Finite Element Analysis. Barna Szabó

in the space as . For the definition of convergence refer to Section A.2 in the appendix.

      This exercise illustrates that restriction imposed on (or higher derivatives of u) at the boundaries will not impose a restriction on . Therefore natural boundary conditions cannot be enforced by restriction. Whereas all functions in are continuous and bounded, the derivatives do not have to be continuous or bounded.

Exercise 1.4 Show that defined on by eq. (1.20) satisfies the properties of linear forms listed in Section A.1.2 if f is square integrable on I. This is a sufficient but not necessary condition for to be a linear form.

Figure 1.2 Exercise 1.3: The function

.

Remark 1.1 defined on by eq. (1.20) satisfies the properties of linear forms listed in Section A.1.2 if the following inequality is satisfied:

      (1.35)

      1.2.2 The principle of minimum potential energy

Theorem 1.2 The function that satisfies for all minimizes the quadratic functional⁸ , called the potential energy;

      (1.36)

      on the space .

      Proof: For any , we have:

      (1.37)

      where unless . Therefore any admissible nonzero perturbation of u will increase .

      This important theorem, called the theorem or principle of minimum potential energy, will be used in Chapter 7 as our starting point in the formulation of mathematical models for beams, plates and shells.

      Given the potential energy and the space of admissible functions, it is possible to determine the strong form. This is illustrated by the following example.

Example 1.2 Let us determine the strong form corresponding to the potential energy defined by

(1.38)

      with .

      Since u minimizes ,