2. Obtaining a variational formulation#

We can obtain the variational formulation of a problem from partial differential equations, by multiplying them by test functions and by integrating by parts. In solid mechanics, the weak formulation then obtained is identical to that given by the Principle of Virtual Works and in the conservative case, the minimization of the total potential energy of the structure. However, it should be noted that for some problems, the equations of the model are easier to establish in the variational framework (case of plates and shells for example).

2.1. Physical problem modeling — Principles and notation#

A physical system is most often modeled by partial differential equations that act on unknowns \(u\) that can be:

  • A scalar like temperature in thermal problems;

  • A vector such as displacements in mechanical problems;

  • A tensor such as stresses in mechanical problems;

It is also possible to use several unknown fields simultaneously, linked by partial differential equations. These are couple problems. In Code_Aster, we can cite as an example the thermo-hydro-mechanical problems that combine movements, pressure and temperature.

The unknown fields are set by:

  • Space, which can be described by a Cartesian coordinate system or any other type of parametrization. In the rest of the document, it will be noted \(x\);

  • The weather, noted \(t\);

2.2. System equations#

A continuous physical system can be represented by a system of partial differential equations that will be written in field \(\Omega\):

(2.1)#\[\mathrm{L}(\mathrm{u})+\mathrm{f}\mathrm{=}0\text{dans}\Omega\]

This system is associated with the boundary conditions on border \(\Gamma\) of domain \(\Omega\):

(2.2)#\[C(u)=h\text{sur}\Gamma =\partial \Omega\]

The differential operator can be expressed on several partial differential equations. You could write:

(2.3)#\[\begin{split}\begin{array}{}{L}_{1}(u)+{f}_{1}=0\\ {L}_{2}(u)+{f}_{2}=0\\ \dots \end{array}\end{split}\]

\({L}_{i}(u)\) is a differential operator acting on the unknown vector \(u\). More generally, the differential operator \({L}_{i}(u)\) is written according to the unknowns and their partial derivatives:

(2.4)#\[{L}_{i}(u,\frac{\partial u}{\partial {x}_{1}},\dots ,\frac{{\partial }^{2}u}{\partial {x}_{1}\mathrm{.}\partial {x}_{2}},\dots ,\frac{{\partial }^{m}u}{\partial {x}_{\alpha }^{m}},t,\frac{\partial u}{\partial t},\dots ,\frac{{\partial }^{p}u}{\partial {t}^{p}},)\]

Such an operator is said to be of order \(m\) in space and of order \(p\) in time. If it does not depend on time (and its derivatives), we say that the problem is stationary. In the rest of the document, only stationary problems will be considered.

2.3. Weighted residue method — Strong integral formulation#

We will define residue \(R(u)\) as being the quantity that cancels out when \(u\) is the solution of the physical problem:

(2.5)#\[R(u)=L(u)-f=0\text{dans}\Omega\]

The weighted residue method consists of:

1/ To build an approximate solution \(u\) by the linear combination of well-chosen functions

(2.6)#\[u(x)=\sum _{i=1}^{N}{c}_{i}\mathrm{.}{\phi }_{i}(x)\]

Where \({\phi }_{i}(x)\) are the shape functions of the approximation and \({c}_{i}\) are the coefficients to be identified.

2/ To solve the system in integral form:

(2.7)#\[\begin{split}\begin{array}{c}\text{Trouver}\mathrm{u}\mathrm{\in }{E}_{u}\text{tel que}\mathrm{\forall }\mathrm{P}\mathrm{\in }{E}_{P}\\ \text{Avec}W\mathrm{=}\underset{\Omega }{\mathrm{\int }}\mathrm{R}(\mathrm{u})\mathrm{.}\mathrm{P}(\mathrm{u})\mathrm{.}d\Omega +\underset{\Gamma }{\mathrm{\int }}\mathrm{[}\mathrm{C}(\mathrm{u})\mathrm{-}\mathrm{h}\mathrm{]}\mathrm{.}\mathrm{P}(\mathrm{u})\mathrm{.}d\Omega \mathrm{=}0\end{array}\end{split}\]

We used the same weighting functions for the main system and for the boundary conditions, but this is not a mandatory step. \(P(u)\) are the weighting functions belonging to a set of \({E}_{P}\) functions. Solution \(u\) belongs to the \({E}_{u}\) space of « sufficiently » regular functions (differentiable up to the order m).

The choice of weighting functions \(P(u)\) makes it possible to identify several methods:

  • If function \(P(u)\) is a Dirac distribution, we get the point collocation method.

  • If the \(P(u)\) function is constant on sub-domains, we get the method of collocation by sub-domains.

  • If the weighting functions \(P(u)\) use the same functions in the form \({\phi }_{i}(x)\) as the approximation of the solution (), we get the Galerkin method.

The strong integral form is thus obtained**.

2.4. Weak integral formulation#

The integral formulation () requires differentiable function spaces in order \(m\) for \({E}_{u}\). The weak formulation consists in carrying out an integration by parts (by applying the Green formula) of the system (). In return, the regularity requirements on the \(P(u)\) weighting functions are increased. Green’s formula is as follows:

(2.8)#\[\underset{\Omega }{\mathrm{\int }}\mathrm{u}\mathrm{.}\mathrm{\nabla }\mathrm{.}\mathrm{P}\mathrm{.}d\Omega \mathrm{=}\mathrm{-}\underset{\Omega }{\mathrm{\int }}\mathrm{P}\mathrm{.}\mathrm{\nabla }\mathrm{.}\mathrm{u}\mathrm{.}d\Omega +\underset{\Gamma }{\mathrm{\int }}\mathrm{u}\mathrm{.}\mathrm{P}\mathrm{.}\mathrm{n}\mathrm{.}d\Gamma\]

where \(n\) is the outgoing normal at the domain’s \(\Gamma\) border.