5. Implemented in Code_Aster#

5.1. Thermal analogy#

To solve the Laplace problem under pressure, a thermal analogy is used: it is a question of solving the stationary heat equation with a material whose thermal conductivity is equal to unity. So:

\(\{\begin{array}{c}\mathrm{\Delta }p=0\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\\ (\frac{\partial p}{\partial \mathrm{n}}{)}_{\mathrm{\Gamma }}\text{=}-{\rho }_{f}{\ddot{\mathrm{x}}}_{s}·\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}\mathrm{\Gamma }\end{array}\iff \{\begin{array}{c}\text{div}(\lambda \mathrm{grad}T)=0\phantom{\rule{4em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\Leftarrow \mathrm{\Delta }T=0\phantom{\rule{2em}{0ex}}\text{si}\phantom{\rule{2em}{0ex}}\lambda =1\\ (\frac{\partial T}{\partial \mathrm{n}}{)}_{\mathrm{\Gamma }}={\mathrm{\varphi }}_{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}\mathrm{\Gamma }\end{array}\)

\(T\) represents the temperature in the medium, it plays the role of the pressure in the fluid medium. \({\phi }_{n}\) is the normal heat flow to the wall, it plays the role of the term \(-{\rho }_{f}{\ddot{\mathrm{x}}}_{s}\mathrm{.}\mathrm{n}\) which is assimilated to the variation over time in the flow of mass (fluid) at the wall of the structure. This quantity \(-{\rho }_{f}{\ddot{\mathrm{x}}}_{s}\mathrm{.}\mathrm{n}\) is in fact homogeneous to a mass divided by an area and a time squared.

5.2. Practical implementation#

The operator CALC_MATR_AJOU [U4.66.01] was developed to take into account inertial coupling (added mass: OPTION = “MASS_AJOU”) between structures immersed in the same perfect, incompressible fluid and at rest. The fluid is described by equivalent thermal characteristics (operator DEFI_MATERIAU [U4.43.01]) and the portion of the representing mesh is affected by thermal elements (operator AFFE_MODELE [U4.41.01]). This operator CALC_MATR_AJOU also allows you to calculate the stiffness or damping added. In order to facilitate its use in some cases, there is also the macro command MACRO_MATR_AJOU [U4.66.11].

The operator uses five mandatory keywords:

  • the keyword MODELE_FLUIDE: it is on this model that we solve the Laplace problem with Von Neumann limit conditions (or its equivalent thermal problem),

  • the keyword MODE_MECA (or CHAM_NO, or MODELE_GENE): this keyword makes it possible to calculate the flow-type boundary conditions at the wall of the structure,

  • the keyword MODELE_INTERFACE: it is on this model, which includes all the thermal edge elements of the fluid/structure interface, that we calculate the dot product mentioned in paragraph [§4.2],

  • the keyword CHAM_MATER: it is the fluid material (described by equivalent thermal characteristics),

  • the keyword CHARGE: it is a thermal load (temperature imposed at any node of the fluid mesh) that corresponds to the Dirichlet limit condition to remove the singularity of the Laplace problem (see [§4.1]).

A generalized added mass matrix is thus obtained. This matrix having a sky line profile but full (operator NUME_DDL_GENE [U4.65.03]) can be summed to the generalized mass matrix of the structure using the COMB_MATR_ASSE [U4.72.01] operator. This makes it possible to calculate the coupled fluid/structure modes of submerged structures (« wet » modes) (operator CALC_MODES [U4.52.02]).

It is recommended to consult the test case FDLV114 - Seismic response of a cylindrical tank [V8.01.114].