element object of the specified type. AutoPtr<FEBase> fe (FEBase::build(dim, fe_type)); AutoPtr<FEBase> fe_face (FEBase::build(dim, fe_type)); // A Gauss quadrature rule for numerical integration. // Let the \p FEType object decide what order rule is appropriate. QGauss qrule (dim, fe_type.default_quadrature_order()); QGauss qface (dim1, fe_type.default_quadrature_order()); // Tell the finite element object to use our quadrature rule. fe>attach_quadrature_rule (&qrule); fe_face>attach_quadrature_rule (&qface); // Here we define some references to cellspecific data that // will be used to assemble the linear system. We will start // with the element Jacobian * quadrature weight at each integration point. const std::vector<Real>& JxW = fe>get_JxW(); const std::vector<Real>& JxW_face = fe_face>get_JxW(); // The element shape functions evaluated at the quadrature points. const std::vector<std::vector<Real> >& N = fe>get_phi(); const std::vector<std::vector<Real> >& N_face = fe_face>get_phi(); // The element shape function gradients evaluated at the quadrature // points. const std::vector<std::vector<RealGradient> >& B = fe>get_dphi(); // The XY locations of the quadrature points used for face integration const std::vector<Point>& qface_points = fe_face>get_xyz(); // A reference to the \p DofMap object for this system. const DofMap& dof_map = get_dof_map(); // Define data structures to contain the element matrix // and righthandside vector contribution. DenseMatrix<Number> Me; // element mass matrix //DenseMatrix<Number> Ke; // element stiffness matrix // Storage for the degree of freedom indices std::vector<unsigned int> dof_indices; // Get the system time Real t = this>time; // Loop over all the elements in the mesh that are on local processor MeshBase::const_element_iterator el = mesh.active_local_elements_begin(); const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end(); for ( ; el != end_el; ++el){ // Pointer to the element current element const Elem* elem = *el; // Get the degree of freedom indices for the current element dof_map.dof_indices(elem, dof_indices); // Compute the elementspecific data for the current element fe>reinit (elem); // Extract a vector of quadrature x,y,z coordinates const vector<Point> qp_vec = fe>get_xyz(); // Zero the element matrices and vectors Me.resize (dof_indices.size(), dof_indices.size()); //Ke.resize (dof_indices.size(), dof_indices.size()); // Compute the RHS and mass and stiffness matrix for this element (Me) for (unsigned int qp = 0; qp < qrule.n_points(); qp++){ // Extract the velocity vector for the current point VectorValue<Number> v(dim); velocity(v, qp_vec[qp], t); for (unsigned int i = 0; i < N.size(); i++){ for (unsigned int j = 0; j < N.size(); j++){ Me(i,j) += JxW[qp]*(N[i][qp] * N[j][qp]); // mass matrix //Ke(i,j) += JxW[qp]*(N[i][qp] * (v*B[j][qp])); // stiffness matrix } } } printf("Me:\n"); Me.print(std::cout); // Apply the local components to the global mass matrix request_matrix("mass")>add_matrix(Me, dof_indices); //this>matrix>add_matrix(Ke, dof_indices); } // (end) for ( ; el != end_el; ++el) // Indicate that the mass matrix is complete request_matrix("mass")>close(); //this>matrix>close(); } // (end) assemble()  Andrew E. Slaughter, PhD andrew.e.slaughter@... Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 148533801 (607) 2291829 http://aeslaughter.github.com/ 