Minimum Potential Energy for Finite Element Analysis

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The Finite Element Analysis can also be executed with a variation principle. In the case of one-dimensional elastostatics, the minimum of potential energy is resilient for conservative systems. The equilibrium position is stable if the potential energy of the system \Pi is minimum. Every infinitesimal disturbance of the stable position leads to an energetic unfavorable state and implies a restoring reaction. The total potential energy \Pi of a system consists of the work of the inner forces (strain energy)

A_i = \int_0^l \underbrace{\frac{1}{2} E(x)A(x) \left(\frac{du}{dx} \right)^2}_{\frac{1}{2}\sigma\epsilon A(x)} dx

and the work of the external forces

A_a = A(x)\overline{t}(x)u(x)|_{\Gamma _t}

The total energy is:

\Pi = A_i - A_a

The condition for a minimum of the total potential energy is:

\Pi(u) - \Pi(u^*) \geq 0

where u is an arbitrary solution which is compatible with the boundary conditions. u^* is the displacement field for which a minimum will be generated.

When disturbing the solution we get:

u = u^* + \lambda w(x)

where \lambda \in \mathbb{R} is the amplitude of the disturbance and w(x) the test function with w=0 on the Dirichlet boundary. The test functions w(x) have to be from the Sobolev space \mathcal{H}_0^1.

Since \Pi is minimal for u^* we have:

\Pi(u^* + \lambda w) - \Pi(u^*) \geq 0

This also has to be resilient for infinitesimal disturbances \lambda \rightarrow 0, \lambda \neq 0. The necessary condition for a minimum is then:

\lim\limits_{n \rightarrow 0}{\frac{(\Pi(u^* + \lambda w) - \Pi(u^*))}{\lambda}} = 0 \tag{1}

The expression above is also known as a Gâteaux derivative.

A_i(u^* + \lambda w) = \int_0^l \frac{1}{2} E(x)A(x) \left(\frac{d(u^* +\lambda w)}{dx}\right)^2 dx

which gives us the binomial formula:

\int_0^l \frac{1}{2} E(x)A(x) \left[\left(\frac{du*}{dx}\right)^2 + 2\lambda\frac{du^*}{dx}\frac{dw}{dx} +\lambda^2 \left(\frac{dw}{dx}\right)^2\right] dx

Formula (1) with \lambda \rightarrow 0 gives us:

0 \overset{!}{=} \lim\limits_{\lambda \rightarrow 0}{\Pi (u^* + \lambda w) - \Pi (u^*)} = 0

which is after some easy calculations:

0\overset{!}{=} \lim_{\lambda\rightarrow 0}\left[{\int_0^lE(x)A(x)\frac{du^*}{dx}\frac{dw}{dx}dx-A(x)\overline{t}(x)w(x)\vert_{\Gamma_{t}}- \int_o^lb(x)w(x)dx}\right]

Since \lambda \rightarrow 0 and \lambda \neq 0 apply at the same time, we have an equivalence to the strong form. So if a function u^*(x) is solving the minimization problem, it automatically fulfills the weak form and vice versa.

If you want to read more, please go to our SimWiki article about FEA to learn more about the reason for finding the Minimum Potential Energy.

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