4. Solving the cellular problem

4.1. Problem to be solved

On the two-dimensional elementary cell (see [fig 4.1-a]), we seek to calculate the functions \({\chi }_{\alpha }(\alpha =\mathrm{1,2})\) verifying:

\(\begin{array}{c}{\mathrm{\int }}_{{Y}^{\text{*}}}\mathrm{\nabla }{\chi }_{\alpha }\mathrm{.}\mathrm{\nabla }v\mathrm{=}{\mathrm{\int }}_{\gamma }{n}_{\alpha }v\mathrm{\forall }v\mathrm{\in }V\\ {\mathrm{\int }}_{{Y}^{\text{*}}}{\chi }_{\alpha }\mathrm{=}0(\text{pour avoir une solution unique})\end{array}\) eq 4.1-1

where:

\(V=\text{{}v\in {H}_{1}({Y}^{\text{*}}),v(y)\text{périodique en}y\text{de période 1}\text{}}\)

Figure 4.1-a

After having determined the functions \({\chi }_{\alpha }(\alpha =\mathrm{1,2})\), the homogenized coefficients defined by \({b}_{\alpha \beta }(\alpha =\mathrm{1,2};\beta =\mathrm{1,2})\) the formula are calculated:

\({b}_{\alpha \beta }\mathrm{=}{\mathrm{\int }}_{{Y}^{\text{*}}}\frac{\mathrm{\partial }{\chi }_{\alpha }}{\mathrm{\partial }{y}_{B}}\) eq 4.1-2

Using Green’s formula and the periodicity of \({\chi }_{\alpha }\), we show that the coefficients \({b}_{\alpha \beta }\) can be written differently:

\({b}_{\alpha \beta }={\int }_{\gamma }{\chi }_{\alpha }{n}_{\beta }\) eq 4.1-3

To estimate this quantity, during finite element discretization, it is necessary to determine the outgoing normal for each element, which can be tedious. We then operate in another way; taking into the equation [éq 4.1-1] \(v\mathrm{=}{\chi }_{\beta }\), we get:

\({b}_{\alpha \beta }\mathrm{=}{\mathrm{\int }}_{{Y}^{\text{*}}}\mathrm{\nabla }{\chi }_{\alpha }\mathrm{\nabla }{\chi }_{\beta }\) eq 4.1-4

From the potential energy function defined by the classical formula:

\(W(v)\mathrm{=}\mathrm{-}\frac{1}{2}{\mathrm{\int }}_{{Y}^{\text{*}}}\mathrm{\nabla }v\mathrm{.}\mathrm{\nabla }v\) eq 4.1-5

we can rewrite the homogenized coefficients in the form:

\({b}_{\alpha \beta }=-(W({\chi }_{\alpha }+{\chi }_{\beta })-W({\chi }_{\alpha })-W({\chi }_{\beta }))\) eq 4.1-6

In the general two-dimensional case, we have to calculate three coefficients of the homogenized problem (we \({b}_{\mathrm{11,}}{b}_{12}={b}_{\mathrm{21,}}{b}_{22}\) know that the matrix \(B=({b}_{\alpha \beta })\) is symmetric). The following two problems need to be resolved:

\(\{\begin{array}{}\text{Calculer}{\chi }_{1}\in V/\underset{{Y}^{\text{*}}}{\int }\nabla {\chi }_{1}\nabla v=\underset{{Y}^{\text{*}}}{\int }{n}_{1}v\\ \text{Calculer}{\chi }_{2}\in V/\underset{{Y}^{\text{*}}}{\int }\nabla {\chi }_{2}\nabla v=\underset{{Y}^{\text{*}}}{\int }{n}_{2}v\\ \text{Calculer}{\chi }^{\text{*}}\in V/{\chi }^{\text{*}}={\chi }_{1}+{\chi }_{2}\end{array}\) eq 4.1-7

We then have:

\(\{\begin{array}{}{b}_{11}=-2W({\chi }_{1})\\ {b}_{22}=-2W({\chi }_{2})\\ {b}_{12}={b}_{21}=-(W({\chi }^{\text{*}})-W({\chi }_{1})-W({\chi }_{2}))\end{array}\) eq 44.1-8

Note:

If the elementary cell has symmetries, this makes it possible to solve the problem on a part of the cell with very appropriate boundary conditions and to calculate only certain coefficients of the homogenized problem. For example for the cell in figure No. 4.1-a we have: \({b}_{11}={b}_{\mathrm{22,}}{b}_{12}={b}_{21}=0\) .

4.2. Equivalent problem to define \({\chi }_{\alpha }\)

In equation [éq 4.1-1], calculating the second member requires determining the normal at the edge. To avoid determining the normal, we can write an equivalent problem, verified by functions \({\chi }_{\alpha }\).

Let’s say vectors \({G}_{1}=(\begin{array}{c}1\\ 0\end{array}),{G}_{2}=(\begin{array}{c}0\\ 1\end{array})\text{et}{G}^{\text{*}}=(\begin{array}{c}1\\ 1\end{array})\), we’re looking for functions \({\chi }^{\text{*}},{\chi }_{\mathrm{1,}}{\chi }_{2}\in V\) such as:

\(\{\begin{array}{}{\int }_{{Y}^{\text{*}}}\nabla {\chi }_{1}\mathrm{.}\nabla v={\int }_{{Y}^{\text{*}}}{G}_{1}\mathrm{.}v\forall v\in V\\ {\int }_{{Y}^{\text{*}}}\nabla {\chi }_{2}\mathrm{.}\nabla v={\int }_{{Y}^{\text{*}}}{G}_{2}\mathrm{.}v\forall v\in V\\ {\int }_{{Y}^{\text{*}}}\nabla {\chi }^{\text{*}}\mathrm{.}\nabla v={\int }_{{Y}^{\text{*}}}{G}^{\text{*}}\mathrm{.}v\forall v\in V\end{array}\) eq 4.2-1

Using Green’s formula and the anti-periodicity of normal \(n\), we show that the problems [éq4.1-1] and [éq 4.2-1] are equivalent.

4.3. Put into practice in Code_Aster

In the*Code_Aster*, to solve the problem [éq 4.2-1], thermal analogy by defining a material having a coefficient \({c}_{p}\) equal to zero and a coefficient \(\lambda\) equal to one is used. To impose the calculation of the second member involving the term in \({G}_{\alpha }\), the PRE_GRAD_TEMP keyword in the AFFE_CHAR_THER command is selected. The thermal issue is resolved using the THER_LINEAIRE command. The calculation of the potential energy \(W\) is provided by the command POST_ELEM with the option ENER_POT. In the general case, three calculations are performed to determine the \(W({\chi }_{1}),W({\chi }_{2}),W({\chi }^{\text{*}})\) values and then, the values of the coefficients of the homogenized problem are deduced manually. To impose the periodic nature of the space in which the solution is sought, the LIAISON_GROUP keyword in the AFFE_CHAR_THER command is used.