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


Geometry of the problem
---------------------

It is a column with height :math:`\mathit{LZ}=5m`, length :math:`\mathit{LX}=1m`, and width :math:`\mathit{LY}=1m`. In :math:`Z={L}_{d}`, this column has an interface-type discontinuity (a non-meshed interface that is introduced into the model through level-sets using the DEFI_FISS_XFEM operator). The column is thus entirely crossed by the discontinuity (in terms of approximating the displacement field, only Heaviside enrichment is taken into account). The height of the :math:`Y={L}_{d}` interface will be different in the c and d models.

The geometry of the column is shown in the figure.


.. image:: images/100000000000031F00000322B0A8D7EC6E3AE2DF.png
    :width: 3.6008in
    :height: 3.4866in


.. _RefImage_100000000000031F00000322B0A8D7EC6E3AE2DF.png:

**Figure** 1.1-a **: Problem geometry** **3D**


Material properties
--------------------

Young's module: :math:`E=5800\mathit{MPa}`

Poisson's ratio: :math:`\nu =0`

Coefficient of thermal expansion: :math:`\alpha =0{K}^{\text{-1}}`


Boundary conditions
----------------------

*2D case*

The boundary conditions that are applied are of the Neuman type. We impose:


* Across the field :math:`{u}_{y}(x,y)={f}_{y}(x,y)` with :math:`{f}_{y}(x,y)=\{\begin{array}{c}-0.01\ast y\mathit{si}Y<{L}_{d}\\ 0.01\ast (\mathit{LY}-y)\mathit{si}Y>{L}_{d}\end{array}`


* On the [AD] :math:`{u}_{x}(x,y)={f}_{\mathit{null}}` side with :math:`{f}_{\mathit{null}}` the zero value constant function


* On the side [BC] :math:`{u}_{x}(x,y)={f}_{x}(x,y)`

In modeling A (linear) :math:`{f}_{x}(x,y)=\{\begin{array}{c}0.01\ast y\mathit{si}Y<{L}_{d}\\ -0.01\ast (\mathit{LY}-y)\mathit{si}Y>{L}_{d}\end{array}`

In B modeling (quadratic) :math:`{f}_{x}(x,y)=\{\begin{array}{c}0.01\ast {y}^{2}\mathit{si}Y<{L}_{d}\\ -0.01\ast {(\mathit{LY}-y)}^{2}\mathit{si}Y>{L}_{d}\end{array}`

*3D case*

The boundary conditions that are applied are of the Neuman type. We impose:


* Across the field :math:`{u}_{z}(x,y,z)={f}_{z}(x,y,z)` with :math:`{f}_{z}(x,y,z)=\{\begin{array}{c}-0.01\ast z\mathit{si}Z<{L}_{d}\\ 0.01\ast (\mathit{LZ}-z)\mathit{si}Z>{L}_{d}\end{array}`


* On the [ADEH] :math:`{u}_{x}(x,y,z)={f}_{\mathit{null}}` face with :math:`{f}_{\mathit{null}}` the zero value constant function


* On the face [BCGF] :math:`{u}_{x}(x,y,z)={f}_{x}(x,y,z)`

In C modeling (linear) :math:`{f}_{x}(x,y,z)=\{\begin{array}{c}0.01\ast z\mathit{si}Z<{L}_{d}\\ -0.01\ast (\mathit{LZ}-z)\mathit{si}Z>{L}_{d}\end{array}`

In D modeling (quadratic) :math:`{f}_{x}(x,y,z)=\{\begin{array}{c}0.01\ast {z}^{2}\mathit{si}Z<{L}_{d}\\ -0.01\ast {(\mathit{LZ}-z)}^{2}\mathit{si}Z>{L}_{d}\end{array}`


* Throughout the :math:`{u}_{y}(x,y,z)={f}_{\mathit{null}}` domain