Welcome everybody, good morning for our today's lecture, Finite Element Method.

First of all I would like to recapitulate what we did last time.

We started to discuss the Finite Element Method in 2 and 3D for a linear elastic mechanical problem.

First of all we formulated the actual model problem leading to the strong form which is given here.

So we have the divergence of the stress tensor plus the body force vector equals zero plus boundary conditions

which is given displacement on the Dirichlet boundary and given surface traction on the Neumann boundary.

This term here, the fourth order elasticity tensor times the symmetric displacement gradient which is the strain tensor.

This can be written as the stress tensor.

Then we considered some basics of mechanics here.

We formulated the stress and the strain tensors.

We considered the Fugt notation to facilitate a little bit the writing here.

Then we addressed the distinction between plane strain and plane stress in the two-dimensional case.

And after that we formulated the weak form.

Here we started with the residuum which is the divergence of the stress tensor plus the body force vector has to vanish.

Then we introduced the test function which is now a vectorial quantity.

So we multiplied the test function with the residuum and integrate over the entire domain.

Then we applied integration by parts, Gauss theorem and did some term manipulations and we eventually ended up with this formulation here.

This is the weak form.

Here we have the symmetric gradient of the test function times the fourth order elasticity tensor times the symmetric displacement gradient integrated over the domain.

Then minus a term where we have the surface traction vector minus another term where we have the body force vector.

The surface traction tensor is an integral or is part of an integral over the surface and the body force vector is part of an integral over the domain.

And very important, this has to hold for all test functions of the test function space plus boundary conditions which we already addressed.

And here we can also use Fock notation which leads to this formulation.

I just give it here.

And we specified the function spaces based on a Sobolev space H1.

After that we considered the finite element approximation.

So the test, sorry, the shape functions are the same as we introduced in case of the heat conduction problem.

So we have triangular elements, we have quadrilateral elements, then of course we can have different orders of the polynomials appearing there.

So here are the linear and bilinear elements given, but we already addressed, for instance, bi-quadratic elements.

After that we reconsidered the approximation of the geometry which is the same like in the heat conduction problem.

So we have for all coordinate directions of x, we have the same approximation and this can be subsumed like here.

So the vector x is this matrix of shape functions times the vector of nodal quantities of x.

And this can be written in this form here.

Likewise, then I just jump over the Jacobian and the relation between line area and volume elements.

Likewise, we can now approximate the solution in contrast to the heat conduction problem.

Our unknown is not a scalar field like the temperature before.

Now it's the displacement field which has a dimension of three.

So we apply the same procedure as for the geometry.

You remember this is the isoparametric concept that we use the same approximation for the test function for the geometry and for the unknown.

And again we have the same matrix of shape functions times here now the vector of nodal displacements.

And this can be written in this form.

The same way we approximate the test function.

Okay, so this is more or less what we did so far.

And now let me please take this version of the weak form.

And with that we continue because what we are now aiming for is to do the same actually like in the one dimensional case and in the heat conduction problem.

We want to rewrite this integral as the integrals over the individual element domains.

And then we want to introduce the approximations of the quantities given here.

So we will need an approximation of the symmetric gradient of the test function which is here called delta epsilon tilde.

And we need an approximation of the strain tensor and folk notation epsilon tilde here.

And for the test function we already know the approximation.