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Solving Efficiently Neumann Problems

tlaverne's picture

I would like to know simply how to solve efficiently elasticity problems with Neumann boundary conditions? Neumann bc implies that I have 6 null eigenvalues in 3D (3 in 2D) associated to rigid body modes  and iterative methods such has conjugate gradient are not likely to work (because the problem is no longer positive definite, but only semi-definite). Is there any way to use iterative solver anyway by filtering rigid body motions or something like that ? However has anyone a good idea to solve such problem? For now, fixing aribrarily the displacement of some nodes (3 in 3D but not aligned, 2in 2D) is the only way I see, but I would like to have your smart advise on the subject...

Thank you !


Rich Lehoucq's picture

Specifying nodes to specific values removes the singularity of the stiffness matrix but leads to a larger condition number. For a 3D problem, refining the mesh leads to a condition number is inversely porportional the cube of the element size instead of the square. This leads to a much slower rate of convergence for an unpreconditioned iterative method. The reason is that as you refine the mesh, the discretization is converging to a continuous problem that is not well-posed, equivalently one with an infinite stress where you specified the nodes. A simple and effective scheme is to orthongonalize the discrete load against the discrete rigid body modes before you use an iterative solver, and then occasionaly orthogonalizing the discrete estimate of the displacement.

To learn more, the pure Neumann problem is a good case study; see On finite element discretizations of the pure Neumann problem, SIAM Review, Volume 47, Number 1, pp. 50-66, 2005 DOI:10.1137/S0036144503426074.


tlaverne's picture

Dear Mr. Lehoucq,

Thank you very much ! I could not dream about a clearer answer ! However I'm a bit surprised that fe textbooks or even pdes or iterative solver textbooks don't discuss this issue more extensively, since pure Neumann problems are (with Dirichlet problems) corner-stone of pdes theory and practice. 


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