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aka_optimize.hh
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/**
* @file aka_optimize.hh
*
* @author Alejandro M. Aragón <alejandro.aragon@epfl.ch>
*
* @date creation: Fri Jan 04 2013
* @date last modification: Sun Oct 19 2014
*
* @brief Objects that can be used to carry out optimization
*
* @section LICENSE
*
* Copyright (©) 2014, 2015 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu 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 3 of the License, or (at your option) any
* later version.
*
* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
#ifndef __AKANTU_OPTIMIZE_HH__
#define __AKANTU_OPTIMIZE_HH__
#include "aka_config.hh"
#include "aka_common.hh"
#include "aka_point.hh"
#include "solid_mechanics_model.hh"
#include <iostream>
#include <nlopt.hpp>
#include <array/expr.hpp>
//#define DEBUG_OPTIMIZE 1
__BEGIN_AKANTU__
using std::cout;
using std::endl;
typedef array::Array<1,Real> vector_type;
typedef array::Array<2,Real> matrix_type;
std::ostream& operator<<(std::ostream&, nlopt::result);
enum Optimizator_type { Min_t, Max_t };
//! Class used for optimization
/*! This class is a convenience object that inherits from nlopt::opt and carries
* some routines that are common to a nonlinear optimization. The objects sets
* the optimization algorithm as nlopt::LD_SLSQP, a sequential quadratic programming
* (SQP) algorithm for nonlinearly constrained gradient-based optimization
* (supporting both inequality and equality constraints), based on the
* implementation by Dieter Kraft.
*/
class Optimizator : public nlopt::opt {
typedef std::vector<Real> point_type;
typedef nlopt::opt base_type;
point_type& x_; //!< Initial guess for the optimization
Real val_; //!< Function value
public:
//! Parameter constructor that takes an initial guess and a functor
template <class functor_type>
Optimizator(point_type& x0,
functor_type& fn,
Optimizator_type t = Min_t,
nlopt::algorithm alg = nlopt::LD_SLSQP) :
base_type(alg, x0.size()), x_(x0) {
if (t == Min_t)
this->set_min_objective(functor_type::wrap, &fn);
else
this->set_max_objective(functor_type::wrap, &fn);
this->set_xtol_rel(1e-4);
}
//! Carry out the optimization and print result
Real result() {
optimize(x_, val_);
cout<<"Optimum value found at location";
for (size_t i=0; i<x_.size(); ++i)
cout<<" "<<x_[i];
cout<<"\nFunction value: "<<val_;
return val_;
}
};
//! Traits class used as a base class for the Distance_minimizator class template
template <ElementType>
struct Distance_minimizator_traits;
//! Partial template specialization for a segment
template <>
struct Distance_minimizator_traits<_segment_2> {
//! Set lower and upper bounds for the master coordinate
static void set_bounds(nlopt::opt &opt) {
std::vector<Real> lb(1,-1), ub(1,1);
opt.set_lower_bounds(lb);
opt.set_upper_bounds(ub);
}
//! Select the start point between the center or the ends of the segmet
template <class object_type>
static void start(object_type &obj) {
Real min = std::numeric_limits<Real>::infinity();
std::vector<Real> xstart = { -1., 0., 1. }; // check center and extremes of element
int idx = -1;
for (size_t i=0; i<xstart.size(); ++i) {
obj.xi_[0] = xstart[i];
std::vector<Real> grad; // empty vector
Real new_dist = obj(obj.xi_, grad);
if (new_dist < min) {
min = new_dist;
idx = i;
}
}
obj.xi_[0] = xstart[idx];
}
};
//! Constraint function for a triangle
static Real constrain_triangle_3(const std::vector<Real> &xi, std::vector<Real> &grad, void *data) {
if (!grad.empty()) {
grad[0] = 1.;
grad[1] = 1.;
}
return (xi[0] + xi[1] - 1.);
}
//! Helper class used for the definition of triangle partial template specializations
struct Triangle_minimizator_traits {
//! Set lower and upper bounds for the master coordinate
static void set_bounds(nlopt::opt &opt) {
std::vector<Real> lb(2, Real());
opt.set_lower_bounds(lb);
opt.add_inequality_constraint(constrain_triangle_3, NULL, 1e-4);
}
//! Select the start point between the center or the vertices of the triangle
template <class object_type>
static void start(object_type &obj) {
Real min = std::numeric_limits<Real>::infinity();
Real xstart[4][2] = { {0.,0.}, {1.,0.}, {0.,1.}, {1./3.,1./3.} }; // check center and corners of element
int idx = -1;
for (int i=0; i<4; ++i) {
obj.xi_[0] = xstart[i][0];
obj.xi_[1] = xstart[i][1];
std::vector<Real> grad; // empty vector
Real new_dist = obj(obj.xi_, grad);
if (new_dist < min) {
min = new_dist;
idx = i;
}
}
obj.xi_[0] = xstart[idx][0];
obj.xi_[1] = xstart[idx][1];
}
};
//! Partial template specialization for a 3-node triangle
template <>
struct Distance_minimizator_traits<_triangle_3> : public Triangle_minimizator_traits {};
//! Partial template specialization for a 6-node triangle
template <>
struct Distance_minimizator_traits<_triangle_6> : public Triangle_minimizator_traits {};
/*! The Distance_minimizator class template can be used to obtain the closest point
* to a a finite element using the NLopt optimization library.
* \tparam d - The dimension of the problem
* \tparam element_policy - The element type to which the closest point is sought
* The class inherits from Distance_minimizator_traits to take care of functionality
* specific to elements of certain type. The code can be used for elements of type
* _segment_2, _triangle_3, and _triangle_6. The optimization stage is done during the
* constructor by calling the function constructor_common. The closest point is then
* obtained by calling the function point.
*/
template <int d, ElementType element_policy>
class Distance_minimizator : public Distance_minimizator_traits<element_policy> {
friend class Triangle_minimizator_traits;
friend class Distance_minimizator_traits<element_policy>;
const UInt nb_nodes = ElementClass<element_policy>::getNbNodesPerElement();
typedef Distance_minimizator_traits<element_policy> traits_type;
typedef Point<d> point_type;
nlopt::opt opt_; //!< Optimizator reference
std::vector<Real> xi_; //!< Master coordinate closest to point
vector_type p_; //!< Point to which the distance is minimized
matrix_type XX_; //!< Triangle coordinates
UInt counter_; //!< Optimization iteration counter
Real fmin_; //!< Minimum distance value
//! Common function called during construction to carry out the minimization
void constructor_common() {
traits_type::set_bounds(opt_);
opt_.set_min_objective(wrap, this);
opt_.set_ftol_abs(1e-4);
// compute start point
traits_type::start(*this);
// optimize
#ifdef DEBUG_OPTIMIZE
nlopt::result result = opt_.optimize(xi_, fmin_);
if (result > 0)
cout<<"Optimium found in "<<counter_<<" iterations: "<<fmin_<<endl;
cout<<"Point at master coordinate "<<xi_[0]<<": "<<point()<<endl;
cout<<result<<endl;
#else
opt_.optimize(xi_, fmin_);
#endif
}
public:
//! Parameter constructor
/*! This parameter constructor takes the point to which the minimum distance is
* sought, and a container of points points that are the coordinates of the finite
* element
* \param r - Point coordinates
* \param pts - Container of triangle points
*/
template <class point_type, class point_container>
Distance_minimizator(const point_type& p, const point_container& pts)
: opt_(nlopt::LD_SLSQP, d-1), xi_(d-1), p_(d), XX_(nb_nodes,d), counter_() {
// get triangle and point coordinates
for (UInt i=0; i<d; ++i) {
p_[i] = p[i];
for (UInt j=0; j<nb_nodes; ++j)
XX_(j,i) = pts[j][i];
}
// common constructor operation
constructor_common();
}
//! Parameter constructor
/*! This parameter constructor uses a pointer to the coordinates of the point,
* an Element pointer, and the SolidMechanicsModel.
* \param r - Point coordinates
* \param el - Finite element to which the distance is minimized
* \param model - Solid mechanics model
*/
Distance_minimizator(const Real *r, const Element *el, SolidMechanicsModel &model)
: opt_(nlopt::LD_SLSQP, d-1), xi_(d-1), p_(d), XX_(nb_nodes,d), counter_() {
Mesh& mesh = model.getMesh();
const Array<Real> &X = model.getCurrentPosition();
const Array<UInt> &conn = mesh.getConnectivity(el->type);
for (UInt i=0; i<nb_nodes; ++i) {
p_(i) = r[i];
for (UInt j=0; j<d; ++j)
XX_(i,j) = X(conn(el->element,i),j);
}
constructor_common();
}
vector_type operator()(const std::vector<Real> &xi)
{
vector_type N(nb_nodes);
vector_type xi2(xi.size(), const_cast<Real*>(&xi[0]));
ElementClass<element_policy>::computeShapes(xi2, N);
return transpose(XX_)*N;
}
//! Evaluation of the function and its gradient
Real operator()(const std::vector<Real> &xi, std::vector<Real> &grad)
{
// increment function evaluation counter
++counter_;
vector_type x = (*this)(xi);
vector_type diff = x-p_;
if (!grad.empty()) {
// compute shape function derivatives
matrix_type DN(d-1,nb_nodes);
vector_type xi2(xi.size(), const_cast<Real*>(&xi[0]));
ElementClass<element_policy>::computeDNDS(xi2, DN);
DN = transpose(DN);
// compute jacobian
matrix_type J = transpose(XX_)*DN;
// compute function gradient
vector_type gradF = transpose(J) * diff;
for (UInt i=0; i<gradF.size(); ++i)
grad[i] = gradF[i];
}
// return function value
return 0.5 * transpose(diff)*diff;
}
//! Return point at current master coordinate
point_type point() {
vector_type x = (*this)(xi_);
point_type p;
for (UInt i=0; i<x.size(); ++i)
p[i] = x[i];
return p;
}
//! Return the number of function evaluations
UInt iterations() const
{ return counter_; }
//! Return the master coordinate
const std::vector<Real>& master_coordinates()
{ return xi_; }
//! Function that is used to have this class working with nlopt
static double wrap(const std::vector<double> &x, std::vector<double> &grad, void *data) {
return (*reinterpret_cast<Distance_minimizator<d,element_policy>*>(data))(x, grad); }
};
__END_AKANTU__
#endif /* __AKANTU_OPTIMIZE_HH__ */

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