Reduced Basis

With this application reduced basis solutions to Stokes flow with linear rheolgy solved by LaMEM can be created. At the moment it is important, that the src files of LaMEM which are stored in the LaMEM_src_files are used. It is feasible for 2D setups with linear rheology.

Workflow

LaMEM input file

• FSSA needs to be set to 0
• in the Petsc options sections the following option must be enabled:
  • -objects_dump
  • -dump_sol
  • -dump_precondition_matrixes
  • -dump_precondition_matrixes_prefix Ass
  • -dump_rhs
  • -dump_rho

Create RB components

• All ingredients that are needed to access a reduced solution can be produced with the file createRB.m.

• Steps:

  • specify LaMEM input file / filename of marker routine (e.g setup2D.m) / number of cells in every direction / folder in which RB components should be stored
  • specify input parameter with appropriate ranges

  % parameter 2
  st   = 30;  % smallest parameter value
  en   = 40;  % largest parameter value
  n    = 14;  % parameter spacing
  par2 = linspace(st,en,n);
  

  • add parameter to list

  par = allcomb(par1,par2);
  

  • marker file must be adapted according to the number and type of input parameters (here for file setup2D.m). par is the list that was created by the previous step. In this example par(1) belongs to par1 and is in setup2D.m treated as the variable SubductionAngle1

  function [] = setup2D (par)
  
  SubductionAngle1 = par(1);
  SubductionAngle2 = par(2);
  .
  .
  .
  end
  

  • specify tolerance for greedy algorithm

  tol  = 1e-6;    % tolerance for Greedy algorithm
  

access RB solution

• the file accessRB.m help to access a RB solution with the previously calculated components
• Steps:
  • specify LaMEM input file / filename of marker routine (e.g setup2D.m) / number of cells in every direction / folder in which RB components were stored
  • specify if simulation is nondimensional or geo units were used.

    % units: geo or none
    unit = 'geo';
    

  • specify values for the parameters that were selceted in createRB.m

    par1 = 32.456;
    par2 = 36.342;
    

  • creates truth solution, RB solution (creaqted without DEIM) and DEIM solution

    % whole solutions
    u_t    % truth solution   
    u_RB   % RB solution      
    u_DEIM % DEIM solution
    
    % only velocity components
    v_t    % truth solution   
    v_RB   % RB solution      
    v_D    % DEIM solution
    
    

  • some plotting routines are added. Example:

    [V3d_t] = arrange_vel (nel_x, nel_y, nel_z, coordx, coordy, coordz, v_t,'z','xz');  % z velocity is plotted in xz plane
    x     = linspace(0,coordx,nel_x);
    y     = linspace(0,coordz,nel_z+1);         % nel_z +1 velocity nodes in z direction
    [X,Y] = meshgrid(x,y);
    pcolor(X,Y,V3d_t(:,:,2).'); c = colorbar;   %   V3d_t(:,:,n) n decides about the number of the plane which should be plotted 
    

  2D example: double subduction zone