resolution_augmented_lagrangian.cc
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resolution_augmented_lagrangian.cc
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	/**
	* @file resolution_augmented_lagrangian.cc
	*
	* @author Alejandro M. Aragón <alejandro.aragon@epfl.ch>
	*
	* @date creation: Mon Sep 15 2014
	* @date last modification: Wed Sep 17 2014
	*
	* @brief contact resolution classes
	*
	* @section LICENSE
	*
	* Copyright (©) 2014 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/>.
	*
	*/

	//#include "resolution_augmented_lagrangian.hh"

	#include "resolution_augmented_lagrangian.hh"

	#define COUT(name) std::cout << std::string(#name) << ": " << name << std::endl;

	__BEGIN_AKANTU__

	template <int Dim>
	void ContactResolution<Dim, _static, _augmented_lagrangian>::initialize() {

	// give priority to command line arguments instead of those in a file
	if (contact_argparser.has("aka_penalty"))
	options_[Epsilon] = contact_argparser["aka_penalty"];
	else
	flags_[Automatic_penalty_parameter] = true;
	if (contact_argparser.has("aka_alpha"))
	options_[Alpha] = contact_argparser["aka_alpha"];
	if (contact_argparser.has("aka_utol"))
	options_[Multiplier_tol] = contact_argparser["aka_utol"];
	if (contact_argparser.has("aka_ntol"))
	options_[Newton_tol] = contact_argparser["aka_ntol"];
	if (contact_argparser.has("aka_usteps"))
	options_[Multiplier_max_steps] = contact_argparser["aka_usteps"];
	if (contact_argparser.has("aka_nsteps"))
	options_[Newton_max_steps] = contact_argparser["aka_nsteps"];
	if (contact_argparser.has("aka_verbose"))
	flags_[Verbose] = true;
	}

	template <int Dim>
	ContactResolution<Dim, _static, _augmented_lagrangian>::ContactResolution(
	model_type &m)
	: Parsable(_st_contact), model_(m),
	multiplier_dumper_(m.getMesh().getNbNodes(), 3),
	pressure_dumper_(m.getMesh().getNbNodes(), 3) {
	// register dumpers
	m.getMesh().addDumpFieldExternal("multipliers", multiplier_dumper_);
	m.getMesh().addDumpFieldExternal("pressure", pressure_dumper_);

	// register parameters from the file
	registerParam("penalty", options_[Epsilon], _pat_parsable,
	"Penalty parameter for Augmented-Lagrangian formulation");
	registerParam("alpha", options_[Alpha], 1., _pat_parsable,
	"Multiplier for values of the penalty parameter");
	registerParam("utol", options_[Multiplier_tol], 1e-4, _pat_parsable,
	"Tolerance used for multipliers in the Uzawa method");
	registerParam("ntol", options_[Newton_tol], 1e-4, _pat_parsable,
	"Tolerance used in the Newton-Raphson inner convergence loop");
	registerParam("usteps", options_[Multiplier_max_steps], 100., _pat_parsable,
	"Maximum number of steps allowed in the Uzawa loop");
	registerParam("nsteps", options_[Newton_max_steps], 100., _pat_parsable,
	"Maximum number of steps allowed in the Newton-Raphson loop");

	// register parameters from the command line
	contact_argparser.addArgument(
	"--aka_penalty", "Penalty parameter for Augmented-Lagrangian formulation",
	1, cppargparse::_float);
	contact_argparser.addArgument(
	"--aka_alpha", "Multiplier for values of the penalty parameter", 1,
	cppargparse::_float);
	contact_argparser.addArgument(
	"--aka_utol", "Tolerance used for multipliers in the Uzawa method", 1,
	cppargparse::_float);
	contact_argparser.addArgument(
	"--aka_ntol",
	"Tolerance used in the Newton-Raphson inner convergence loop", 1,
	cppargparse::_float);
	contact_argparser.addArgument(
	"--aka_usteps", "Maximum number of steps allowed in the Uzawa loop", 1,
	cppargparse::_float);
	contact_argparser.addArgument(
	"--aka_nsteps",
	"Maximum number of steps allowed in the Newton-Raphson loop", 1,
	cppargparse::_float);
	contact_argparser.addArgument("--aka_verbose", "Verbose output flag", 0,
	cppargparse::_boolean);
	}

	template <int Dim>
	void
	ContactResolution<Dim, _static, _augmented_lagrangian>::solveContactStepImpl(
	SearchBase *sp, Int2Type<_generalized_newton> gn) {

	ContactResolution &cd = *this;

	model_.implicitPred();
	model_.updateResidual();

	AKANTU_DEBUG_ASSERT(model_.stiffness_matrix != NULL,
	"You should first initialize the implicit solver and "
	"assemble the stiffness matrix");

	//** comments that start like this will comment on the work that has to be
	//** done for the implementation of the frictional terms in the code

	UInt k = 0;
	UInt ntotal = 0;

	std::list<int> nt;

	// get global stiffness matrix and force vector
	SparseMatrix &K = model_.getStiffnessMatrix();
	Array<Real> &F = model_.getResidual();

	// get size of the whole system
	UInt original = model_.increment->getSize() * Dim;
	UInt size = original + sm_.size();

	//** the size variable at this point is computed by adding only the number
	//** of slave nodes because for each slave node there's a lagrangian
	//** multiplier for the normal contact assigned to it. In the case of
	//** frictional contact, this variable will have to account for 2 (3)
	//** multipliers for each in slave node in 2D (3D), accounting for the
	//** tangential components.

	contact_status_ = std::map<UInt, bool>();
	status_change_ = std::map<UInt, int>();

	size_t ccc = 0;
	for (auto g : gaps_) {
	if (std::abs(g.second) <= 1.e-10) {
	contact_status_[g.first] = true;
	++ccc;
	}
	}

	Array<Real> solution(size);
	Array<Real> rhs(size);
	rhs.clear();

	// extend data structures to consider Lagrange multipliers
	K.resize(size);

	//** with the right value of the 'size' variable, all these data structures
	//** will be resized accordingly

	cout << std::boolalpha;

	if (cd[Verbose])
	cout << "- Start Generalized Newton:" << endl;

	UInt j = 0;

	bool converged = false;
	bool converged_multiplier = false;

	Real fnorm = 0;

	do {
	Real nerror = 0.;

	cd.niter_ = j;

	// assemble material matrix
	model_.assembleStiffnessMatrix();

	// copy residual to rhs
	std::copy_n(F.storage(), original, rhs.storage());

	// compute contribution to tangent matrix and residual
	Real dummy;
	computeTangentAndResidual(solution, rhs, sp, dummy, gn);

	//** The computeTangentAndResidual is where the major development for the
	//** frictional part will take place.
	//** All the terms coded in this function into the stiffness matrix and the
	//** force vector take into account only the normal contact component. The
	//** implementation of the frictional part will include terms for the
	//** tangential multipliers. The process I followed for the implementation
	//** was to code the stiffness matrix terms from the book by Laursen
	//** (Computational Contact and Impact Mechanics). I then took the terms
	//** involving the lagrangian multiplier part from the thesis by Grzegorz
	//** Pietrzak (Continuum mechanics modelling and augmented Lagrangian
	//** formulation of large deformation frictional contact problems) as
	//** these terms were missing in the Laursen book. Some little changes had
	//** to be made into these terms, as Laursen and Pietrzak use different
	//** conventions for the sign of the gap function. In Pietrzak's thesis,
	//** the residual terms are given following Eqn. 6.20 (page 146), and the
	//** stiffness terms in following equation 6.25 (page 149).
	//** I suggest to start by the implementation of the Uzawa method, as it is
	//** not required to code the tangential lagrangian multiplier terms of
	//** Pietrzak

	// solve
	model_.template solve<IntegrationScheme2ndOrder::_displacement_corrector>(
	solution, 1., true, true, rhs);

	// copy the solution of the system to increment for the primal variable only
	std::copy_n(solution.begin(), model_.increment->getSize() * Dim,
	model_.increment->storage());

	// copy the solution of the system to increment for the lagrange multiplier
	size_t m = 0;
	std::vector<Real> multiplier_check(multipliers_.size());
	vector_type v_old(multipliers_.size());
	vector_type v_new(v_old);
	for (auto &pair : multipliers_) {

	v_old[m] = pair.second;
	v_new[m] = v_old[m] + solution[original + m];

	multiplier_check[m] = pair.second;
	multipliers_[pair.first] += solution[original + m];
	multiplier_check[m] -= pair.second;
	++m;
	}

	Real sum_multiplier = 0.;
	for (auto m : multiplier_check)
	sum_multiplier += m * m;

	//** this check basically computes the L_2 norm of the lagrange multiplier
	//** difference, so this test may change when implementing the frictional
	//** part

	Real mnorm = sqrt(sum_multiplier);
	Real abs_tol = cd[Multiplier_tol];

	if (j == 0)
	fnorm = mnorm;

	converged_multiplier =
	(mnorm <= abs_tol \|\| mnorm <= abs_tol * abs_tol * fnorm);

	model_.implicitCorr();
	model_.updateResidual();

	converged =
	model_.template testConvergence<_scc_increment>(cd[Newton_tol], nerror);

	if (cd[Verbose]) {

	size_t w = 10;
	cout << std::setw(2) << j << ": Primal: " << std::setw(w)
	<< std::setprecision(4) << std::right << nerror
	<< " <= " << cd[Newton_tol] << " = " << std::left << std::setw(5)
	<< (nerror < cd[Newton_tol]) << " \tDual: " << std::setw(w)
	<< std::setprecision(4) << std::right << mnorm
	<< " <= " << cd[Multiplier_tol] << " = "
	<< (mnorm <= cd[Multiplier_tol]) << ", " << std::setw(w) << mnorm
	<< " <= " << abs_tol * abs_tol * fnorm << " = "
	<< (mnorm <= abs_tol * abs_tol * fnorm) << endl;
	}

	++j;
	AKANTU_DEBUG_INFO("[" << _scc_increment << "] Convergence iteration "
	<< std::setw(std::log10(cd[Newton_max_steps])) << j
	<< ": error " << nerror << (converged ? " < " : " > ")
	<< cd[Newton_tol] << std::endl);
	} while (!(converged && converged_multiplier) && j < cd[Newton_max_steps]);

	if (j == cd[Newton_max_steps]) {
	cout << "* ERROR * Newton-Raphson loop did not converge within max "
	"number of iterations: " << cd[Newton_max_steps] << endl;
	exit(1);
	}

	nt.push_back(j);
	ntotal += j;

	AKANTU_DEBUG_INFO("[" << _scc_increment << "] Uzawa convergence iteration "
	<< std::setw(std::log10(cd[Newton_max_steps])) << k
	<< std::endl);

	cout << "Generalized Newton iterations: " << j << endl;

	// dump vtk files
	this->dump();
	}

	template <int Dim>
	void
	ContactResolution<Dim, _static, _augmented_lagrangian>::solveContactStepImpl(
	SearchBase *sp, Int2Type<_uzawa> uz) {

	ContactResolution &cd = *this;

	model_.implicitPred();
	model_.updateResidual();

	AKANTU_DEBUG_ASSERT(model_.stiffness_matrix != NULL,
	"You should first initialize the implicit solver and "
	"assemble the stiffness matrix");

	//** comments that start like this will comment on the work that has to be
	//** done for the implementation of the frictional terms in the code

	// implementation of the Uzawa method for solving contact
	bool uzawa_converged = false;
	static UInt step = 0;
	UInt k = 0;
	UInt ntotal = 0;

	std::list<int> nt;

	std::ofstream ofs;
	ofs.open("iterations.out", std::ofstream::out \| std::ofstream::app);

	// initialize Lagrange multipliers
	// NOTE: It doesn't make any difference to start from the previous
	// converged solution of Lagrange multipliers
	real_map lambda_new;

	cout << std::boolalpha;

	if (cd[Verbose])
	cout << "- Start Uzawa:" << endl;

	do {
	Real uerror = 0.;

	bool converged = false;
	UInt j = 0;

	cd.uiter_ = k;

	do {
	Real nerror = 0.;

	cd.niter_ = j;

	// assemble material matrix
	model_.assembleStiffnessMatrix();

	// compute contribution to tangent matrix and residual
	uzawa_converged = computeTangentAndResidual(lambda_new, sp, uerror, uz);

	//** The computeTangentAndResidual is where the major development for the
	//** frictional part will take place.
	//** All the terms coded in this function into the stiffness matrix and
	//** the force vector take into account only the normal contact component.
	//** The implementation of the frictional part will include terms for the
	//** tangential multipliers. The process I followed for the implementation
	//** was to code the stiffness matrix terms from the book by Laursen
	//** (Computational Contact and Impact Mechanics).
	//** I suggest you start by the implementing the Uzawa method first,
	//** before jumping to the more involved implementation of the tangential
	//** lagrangian multiplier terms of Pietrzak

	// solve
	model_.template solve<IntegrationScheme2ndOrder::_displacement_corrector>(
	*model_.increment, 1., true, true);

	model_.implicitCorr();
	model_.updateResidual();

	converged = model_.template testConvergence<_scc_increment>(
	cd[Newton_tol], nerror);

	if (cd[Verbose])
	cout << " Newton: " << j << ", " << nerror << " < " << cd[Newton_tol]
	<< " = " << (nerror < cd[Newton_tol]) << endl;

	++j;
	AKANTU_DEBUG_INFO("[" << _scc_increment << "] Convergence iteration "
	<< std::setw(std::log10(cd[Newton_max_steps])) << j
	<< ": error " << nerror
	<< (converged ? " < " : " > ") << cd[Newton_tol]
	<< std::endl);
	} while (!converged && j < cd[Newton_max_steps]);

	if (cd[Verbose])
	cout << " Uzawa: " << k << ", " << uerror << " < " << cd[Multiplier_tol]
	<< " = " << (uerror < cd[Multiplier_tol]) << endl;

	if (j == cd[Newton_max_steps]) {
	cout << "* ERROR * Newton-Raphson loop did not converge within max "
	"number of iterations: " << cd[Newton_max_steps] << endl;
	exit(1);
	}

	nt.push_back(j);
	ntotal += j;

	// increment uzawa loop counter
	++k;

	AKANTU_DEBUG_INFO("[" << _scc_increment << "] Uzawa convergence iteration "
	<< std::setw(std::log10(cd[Newton_max_steps])) << k
	<< std::endl);

	// update lagrange multipliers
	cd.multipliers_ = lambda_new;
	} while (!uzawa_converged && k < cd[Multiplier_max_steps]);

	if (k == cd[Multiplier_max_steps]) {
	cout << "* ERROR * Uzawa loop did not converge within max number of "
	"iterations: " << cd[Multiplier_max_steps] << endl;
	exit(1);
	}

	cout << "Summary: Uzawa [" << k << "]: Newton [" << ntotal << "]:";
	for (int n : nt)
	cout << " " << n;
	cout << endl;

	ofs << std::setw(10) << ++step << std::setw(10) << k << std::setw(10)
	<< ntotal << endl;
	ofs.close();

	this->dump();
	}

	template <int Dim>
	void ContactResolution<Dim, _static, _augmented_lagrangian>::dump() {

	multiplier_dumper_.clear();
	pressure_dumper_.clear();

	for (auto v : multipliers_) {
	element_type &el = sm_[v.first];

	if (el == element_type())
	continue;

	auto n = el.normal();

	Real lambda = v.second;

	for (size_t i = 0; i < n.size(); ++i)
	multiplier_dumper_(v.first, i) = lambda * n[i];

	// dump pressures only if area is associated with node
	auto it = areas_.find(v.first);
	if (it != areas_.end())
	for (size_t i = 0; i < n.size(); ++i) {
	Real a = it->second;
	assert(a != 0.);
	pressure_dumper_(v.first, i) = lambda * n[i] / a;
	}
	else
	cout << "* WARNING * Zero area for slave node " << v.first << endl;
	}
	model_.dump();
	}

	template <int Dim>
	void
	ContactResolution<Dim, _static, _augmented_lagrangian>::getPenaltyValues() {
	cout << "* INFO * Obtaining penalty parameters automatically. ";

	const SparseMatrix &Kconst = model_.getStiffnessMatrix();

	Real ave = 0.;
	size_t k = 0;

	// loop over pairs
	for (auto it = sm_.begin(); it != sm_.end(); ++it) {
	auto slave = it->first;
	auto master = it->second;

	if (master != element_type()) {
	std::vector<UInt> conn(master.numNodes() + 1); // 1 slave (not hardcoded)

	conn[0] = slave;
	for (UInt i = 0; i < master.numNodes(); ++i)
	conn[1 + i] = master.node(i);

	// compute normal
	vector_type nu = master.normal();

	// carry out stiffness multiplication with the normal
	// the product Kij*nj would give the force for a unit displacement
	// (i.e., the stiffness needed to move the node by 1)
	matrix_type r(Kconst.getSize(), master.numNodes() + 1);

	// loop over stifness matrix dimension
	for (size_t i = 0; i < Kconst.getSize(); ++i)
	// loop over problem dimensions
	for (int j = 0; j < Dim; ++j)
	// loop over nodes considered
	for (size_t k = 0; k < master.numNodes() + 1; ++k)
	r(i, k) += Kconst(i, conn[k] + j) * nu(j);

	// get results (norm of each column in r)
	vector_type rsum(master.numNodes() + 1);

	for (size_t i = 0; i < rsum.size(); ++i)
	for (size_t j = 0; j < r.rows(); ++j)
	rsum(i) += r(j, i) * r(j, i);

	// get average value as the penalty parameter
	Real epsilon = 0.;
	for (size_t i = 0; i < rsum.size(); ++i)
	epsilon += sqrt(rsum(i));

	epsilon /= master.numNodes() + 1;
	penalty_[slave] = epsilon;

	ave += penalty_[slave];
	++k;
	}
	// dummy master
	else {
	// carry out stiffness multiplication with the normal
	// the product Kij*nj would give the force for a unit displacement
	// (i.e., the stiffness needed to move the node by 1)
	vector_type r(Kconst.getSize());

	// loop over stifness matrix dimension
	for (size_t i = 0; i < Kconst.getSize(); ++i)
	// loop over problem dimensions
	for (int j = 0; j < Dim; ++j)
	// loop over nodes considered
	r(i) += Kconst(i, slave + j) * 1. / Dim;

	// get results (norm of each column in r)
	Real epsilon = 0;
	for (size_t i = 0; i < r.size(); ++i)
	epsilon += r(i) * r(i);

	epsilon = sqrt(epsilon);
	penalty_[slave] = epsilon;

	ave += penalty_[slave];
	++k;
	}
	}
	cout << "Average value: " << (this)[Alpha] ave / k << endl;
	}

	template <int dim> struct TangentTraits;

	template <> struct TangentTraits<2> {

	constexpr static UInt dim = 2;
	constexpr static ElementType master_type = _segment_2;
	constexpr static InterpolationType interpolation_type =
	_itp_lagrange_segment_2;

	typedef Point<dim> point_type;
	typedef array::Array<1, Real> vector_type;
	typedef array::Array<2, Real> matrix_type;
	typedef SolidMechanicsModel model_type;

	template <class element_type>
	static bool projects(const point_type &s, const element_type &master,
	const Array<Real> &position) {

	return has_projection(s, point_type(&position(master.node(0))),
	point_type(&position(master.node(1))));
	}

	template <class real_tuple, class element_type, class vector_type>
	static std::tuple<matrix_type, vector_type>
	computeTangentAndResidual(model_type &model, real_tuple t,
	element_type &master, const vector_type &sh,
	const matrix_type &dsh, const vector_type &N) {

	const Array<Real> &position = model.getCurrentPosition();

	Real gap = std::get<0>(t);
	Real s1 = std::get<1>(t);
	Real s2 = std::get<2>(t);

	// compute the point on the surface
	point_type a(&position(master.node(0)));
	point_type b(&position(master.node(1)));

	vector_type nu = master.normal();

	// compute vector T
	point_type tau = dsh(0, 0) * a + dsh(0, 1) * b;
	vector_type T(dim * (master.numNodes() + 1));

	for (UInt i = 0; i < dim; ++i) {
	T[i] = tau[i];
	for (UInt j = 0; j < master.numNodes(); ++j)
	T[(1 + j) * dim + i] = -tau[i] * sh[j];
	}

	// compute N1
	vector_type N1(dim * (master.numNodes() + 1));

	for (UInt i = 0; i < dim; ++i) {
	for (UInt j = 0; j < master.numNodes(); ++j)
	N1[(1u + j) * dim + i] = -nu[i] * dsh(0u, j);
	}

	// compute m11
	Real m11 = tau * tau;

	// compute D1
	vector_type D1 = T + gap * N1;
	D1 *= 1. / m11;

	// Note: N1bar = N1 - k11*D1, but since k11 = 0 for 2D, then
	// N1bar = N1
	vector_type &N1bar = N1;

	// stiffness matrix (only non-zero terms for 2D implementation)
	matrix_type kc = s1 * N * transpose(N); // first term
	kc += (s2 * gap * m11) * N1bar * transpose(N1bar); // second term
	kc -= s2 * D1 * transpose(N1); // sixth term
	kc -= s2 * N1 * transpose(D1); // eight term

	// residual vector
	vector_type fc = s2 * N;

	assert(kc.rows() == fc.size());

	return std::make_tuple(kc, fc);
	}

	template <class element_type>
	static std::tuple<point_type, vector_type>
	compute_projection(const point_type &s, element_type &master) {

	Distance_minimizator<dim, master_type> dm(s, master.coordinates());
	vector_type xi(1, dm.master_coordinates()[0]);
	return std::make_tuple(dm.point(), xi);
	}
	};

	template <> struct TangentTraits<3> {

	constexpr static UInt dim = 3;
	constexpr static ElementType master_type = _triangle_3;
	constexpr static InterpolationType interpolation_type =
	_itp_lagrange_triangle_3;

	typedef Point<dim> point_type;
	typedef array::Array<1, Real> vector_type;
	typedef array::Array<2, Real> matrix_type;
	typedef SolidMechanicsModel model_type;

	template <class element_type>
	static bool projects(const point_type &s, const element_type &master,
	const Array<Real> &position) {

	return point_has_projection_to_triangle(
	s, point_type(&position(master.node(0))),
	point_type(&position(master.node(1))),
	point_type(&position(master.node(2))));
	}

	template <class real_tuple, class element_type, class vector_type>
	static std::tuple<matrix_type, vector_type>
	computeTangentAndResidual(model_type &model, real_tuple t,
	element_type &master, const vector_type &sh,
	const matrix_type &dsh, const vector_type &N) {

	const Array<Real> &position = model.getCurrentPosition();

	Real gap = std::get<0>(t);
	Real s1 = std::get<1>(t);
	Real s2 = std::get<2>(t);
	Real s3 = std::get<3>(t);

	// compute the point on the surface
	point_type a(&position(master.node(0)));
	point_type b(&position(master.node(1)));
	point_type c(&position(master.node(2)));

	vector_type nu = master.normal();

	point_type tau1 = dsh(0, 0) * a + dsh(0, 1) * b + dsh(0, 2) * c;
	point_type tau2 = dsh(1, 0) * a + dsh(1, 1) * b + dsh(1, 2) * c;

	vector_type nucheck(3);

	Math::vectorProduct3(&tau1[0], &tau2[0], &nucheck[0]);
	Math::normalize3(&nucheck[0]);

	if ((nucheck - nu)().norm() > 1.0e-10) {
	cout << "* ERROR * Normal failed" << endl;
	cout << "nu1: " << nu << endl;
	cout << "nu2: " << nucheck << endl;
	exit(1);
	}

	// compute vectors T1, T2, N1, N2
	size_t vsize = dim * (master.numNodes() + 1);
	vector_type T1(vsize), T2(vsize), N1(vsize), N2(vsize);
	for (UInt i = 0; i < dim; ++i) {
	T1[i] = tau1[i];
	T2[i] = tau2[i];
	for (UInt j = 0; j < master.numNodes(); ++j) {
	T1[(1 + j) * dim + i] = -tau1[i] * sh[j];
	T2[(1 + j) * dim + i] = -tau2[i] * sh[j];
	N1[(1 + j) * dim + i] = -nu[i] * dsh(0u, j);
	N2[(1 + j) * dim + i] = -nu[i] * dsh(1u, j);
	}
	}

	// compute matrix A = m + k*g (but kappa is zero for linear elements)
	Real A11 = tau1 * tau1;
	Real A12 = tau1 * tau2;
	Real A22 = tau2 * tau2;
	Real detA = A11 * A22 - A12 * A12;

	// compute vectors D1, D2
	vector_type D1 =
	(1 / detA) * (A22 * (T1 + gap * N1)() - A12 * (T2 + gap * N2)())();
	vector_type D2 =
	(1 / detA) * (A11 * (T2 + gap * N2)() - A12 * (T1 + gap * N1)())();

	// Note: N1bar = N1 - k12*D2, but since k12 = 0 for linear elements, then
	// N1bar = N1, N2bar = N2
	vector_type &N1bar = N1;
	vector_type &N2bar = N2;

	// stiffness matrix (only non-zero terms for 3D implementation with linear
	// elements)

	// get covariant terms (det(A) = det(inv(A))
	Real m11 = A22 / detA;
	Real m12 = -A12 / detA;
	Real m22 = A11 / detA;

	// 1st term:
	// epsilon * Heaviside(lambda + epsilon gap) * N * N' = s1 * N * N'
	matrix_type kc = s1 * N * transpose(N);
	// 2nd term:
	// t_N * gap * m_11 * N1_bar * N1_bar', where t_N = <lambda + epsilon*gap>
	kc += (s3 * m11) * N1bar * transpose(N1bar);
	// 3rd and 4th terms:
	// t_N * gap * m_12 * (N1_bar * N2_bar' + N2_bar * N1_bar')
	matrix_type tmp = N1bar * transpose(N2bar);
	tmp += N2bar * transpose(N1bar);
	kc += (s3 * m12) * tmp;
	// 5th term:
	// t_N * gap * m_22 * N2_bar * N2_bar'
	kc += (s3 * m22) * N2bar * transpose(N2bar);
	// 6th term:
	// - t_N * D1 * N1'
	kc -= s2 * D1 * transpose(N1);
	// 7th term:
	// - t_N * D2 * N2'
	kc -= s2 * D2 * transpose(N2);
	// 8th term:
	// - t_N * N1 * D1'
	kc -= s2 * N1 * transpose(D1);
	// 9th term:
	// - t_N * N2 * D2'
	kc -= s2 * N2 * transpose(D2);

	// residual vector
	vector_type fc = s2 * N;
	assert(kc.rows() == fc.size());

	return std::make_tuple(kc, fc);
	}

	//! Function template specialization for inversion of a \f$ 3 \times 3 \f$
	// matrix.
	template <class matrix_type>
	static std::pair<matrix_type, Real> invert(matrix_type &A) {
	// obtain determinant of the matrix
	Real det = A[0][0] * (A[1][1] * A[2][2] - A[1][2] * A[2][1]) -
	A[0][1] * (A[1][0] * A[2][2] - A[1][2] * A[2][0]) +
	A[0][2] * (A[1][0] * A[2][1] - A[1][1] * A[2][0]);

	// compute inverse
	matrix_type inv(3, 3, 1. / det);

	inv[0][0] = A[1][1] A[2][2] - A[1][2] * A[2][1];
	inv[0][1] = A[0][2] A[2][1] - A[0][1] * A[2][2];
	inv[0][2] = -A[0][2] A[1][1] + A[0][1] * A[1][2];
	inv[1][0] = A[1][2] A[2][0] - A[1][0] * A[2][2];
	inv[1][1] = A[0][0] A[2][2] - A[0][2] * A[2][0];
	inv[1][2] = A[0][2] A[1][0] - A[0][0] * A[1][2];
	inv[2][0] = -A[1][1] A[2][0] + A[1][0] * A[2][1];
	inv[2][1] = A[0][1] A[2][0] - A[0][0] * A[2][1];
	inv[2][2] = -A[0][1] A[1][0] + A[0][0] * A[1][1];

	return std::make_pair(inv, det);
	}

	template <class vector_type, class point_type>
	static vector_type invert_map(const point_type &s, const point_type &a,
	const point_type &b, const point_type &c) {
	typedef array::Array<2, Real> matrix_type;

	// matrix for inverse
	matrix_type A = { { b[0] - a[0], c[0] - a[0], a[0] },
	{ b[1] - a[1], c[1] - a[1], a[1] },
	{ b[2] - a[2], c[2] - a[2], a[2] } };

	std::pair<matrix_type, Real> Ainv = invert(A);
	vector_type x = { s[0], s[1], s[2] };
	vector_type r1 = Ainv.first * x;

	return vector_type{ r1[0],
	r1[1] }; // return only the first two components of r1
	}

	template <class element_type>
	static std::tuple<point_type, vector_type>
	compute_projection(const point_type &s, element_type &master) {

	auto coord = master.coordinates();

	// compute the point on the surface
	point_type a(coord[0]);
	point_type b(coord[1]);
	point_type c(coord[2]);

	point_type p = closest_point_to_triangle(s, a, b, c);
	vector_type xi = invert_map<vector_type, point_type>(p, a, b, c);

	// Distance_minimizator<dim, TangentTraits<dim>::master_type> dm(
	// s, master.coordinates());
	// xi = vector_type(dim - 1);
	// for (int i = 0; i < dim - 1; ++i)
	// xi[i] = dm.master_coordinates()[i];
	// point_type p = dm.point();
	return std::make_tuple(p, xi);
	}
	};

	template <int dim>
	bool ContactResolution<dim, _static, _augmented_lagrangian>::
	computeTangentAndResidual(real_map &lambda_new, SearchBase *cp, Real &error,
	Int2Type<_uzawa>) {

	const Array<Real> &position = model_.getCurrentPosition();
	const Real tol = (*this)[Multiplier_tol];

	// get global stiffness matrix and force vector
	SparseMatrix &K = model_.getStiffnessMatrix();
	Array<Real> &F = model_.getResidual();
	const Array<Int> &eqnum =
	model_.getDOFSynchronizer().getLocalDOFEquationNumbers();

	static bool auto_flag = true;
	if (auto_flag) {
	auto_flag = false;
	if (!(*this)[Automatic_penalty_parameter]) {
	Real epsilon = (*this)[Epsilon];
	for (auto it = sm_.begin(); it != sm_.end(); ++it)
	penalty_[it->first] = epsilon;
	cout << "* INFO * Uniform penalty parameter used for all slaves: "
	<< epsilon << endl;
	;
	}
	// else get penalty values automatically
	else
	getPenaltyValues();
	}

	Real lm_diff = 0;
	Real lm_max = 0;

	auto it = sm_.begin();
	while (it != sm_.end()) {
	auto slave = it->first;

	Real epsilon = (this)[Alpha] penalty_[slave];
	AKANTU_DEBUG_ASSERT(epsilon != 0, "Penalty value cannot be zero");

	// get slave point
	point_type s(&position(slave));

	auto master = it->second;
	bool no_master = master == element_type();

	// if node lies outside triangle
	if (no_master \|\| !TangentTraits<dim>::projects(s, master, position)) {

	auto r = cp->search(&position(slave));

	// try to find a new master
	if (r != -1) {
	it->second = master =
	element_type(model_, TangentTraits<dim>::master_type, r);
	}
	// else remove master-slave pair from simulation
	else {
	master = element_type();

	gaps_.erase(slave);
	lambda_new.erase(slave);
	++it;
	continue;
	}
	}

	assert(master.type == TangentTraits<dim>::master_type);

	Distance_minimizator<dim, TangentTraits<dim>::master_type> dm(
	s, master.coordinates());

	vector_type xi = vector_type(dim - 1);
	for (int i = 0; i < dim - 1; ++i)
	xi[i] = dm.master_coordinates()[i];

	point_type p = dm.point();

	// compute normal
	vector_type nu = master.normal();
	point_type nup(static_cast<const Real *>(nu.data()));

	// compute and save gap
	Real gap = -(nup * (s - p));
	gaps_[slave] = gap;

	Real lambda_hat = multipliers_[slave] + epsilon * gap;

	if (lambda_hat < 0) {
	// increase iterator
	++it;
	// save value of lambda
	lambda_new[slave] = 0;
	continue;
	}

	Real s1 = epsilon * Heaviside(lambda_hat);
	Real s2 = Macauley(lambda_hat); // max(0,lambda_hat)
	Real s3 = s2 * gap;

	std::vector<UInt> conn(master.numNodes() + 1); // 1 slave (not hardcoded)
	conn[0] = slave;
	for (UInt i = 0; i < master.numNodes(); ++i)
	conn[1 + i] = master.node(i);

	// evaluate shape functions at slave master coordinate
	vector_type sh(master.numNodes());
	InterpolationElement<TangentTraits<dim>::interpolation_type>::computeShapes(
	xi, sh);

	// compute vector N
	vector_type N(dim * (master.numNodes() + 1));
	for (UInt i = 0; i < dim; ++i) {
	N[i] = nu[i];
	for (UInt j = 0; j < master.numNodes(); ++j)
	N[(1 + j) * dim + i] = -nu[i] * sh[j];
	}

	matrix_type dsh(dim - 1, master.numNodes());
	InterpolationElement<TangentTraits<dim>::interpolation_type>::computeDNDS(
	xi, dsh);

	// obtain contribution to stiffness matrix and force vector depending on
	// the dimension
	auto t = TangentTraits<dim>::computeTangentAndResidual(
	model_, std::make_tuple(gap, s1, s2, s3), master, sh, dsh, N);

	matrix_type &kc = std::get<0>(t);
	vector_type &fc = std::get<1>(t);

	// assemble local components into global matrix and vector
	std::vector<UInt> eq;
	for (UInt i = 0; i < conn.size(); ++i)
	for (UInt j = 0; j < dim; ++j)
	eq.push_back(eqnum(conn[i] * dim + j));

	for (UInt i = 0; i < kc.rows(); ++i) {
	F[eq[i]] += fc(i);
	for (UInt j = i; j < kc.columns(); ++j) {
	K.addToProfile(eq[i], eq[j]);
	K(eq[i], eq[j]) += kc(i, j);
	}
	}

	// update multiplier
	lambda_new[slave] = s2;

	Real lm_old = multipliers_[slave];
	lm_max += lm_old * lm_old;
	lm_old -= s2;
	lm_diff += lm_old * lm_old;

	// increase iterator
	++it;
	}

	if (lm_max < tol) {
	error = sqrt(lm_diff);
	return sqrt(lm_diff) < tol;
	}

	error = sqrt(lm_diff / lm_max);
	return sqrt(lm_diff / lm_max) < tol;
	}

	template <typename T>
	Point<2, T>
	closest_point_to_triangle(const Point<2, T> &p, const Point<2, T> &a,
	const Point<2, T> &b, const Point<2, T> &c) {
	return Point<2, T>();
	}

	template <int dim>
	bool ContactResolution<dim, _static, _augmented_lagrangian>::
	computeTangentAndResidual(Array<Real> &solution, Array<Real> &F,
	SearchBase *cp, Real &error,
	Int2Type<_generalized_newton>) {

	const Array<Real> &position = model_.getCurrentPosition();

	// get global stiffness matrix and force vector
	SparseMatrix &K = model_.getStiffnessMatrix();
	const Array<Int> &eqnum =
	model_.getDOFSynchronizer().getLocalDOFEquationNumbers();

	static bool auto_flag = true;
	if (auto_flag) {
	auto_flag = false;
	if (!(*this)[Automatic_penalty_parameter]) {
	Real epsilon = (*this)[Epsilon];
	for (auto it = sm_.begin(); it != sm_.end(); ++it)
	penalty_[it->first] = epsilon;
	cout << "* INFO * Uniform penalty parameter used for all slaves: "
	<< epsilon << endl;
	;

	}
	// else get penalty values automatically
	else
	getPenaltyValues();
	}

	// size of original system
	UInt original = model_.increment->getSize() * dim;

	// multiplier count
	size_t kk = 0;

	auto it = sm_.begin();
	while (it != sm_.end()) {
	auto slave = it->first;

	Real epsilon = (this)[Alpha] penalty_[slave];

	if (status_change_[slave] != 0) {

	;
	epsilon *= (status_change_[slave] + 1.);
	}

	AKANTU_DEBUG_ASSERT(epsilon != 0, "Penalty value cannot be zero");

	// get slave point
	point_type s(&position(slave));

	auto master = it->second;
	bool no_master = master == element_type();

	static std::map<UInt, bool> excluded;

	// if node lies outside triangle
	if (no_master \|\| !TangentTraits<dim>::projects(s, master, position)) {

	auto r = cp->search(&position(slave));

	// try to find a new master
	if (r != -1) {
	it->second = master =
	element_type(model_, TangentTraits<dim>::master_type, r);
	no_master = false;
	}
	// else remove master-slave pair from simulation
	else {
	master = element_type();
	no_master = true;
	excluded[slave] = true;
	}
	}

	Real gap;
	vector_type xi;

	if (!no_master) {

	assert(master.type == TangentTraits<dim>::master_type);

	auto tuple = TangentTraits<dim>::compute_projection(s, master);
	point_type &p = std::get<0>(tuple);
	xi = std::get<1>(tuple);

	// compute normal
	vector_type nu = master.normal();
	point_type nup(static_cast<const Real *>(nu.data()));

	// Real old_gap = gaps_[slave];

	// compute and save gap
	gap = -(nup * (s - p));
	gaps_[slave] = gap;

	// track status
	// if node in contact
	if (contact_status_[slave]) {
	if (gap < -1.e-10) {
	contact_status_[slave] = false;
	++status_change_[slave];
	// cout<<"["<<status_change_[slave]<<"] changing to
	// non-contact status for node "<<slave<<". Gap from "<<old_gap<<" to
	// "<<gap<<endl;
	}

	} else {
	if (gap >= -1.e-10) {
	contact_status_[slave] = true;
	++status_change_[slave];
	// cout<<"["<<status_change_[slave]<<"] changing to contact
	// status for node "<<slave<<". Gap from "<<old_gap<<" to
	// "<<gap<<endl;
	}
	}
	}

	Real lambda_hat = multipliers_[slave] + epsilon * gap;

	// no contact
	if (lambda_hat < 0 \|\| excluded[slave]) {

	size_t ii = original + kk;

	// add contribution to stiffness matrix and residual vector
	F[ii] = multipliers_[slave] / epsilon;

	K.addToProfile(ii, ii);
	K(ii, ii) += -1 / epsilon;
	}
	// contact
	else {

	Real s1 = epsilon * Heaviside(lambda_hat);
	Real s2 = Macauley(lambda_hat); // max(0,lambda_hat)
	Real s3 = s2 * gap;

	std::vector<UInt> conn(master.numNodes() + 1); // 1 slave (not hardcoded)
	conn[0] = slave;
	for (UInt i = 0; i < master.numNodes(); ++i)
	conn[1 + i] = master.node(i);

	// evaluate shape functions at slave master coordinate
	vector_type nu = master.normal();
	vector_type sh(master.numNodes());
	InterpolationElement<
	TangentTraits<dim>::interpolation_type>::computeShapes(xi, sh);

	// compute vector N
	vector_type N(dim * (master.numNodes() + 1));
	for (UInt i = 0; i < dim; ++i) {
	N[i] = nu[i];
	for (UInt j = 0; j < master.numNodes(); ++j)
	N[(1 + j) * dim + i] = -nu[i] * sh[j];
	}

	matrix_type dsh(dim - 1, master.numNodes());
	InterpolationElement<TangentTraits<dim>::interpolation_type>::computeDNDS(
	xi, dsh);

	// obtain contribution to stiffness matrix and force vector depending on
	// the dimension
	auto t = TangentTraits<dim>::computeTangentAndResidual(
	model_, std::make_tuple(gap, s1, s2, s3), master, sh, dsh, N);

	matrix_type &kc = std::get<0>(t);
	vector_type &fc = std::get<1>(t);

	Array<bool> &boundary = model_.getBlockedDOFs();

	// assemble local components into global matrix and vector not taking into
	// account fixed dofs
	std::vector<UInt> eq(conn.size() * dim);
	std::vector<bool> fixed(conn.size() * dim, false);
	for (UInt i = 0; i < conn.size(); ++i)
	for (UInt j = 0; j < dim; ++j) {
	eq.at(i dim + j) = eqnum(conn[i] dim + j);
	fixed.at(i *dim + j) = boundary(conn[i], j);
	}

	for (UInt i = 0; i < kc.rows(); ++i) {
	// if dof is blocked, don't add terms
	if (fixed.at(i))
	continue;
	F[eq[i]] += fc(i);
	for (UInt j = i; j < kc.columns(); ++j) {
	K.addToProfile(eq[i], eq[j]);
	K(eq[i], eq[j]) += kc(i, j);
	}
	}

	// terms corresponding to lagrangian multiplier contribution
	size_t ii = original + kk;

	// assemble contribution to force vector
	F[ii] = -gap;

	// assemble contribution to stiffness matrix (only upper-triangular)
	for (UInt i = 0; i < N.size(); ++i) {
	K.addToProfile(eq[i], ii);
	K(eq[i], ii) -= N[i];
	}
	}

	// increment multiplier counter
	++kk;

	// increase iterator
	++it;
	}

	return true;
	}

	template <int Dim>
	std::ostream &operator<<(
	std::ostream &os,
	const ContactResolution<Dim, _static, _augmented_lagrangian> &cr) {

	typedef typename ContactResolution<
	Dim, _static, _augmented_lagrangian>::element_type element_type;

	os << "Augmented-Lagrangian resolution type. Parameters:" << endl;
	if (cr[Automatic_penalty_parameter])
	cout << "\tpenalty = auto" << endl;
	else
	cout << "\tpenalty = " << cr[Epsilon] << endl;

	cout << "\talpha = " << cr[Alpha] << endl;
	cout << "\tutol = " << cr[Multiplier_tol] << endl;
	cout << "\tntol = " << cr[Newton_tol] << endl;
	cout << "\tusteps = " << cr[Multiplier_max_steps] << endl;
	cout << "\tnsteps = " << cr[Newton_max_steps] << endl;
	cout << "\tverbose = " << cr[Verbose] << endl;

	cout << "\n Slave nodes: ";
	for (auto it = cr.sm_.begin(); it != cr.sm_.end(); ++it)
	os << it->first << " ";
	os << endl;

	// loop over pairs
	cout << "\n Slave master pairs" << endl;
	for (auto it = cr.sm_.begin(); it != cr.sm_.end(); ++it) {
	auto slave = it->first;
	auto master = it->second;
	os << "\tslave: " << slave << ", Master: ";
	if (master == element_type())
	os << "none" << endl;
	else
	os << master << endl;
	}
	return os;
	}

	template std::ostream &operator<<(
	std::ostream &,
	const ContactResolution<2, _static, _augmented_lagrangian> &);

	template std::ostream &operator<<(
	std::ostream &,
	const ContactResolution<3, _static, _augmented_lagrangian> &);

	template class ContactResolution<2, _static, _augmented_lagrangian>;
	template class ContactResolution<3, _static, _augmented_lagrangian>;

	__END_AKANTU__

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