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,

$$K = \int_{\Omega_e} B^T D B$$

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:

$$K = \frac{t}{4\Delta} B_a^T D B_b$$

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:

$$A_{ii}\Delta U_i^{k+1} = R_i - \sum_{i\neq j}^N A_{ij}\Delta U_j^k$$