H(curl)? 3

[WARose]

Sometimes I hear FEA analyst(s) throw around the term H(curl) [or H(div)] when discussing “domains/function spaces”…..I saw it referenced in a Higher-order FEA text I was flipping through this weekend (but didn’t follow what they were talking about). Does anybody know what this is?

 

[IRstuff]

div is short for divergence:

http://en.wikipedia.org/wiki/Divergence

curl is a measure of rotation:

http://en.wikipedia.org/wiki/Curl_(mathematics)

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[WARose]

Ok but how does this relate to FEA.....is it an alternate type of shape function or a means to manipulate a shape function?

 

[IRstuff]

What equations does FEA discretize?

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[WARose]

What equations does FEA discretize?

The interpolation functions that describe the displacements of an element. (Or (in the case of an isoparametric element): the geometry as well.)

 

[WARose]

Anyone else know?

 

[96bulls]

H(div) and H(curl) refer to function spaces whose div/curl are square integrable in Lebesque sense. Basically function spaces that have continuous function values but whose first derivatives (curl and div operators involve first derivatives) have isolated discontinuities at element boundaries. The basis functions in almost all classical finite element methods belong to these spaces.

You might also see these referred to as H^1 spaces.

 

[flash3780]

MIT has the lectures for their Mathematical Methods for Engineers course online.

http://academicearth.org/courses/mathematical-methods-for-engineers-ii

You may find the lecture on the calculus of variations interesting.

 

[3doorsdwn]

H(div) and H(curl) refer to function spaces whose div/curl are square integrable in Lebesque sense. Basically function spaces that have continuous function values but whose first derivatives (curl and div operators involve first derivatives) have isolated discontinuities at element boundaries. The basis functions in almost all classical finite element methods belong to these spaces.

You might also see these referred to as H^1 spaces.

Ok so this is something most shape functions conform to. What is the significance of it in higher order FEA techniques? Is the measure of this significant to the accuracy of results?

 

[96bulls]

The significance (from what I understand) is that it allows you to derive things like error estimates and convergence rates based on these spaces. Then when you go to apply these measures to FEA you know it's going to be valid since your basis functions are a part of that space.

You might take a look at:

http://en.wikipedia.org/wiki/Hilbert_space#Partial_differential_equations

The H in H(div/curl) stands for Hilbert space

 

[WARose]

Thanks for the replies all. 96Bulls, I appreciate the link on Hilbert space....not sure I follow it 100%, but appreciate it none the less.