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sylvain October 28, 2001 14:25

Augmented Lagrangian method for FEM-CFD
 
Hi, i am looking for someone who uses the augmented lagrangian method with a finite element method to simulate incompressible fluid flow. I am actualy using a penalty method and i am wondering if this method is better. Thank you best regards

Anthony Wachs October 29, 2001 05:08

Re: Augmented Lagrangian method for FEM-CFD
 
Hi,

I have used Augmented Lagrangian Method (ALM) for solving incompressible fluid flows (especially non-Newtonian) in the past. The discretization method was a Finite Volume method, but that is likely not to make any difference with your FEM since at the end you get symmetric matrix system that looks like

[ A B ](v)=(f)

[ Bt 0 ](p) (g)

(the two lines are together and represent a single matrix system, e.g. it corresponds to : A.v+B.p=f and Bt.v=g)

where A is the velocity matrix, B pressure matrix, v velocity vector, p pressure vector, f and g the second members of your system

With the ALM, you replace this problem by the following one

[ A+rBBt B ](v)=(f+rBg)

[ Bt 0 ](p) (g)

where r stands for the Lagrangian parameter.

The larger r, the faster the convergence is achieved. But since in the algorithm, you are to solve matrix sub-systems like :

(A+rBBt)(x)=b

Those systems become bad-conditioned as r increases. Therefore, if u use a direct method to solve this sub-system (as for instance Cholevski factorization), then you don't have to take care of the magnitude of r. On the other hand, if u choose to solve this sub-system using iterative solver (conjugate gradient), then the larger r, the poorer the convergence rate.

Finally, the ALM is a very powerful method provided you can factortize the augmented matrix A+rBBt once at the beginning of the procedure in its Cholevski form. Otherwise, you can still use it but it requires to pay attention to the magnitude of r.

Do not hesitate to contact me if you need additional informations

I hope it might help u

Best regards, Anthony


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