AdjointStokes_GradientBetaSolver.md

Adjoint Stokes : Slip Coefficient Gradient Solver {#stokes_gradientBeta_solver}

Module name: AdjointStokes_GradientBetaSolver.F90 Module subroutines: AdjointStokes_GradientBetaSolver Module authors: Fabien Gillet-Chaulet (IGE-Grenoble) Document authors: Fabien Gillet-Chaulet Document edited: 17/06/2020

Required input variables:

• Variable solution of the direct problem (Flow Solution)

• The adjoint variable computed from the adjoint linear system

output variables:

• Nodal derivatives with respect to the Slip Coefficient

Introduction

This is the adjoint code corresponding to the application of the slip coefficients in the Stokes Solver.

The sequence in the .sif is usually as follow:

1. Compute the velocity using the Stokes solver

2. Compute a cost function that measures the difference between the model and some observations

3. Compute the solution of the adjoint linear system

4. Compute the gradient of your cost function with respect to your slip coefficient

If a change of variable is used for the input slip coefficient, the derivative of the friction coefficient with respect to your input variable can directly be provided in the Boundary Condition section.

Be careful:

• This solver must be executed on the bottom boundary where the slip condition is applied

• The NormalTangential system has to be used

• In 3D Slip Coefficient 2 and 3 have to be equal

• It will initialise the derivative to 0 by default. Use Reset Gradient Variable=Logical False if the gradient should be added to the values computed in another solver.

Keywords

Solver section

  Solver *id*
   
  procedure = "ElmerIceSolvers" "AdjointStokes_GradientBetaSolver"


  !# name of the variable solution of the direct problem 
   Flow Solution Name = String ... *[default:ssavelocity]*

  !# name of the variable solution of the adjoint problem 
   Adjoint Solution Name = String ... *[default:adjoint]*

  !# name of the variable that contains the derivative with respect to the slip coefficient
   Gradient Variable Name = String .... *[default: DJDB]*

  !# Should the gradient be initialised to 0
  Reset Gradient Variable = Logical ... *[default: True]*

  End

Boundary Conditionds

By default the output for the gradient is the derivative with respect to the slip coefficient. If a change of variable is used, you can directly provide the derivative here:

e.g.

Boundary Condition *id*
 ! This boundary must be a body to execute the solver
  Body Id = *BdId*

  Normal-tangential Velocity = True
  Velocity 1 = Real 0.0e0

  Slip Coefficient 2 = Variable alpha
    REAL procedure "ElmerIceUSF" "TenPowerA"

  Slip Coefficient 3 = Variable alpha
    REAL procedure "ElmerIceUSF" "TenPowerA"

  Slip Coefficient derivative = Variable alpha
    REAL procedure "ElmerIceUSF" "TenPowerA_d"
End

Tests and Examples

• See examples for the Stokes inverse methods