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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__

rAKA/src/contact/resolution/resolution_augmented_lagrangian.cc848f0fd8c83bfeatures/friction_examp…

resolution_augmented_lagrangian.ccView Options

rAKA/src/contact/resolution/resolution_augmented_lagrangian.cc
848f0fd8c83bfeatures/friction_examp…

resolution_augmented_lagrangian.cc
View Options