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/* rbOOmit: An implementation of the Certified Reduced Basis method. */
/* Copyright (C) 2009, 2010 David J. Knezevic */
/* This file is part of rbOOmit. */
/* rbOOmit is free software; you can redistribute it and/or */
/* modify it under the terms of the GNU Lesser General Public */
/* License as published by the Free Software Foundation; either */
/* version 2.1 of the License, or (at your option) any later version. */
/* rbOOmit is distributed in the hope that it will be useful, */
/* but WITHOUT ANY WARRANTY; without even the implied warranty of */
/* Lesser General Public License for more details. */
/* You should have received a copy of the GNU Lesser General Public */
/* License along with this library; if not, write to the Free Software */
/* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
// <h1>Reduced Basis Example 2 - Successive Constraint Method</h1>
// In this example we extend reduced_basis_ex1 to solve a steady convection-diffusion
// problem on the unit square via the Reduced Basis Method. In this case, we modify the
// PDE so that it no longer has a parameter-independent coercivity constant. Therefore,
// in order to obtain an error bound, we need to employ the Successive Constraint
// Method (SCM) implemented in RBSCMConstruction/RBSCMEvaluation to obtain a
// parameter-dependent lower bound for the coercivity constant.
// The PDE being solved is div(k*grad(u)) + Beta*grad(u) = f
// k is the diffusion coefficient :
// - constant in the domain 0<=x<0.5 , its value is given by the first parameter mu[0]
// - constant in the domain 0.5<=x<=1 , its value is given by the second parameter mu[1]
// Beta is the convection velocity :
// - constant in the whole domain
// - equal to zero in the y-direction
// - its value in the x-direction is given by the third (and last) parameter mu[2]
// Boundary conditions :
// - dyu=0 on top and bottom
// - u=0 on the left side
// - dxu + Beta*u = 0 on the right side
// C++ include files that we need
#include <iostream>
#include <algorithm>
#include <cstdlib> // *must* precede <cmath> for proper std:abs() on PGI, Sun Studio CC
#include <cmath>
#include <set>
// Basic include file needed for the mesh functionality.
#include "libmesh/libmesh.h"
#include "libmesh/mesh.h"
#include "libmesh/mesh_generation.h"
#include "libmesh/exodusII_io.h"
#include "libmesh/equation_systems.h"
#include "libmesh/dof_map.h"
#include "libmesh/getpot.h"
#include "libmesh/elem.h"
// local includes
#include "rb_classes.h"
#include "assembly.h"
// Bring in everything from the libMesh namespace
using namespace libMesh;
// The main program.
int main (int argc, char** argv)
// Initialize libMesh.
LibMeshInit init (argc, argv);
// This example requires SLEPc and GLPK
#if !defined(LIBMESH_HAVE_SLEPC) || !defined(LIBMESH_HAVE_GLPK)
libmesh_example_assert(false, "--enable-slepc --enable-glpk");
#if !defined(LIBMESH_HAVE_XDR)
// We need XDR support to write out reduced bases
libmesh_example_assert(false, "--enable-xdr");
// XDR binary support requires double precision
libmesh_example_assert(false, "--disable-singleprecision");
// FIXME: This example currently segfaults with Trilinos?
libmesh_example_assert(libMesh::default_solver_package() == PETSC_SOLVERS, "--enable-petsc");
// Skip this 2D example if libMesh was compiled as 1D-only.
libmesh_example_assert(2 <= LIBMESH_DIM, "2D support");
// Parse the input file ( using GetPot
std::string parameters_filename = "";
GetPot infile(parameters_filename);
unsigned int n_elem = infile("n_elem", 1); // Determines the number of elements in the "truth" mesh
const unsigned int dim = 2; // The number of spatial dimensions
bool store_basis_functions = infile("store_basis_functions", true); // Do we write the RB basis functions to disk?
// Read the "online_mode" flag from the command line
GetPot command_line (argc, argv);
int online_mode = 0;
if (, "-online_mode") )
online_mode =;
// Build a mesh.
Mesh mesh (dim);
MeshTools::Generation::build_square (mesh,
n_elem, n_elem,
0., 1.,
0., 1.,
// Create an equation systems object.
EquationSystems equation_systems (mesh);
// We override RBConstruction with SimpleRBConstruction in order to
// specialize a few functions for this particular problem.
SimpleRBConstruction & rb_con =
equation_systems.add_system<SimpleRBConstruction> ("RBConvectionDiffusion");
// Initialize the SCM Construction object
RBSCMConstruction & rb_scm_con =
equation_systems.add_system<RBSCMConstruction> ("RBSCMConvectionDiffusion");
rb_scm_con.add_variable("p", FIRST);
// Initialize the data structures for the equation system.
equation_systems.init ();
// Print out some information about the "truth" discretization
// Set parameters for the eigenvalue problems that will be solved by rb_scm_con
equation_systems.parameters.set<unsigned int>("eigenpairs") = 1;
equation_systems.parameters.set<unsigned int>("basis vectors") = 3;
equation_systems.parameters.set<unsigned int>
("linear solver maximum iterations") = 1000;
// Build a new RBEvaluation object which will be used to perform
// Reduced Basis calculations. This is required in both the
// "Offline" and "Online" stages.
SimpleRBEvaluation rb_eval;
// We need to give the RBConstruction object a pointer to
// our RBEvaluation object
// We also need a SCM evaluation object to perform SCM calculations
RBSCMEvaluation rb_scm_eval;
rb_scm_eval.set_rb_theta_expansion( rb_eval.get_rb_theta_expansion() );
// Tell rb_eval about rb_scm_eval
rb_eval.rb_scm_eval = &rb_scm_eval;
// Finally, need to give rb_scm_con and rb_eval a pointer to the
// SCM evaluation object, rb_scm_eval
if(!online_mode) // Perform the Offline stage of the RB method
// Read in the data that defines this problem from the specified text file
// Print out info that describes the current setup of rb_con
// Prepare rb_con for the Construction stage of the RB method.
// This sets up the necessary data structures and performs
// initial assembly of the "truth" affine expansion of the PDE.
// Perform the SCM Greedy algorithm to derive the data required
// for rb_scm_eval to provide a coercivity lower bound.
// Compute the reduced basis space by computing "snapshots", i.e.
// "truth" solves, at well-chosen parameter values and employing
// these snapshots as basis functions.
// Write out the data that will subsequently be required for the Evaluation stage
// If requested, write out the RB basis functions for visualization purposes
// Write out the basis functions
else // Perform the Online stage of the RB method
// Read in the reduced basis data
// Read in online_N and initialize online parameters
unsigned int online_N = infile("online_N",1);
Real online_mu_0 = infile("online_mu_0", 0.);
Real online_mu_1 = infile("online_mu_1", 0.);
Real online_mu_2 = infile("online_mu_2", 0.);
RBParameters online_mu;
online_mu.set_value("mu_0", online_mu_0);
online_mu.set_value("mu_1", online_mu_1);
online_mu.set_value("mu_2", online_mu_2);
// Now do the Online solve using the precomputed reduced basis
// Print out outputs as well as the corresponding output error bounds.
std::cout << "output 1, value = " << rb_eval.RB_outputs[0]
<< ", bound = " << rb_eval.RB_output_error_bounds[0]
<< std::endl;
std::cout << "output 2, value = " << rb_eval.RB_outputs[1]
<< ", bound = " << rb_eval.RB_output_error_bounds[1]
<< std::endl;
std::cout << "output 3, value = " << rb_eval.RB_outputs[2]
<< ", bound = " << rb_eval.RB_output_error_bounds[2]
<< std::endl;
std::cout << "output 4, value = " << rb_eval.RB_outputs[3]
<< ", bound = " << rb_eval.RB_output_error_bounds[3]
<< std::endl << std::endl;
// Read in the basis functions
// Plot the solution
ExodusII_IO(mesh).write_equation_systems ("RB_sol.e",equation_systems);
// Plot the first basis function that was generated from the train_reduced_basis
// call in the Offline stage
ExodusII_IO(mesh).write_equation_systems ("bf0.e",equation_systems);
return 0;
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