ABSTRACT
In this section we introduce the central notion of the variation. Recall that Chapter 1 described one step in FEA as expressing equilibrium equations as integral equations using variational calculus.
Let u(x) be a vector-valued function of position vector x, and consider a vectorvalued functional F(u(x),u0(x),x), in which u0(x)¼ @u=@x. (Just like the definite integral, a functional maps functions, say of x, into numbers.) Next, let v(x) be a function such that v(x)¼ 0 whenever u(x)¼ 0, and also v0(x)¼ 0 when u0(x)¼ 0. Otherwise, v(x) is arbitrary. The differential dF measures how much F changes if x changes.
