Enforcing Global Linearity

If some of the interpolating functions do not satisfy the same linearity conditions that the convex shape functions satisfy, they can be altered. Given the entire set of interpolating functions corresponding to the nodal points and the set of functions that correspond to the points on the convex hull, the following requirement can be imposed:

When the function is either 1, , or , the convex interpolant satisfies the requirement exactly over the convex hull points:

The functions can be modified to ensure linearity:

When the points are not the convex hull points and , boundedness is not necessarily preserved.

For example, construct linearized interpolations on a quadrilateral.

Figure 10. Linearized interpolants.