~~NOTOC~~
===== Solver Shallow Stream/Shelf Approximation (SSA) =====

==== General Informations ====
:!: **Important changes have been made in ''SSABasalSolver''. This doc applies from Rev. 6480.**

  * **Solver Fortran File:** ''SSASolver.f90''
  * **Solver Name:** (1) ''SSABasalSolver'', (2) ''GetMeanValueSolver'' and (3) ''SSASolver''
  * **Required Output Variable(s):** 
    * (1) ''SSAVelocity'' 
    * (2) ''Mean Viscosity'' and ''Mean Density''
    * (3) ''SSAFlow''
  * **Required Input Variable(s):** 
    * (1) ''Zb'', ''Zs'' and ''Effective Pressure'' when using the Coulomb type friction law
    * (2) ''Depth'' 
    * (3) ''Depth'', ''FreeSurfGrad1'', ''FreeSurfGrad2'' and ''SSABasalFlow''
  * **Optional Output Variable(s):** None  
  * **Optional Input Variable(s):** None


==== General Description ====


=== Ice flow ===

The ''SSABasalSolver'' solve the classical SSA equation, it has been modified in Rev. 6440 to be executed either on a grid of dimension lower than the problem dimension itself (i.e. the top or bottom grid of a 2D or 3D mesh for a SSA 1D or 2D problem), or on a grid of the same dimension of the problem (i.e. 2D mesh for a 2D plane view SSA solution). \\

It will work on a 3D mesh only if the mesh as been extruded along the vertical direction and if the base line boundary conditions have been preserved (to impose neumann conditions). \\

The mandatory input variables are the bottom surface elevation and top surface elevation variables called ''Zb'' and ''Zs'', respectively. \\

For the Flow law the SSA solver use a "power-law" formulation and use the keywords ''Viscosity Exponent'', ''Critical Shear Rate'', and ''Mean Viscosity''. It Doesn't work with the build-in Glen's flow law (TODO).\\
Newton linearisation of the viscosity can be used using the keywords ''Nonlinear System Newton After Tolerance'' and/or
''Nonlinear System Newton After Iterations''. It is automatically reset to False at the beginning of a new iteration.
 
The ''Mean Density'' and ''Mean Viscosity'', if not uniform along the vertical direction, can be computed using the ''GetMeanValueSolver'' routine or the ''StucturedProjectToPlan'' solver (preferred solution).

Contrary to the NS solver, the gravity must be orientated along the z-axis and is taken from the value of
''Flow BodyForce 2'' for a SSA-1D problem or ''Flow BodyForce 3'' for a SSA-2D problem.

A Neumann condition on the lateral boundaries can be applied with the keyword ''Calving front = Logical True'' in the Bounadry condition section. The condition is : //0.5 * g * (rho_ice * h^2 - rho_water * h_im^2)//
where\\
  * //g// is the absolute value of the gravity taken from ''Flow BodyForce i''\\
  * //rho_ice// is the ice ''Mean Density''\\
  *  //rho_water// is ''water density'' taken from the ''constants'' section (or default=1.03225e-18)\\
  * //h// is the front thickness computed as ''Zs-Zb''\\
  * //h_im// is the thickness below sea level computed as ''Sea Level - Zb'', where ''Sea Level'' is taken from the ''constants'' section (or default=0.0). \\
Note that in the absence of explicit boundary condition (no dirichlet condition or ''Calving front = Logical True'' not found) the natural boundary condition is force equilibrium (//rho_ice * h^2 = rho_water * h_im^2//).

The SSA velocities and pressure can be used, for example, as initial conditions for the Stokes Solver. 


When the SSA solution is computed on a boundary of a mesh of dimension larger than the SSA problem (e.g. a 3D mesh for a SSA-2D problem), the SSA solution computed on the boundary can be
  * exported on the whole mesh using  the ''StructuredProjectToPlane'' solver (preffered solution) or  the  ''SSASolver'' routine \\
  *  used as a Dirichlet condition for the SIA velocity (see the [[solvers:sia|SIA Solver]]).

=== Basal friction ===

Since version 6480, there are three friction laws implemented in the SSA solver: 

  * a linear friction law
  <m>  tau_b = beta . u  </m>\\
  * a Weertman type friction law
  <m>  tau_b = beta.{u_b}^{m - 1} . u  </m>\\
  * a Coulomb type friction law 
<m>  tau_b = 1/{A_s}^{1/n} {[{ 1/ {(1 + alpha . chi^q)} }]}^{1/n} . {u_b}^{1/n-1}. u  </m>\\
where 
<m>  alpha = {(q - 1)^{q-1}}/{q^q}  </m>
and
<m>  chi = {u_b}/{C^n N^n A_s}  </m>\\ 
 
The two latests are non-linear and a Newton linearisation can be used. The friction law is chosen using the keyword ''SSA Friction Law'', which takes the value ''Linear'', ''Weertman'' or ''Coulomb''. The other keywords are:\\
  * a linear friction law
    * ''SSA Friction Parameter'' -> <m>beta</m>   
  * a Weertman type friction law
    * ''SSA Friction Parameter'' -> <m>beta</m>
    * ''SSA Friction Exponent'' -> <m>m</m>
    * ''SSA Friction Linear Velocity'' -> <m>u_{t0}</m>
  * a Coulomb type friction law     
    * ''SSA Friction Parameter'' -> <m>beta= {A_s}^{-m}</m>
    * ''SSA Friction Exponent'' -> <m>m = 1/n</m>
    * ''SSA Friction Linear Velocity'' -> <m>u_{t0}</m>
    * ''SSA Friction Post-Peak'' -> <m>q >= 1</m>
    * ''SSA Friction Maximum Value'' -> <m>C</m> ~ max bed slope
    * ''Effective Pressure'' (variable) -> <m>N</m>
    * '' SSA Min Effective Pressure'' -> <m>N_{min}</m>, such that <m>N >= N_{min}</m>

When  <m>u_b = (u^2+v^2)^{1/2}< u_{t0}</m>, <m>u_b</m> in the previous equations is replaced by <m>u_{t0}</m>.




==== SIF contents ====
**Solver section:**
<code>
Solver 1
  Equation = "SSA"
  Procedure = File "ElmerIceSolvers" "SSABasalSolver"
  Variable = String "SSAVelocity"
  Variable DOFs = 2   ! 1 in SSA 1-D or 2 in SSA-2D

  Linear System Solver = Direct
  Linear System Direct Method = umfpack

  Nonlinear System Max Iterations = 100
  Nonlinear System Convergence Tolerance  = 1.0e-08
  Nonlinear System Newton After Iterations = 5
  Nonlinear System Newton After Tolerance = 1.0e-05

  Nonlinear System Relaxation Factor = 1.00

  Steady State Convergence Tolerance = Real 1.0e-3
End
</code>


**Material Properties:** 
<code>
Material 1

! Material properties
  Viscosity Exponent = Real $1.0/n
  Critical Shear Rate = Real 1.0e-10

  SSA Mean Viscosity = Real $eta
  SSA Mean Density = Real $rhoi

! Needed for Linear, Weertman and Coulomb 
  ! Which law are we using (linear, weertman or coulomb)
  SSA Friction Law = String "Coulomb"
  ! beta parameter (beta = 1/As^m)
  SSA Friction Parameter = Variable coordinate 1 , Coordinate 2
     Real  MATC "1.0e-3*(1.0 + sin(2.0*pi* tx(0) / L)*sin(2.0*pi* tx(1) / L))

! Needed for Weertman and Coulomb
  ! Exponent m 
  SSA Friction Exponent = Real $1.0/n
  
  ! Min velocity for linearisation where ub=0
  SSA Friction Linear Velocity = Real 0.0001

! Needed for Coulomb only
  ! post peak exponent in the Coulomb law (q, in Gagliardini et al., 2007)
  SSA Friction Post-Peak = Real 1.0
  ! Iken's bound  tau_b/N < C (see Gagliardini et al., 2007)
  SSA Friction Maximum Value = Real 0.5
End
</code>


**Body Forces:**
<code>
Body Force 1
  Flow BodyForce 1 = Real 0.0
  Flow BodyForce 2 = Real 0.0
  Flow BodyForce 3 = Real $gravity
End
</code>


**Constants:**
<code>
Constants
! Used for Neumann condition
  Water Density = Real ....
  Sea Level = Real ...
End
</code>


**Boundary Conditions:**
<code>
Boundary Condition 1 
! Dirichlet condition
  SSAVelocity 1 = Real ...
  SSAVelocity 2 = Real ...
End
Boundary Condition 1 
! Neumann Condition
  Calving Front = Logical True
End
</code>


**For the "GetMeanValueSolver" routine, the required keywords in the SIF file for this solver are:**
<code>
Solver 1
  Equation = "SSA-IntValue"
  Procedure = File "ElmerIceSolvers" "GetMeanValueSolver"
  Variable = -nooutput String "Integrated variable"
  Variable DOFs = 1

  Exported Variable 1 = String "Mean Viscosity"
  Exported Variable 1 DOFs = 1
  Exported Variable 2 = String "Mean Density"
  Exported Variable 2 DOFs = 1

  Linear System Solver = Direct
  Linear System Direct Method = umfpack

  Steady State Convergence Tolerance = Real 1.0e-3
End

!!! Upper free surface
Boundary Condition 1

  Depth = Real 0.0
  Mean Viscosity = Real 0.0
  Mean Density = real 0.0
End
</code>

**For the "SSASolver" routine, the required keywords in the SIF file for this solver are:**
<code>
Solver 4
  Equation = "SSA Velocity"
  Procedure = File "ElmerIceSolvers" "SSASolver"
  Variable = -nooutput String "varSSA"
  Variable DOFs = 1

  Exported Variable 1 = String "SSAFlow"
  Exported Variable 1 DOFs = 4  ! 3 in 2D, 4 in 3D

  Linear System Solver = Direct
  Linear System Direct Method = umfpack

  Steady State Convergence Tolerance = Real 1.0e-3
End

!!! bedrock
Boundary Condition 1
  
  SSAFlow 1 = Equals SSAVelocity 1
  SSAFlow 2 = Equals SSAVelocity 2
  SSAFlow 3 = Real 0.0e0
End

!!! Upper free surface
Boundary Condition 2
 
  Depth = Real 0.0
 
  SSAFlow 4 = Real 0.0     ! p=0 at the surface
End
</code>

If one wants to solve the SSA + SIA, the sif will read:
<code>
Solver 4
  Equation = "SIA Velocity"
  Procedure = File "SIASolver" "SIASolver"
  Variable = -nooutput String "varSIA"
  Variable DOFs = 1

  Exported Variable 1 = String "SIAFlow"
  Exported Variable 1 DOFs = 4  ! 3 in 2D, 4 in 3D

  Linear System Solver = Direct
  Linear System Direct Method = umfpack

  Steady State Convergence Tolerance = Real 1.0e-3
End

!!! bedrock
Boundary Condition 1
  
 ...
  SIAFlow 1 = Equals SSAVelocity 1
  SIAFlow 2 = Equals SSAVelocity 2
  SIAFlow 3 = Real 0.0e0
  ...
End

!!! Upper free surface
Boundary Condition 2
 
 ...  
  SIAFlow 4 = Real 0.0     ! p=0 at the bottom
End
</code>


==== Examples ====
For examples look in your elmer source distribution under ''[ELMER_TRUNK]/elmerice/Tests/SSA'' and under ''[ELMER_TRUNK]/elmerice/examples/Test_SSA''.