Let us consider discretization of the problem domain into set of nonoverlapping subdomains , called elements.
Next we will consider the approximation of the unknown displacement field, defined on individual subdomains. Note that the approximation is not arbitrary:
- The weak form kontains only first derivatives of the unknown and test functions, thus only continuity is required.
The element approxiamtion of the arbitrary function has the form
where are so called shape or approximation functions and are nodal values.
Note that for the approximation functions to be interpolatory, the shape functions have to satisfy Kronecker-delta property, i.e.,
, where
is the position vector of the i-th node. Also, the shape functions have to satisfy the condition
, which follows from the requirement to approximate the constant function.
The required continuity of element approximations have to be satisfied. This is typically achieved by enforcing the continuity at the nodal points.
In our case, the approximation of displacements and test functions is
We will use the weak form 1.5, which using Voight's notation has the form
We will need also the derivatives of the displacement and test functions
where matrix contains the first partial derivatives of the shape functions.
By substiituting into the weak form 1.6 we obtain
After introducing a mappig between element displacement vectors
, nodal vectors of test function values
and their global counterparts
one can obtain
|
(1.12) |
By taking into account that the test fuctions are arbitrary (i.e.
), one finnaly obtains the following set of linear algebraic equations for unknonwn nodal displacements
:
|
(1.13) |
Borek Patzak
2017-12-30