This is what Roy discussed. The divergence theorem has been applied to that term.
-J li pan writes: > thanx John, > there is another point. I read the Navier-Stokes > equation in wiki > (http://en.wikipedia.org/wiki/Navier-Stokes_equations). > There is a grad(p) term. But it doesn't appear in the > equaiton of ex13. > > pan > > > --- John Peterson <[EMAIL PROTECTED]> > wrote: > > > You mean how do you linearize it? Newton's method > > and iterate > > within each timestep. > > -J > > > > > > li pan writes: > > > hi Roy, > > > thanx for the explaination. But how did you solve > > ((u > > > * grad)u, v)_Omega? It's a square term. I heard > > there > > > are some other methods, streamline, least square > > FEM > > > ... I would like to hear your comments. > > > > > > pan > > > > > > > > > > > > --- Roy Stogner <[EMAIL PROTECTED]> wrote: > > > > > > > > > > > On Thu, 24 Jan 2008, li pan wrote: > > > > > > > > > I've worked with Newton type flow equation. > > To > > > > make it > > > > > sure, I would like to know the exact > > expression of > > > > > equation in ex13. Can you tell me? > > > > > > > > The system of equations with variables (u,p) is > > : > > > > (partial u)/(partial t) = - (u * grad)u - > > div(sigma) > > > > div(u) = 0 > > > > > > > > Where sigma is the stress tensor (normalized to > > have > > > > unit viscosity) > > > > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) > > > > > > > > Then the weak form we use in ex13 and ex18, > > with > > > > test functions (v,q) > > > > is: > > > > ((partial u)/(partial t), v)_Omega = - ((u * > > grad)u, > > > > v)_Omega > > > > + (sigma, grad v)_Omega + (sigma * n, > > v)_dOmega > > > > (div(u), q) = 0 > > > > > > > > In ex13 we use Dirichlet boundaries everywhere, > > so v > > > > = 0 on the > > > > boundary and we drop the dOmega term. > > Otherwise, > > > > you'd substitute > > > > into that term the natural boundary condition: > > > > sigma * n = 0 > > > > > > > > which is actually what David wanted in the > > first > > > > place. ;-) > > > > > > > > You know, we probably ought to have something > > like > > > > this in the > > > > comments heading examples 13 and 18. "The > > > > Navier-Stokes equations" is > > > > definitive enough, but the fact that we > > integrate > > > > all of sigma > > > > (including the pressure term) by parts isn't > > set in > > > > stone. > > > > --- > > > > Roy > > > > > > > > > > > > > > ------------------------------------------------------------------------- > > > > This SF.net email is sponsored by: Microsoft > > > > Defy all challenges. Microsoft(R) Visual Studio > > > > 2008. > > > > > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > > _______________________________________________ > > > > Libmesh-users mailing list > > > > [email protected] > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > > > > > > > > > > > > > > ____________________________________________________________________________________ > > > Be a better friend, newshound, and > > > know-it-all with Yahoo! Mobile. Try it now. > > > http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ > > > > > > > > > > > > > > ------------------------------------------------------------------------- > > > This SF.net email is sponsored by: Microsoft > > > Defy all challenges. Microsoft(R) Visual Studio > > 2008. > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > _______________________________________________ > > > Libmesh-users mailing list > > > [email protected] > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > > ____________________________________________________________________________________ > Looking for last minute shopping deals? > Find them fast with Yahoo! Search. > http://tools.search.yahoo.com/newsearch/category.php?category=shopping ------------------------------------------------------------------------- This SF.net email is sponsored by: Microsoft Defy all challenges. Microsoft(R) Visual Studio 2008. http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ _______________________________________________ Libmesh-users mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/libmesh-users
