Here are the steps to solve Laplacian u(x,y,z) = f(x,y,z): 1. Form A = Assembly( A_elem ) where A_elem = integral(dPhi_i*dPhi_j) 2. Form F_nodal = f(x_i,y_i,z_i) at all mesh nodes 3. Form MassMatrix = Assembly( M_elem ) where M_elem = integral(Phi_k, Phi_j) 4. Form RHS = MassMatrix*F_nodal 5. Solve A U_nodal = RHS for U_nodal
On Fri, Mar 26, 2010 at 5:26 PM, Karen Lee <[email protected]> wrote: > This would be for the interpolation only right? > > So if I use this GlobalRHS, and construct my LHS based on the actual problem > I want to solve (just a Poisson, so I would be using dphi), it would be > equivalent to solving the problem with the RHS being interpolated? > > Karen > > > On Fri, Mar 26, 2010 at 5:23 PM, Rahul Sampath <[email protected]> > wrote: >> >> To be more precise: >> >> GlobalMassMatrix = Sum_over_elements{ >> integral_over_current_element_using_quadrature(phi_i*phi_j) } >> >> GlobalRHS = GlobalMassMatrix*Nodal_F_Vector >> >> On Fri, Mar 26, 2010 at 5:22 PM, Rahul Sampath <[email protected]> >> wrote: >> > This is what I meant: >> > >> > MassMatrix = integral(phi_i*phi_j) >> > >> > RHS = MassMatrix*Nodal_F_Vector >> > >> > >> > On Fri, Mar 26, 2010 at 5:11 PM, Karen Lee <[email protected]> wrote: >> >> Sorry, I think I was probably confused. I guess you just meant that I >> >> can >> >> simply use integral of f_i phi_i phi_j as my RHS, and my original LHS >> >> as my >> >> LHS, and that would already be effectively an interpolation? Please let >> >> me >> >> know if this is the correct way to think about it instead of the long >> >> message I sent with lots of code... >> >> >> >> My 2 questions regarding access to data would still remain: >> >> >> >> 1) I am still having problems accessing MeshData from the assemble >> >> function. I would like to do something like >> >> mesh_data.get_data(el.get_node(i))[0] to get the first data variable >> >> for >> >> node i with an element, i from 0 to 3, but I'm not sure how to access >> >> mesh_data from my System. (I should use Linear Implicit System so I can >> >> have >> >> the matrix on the lhs right?) >> >> >> >> 2) The other question is, for Fe, do I integrate over all the >> >> quadrature >> >> points of phi_i and phi_j with f_i being a constant? >> >> >> >> Thank you so much!!! >> >> Karen >> >> >> >> >> >> On Fri, Mar 26, 2010 at 4:57 PM, Karen Lee <[email protected]> wrote: >> >>> >> >>> Thanks Rahul. Your responses have clarified things for me. >> >>> >> >>> I am still having problems accessing MeshData from the assemble >> >>> function >> >>> (which is only supposed to have 2 arguments right?) >> >>> >> >>> I can do: >> >>> >> >>> void assemble_load(EquationSystems& es, >> >>> const std::string& system_name) >> >>> { >> >>> >> >>> libmesh_assert (system_name == "load"); >> >>> >> >>> >> >>> const MeshBase& mesh = es.get_mesh(); >> >>> printf("mesh_data obtained"); >> >>> >> >>> ...... >> >>> >> >>> But I'm not sure how I can get the MeshData object that I created in >> >>> main >> >>> that's attached to the mesh I created. In the look iterating over the >> >>> elements el, I know that I have to use something like >> >>> mesh_data.get_data(el.get_node(i))[0] to get the first data variable >> >>> for >> >>> node i with an element, i from 0 to 3, but I'm not sure how to access >> >>> mesh_data from my System. (I should use Linear Implicit System so I >> >>> can have >> >>> the matrix on the lhs right?) >> >>> >> >>> My code is based on example 3, and the relevant part is this (not sure >> >>> if >> >>> it's correct): >> >>> >> >>> for ( ; el != end_el ; ++el) >> >>> { >> >>> const Elem* elem = *el; >> >>> >> >>> dof_map.dof_indices (elem, dof_indices); >> >>> >> >>> fe->reinit (elem); >> >>> >> >>> Ke.resize (dof_indices.size(), >> >>> dof_indices.size()); >> >>> >> >>> Fe.resize (dof_indices.size()); >> >>> >> >>> for (unsigned int qp=0; qp<qrule.n_points(); qp++) >> >>> { >> >>> >> >>> for (unsigned int i=0; i<phi.size(); i++) >> >>> for (unsigned int j=0; j<phi.size(); j++) >> >>> { >> >>> Ke(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp]); >> >>> } >> >>> >> >>> { >> >>> for (unsigned int i=0; i<phi.size(); i++) >> >>> { >> >>> for (unsigned int k=0; k<phi.size(); k++) >> >>> { >> >>> Fe(i) += >> >>> JxW[qp]*mesh_data.get_data(el.get_node(k))[0]*phi[i][qp]phi[k][qp]; >> >>> >> >>> } >> >>> } >> >>> } >> >>> .... >> >>> >> >>> The other question is, for Fe, do I integrate over all the quadrature >> >>> points of phi_i and phi_j with f_i being a constant? >> >>> >> >>> Then once I get this solution (the variable name is "R", let's say), I >> >>> hope to use it in place of mesh_data.get_data(el.get_node(k))[0] and >> >>> do >> >>> something like this (of course this is another system that I'm adding >> >>> to >> >>> EquationSystems): >> >>> >> >>> for (unsigned int qp=0; qp<qrule.n_points(); qp++) >> >>> { >> >>> >> >>> for (unsigned int i=0; i<phi.size(); i++) >> >>> >> >>> >> >>> for (unsigned int j=0; j<phi.size(); j++) >> >>> { >> >>> Ke2(i,j) += JxW[qp]*(dphi[i][qp]*dphi[j][qp]); >> >>> >> >>> >> >>> } >> >>> >> >>> { >> >>> const Real x = q_point[qp](0); >> >>> const Real y = q_point[qp](1); >> >>> const Real z = q_point[qp](2); >> >>> >> >>> >> >>> >> >>> >> >>> const Real fxy = R(x,y,z); >> >>> >> >>> for (unsigned int i=0; i<phi.size(); i++) >> >>> >> >>> >> >>> Fe2(i) += JxW[qp]*fxy*phi[i][qp]; >> >>> } >> >>> } >> >>> >> >>> I'm not sure how to access anything in a Variable object though... >> >>> Let's >> >>> say "R" being a a variable I add to the first system to get the RHS >> >>> interpolation, and "u" is the variable I add to the second equation >> >>> system, >> >>> which is the actual solution I'm after... I just know that I can >> >>> output the >> >>> values of the solution at various nodal points in a file, but am not >> >>> sure >> >>> what to do with the data structure and how I can extract values at >> >>> different >> >>> arbitrary locations. >> >>> >> >>> Apologies for the lengthy email... >> >>> >> >>> Thanks, >> >>> Karen >> >>> >> >>> >> >>> On Fri, Mar 26, 2010 at 9:15 AM, Rahul Sampath >> >>> <[email protected]> >> >>> wrote: >> >>>> >> >>>> Hi Karen: >> >>>> >> >>>> Take a look at any Nonlinear example problem. Whenever you want to >> >>>> use >> >>>> any solution vector in your residual computation, you will need to >> >>>> interpolate the nodal values using the FEM shape functions for this >> >>>> element and then do the integration. It is very similar to what you >> >>>> want to do. That is why I suggested the Mass matrix trick. It is very >> >>>> simple to implement and fast too especially if you want to change f >> >>>> often. You can use the same Mass matrix with differen f. The catch is >> >>>> that you are using your FE shape functions for your interpolation. As >> >>>> long as you are happy with linear interpolation for f this should do. >> >>>> If you want to interpolate f with a higher order polynomial than your >> >>>> FE shape function then this wont work. >> >>>> >> >>>> Btw, if I was not clear earlier: >> >>>> You have to form a global Mass matrix by integrating phi_i phi_j over >> >>>> all elements and doing a typically FEM assembly. Then you can simply >> >>>> multiply this global Mass matrix with you global nodal vector for f. >> >>>> >> >>>> On Fri, Mar 26, 2010 at 12:51 AM, Karen Lee <[email protected]> >> >>>> wrote: >> >>>> > Hi Rahul, I'm not completely sure what you mean. >> >>>> > >> >>>> > I would like to form my RHS by integrating f_i Phi_i (I guess >> >>>> > there's >> >>>> > no >> >>>> > need to multiply Phi_j? But you can correct me) for each element. >> >>>> > >> >>>> > In order to do so, I need values of f at various quadrature points. >> >>>> > I >> >>>> > have f >> >>>> > at various nodal values. The question is, how do I get this linear >> >>>> > interpolation... >> >>>> > >> >>>> > Do you mean that, for each element, I form the mass matrix by the >> >>>> > xyz >> >>>> > values >> >>>> > of the nodes (and a constant 1) and solve for the coefficient by >> >>>> > saying >> >>>> > \sum_j A_ij y_j= f_i, where: >> >>>> > >> >>>> > A = [1 x1 y1 z1; >> >>>> > 1 x2 y2 z2; >> >>>> > 1 x3 y3 z3; >> >>>> > 1 x4 y4 z4] and y_j would be my unknown (where j = 1 >> >>>> > corresponds >> >>>> > to >> >>>> > the constant value, and 2, 3, 4 corresponds to the gradient in the >> >>>> > x, >> >>>> > y, z >> >>>> > directions respectively)? >> >>>> > >> >>>> > Thanks, >> >>>> > Karen >> >>>> > >> >>>> > >> >>>> > On Thu, Mar 25, 2010 at 11:44 PM, Rahul Sampath >> >>>> > <[email protected]> >> >>>> > wrote: >> >>>> >> >> >>>> >> If you want to form a RHS by integrating f_i Phi_i Phi_j, You >> >>>> >> could >> >>>> >> form a Mass matrix and then multiply with your vector of nodal >> >>>> >> values. >> >>>> >> >> >>>> >> Rahul >> >>>> >> >> >>>> >> On Thu, Mar 25, 2010 at 11:40 PM, Karen Lee <[email protected]> >> >>>> >> wrote: >> >>>> >> > I'm afraid you misunderstood. I don't have the function that >> >>>> >> > when >> >>>> >> > given >> >>>> >> > x, >> >>>> >> > y, z values gives me the function value. What I do have is just >> >>>> >> > the >> >>>> >> > values >> >>>> >> > at the nodes of the mesh, which need to be linearly interpolated >> >>>> >> > such >> >>>> >> > that I >> >>>> >> > will have something like exact_function. which gives me the >> >>>> >> > value >> >>>> >> > when >> >>>> >> > supplied with any x, y, z. >> >>>> >> > >> >>>> >> > >> >>>> >> > >> >>>> >> > On Thu, Mar 25, 2010 at 10:54 PM, Liang <[email protected]> >> >>>> >> > wrote: >> >>>> >> > >> >>>> >> >> Karen Lee wrote: >> >>>> >> >> >> >>>> >> >>> I guess I'm not clear how to do this: Load data as a solution >> >>>> >> >>> into >> >>>> >> >>> that, >> >>>> >> >>> and >> >>>> >> >>> query >> >>>> >> >>> it when you're integrating your real system. >> >>>> >> >>> >> >>>> >> >>> I have: >> >>>> >> >>> Mesh mesh(3); >> >>>> >> >>> MeshData mesh_data(mesh); >> >>>> >> >>> mesh_data.activate(); >> >>>> >> >>> mesh.read (mesh_file, &mesh_data); >> >>>> >> >>> mesh_data.read(mesh_file); >> >>>> >> >>> EquationSystems equation_systems (mesh); >> >>>> >> >>> >> >>>> >> >>> >> >>>> >> >>> equation_systems.add_system<ExplicitSystem> ("RHS"); >> >>>> >> >>> equation_systems.get_system("RHS").add_variable("R", FIRST); >> >>>> >> >>> >> >>>> >> >>> After that, I'm not clear how exactly to load data as a >> >>>> >> >>> solution >> >>>> >> >>> in >> >>>> >> >>> the >> >>>> >> >>> code. My goal is to get a linearly interpolated function of my >> >>>> >> >>> data on >> >>>> >> >>> the >> >>>> >> >>> nodes (in the form of exact_solution, such that I get the >> >>>> >> >>> function >> >>>> >> >>> value >> >>>> >> >>> out >> >>>> >> >>> when supplying x, y and z). >> >>>> >> >>> >> >>>> >> >>> Hope that clarifies things, and sorry for the multiple >> >>>> >> >>> emails... >> >>>> >> >>> >> >>>> >> >>> Karen >> >>>> >> >>> >> >>>> >> >>> >> >>>> >> >>> >> >>>> >> >>> >> >>>> >> >>> ------------------------------------------------------------------------------ >> >>>> >> >>> Download Intel® Parallel Studio Eval >> >>>> >> >>> Try the new software tools for yourself. Speed compiling, find >> >>>> >> >>> bugs >> >>>> >> >>> proactively, and fine-tune applications for parallel >> >>>> >> >>> performance. >> >>>> >> >>> See why Intel Parallel Studio got high marks during beta. >> >>>> >> >>> http://p.sf.net/sfu/intel-sw-dev >> >>>> >> >>> _______________________________________________ >> >>>> >> >>> Libmesh-users mailing list >> >>>> >> >>> [email protected] >> >>>> >> >>> https://lists.sourceforge.net/lists/listinfo/libmesh-users >> >>>> >> >>> >> >>>> >> >>> >> >>>> >> >>> >> >>>> >> >> so you already have the function, which is obtained from your >> >>>> >> >> discreted >> >>>> >> >> data? >> >>>> >> >> then just put the function as the exact_function. >> >>>> >> >> I think you are trying the 3D case, start from a 2d will be >> >>>> >> >> easier. >> >>>> >> >> >> >>>> >> >> Liang >> >>>> >> >> >> >>>> >> > >> >>>> >> > >> >>>> >> > >> >>>> >> > ------------------------------------------------------------------------------ >> >>>> >> > Download Intel® Parallel Studio Eval >> >>>> >> > Try the new software tools for yourself. Speed compiling, find >> >>>> >> > bugs >> >>>> >> > proactively, and fine-tune applications for parallel >> >>>> >> > performance. >> >>>> >> > See why Intel Parallel Studio got high marks during beta. >> >>>> >> > http://p.sf.net/sfu/intel-sw-dev >> >>>> >> > _______________________________________________ >> >>>> >> > Libmesh-users mailing list >> >>>> >> > [email protected] >> >>>> >> > https://lists.sourceforge.net/lists/listinfo/libmesh-users >> >>>> >> > >> >>>> > >> >>>> > >> >>> >> >> >> >> >> > > > ------------------------------------------------------------------------------ Download Intel® Parallel Studio Eval Try the new software tools for yourself. Speed compiling, find bugs proactively, and fine-tune applications for parallel performance. 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