1. Reference problem

1.1. Geometry

We consider a periodic network of \(4\mathrm{\times }4\) beams [fig 1.1-a]. The domain period is \(\varepsilon Y\). Figure [fig 1.1-b] represents an enlargement of \(1/\varepsilon\) of the period. Each beam is straight and has a square cross section.

Figure 1.1-a: Geometry of the heterogeneous medium - Fluidless beams

Figure 1.1-b: Reference cell \(Y\) **- Magnification of \(\frac{1}{\varepsilon }=10\)

• Characteristics of the period:

• Dimensions:

\(\varepsilon Y=(0.21m,0.21m)\)

\(a=1.5m\)

\(e=0.3m\)

• Characteristics of each beam:

• Section:

\(A\mathrm{=}{(\varepsilon \mathrm{\times }a)}^{2}\mathrm{=}{(0.1\mathrm{\times }1.5)}^{2}\mathrm{=}0.0225{m}^{2}\)

• Length:

\(L=4.1m\)

• Bending moment of inertia:

\({I}_{x}={I}_{y}={(\varepsilon \times a)}^{4}/12{m}^{4}\)

1.2. Material properties

Isotropic linear elastic material:

\(E={10}^{9}\mathrm{Pa}\)

\(\mathrm{NU}=0.3\)

Density:

Beam:

\(\mathrm{Rho}=7641\mathrm{kg}/{m}^{3}\)

Fluid:

\(\mathit{Rho}\mathrm{=}0\mathit{kg}\mathrm{/}{m}^{3}\) (case without fluid)

\(\mathrm{Rho}=1000\mathrm{kg}/{m}^{3}\) (case with fluid)

1.3. Corrective terms

The correction terms are calculated on the reference cell \(Y\) [fig 1.1-b].

B_T= \(0.79{m}^{2}\)

B_N= \(0.79{m}^{2}\)

B_TN= \(0{m}^{2}\)

A_ FLUI = \(2.16{m}^{2}\)

A_ CELL = \(2.25{m}^{2}\)

COEF_ECHELLE =10

1.4. Boundary conditions and loads

Case without fluid:

Bottom surface \(\mathrm{Sb}\): recessed

All degrees of freedom are blocked.

Top surface \(\mathrm{Sh}\): recessed

All degrees of freedom are blocked.

Case with fluid:

Bottom surface \(\mathrm{Sb}\): recessed

All degrees of freedom are blocked.

Top surface \(\mathrm{Sh}\): flat support (bilateral connection)

All rotations are locked.

Longitudinal displacement \(\mathrm{DZ}\) is blocked.

All the nodes in \(\mathrm{Sh}\) have the same transverse displacement \(\mathrm{DX}\) and the same normal displacement \(\mathrm{DY}\).