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/**
* @file newmark-beta.hh
*
* @author David Simon Kammer <david.kammer@epfl.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Tue Oct 05 2010
* @date last modification: Sat Sep 12 2020
*
* @brief implementation of the newmark-@f$\beta@f$ integration scheme. This
* implementation is taken from Méthodes numériques en mécanique des solides by
* Alain Curnier \note{ISBN: 2-88074-247-1}
*
*
* @section LICENSE
*
* Copyright (©) 2010-2021 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 "integration_scheme_2nd_order.hh"
/* -------------------------------------------------------------------------- */
#ifndef AKANTU_NEWMARK_BETA_HH_
#define AKANTU_NEWMARK_BETA_HH_
/* -------------------------------------------------------------------------- */
namespace akantu {
/**
* The three differentiate equations (dynamic and cinematic) are :
* \f{eqnarray*}{
* M \ddot{u}_{n+1} + C \dot{u}_{n+1} + K u_{n+1} &=& q_{n+1} \\
* u_{n+1} &=& u_{n} + (1 - \alpha) \Delta t \dot{u}_{n} + \alpha \Delta t
*\dot{u}_{n+1} + (1/2 - \alpha) \Delta t^2 \ddot{u}_n \\
* \dot{u}_{n+1} &=& \dot{u}_{n} + (1 - \beta) \Delta t \ddot{u}_{n} + \beta
*\Delta t \ddot{u}_{n+1}
* \f}
*
* Predictor:
* \f{eqnarray*}{
* u^{0}_{n+1} &=& u_{n} + \Delta t \dot{u}_n + \frac{\Delta t^2}{2}
*\ddot{u}_n \\
* \dot{u}^{0}_{n+1} &=& \dot{u}_{n} + \Delta t \ddot{u}_{n} \\
* \ddot{u}^{0}_{n+1} &=& \ddot{u}_{n}
* \f}
*
* Solve :
* \f[ (c M + d C + e K^i_{n+1}) w = = q_{n+1} - f^i_{n+1} - C \dot{u}^i_{n+1}
*- M \ddot{u}^i_{n+1} \f]
*
* Corrector :
* \f{eqnarray*}{
* \ddot{u}^{i+1}_{n+1} &=& \ddot{u}^{i}_{n+1} + c w \\
* \dot{u}^{i+1}_{n+1} &=& \dot{u}^{i}_{n+1} + d w \\
* u^{i+1}_{n+1} &=& u^{i}_{n+1} + e w
* \f}
*
* c, d and e are parameters depending on the method used to solve the equations
*\n
* For acceleration : \f$ w = \delta \ddot{u}, e = \alpha \beta \Delta t^2, d =
*\beta \Delta t, c = 1 \f$ \n
* For velocity : \f$ w = \delta \dot{u}, e = 1/\beta \Delta t, d =
*1, c = \alpha \Delta t \f$ \n
* For displacement : \f$ w = \delta u, e = 1, d =
*1/\alpha \Delta t, c = 1/\alpha \beta \Delta t^2 \f$
*/
class NewmarkBeta : public IntegrationScheme2ndOrder {
/* ------------------------------------------------------------------------ */
/* Constructors/Destructors */
/* ------------------------------------------------------------------------ */
public:
NewmarkBeta(DOFManager & dof_manager, const ID & dof_id, Real alpha = 0.,
Real beta = 0.);
/* ------------------------------------------------------------------------ */
/* Methods */
/* ------------------------------------------------------------------------ */
public:
void predictor(Real delta_t, Array<Real> & u, Array<Real> & u_dot,
Array<Real> & u_dot_dot,
const Array<bool> & blocked_dofs) const override;
void corrector(const SolutionType & type, Real delta_t, Array<Real> & u,
Array<Real> & u_dot, Array<Real> & u_dot_dot,
const Array<bool> & blocked_dofs,
const Array<Real> & delta) const override;
void assembleJacobian(const SolutionType & type, Real delta_t) override;
public:
Real getAccelerationCoefficient(const SolutionType & type,
Real delta_t) const override;
Real getVelocityCoefficient(const SolutionType & type,
Real delta_t) const override;
Real getDisplacementCoefficient(const SolutionType & type,
Real delta_t) const override;
private:
template <SolutionType type>
void allCorrector(Real delta_t, Array<Real> & u, Array<Real> & u_dot,
Array<Real> & u_dot_dot, const Array<bool> & blocked_dofs,
const Array<Real> & delta) const;
/* ------------------------------------------------------------------------ */
/* Accessors */
/* ------------------------------------------------------------------------ */
public:
AKANTU_GET_MACRO(Beta, beta, Real);
AKANTU_SET_MACRO(Beta, beta, Real);
AKANTU_GET_MACRO(Alpha, alpha, Real);
AKANTU_SET_MACRO(Alpha, alpha, Real);
/* ------------------------------------------------------------------------ */
/* Class Members */
/* ------------------------------------------------------------------------ */
protected:
/// the \f$\beta\f$ parameter
Real beta;
/// the \f$\alpha\f$ parameter
Real alpha;
Real k;
Real h;
/// last release of M matrix
UInt m_release;
/// last release of K matrix
UInt k_release;
/// last release of C matrix
UInt c_release;
};
/**
* central difference method (explicit)
* undamped stability condition :
* \f$ \Delta t = \alpha \Delta t_{crit} = \frac{2}{\omega_{max}} \leq \min_{e}
*\frac{l_e}{c_e}\f$
*
*/
class CentralDifference : public NewmarkBeta {
public:
CentralDifference(DOFManager & dof_manager, const ID & dof_id)
: NewmarkBeta(dof_manager, dof_id, 0., 1. / 2.){};
std::vector<std::string> getNeededMatrixList() override { return {"M", "C"}; }
};
//#include "integration_scheme/central_difference.hh"
/// undamped trapezoidal rule (implicit)
class TrapezoidalRule2 : public NewmarkBeta {
public:
TrapezoidalRule2(DOFManager & dof_manager, const ID & dof_id)
: NewmarkBeta(dof_manager, dof_id, 1. / 2., 1. / 2.){};
};
/// Fox-Goodwin rule (implicit)
class FoxGoodwin : public NewmarkBeta {
public:
FoxGoodwin(DOFManager & dof_manager, const ID & dof_id)
: NewmarkBeta(dof_manager, dof_id, 1. / 6., 1. / 2.){};
};
/// Linear acceleration (implicit)
class LinearAceleration : public NewmarkBeta {
public:
LinearAceleration(DOFManager & dof_manager, const ID & dof_id)
: NewmarkBeta(dof_manager, dof_id, 1. / 3., 1. / 2.){};
};
/* -------------------------------------------------------------------------- */
} // namespace akantu
#endif /* AKANTU_NEWMARK_BETA_HH_ */
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