Finite element discretization

Let us consider discretization of the problem domain $ \Omega$ into set of nonoverlapping subdomains $ \Omega_e$, 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 element approxiamtion of the arbitrary function $ f$ has the form

$\displaystyle f = \sum N_j($$ x$$\displaystyle )r_j =$   $ N$$ r$

where $ N_j$ are so called shape or approximation functions and $ r_j$ are nodal values. Note that for the approximation functions to be interpolatory, the shape functions have to satisfy Kronecker-delta property, i.e., $ N_j($$ x$$ _i)=\delta_{ij}$, where $ x$$ _i$ is the position vector of the i-th node. Also, the shape functions have to satisfy the condition $ \sum N_i=1$, which follows from the requirement to approximate the constant function. The required continuity of element approximations have to be satisfied. This is typically achieved by enforcing the continuity at the nodal points. In our case, the approximation of displacements and test functions is
$\displaystyle \mbox{\boldmath$u$}$$\displaystyle ^e$ $\displaystyle =$ $\displaystyle \mbox{\boldmath$N$}$$\displaystyle ^e($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle ^e$ (1.7)
$\displaystyle \delta$$\displaystyle \mbox{\boldmath$u$}$$\displaystyle ^t$ $\displaystyle =$ $\displaystyle \mbox{\boldmath$N$}$$\displaystyle ^e($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )\delta$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle ^e$ (1.8)

We will use the weak form 1.5, which using Voight's notation has the form

$\displaystyle \int_\Omega\nabla ^s\delta$$ u$$\displaystyle \cdot$$ C$ $ \varepsilon $$\displaystyle \ d\Omega =
\int_\Omega\delta$$ u$$\displaystyle \cdot$$ F$$\displaystyle \ d\Omega + \int_\Gamma\delta$$ u$$\displaystyle \cdot$$ t$$\displaystyle \ d\Gamma
$

We will need also the derivatives of the displacement and test functions

$\displaystyle \tilde{\varepsilon }^e = =$   $\displaystyle \mbox{\boldmath$B$}$$\displaystyle ^e($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle ^e$     (1.9)
$\displaystyle \delta\tilde{\varepsilon }^e$ $\displaystyle =$ $\displaystyle \mbox{\boldmath$B$}$$\displaystyle ^e($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )\delta$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle ^e$ (1.10)

where $ B^e$ matrix contains the first partial derivatives of the shape functions. By substiituting into the weak form 1.6 we obtain

$\displaystyle \sum_e \delta$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle ^{e,T} \left[ \underbrace{\int_{\Omega^e} \mbox{\boldmath$B$}^{e,...
...ath$t$}}\ d\Gamma}_{\mbox{\boldmath$f$}^e_\Gamma} \right] = \mbox{\boldmath$0$}$ (1.11)

After introducing a mappig between element displacement vectors $ r$$ ^e$, nodal vectors of test function values $ \delta$$ r$ and their global counterparts $ \hat{\mbox{\boldmath $r$}}, \delta\hat{\mbox{\boldmath $r$}}$ one can obtain

$\displaystyle \delta\hat{\mbox{\boldmath$r$}}^{T} \left[ \hat{\mbox{\boldmath$K...
...$f$}}_\Omega - \hat{\mbox{\boldmath$f$}}_{\Gamma} \right] = \mbox{\boldmath$0$}$ (1.12)

By taking into account that the test fuctions are arbitrary (i.e. $ \delta\hat{\mbox{\boldmath $r$}}\ne\mbox{\boldmath $0$}$), one finnaly obtains the following set of linear algebraic equations for unknonwn nodal displacements $ \hat{\mbox{\boldmath $r$}}$:

$\displaystyle \hat{\mbox{\boldmath$K$}}\hat{\mbox{\boldmath$r$}}=\hat{\mbox{\boldmath$f$}}_\Omega + \hat{\mbox{\boldmath$f$}}_{\Gamma}$ (1.13)

Borek Patzak 2017-12-30