| Author | Message | ||
     fearrr (fea) New member Username: fea Post Number: 1 Registered: 02-2005 |
Hello! Consultation is necessary, please. There is 2d boundary problem on Omega domain: D(u)=0 Boundary condition (Neumann?) on border curve G: du/dn = 2*x*y*cos(n,x) + y*y*cos(n,y) where n - outward normal to G; D - Laplace operator; u - unknown function u=u(x,y). In discretization by finite elements method (FEA) the problem is reduced to system equations: A*u=F How to generate these matrixes in this case (especially interests vector F)? | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 312 Registered: 06-2003 |
Have you read the FlexPDE tutorial (User Guide)? The problem you pose is exactly the one used as an example in the tutorial. See the User Guide, or click "Product Info", then "Tutorial" on the web site. [The bottom line is that you don't generate "these matrices". FlexPDE does. You merely state your PDE.] [And incidentally, the problem you have stated is underdetermined. With only derivative boundary conditions, there are an infinite number of solutions, differing by an additive constant.] | ||
| fearrr (fea) New member Username: fea Post Number: 2 Registered: 02-2005 |
I read help, but I make this by fortran... Can I print this matrix by FlexPDE? | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 313 Registered: 06-2003 |
There are a large number of text books that will show you how to construct finite element matrices. Some, like "Application and Implementation of Finite Element Methods" by J.E.Akin (Academic Press, 1982) include FORTRAN source listings. | ||
| Tong Liu (siegfried) Member Username: siegfried Post Number: 4 Registered: 02-2006 |
I am a novice user.Could you tell me which example in the tutorial describes this problem? Thank you very much! | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 555 Registered: 06-2003 |
I don't understand this question. The Tutorial (User Guide in the installed help) gradually builds up a script for a sample problem. The problem used as an example is a Laplace equation. "User Guide | Basic Usage | Putting it all together" shows the first complete listing of this script. Later sections show various modifications to boundary conditions, parameters or geometry, but it's always a Laplace equation. | ||
| Tong Liu (siegfried) Member Username: siegfried Post Number: 6 Registered: 02-2006 |
Please excuse me. I do not describe my problem exactly. I do not remember there is a function could express cos(n,x) and cos(n,y) in the equation du/dn = 2*x*y*cos(n,x) + y*y*cos(n,y) where n - outward normal to the domain. And I think I cannot obtain du/dn everywhere on the domain. I only know that on the natural BC, NATURAL(u)=du/dn. Am I right? | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 557 Registered: 06-2003 |
The NATURAL boundary condition specifies the integrand of the surface terms generated by integrating second-order terms by parts. If the only second-order terms are divergences, this is equivalent to the Divergence Theorem. If the equation is Div(K*Grad(U)) + S = 0, then NATURAL(U) provides the value of the outward normal component of K*Grad(U), which is the outward normal of the negative of the thermal flux, or the INWARD flux. Notice that the sign of this term follows directly from the sign of the divergence in the PDE (when on the left side of the equation). If K=1, then NATURAL(U) = dU/dn. You don't need to know sines and cosines, all you need to know is the normal flux. | ||
| Tong Liu (siegfried) Member Username: siegfried Post Number: 7 Registered: 02-2006 |
I see, thank you. But, can I specify a Cauchy or mixed BC such as {du/dn=0&u=0} or a*du/dn+b*u=0 in flexpde?This condition may be useful in some problem. | ||
| Tong Liu (siegfried) Member Username: siegfried Post Number: 8 Registered: 02-2006 |
I see, thank you. But, can I specify a Cauchy or mixed BC such as {du/dn=0&u=0} or a*du/dn+b*u=0 in flexpde?This condition may be useful in some problem. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 559 Registered: 06-2003 |
You cannot impose two conditions on any boundary (each nodal value has one equation). But you will notice from the documentation of Boundary conditions that the BC statements are of the form NATURAL(<variable>) = <expression> VALUE(<variable>) = <expression> <expression> is very general. This is a scripting language, after all. You can say NATURAL(u) = -b*u/a If your equation is Div(grad(u)), then this BC statement means a*du/dn+b*u=0. The VALUE condition is somewhat more restrictive, so if you say VALUE(u) = -(a/b)*NORMAL(GRAD(u)), which in principle means a*du/dn+b*u=0, you will get a diagnostic on the NORMAL. But in general, you can use implicit dependencies in VALUE conditions. If you use your imagination, you can think of many kinds of boundary conditions that can be imposed using the available structures. Try them and see what happens. You should take a look at the documentation and try some of the example problems, rather than asking me to write you a private tutorial. | ||
| Tong Liu (siegfried) Member Username: siegfried Post Number: 9 Registered: 02-2006 |
I'm so sorry to trouble you. I think I have used the BC structure you mentioned above. But I did not realize that it is a cauchy BC. I will review the guide once more. Thank you for your patience.Thank you. |