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13) where the variation is taken with respect tu u. 13) represent the Primal Variational Form of the Poisson equation. 1. 13) using the rules of Variational Calculus, we obtain ∇ · ρ∇u = s in V as EL equation, and ρ∇u = qˆ on Sq as natural boundary condition. 1. 4. The remaining equations, which pertain to the strong links, are assumed to hold a priori. This happening is not accidental, but can be presented as general rule: The variational principle only reproduces the weak links as EL equations and natural boundary conditions, respectively.

4. Weak form used as departure point to derive the TCPE functional for the Bernoulli-Euler beam. 4. The TCPE (Dual) Functional The Total Complementary Potential Energy (TCPE) functional is mathematically the dual of the primal (TPE) functional. Select the bending moment M as the only master field. 4. Choose the weighting function on the weakened KE to be δ M. The only contribution to the variation of the functional is δ L = 0 4 L (κ M − κ w ) δ M d x = (κ M − w ) δ M d x = 0. 12) 0 Recall that work and energy have opposite signs, since energy is the capacity to produce work.

6. Textbooks & Monographs on Variational Methods There is a very large number of books that focus on variational methods and their applications to engineering and physics. Such material may be also found in supplementary form in books with primary focus on more general subjects, such as mathematical physics, or special ones such as finite elements. The following list was prepared in 1993 for the Variational Methods in Mechanics (VMM) course, but is also relevant to AFEM. 7 It collects only books that the writer has examined, at least superficially.

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