Variational Methods

Abridged section from author's lecture notes in finite elements.

1 Variational methods provide a powerful method to solve complex problems in continuum mechanics (and other fields as well).

2 As shown in Appendix C, there is a duality between the strong form, in which a differential equation (or Euler's equation) is exactly satisfied at every point (such as in Finite Differences), and the weak form where the equation is satisfied in an averaged sense (as in finite elements).

3 Since only few problems in continuum mechanics can be solved analytically, we often have to use numerical techniques, Finite Elements being one of the most powerful and flexible one.

4 At the core of the finite element formulation are the variational formulations (or energy based methods) which will be discussed in this chapter.

5 For illustrative examples, we shall use beams, but the methods is obviously applicable to 3D continuum.

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