Poisson Equation, Energy Function (need help)

Grunf · #1 $Old$ November 5th, 2009, 06:36 AM

Greetings

I don't know is this the right topic form, because I'm new here so please understand me - if I've posted in wrong place, please someone instuct me where to post, whom to ask?

My problem concerns Elastoplastic torsion, which is described in a book Scientific Computing by D.P. O'Leary.

In the book mentioned above stands:
"...the stress function u(x, y) on Omega (interior of the two-dimensional cross section - ellipse) where the quantities -du(x,y)/dx and du(x,y)/dy are the stress components. ...If we set the net force to zero at each point in the cross-section, we obtain (1) and (2). Where G is the shear modulus of the material and theta is the angle of twist per unit length (i.e. G and theta are constants) ...An alternate equivalent formulation is derived from minimizing an energy function (in the next row of the book is shown equation: )
(3)
The magnitude of the gradient (4) is the shear stress at the point (x, y)."

And then, right from the blue sky/ from middle of nowhere in the book stands (I'm saying this because I don't understand this):

"For the sake of generality and in preparation for the more difficult elastoplastic problem, we consider numerical methods. Discretization by finite differences would be a possibility, but the geometry makes the flexibility of finite elements attractive. We can use a finite element package to formulate the matrix K that approximates the operator −∇2u on Omega, and also assemble the right-hand side b so that the solution to the linear system Ku = b is the approximation to u(x, y) at the nodes (xi , yi ) of the finite element mesh. Since the boundary Gamma and the forcing function −2Gθ are smooth, we expect optimal order approximation of the finite element solution to the true solution as the mesh is refined.

Again later in the text (I didn't left nothing crucial) right from the blue sky in the book stands:

"Suppose that the cross-section Omega is the interior of a circle of radius one, G and θ are some constant values G can be 5, and θ can be 1 - that's not important. Use a finite element package to approximate the stress function. Solve problem using a finer mesh and estimate the error in your approximation 1/2u^T Ku−b^T u to E(u)."

I can't find a connection between (1) and (3), how come is (3) alternative to (1)? Using what equations/ formulas can I get (3) from (1)?
Is (5) the same thing as the (3)? How can just those three "letters" (Ku=b) represent the whole equation (3)?

Where does equatiom 1/2u^T Ku−b^T u to E(u). comes from?

How come is possible from (1) to (3) get (5)

Please, someone help me. Please someone, if possible, give me the solution just to some parts if not to whole of my problem.

I thank you in advance for taking the time to consider my problem.

Best regards
G

jeneverboy · #2 $Old$ November 5th, 2009, 09:21 AM

Hi, i am dealing with the same sort of topic. First let me see if I can make something more clear.

(1) is a PDE (partial differential equation) and (3) is the minimization problem which is derived from (1). Finding a solution for (3) will take some numerical methods. But this solution is also a solution for (1).

Now from (3) to (1):
$E(u) = 1/2 \int_{\Omega} \vert \vert \nabla u \vert \vert - 2G\theta u d \Omega$

Let $\hat{u}(x,y)$ be the function minimizing this
$u=u+\epsilon \phi$

Substitute this in (3) minimizing problem
$E(u)=1/2 \int_{\Omega} \vert \vert \nabla (u+\epsilon \phi) \vert \vert - 2 G\theta (u+\epsilon \phi) d \Omega$

Now differentiate with respect to $\epsilon$ , and then E(u) reaches a minimum for $\epsilon=0$ . You should get the following:
$1/2 \int_{\Omega} \nabla u \cdot \nabla \phi - 2 G\theta \phi d \Omega = 0$ . I can't derive this but you will get the following;

Use the divergence theorem to get rid of $\nabla \phi$
$1/2 \int_{\Omega} (-div \nabla u - 2 G\theta) \phi d \Omega + \int_{\Gamma} \phi \nabla u \cdot n d \Gamma= 0$

And use DuBois' Lemma to derive the PDE:
$-div \nabla u - 2 G \theta =0$
with the boundary condition:
$\frac{\partial u}{\partial n} = 0 \; \in \; \Gamma$ Which is the natural boundary condition. The essential boundary condition was already given, u=0 on Gamma.

I skipped some essential steps cause these are not clear to me either, but I am pretty shure this is the way to go from PDE to minimizing problem.

From minimizing to FEM is the next step. Hope someone can contribute to this!!

Bye Alex