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Finite Element Description

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

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,

where where S is a differential operator, and N is the shape function of the element, and 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:

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.

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