Lagamine

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====== MWAT2 ======
Plane or axisymmetric state
===== Description =====
{{  :elements:plxnt.png?400|}}
Complete coupled analysis soil mechanics-3 fluids. Couplings: Mechanical-Water-Air-Temperature in large deformations. \\
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Type: 205 \\
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The element is defined by 3, 4, 6, 8, 15 or 25 nodes indicated in NODES in the order indicated in the figure. \\
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The flow (water, air, temperature) description can be different from the mechanical description: pressure/temperature could be linearly interpolated in a 3 or 4 nodes configurations, while the mechanical DoFs would be parabolically interpolated in a 6 or 8 nodes configuration. In that case, the fluid DoF must be fixed for the non-used nodes. \\ \\
The "fluid" constitutive laws that can be used with this element are, for now:
  * [[laws:wavat|WAVAT]]: MWA coupling: Mechanical - Water - Air
  * [[laws:wapet|WAPET]]: MWP coupling: Mechanical - Water - Oil
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__15-node element (12 I.P.)__ \\

For the element with 15 nodes, the interpolation functions are the following:
{{  :elements:csol2_1.png?400|}}
\[N_1=\zeta(4\zeta-1)(4\zeta-2)(4\zeta-3)/6 \\ N_2=\xi(4\xi-1)(4\xi-2)(4\xi-3)/6 \\ N_3=\eta(4\eta-1)(4\eta-2)(4\eta-3)/6 \\ N_4=4\zeta\xi(4\zeta-1)(4\xi-1) \\ N_5= 4\xi\eta(4\xi-1)(4\eta-1) \\ N_6= 4\eta\zeta(4\eta-1)(4\zeta-1) \\ N_7=\xi\zeta(4\zeta-1)(4\zeta-2)*8/3 \\ N_8=\zeta\xi(4\xi-1)(4\xi-2)*8/3 \\ N_9=\eta\xi(4\xi-1)(4\xi-2)*8/3 \\ N_{10}=\xi\eta(4\eta-1)(4\eta-2)*8/3 \\ N_{11}=\zeta\eta(4\eta-1)(4\eta-2)*8/3 \\ N_{12}=\eta\zeta(4\zeta-1)(4\zeta-2)*8/3 \\ N_{13}=32\eta\xi\zeta(4\zeta-1) \\ N_{14}=32\eta\xi\zeta(4\xi-1) \\ N_{15}=32\eta\xi\zeta(4\eta-1) \]
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__25-node element (16 I.P.)__ \\

For the element with 25 nodes, the interpolation functions are expressed from the following functions:
{{  :elements:csol2_2.png?300|}}
\[\begin{align*}
N_1&=N_1(\xi)N_1(\eta)    &    N_{14}&=N_1(\xi)N_4(\eta)  \\
N_2&=N_2(\xi)N_1(\eta)    &    N_{15}&=N_1(\xi)N_3(\eta)  \\
N_3&=N_3(\xi)N_1(\eta)    &    N_{16}&=N_1(\xi)N_2(\eta)  \\
N_4&=N_4(\xi)N_1(\eta)    &    N_{17}&=N_2(\xi)N_2(\eta)  \\
N_5&=N_5(\xi)N_1(\eta)    &    N_{18}&=N_3(\xi)N_2(\eta)  \\
N_6&=N_5(\xi)N_2(\eta)    &    N_{19}&=N_4(\xi)N_2(\eta)  \\ 
N_7&=N_5(\xi)N_3(\eta)    &    N_{20}&=N_4(\xi)N_3(\eta)  \\
N_8&=N_5(\xi)N_4(\eta)    &    N_{21}&=N_4(\xi)N_4(\eta)  \\
N_9&=N_5(\xi)N_5(\eta)    &    N_{22}&=N_3(\xi)N_4(\eta)  \\ 
N_{10}&=N_4(\xi)N_5(\eta)    &    N_{23}&=N_2(\xi)N_4(\eta)  \\
N_{11}&=N_3(\xi)N_5(\eta)    &    N_{24}&=N_2(\xi)N_3(\eta)  \\
N_{12}&=N_2(\xi)N_5(\eta)    &    N_{25}&=N_3(\xi)N_3(\eta)  \\ 
N_{13}&=N_1(\xi)N_5(\eta)      
\end{align*}\]

With:
\[N_1(X)=1/6X-1/6X^2-2/3X^3+2/3X^4 \\ N_2(X)=-4/3X+8/3X^2+4/3X^3-8/3X^4 \\ N_3(X)=1-5X^2+X^4 \\ N_4(X)=4/3X+8/3X^2-4/3X^3-8/3X^4 \\ N_5(X)=-1/6X-1/6X^2+2/3X^3+2/3X^4 \]
\\
Implmented by: J.P. Radu & J.D. Barnichon, 1986
==== Files ====
Prepro: MWAT2A.F  \\
Lagamine: MWAT2B.F

===== Input file =====
^Title (A5)^^
|TITLE|"MWAT2" in the first 5 columns|
^Control data (4I5)^^
|NELEM|Number of elements|
|ISPSMAS|= 0 → Nothing \\ = 1 → Take into account the specific mass if and only if NTANA<0|
|INSIG|= 0 → No initial stress \\ = 1 or 2 → Initial stresses|
|INBIO|= 0 → No Biot coefficient \\ = 1 → Isotropic Biot coefficient |
^Specific mass in dynamic analysis - Only if ISPMAS = 1 (1G10.0)^^
|SPEMAS|Specific mass|
^Initial stresses - Only if INSIG > 0 (4G10.0)^^
|If INSIG=1: $\sigma_y=\sigma_{y0}+yd\sigma_{y}$ \\ If INSIG=2: $\sigma_y=min(\sigma_{y0}+yd\sigma_y,0)$||
|SIGY0| $\sigma_{y0}$ effective stress $\sigma_y$ at the axes origin|
|DSIGY|Effective stress gradient along Y axis|
|AK0X|$k_0$ ratio $\sigma_x/\sigma_y$|
|AK0Z|$k_0$ ratio $\sigma_z/\sigma_y$ (if AK0Z=0, AK0Z=AK0X)|
|The computation of SIGY0 and DSIGY must take into account the apparent specific mass, defined as \[\rho_a'=[(1-n)\rho_s+nS_w\rho_w]-\rho_w\] where: \\ $\rho_s$ is the solid specific mass - this represents the specific mass of a fictive sample where ther is no porosity, i.e. where the grains occupy the whole volume of the sample \\  $\rho_w$ is the fluid specific mass \\ $n$ is the porosity defined in the flow law related to the element \\ $S_w$ fluid saturation, ∈ [0,1]||
^Biot coefficient - Only if INBIO = 1 (1G10.0)^^
|CBIOT|Biot coefficient|
^Definition of the elements (6I5/16I5(/9I5)) ^^
|NNODM|Number of nodes for the mechancial description: 3, 4, 6, 8, 15, or 25|
|NINTM|Number of integration point (1, 3, 4, 7, 9, 12, or 16) for the mechanical description|
|LMATM|Mechanical material|
|NNODP|Number of nodes for the flow description: 3, 4, 6, 8, 15, or 25|
|NINTP|Number of integration points (1, 3, 4, 7, 9, 12, or 16) for the flow description \\ Must be equal to NINTM|
|LMATF|Flow material|
|NODES(NNODEM)|List of nodes|

===== Results =====
  * Stresses (in global axes)
    * Mechanical stresses $\sigma_x$, $\sigma_y$, $\sigma_{xy}$, $\sigma_z$ 
    * Flow in water $f_{wx}$, $f_{wy}$, $f_{w,stored}$, 0
    * Flow in air $f_{ax}$, $f_{ay}$, $f_{a,stored}$, 0
    * Thermal flow $f_{tx}$, $f_{ty}$, $f_{t,stored}$, 0
  * Internal variables:
    * Internal variables of the mechanical law
    * Internal variables of the flow law
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