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From: John Peterson <peterson@cf...>  20080124 16:44:38

Roy Stogner writes: > > 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) I don't think it should have a 1/2. The stress tensor is typically defined as (see e.g. Panton's fluid mechanics book, http://www.cs.otago.ac.nz/postgrads/alexis/FluidMech/node10.html, http://en.wikipedia.org/wiki/Newtonian_fluid ): sigma = pI + 2*mu*epsilon(u) where epsilon(u) = (1/2)*(grad(u) + grad(u)^t) is known as the "symmetric part" of the velocity gradient. http://mathworld.wolfram.com/SymmetricMatrix.html Now I think this is confusing because the 2 and the 1/2 always cancel, but whatever, this is how people define things. I agree that this email should be in the introductory comments of ex13, a lot of people have asked the same question. J 
From: li pan <li76pan@ya...>  20080124 10:38:43

Dear developers, 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? thanx pan ____________________________________________________________________________________ Be a better friend, newshound, and knowitall with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ 
From: Roy Stogner <roystgnr@ic...>  20080124 16:23:41

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 NavierStokes 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 
From: John Peterson <peterson@cf...>  20080124 16:44:38

Roy Stogner writes: > > 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) I don't think it should have a 1/2. The stress tensor is typically defined as (see e.g. Panton's fluid mechanics book, http://www.cs.otago.ac.nz/postgrads/alexis/FluidMech/node10.html, http://en.wikipedia.org/wiki/Newtonian_fluid ): sigma = pI + 2*mu*epsilon(u) where epsilon(u) = (1/2)*(grad(u) + grad(u)^t) is known as the "symmetric part" of the velocity gradient. http://mathworld.wolfram.com/SymmetricMatrix.html Now I think this is confusing because the 2 and the 1/2 always cancel, but whatever, this is how people define things. I agree that this email should be in the introductory comments of ex13, a lot of people have asked the same question. J 
From: li pan <li76pan@ya...>  20080124 17:15:49

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 <roystgnr@...> 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 > NavierStokes 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/ > _______________________________________________ > Libmeshusers mailing list > Libmeshusers@... > https://lists.sourceforge.net/lists/listinfo/libmeshusers > ____________________________________________________________________________________ Be a better friend, newshound, and knowitall with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ 
From: John Peterson <peterson@cf...>  20080124 17:23:52

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 <roystgnr@...> 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 > > NavierStokes 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/ > > _______________________________________________ > > Libmeshusers mailing list > > Libmeshusers@... > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > > > > > > ____________________________________________________________________________________ > Be a better friend, newshound, and > knowitall 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/ > _______________________________________________ > Libmeshusers mailing list > Libmeshusers@... > https://lists.sourceforge.net/lists/listinfo/libmeshusers 
From: li pan <li76pan@ya...>  20080124 17:34:18

thanx John, there is another point. I read the NavierStokes equation in wiki (http://en.wikipedia.org/wiki/NavierStokes_equations). There is a grad(p) term. But it doesn't appear in the equaiton of ex13. pan  John Peterson <peterson@...> 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 <roystgnr@...> 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 > > > NavierStokes 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/ > > > _______________________________________________ > > > Libmeshusers mailing list > > > Libmeshusers@... > > > > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > > > > > > > > > > > > ____________________________________________________________________________________ > > Be a better friend, newshound, and > > knowitall 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/ > > _______________________________________________ > > Libmeshusers mailing list > > Libmeshusers@... > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > ____________________________________________________________________________________ Looking for last minute shopping deals? Find them fast with Yahoo! Search. http://tools.search.yahoo.com/newsearch/category.php?category=shopping 
From: John Peterson <peterson@cf...>  20080124 17:36:07

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 NavierStokes > equation in wiki > (http://en.wikipedia.org/wiki/NavierStokes_equations). > There is a grad(p) term. But it doesn't appear in the > equaiton of ex13. > > pan > > >  John Peterson <peterson@...> > 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 <roystgnr@...> 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 > > > > NavierStokes 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/ > > > > _______________________________________________ > > > > Libmeshusers mailing list > > > > Libmeshusers@... > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > > > > > > > > > > > > > > > > > > > ____________________________________________________________________________________ > > > Be a better friend, newshound, and > > > knowitall 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/ > > > _______________________________________________ > > > Libmeshusers mailing list > > > Libmeshusers@... > > > > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > > > > > > ____________________________________________________________________________________ > Looking for last minute shopping deals? > Find them fast with Yahoo! Search. http://tools.search.yahoo.com/newsearch/category.php?category=shopping 
From: li pan <li76pan@ya...>  20080124 17:44:22

ok, I see it. thanx a lot pan  John Peterson <peterson@...> wrote: > 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 NavierStokes > > equation in wiki > > > (http://en.wikipedia.org/wiki/NavierStokes_equations). > > There is a grad(p) term. But it doesn't appear in > the > > equaiton of ex13. > > > > pan > > > > > >  John Peterson <peterson@...> > > 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 <roystgnr@...> > 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 > > > > > NavierStokes 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/ > > > > > > _______________________________________________ > > > > > Libmeshusers mailing list > > > > > Libmeshusers@... > > > > > > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > > > > > > > > > > > > > > > > > > > > > > > > > > > ____________________________________________________________________________________ > > > > Be a better friend, newshound, and > > > > knowitall 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/ > > > > > _______________________________________________ > > > > Libmeshusers mailing list > > > > Libmeshusers@... > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmeshusers > > > > > > > > > > > > ____________________________________________________________________________________ > > 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/ > _______________________________________________ > Libmeshusers mailing list > Libmeshusers@... > https://lists.sourceforge.net/lists/listinfo/libmeshusers > ____________________________________________________________________________________ Never miss a thing. Make Yahoo your home page. http://www.yahoo.com/r/hs 
From: Benjamin Kirk <benjamin.kirk@na...>  20080124 18:43:52

> 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. By square I guess you mean asymmetric? The convection term is asymmetric and causes problems when convection is stronger than diffusion, e.g. Cell Reynolds numbers (u*h/nu) exceed 2. In that case you need to do something line the streamline/upwind PetrovGalerkin method, GalerkinLeast squares, etc... We sidestep that problem in ex13 by keeping the cell Reynolds numbers low enough that it is not an issue. Ben 
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