A General natural boundary condition?

  • Thread starter maistral
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196
9
Summary
A little clarification with regard to the natural boundary condition.
Hi, I'd like to be clarified regarding the general natural/Neumann boundary condition for a PDE.

1. The natural boundary condition is generally defined as:
242655
(1)
and can be expressed as, according to this resource:
242656
(2)

But apparently, according to this resource (posted by Dr. Fan):
242657
(3)

Which is which? Is it supposed to be positive, or negative? When should it be positive or negative?

2. If I apply the derivative boundary condition, on say, the bottom of a square plate, I can state:
242658
(4)
or for simplicity,
242659
(5)

Obviously, comparing it with (2), ny is equal to 0. Why is this so? Why is ny = 0?

Thanks!
 

Orodruin

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1. The minus sign is an obvious typo.
2. ##\vec n## is the normal vector. The normal vector's y-component on a coordinate surface of ##y## is zero.
 
196
9
Hi! Thanks for replying. Thank you very much, it did clarify a lot of things. I still have two remaining problems, however.

1. Where is the vector n exactly pointed to? Is it going towards the region, or is it going away from the region?

2. I am assuming that the complete form of the Neumann condition is used for irregular boundaries (ie. curved ones, so on and forth). As stated by the first resource, nx = cos θ, and ny = sin θ.

If I use finite element analysis, I can just take the normal vector perpendicular to the edge of the element which is definitely a no-brainer as it's a straight line.

My problem begins to appear if I begin using finite differences. How do I exactly implement this kind of boundary condition on nodes? I'm confused - nodes are, well, nodes; just a point. I'm under the impression that I can get the angle required for calculating nx and ny by taking the tangent of the irregular boundary curve at the node, then I draw the perpendicular line needed to calculate the angle. Is this even correct?
 

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