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mdn

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In summary, the speaker is knowledgeable about applying boundary conditions in finite element methods, but struggles with implementing a divergence free condition for Maxwell's or Stokes equations. They mention a special element called an edge element that can be used, but they prefer to use a nodal base element. The speaker also explains that divergence free is not a boundary condition, but can be enforced by expressing the vector field in a certain way. They suggest using a stream function for 2D problems and a toroidal-poloidal decomposition for 3D problems. They also ask for an example of using a stream function for the Navier-Stokes equation in fluid flow.

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mdn

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pasmith

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For 2D problems, use a stream function.

For 3D problems, you can use a toroidal-poloidal decomposition.

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mdn

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mdn

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Would you give me the example of NS equation using stream function in fem?

In order to ensure divergence-free solutions in nodal finite element method, you need to apply the divergence-free condition (∇.v=0) to the displacement field. This can be achieved by using a suitable basis function for the displacement field that satisfies the divergence-free condition.

The divergence-free condition is important in nodal finite element method because it ensures that the solution is physically meaningful and satisfies the conservation of mass. It also helps to avoid numerical instabilities and inaccuracies in the solution.

Yes, the divergence-free condition can be applied to any type of element in nodal finite element method. However, for some elements, it may be more challenging to satisfy the condition due to the complexity of their shape functions.

You can check if your solution satisfies the divergence-free condition by calculating the divergence of the displacement field at each node and ensuring that it is equal to zero. This can be done using numerical methods or by using analytical solutions for simpler problems.

One limitation of using the divergence-free condition in nodal finite element method is that it may increase the computational cost and complexity of the solution. This is because it requires the use of specialized basis functions and may also require additional constraints to be imposed on the solution.

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