4. Ritz method

The Galerkin method, in some cases, is equivalent to making a functional one stationary. This is the case if the bilinear form \(a(\mathrm{u},\mathrm{v})\) is symmetric and positive:

(4.1)\[\mathrm{\forall }\mathrm{u},\mathrm{v}\mathrm{\in }{E}_{u}a(\mathrm{u},\mathrm{v})\mathrm{=}a(\mathrm{v},\mathrm{u})\text{et}a(\mathrm{u},\mathrm{u})\mathrm{\ge }0\]

In this case the problem () has one and only one solution \(\mathrm{u}\) minimizes the following functional on \({E}_{u}\):

(4.2)\[\pi (\mathrm{u})\mathrm{=}\frac{1}{2}\mathrm{.}a(\mathrm{u},\mathrm{u})\mathrm{-}f(\mathrm{u})\]

From a mechanical point of view, this means that the principle of virtual powers can also be written as the minimization of a scalar quantity: the total energy of the structure. This way of writing balance is very frequently used. Here we are going to present some of the results.

First, we recall that a functional is a function of a set of functions (and their derivatives). We will write \(\pi\) this functional. We will limit ourselves to formulations on the go, knowing that there are others. In this case, functional \(\pi\) will be written as:

(4.3)\[\pi (u)=\pi (u,\frac{\partial u}{\partial x})\]

For conservative problems, we can demonstrate that writing that the first variation of \(\pi\) is zero (condition of stationarity of the functional) is equivalent to applying the principle of virtual work, or even to using the Galerkine method by taking virtual displacements as a weighting function. It is called that the Galerkin method consists in starting from the problem with partial derivatives establishing the balance of the structure, i.e.:

(4.4)\[L(u)+f=0\text{dans}\Omega \text{avec}\sigma \mathrm{.}n=g\text{sur}{\Gamma }_{N}\text{et}u={u}^{D}\text{sur}{\Gamma }_{D}\]

The aim is then to solve the problem in integral form by using weighting functions that are of the same nature as the approximate solution:

(4.5)\[\begin{split}\begin{array}{}W=\underset{\Omega }{\int }[L(u)+f]\mathrm{.}\psi (u)\mathrm{.}d\Omega =0\\ \text{Avec}\sigma \mathrm{.}n=g\text{sur}{\Gamma }_{N}\text{et}u={u}^{D}\text{sur}{\Gamma }_{D}\end{array}\end{split}\]

If we choose the variation of the unknowns \(\psi =\delta u\) as a weighting function and after integrating by parts once, we obtain:

(4.6)\[\delta \pi (u)=W(u)=0\text{avec}u={u}^{D}\text{sur}{\Gamma }_{D}\]

Finding the exact form of the functional is not an immediate step in the general case. In mechanics, for conservative cases, it turns out that this functional is equivalent to the total potential energy of the system. After discretization of the functional (by a finite element approximation), we end up with a matrix system strictly equivalent to that of the Galerkin method (or its equivalent mechanical principle, the virtual power method).

Intuitively, we understand that a small variation \(\delta u\) of the solution is a field that can be kinematically admissible and that therefore corresponds well to the hypotheses of the virtual power method.