Why is Finite Element Analysis an Upper Bound Method
Why is Finite Element Analysis an Upper Bound Method
(OP)
Here's a pretty easy one I can't wrap my head around:
For finite element analysis of a mechanical stress problem why does deflection and stress always increase towards an exact solution?
I understand that the behavior of elements in between nodes are approximated by linear or quadratic functions and cannot exactly represent an irregular shape, so finite elements must be made smaller until the deflection profile required between the nodes can adequately represented by these shape functions; however, I don't understand why this increasingly more accurate shape approximation always produces higher stress/deflections.
“The most successful people in life are the ones who ask questions. They’re always learning. They’re always growing. They’re always pushing.” Robert Kiyosaki
For finite element analysis of a mechanical stress problem why does deflection and stress always increase towards an exact solution?
I understand that the behavior of elements in between nodes are approximated by linear or quadratic functions and cannot exactly represent an irregular shape, so finite elements must be made smaller until the deflection profile required between the nodes can adequately represented by these shape functions; however, I don't understand why this increasingly more accurate shape approximation always produces higher stress/deflections.
“The most successful people in life are the ones who ask questions. They’re always learning. They’re always growing. They’re always pushing.” Robert Kiyosaki
RE: Why is Finite Element Analysis an Upper Bound Method
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Why is Finite Element Analysis an Upper Bound Method
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Why is Finite Element Analysis an Upper Bound Method
RE: Why is Finite Element Analysis an Upper Bound Method
RE: Why is Finite Element Analysis an Upper Bound Method
"the strain energy in FEM solution is always smaller or equal to"
Only one of those statements can be correct.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Why is Finite Element Analysis an Upper Bound Method
RE: Why is Finite Element Analysis an Upper Bound Method
“The most successful people in life are the ones who ask questions. They’re always learning. They’re always growing. They’re always pushing.” Robert Kiyosaki
RE: Why is Finite Element Analysis an Upper Bound Method
RE: Why is Finite Element Analysis an Upper Bound Method
RE: Why is Finite Element Analysis an Upper Bound Method
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Why is Finite Element Analysis an Upper Bound Method
Adapted from Hughes "The Finite Element Method", Chapter 4.
Consider the exact problem written in weak form:
Find u ∈ S such that for all w ∈ V :
a(w, u) = (w, f) + (w, g)Γ
The approximate finite element problem is:
Find uh ∈ Sh such that for all wh ∈ Vh :
a(wh, uh) = (wh, f) + (wh, g)Γ
And assume the following:
- Sh ⊂ S
- Vh ⊂ V
- a(·,·), (·,·), and (·,·)Γ are symmetric and bilinear
- a(·,·) and ||·||m define equivalent norms on V
Theorem: Let e = uh-u denote the error in the finite element approximation. Then,- a(wh, e) = 0, for all wh in Vh
- a(e, e) ≤ a(Uh-u, Uh-u), for all Uh ∈ Sh
Part (a) means that the error is orthogonal to the subspace Vh ⊂ V, in other words "uh is the projection of u onto Sh with respect to a(·,·) "Part (b) is the best approximation property: "there is no member of Sh that is a better approximation to u (with respect to the energy norm a(·,·) ) than uh "
From here you can prove the corollary:
a(u, u) = a(uh, uh) + a(e, e)
And rearranging to
a(e, e) = a(u, u) - a(uh, uh)
reveals that the energy of the error equals (minus) the error of the energy
It is also a direct consequence of the corollary that
a(uh,uh)≤ a(u, u)
That is that the approximate solution underestimates the strain energy