Reference problem
=====================


Geometry
---------

Software MISS3D uses the frequency coupling method to take into account soil-structure interaction. This method, based on dynamic substructuring, consists in dividing the field of study into three sub-areas:


* the ground (which is discretized into border elements),


* the foundation (which is meshed in EF),


* the structure (which is meshed in EF and represents the building and/or an area bounded by ground).

**The floor**

The soil corresponds to a homogeneous semi-infinite layer of medium.

**The foundation**

The rectangular foundation is shown on [:ref:`Figure 1.1-a <Figure 1.1-a>`] below. Its dimensions are :math:`24m` in the :math:`X` direction, a width of :math:`12m` in the :math:`Y` direction, and an indentation height of :math:`8m`. It is then modelled by 108 surface elements. To speed up the calculations, we deliberately de-refined the height: in practice, it would be necessary to use square-shaped meshes and therefore twice as many meshes in height.


.. image:: images/1000000000000584000002BFA56328C5AB42CCF7.jpg
    :width: 5.122in
    :height: 3.4291in


.. _RefImage_1000000000000584000002BFA56328C5AB42CCF7.jpg:

**0**

**z**


**Figure 1.1-a: Foundation surface mesh**

**The structure**

The structure consists of massive elements representing a homogeneous volume of soil surrounding the foundation. Its dimensions are :math:`72m` in the :math:`X` direction, a width of :math:`72m` in the :math:`Y` direction, and an indentation height of :math:`36m`. It is then modelled by 11520 massive elements. The structure is shown in [:ref:`Figure 1.1-b <Figure 1.1-b>`] below.


.. image:: images/1000000000000584000002BFE70BD21F52F4858D.jpg
    :width: 4.4409in
    :height: 3.4291in


.. _RefImage_1000000000000584000002BFE70BD21F52F4858D.jpg:

Z

**Figure 1.1-b: Volume mesh of the structure**

**The floor**

The mechanical characteristics of the elastic ground model that were used are those indicated below. They make it possible to obtain a shear wave speed of :math:`800m/s` for homogeneous soil.


.. csv-table::

    ":math:`E` (:math:`N/{m}^{2}`)", "4.13 E9"
    ":math:`\nu` ", "0.333"
    ":math:`\rho` (:math:`\mathrm{kg}/{m}^{3}`)", "2420"
    ":math:`\text{AMOR\_HYST}` ", "0.1"


**The foundation and the structure**

The mechanical characteristics of the foundation and the structure that were used are the same as those of the soil described above.


Boundary conditions and mechanical loads
------------------------------------------------

To calculate the 6 static rigid body modes of the foundation and the natural modes, we block the 6 degrees of freedom of translation and rotation of the central node at the base of the foundation by imposing a solid connection relationship. In this central node, 6 nodal force loads of module 1.E6 are also applied to each of the 6 translation and rotation components.

As responses to these 6 requests, the inverse transfer functions of the impedances are then obtained by two harmonic calculation paths: either a modal calculation based on static foundation modes by directly inverting the impedances obtained by the chaining between *Code_Aster* and MISS3D on a model reduced to the foundation, or a calculation on a physical basis by another method using an exclusive modeling by *Code_Aster* of a large volume of soil. homogeneous surrounding the foundation: with this modeling, we represents the condition of an infinite medium by absorbent surface elements assigned to the borders of this volume.