Thanks for the comments. I'm not sure whether I can pull it off, but I'll try.
K. Lee. On Mon, Jan 28, 2013 at 5:25 AM, David Knezevic <[email protected]>wrote: > Hi K, > > It doesn't matter how you get the lift function. If you can construct u0 > "by hand" then that's fine, but it may not always be so easy, e.g. if > the domain is non-trivial. Solving a laplace problem will work in > general. To do this just follow introduction_ex4 (which already has > non-homogeneous Dirichlet BCs). > > Re this: > > On 01/26/2013 07:23 PM, Kyunghoon Lee wrote: > > Overall it looks like a nested problem --- solving a Laplace equation > > inside of a reduced basis model construction. I wonder if you'd suggest > > some relevant examples/codes regarding the lift function creation. > > Well, you solve the Laplace problem (as in introduction_ex4), store the > result, then do the RB stuff. > > David > > > > > > > > > Best, > > K. Lee. > > > > On Sun, Jan 27, 2013 at 1:43 AM, David Knezevic > > <[email protected]>wrote: > > > >> To create the lift function, probably the simplest thing to do is solve > >> a Laplace equation (\Laplacian u = 0) with the Dirichlet boundary > >> conditions that you want, and then use the solution as your u0. This > >> process is called "elliptic lifting" since the Laplace equation is an > >> elliptic PDE. > >> > >> Then to assemble a(u0,v), you need to do something like what you wrote. > >> But the code can be simplified a bit; you don't need to compute Ke and > >> then multiply. You can just get the gradient of u0 directly (by > >> multiplying the coefficients of u0 with c.interior_gradient) and then > >> integrate. > >> > >> David > >> > >> > >> > >> On 01/26/2013 06:14 AM, Kyunghoon Lee wrote: > >>> Thanks for the reply. Now I'd appreciate if you'd help me with the > >>> implementation of a(u0,v). I was thinking of computing Ke*u0 where Ke > is > >>> the stiffness matrix and u0 is the lift function as below: > >>> > >>> //inhomogeneous Dirichlet BC > >>> struct IDBCAssembly : ElemAssembly { > >>> > >>> short unsigned int sbd_id; > >>> IDBCAssembly(short unsigned int sbd_id_in) : sbd_id(sbd_id_in) {} > >>> > >>> virtual void interior_assembly(FEMContext &c) { > >>> > >>> const unsigned int u_var = 0; > >>> const std::vector<Real> &JxW = > >>> c.element_fe_var[u_var]->get_JxW(); > >>> const std::vector<std::vector<RealGradient> >& dphi = > >>> c.element_fe_var[u_var]->get_dphi(); > >>> > >>> const unsigned int n_u_dofs = > >> c.dof_indices_var[u_var].size(); > >>> unsigned int n_qpoints = c.element_qrule->n_points(); > >>> > >>> // stiffness matrix Ke > >>> std::vector<std::vector<Number> > Ke(n_u_dofs, > >>> std::vector<Number>(n_u_dofs)); > >>> for (unsigned int qp=0; qp != n_qpoints; qp++) > >>> for (unsigned int i=0; i != n_u_dofs; i++) > >>> for (unsigned int j=0; j != n_u_dofs; j++) > >>> Ke[i][j] += JxW[qp] * dphi[j][qp] * > dphi[i][qp]; > >>> > >>> // lift function u0 > >>> std::vector<Number> u0(n_u_dofs, 0.0); > >>> > >>> // multiply stiffness matrix by lift function > >>> for (unsigned int qp=0; qp != n_qpoints; qp++) > >>> for (unsigned int i=0; i != n_u_dofs; i++) > >>> for (unsigned int j=0; j != n_u_dofs; j++) > >>> c.elem_residual(i) += Ke[i][j] * u0[j]; > >>> } // end of interior_assembly > >>> }; > >>> > >>> However, I'm not sure of how to create the lift function u0 such that > it > >>> has values for given boundary ID and zeros for elsewhere. I think I > need > >>> to the following: > >>> > >>> iterate nodes > >>> if node belongs to the given boundary ID, set u0(i) = u0_given; > otherwise > >>> u(i) = 0.0 > >>> > >>> but I'm not sure of how to check whether a node is associated with > >> boundary > >>> ID. Can you help me with this problem, plz? (or are there examples > >> related > >>> to this issue?) > >>> > >>> Best, > >>> K. Lee. > >>> > >>> > >>> On Fri, Jan 25, 2013 at 12:38 PM, David Knezevic < > >> [email protected] > >>>> wrote: > >>>> Yep, that's right, use the RB method to solve for u'. > >>>> > >>>> Note that you put a(u0,v) on the right-hand side, since u0 is known > >>>> (it's the "lifting function"). > >>>> > >>>> David > >>>> > >>>> > >>>> > >>>> On 01/24/2013 11:34 PM, Kyunghoon Lee wrote: > >>>>> Thanks for clearing it up. Then I guess we just solve for u' > >>>>> > >>>>> a(u',v) + a(u0,v) = f(v) > >>>>> > >>>>> and attach assemblies a(u',v), a(u0,v), and f(v) as usual, then > >> restore u > >>>>> by u = u' + u0. > >>>>> > >>>>> K. Lee. > >>>>> > >>>>> On Fri, Jan 25, 2013 at 12:17 PM, David Knezevic < > >>>> [email protected] > >>>>>> wrote: > >>>>>> Hi K, > >>>>>> > >>>>>> heterogeneously_constrain_element_matrix_and_vector is not relevant > to > >>>>>> Reduced Basis stuff. For Reduced Basis formulations, you have to > >>>>>> transform the problem using a lifting function so that it has zero > >>>>>> Dirichlet BC's --- this is essential since you want your Reduced > Basis > >>>>>> space to be a vector space, i.e. it must contain 0 (which would be > not > >>>>>> be the case with non-zero Dirichlet BCs). This lifting function > >> approach > >>>>>> is what you described in your email already, so that's fine. > >>>>>> > >>>>>> Once you've transformed your problem using a lifting function, then > >> you > >>>>>> just proceed as normal, e.g. as in reduced_basis_ex1. The only trick > >> is > >>>>>> you have to add your lifting function back on at the end to recover > u > >>>>>> from u'. > >>>>>> > >>>>>> David > >>>>>> > >>>>>> > >>>>>> > >>>>>> On 01/24/2013 11:10 PM, Kyunghoon Lee wrote: > >>>>>>> Hi all, > >>>>>>> > >>>>>>> In my previous email regarding inhomogeneous Dirichlet boundary > >>>>>> conditions, > >>>>>>> David suggested using > >>>> heterogenously_constrain_element_matrix_and_vector > >>>>>> in > >>>>>>> introduction_ex4, but I'm not sure of how to deal with > inhomogeneous > >>>>>>> Dirichlet BCs in connection with reduced basis models. Suppose we > >>>> have a > >>>>>>> simple steady state heat conduction model whose BCs are u = T on > >> \Gamma > >>>>>> and > >>>>>>> u = 0 on the rest surfaces. After variable change, we solve > >>>>>>> > >>>>>>> a(u',v) = f(v) - a(u0,v) > >>>>>>> > >>>>>>> where 1) u' = u - T on \Gamma and u' = u on the rest surfaces; and > 2) > >>>> u0 > >>>>>> = > >>>>>>> T on \Gamma and u0 = zero on the rest surfaces. I thought we build > >> the > >>>>>> LHS > >>>>>>> then call attach_F_assembly to attach it, but in that case, I'm not > >>>> sure > >>>>>>> how heterogenously_constrain_element_matrix_and_vector can be used. > >> Or > >>>>>>> should we attach a(u',v) and f(v) as usual then call > >>>>>>> heterogenously_constrain_element_matrix_and_vector to impose - > >> a(u0,v) > >>>> on > >>>>>>> the LHS? I'd appreciate if someone can briefly describe how the > >>>> function > >>>>>>> work. > >>>>>>> > >>>>>>> Regards, > >>>>>>> K. Lee. > >>>>>>> > >> > ------------------------------------------------------------------------------ > >>>>>>> Master Visual Studio, SharePoint, SQL, ASP.NET, C# 2012, HTML5, > CSS, > >>>>>>> MVC, Windows 8 Apps, JavaScript and much more. Keep your skills > >> current > >>>>>>> with LearnDevNow - 3,200 step-by-step video tutorials by Microsoft > >>>>>>> MVPs and experts. ON SALE this month only -- learn more at: > >>>>>>> http://p.sf.net/sfu/learnnow-d2d > >>>>>>> _______________________________________________ > >>>>>>> Libmesh-users mailing list > >>>>>>> [email protected] > >>>>>>> https://lists.sourceforge.net/lists/listinfo/libmesh-users > >>>>>> > >> > ------------------------------------------------------------------------------ > >>>>>> Master Visual Studio, SharePoint, SQL, ASP.NET, C# 2012, HTML5, > CSS, > >>>>>> MVC, Windows 8 Apps, JavaScript and much more. 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