User:Arthurv~enwiki/finitelement

Finite Element Description

In static linear elasticity problems, the problem to be solved is:

${\vec {F}}=K{\vec {x}}$

The K matrix is known as the static stiffness matrix, and F and x are the force and displacement vectors respectively.

Several element types can be used to construct the K matrix. In this case, 2-D plane strain problems, triangular elements with linear shape functions were used.

Generally,

$K=\int _{\Omega _{e}}B^{T}DB$

where $B=SN$ where S is a differential operator, and N is the shape function of the element, and $\Omega _{e}$ is the area or volume of the element (depending whether it is a 2-D or 3-D problem).

For the linear triangular case, the B matrix is constant, so K is given as:

$K={\frac {t}{4\Delta }}B_{a}^{T}DB_{b}$

If a=b, then the resulting stiffness is for a node, otherwise, it is for the edge. This paves the way for using the geometry to store the stiffness matrix.

$K_{ii}u_{i}^{k+1}=F_{i}-\sum _{i\neq j}^{N}K_{ij}u_{j}^{k}$

But to make it stable, the jacobi iterations must be transformed into an iterative-refinement method:

$A_{ii}\Delta U_{i}^{k+1}=R_{i}-\sum _{i\neq j}^{N}A_{ij}\Delta U_{j}^{k}$