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rAKA/src/model/integration_scheme/newmark-beta.hh
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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: Thu Jun 05 2014
*
* @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 (©) 2010-2012, 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/>.
*
*/
/* -------------------------------------------------------------------------- */
#ifndef __AKANTU_NEWMARK_BETA_HH__
#define __AKANTU_NEWMARK_BETA_HH__
/* -------------------------------------------------------------------------- */
#include "integration_scheme_2nd_order.hh"
/* -------------------------------------------------------------------------- */
__BEGIN_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(Real alpha, Real beta) : beta(beta), alpha(alpha), k(0.), h(0.) {};
~NewmarkBeta(){};
/* ------------------------------------------------------------------------ */
/* Methods */
/* ------------------------------------------------------------------------ */
public:
inline void integrationSchemePred(Real delta_t,
Array<Real> & u,
Array<Real> & u_dot,
Array<Real> & u_dot_dot,
Array<bool> & blocked_dofs) const;
inline void integrationSchemeCorrAccel(Real delta_t,
Array<Real> & u,
Array<Real> & u_dot,
Array<Real> & u_dot_dot,
Array<bool> & blocked_dofs,
Array<Real> & delta) const;
inline void integrationSchemeCorrVeloc(Real delta_t,
Array<Real> & u,
Array<Real> & u_dot,
Array<Real> & u_dot_dot,
Array<bool> & blocked_dofs,
Array<Real> & delta) const;
inline void integrationSchemeCorrDispl(Real delta_t,
Array<Real> & u,
Array<Real> & u_dot,
Array<Real> & u_dot_dot,
Array<bool> & blocked_dofs,
Array<Real> & delta) const;
public:
template<IntegrationSchemeCorrectorType type>
Real getAccelerationCoefficient(Real delta_t) const;
template<IntegrationSchemeCorrectorType type>
Real getVelocityCoefficient(Real delta_t) const;
template<IntegrationSchemeCorrectorType type>
Real getDisplacementCoefficient(Real delta_t) const;
private:
template<IntegrationSchemeCorrectorType type>
void integrationSchemeCorr(Real delta_t,
Array<Real> & u,
Array<Real> & u_dot,
Array<Real> & u_dot_dot,
Array<bool> & blocked_dofs,
Array<Real> & delta) const;
/* ------------------------------------------------------------------------ */
/* Accessors */
/* ------------------------------------------------------------------------ */
public:
AKANTU_GET_MACRO(Beta, beta, Real);
AKANTU_GET_MACRO(Alpha, alpha, Real);
/* ------------------------------------------------------------------------ */
/* Class Members */
/* ------------------------------------------------------------------------ */
protected:
/// the @f$\beta@f$ parameter
const Real beta;
/// the @f$\alpha@f$ parameter
const Real alpha;
const Real k;
const Real h;
};
/* -------------------------------------------------------------------------- */
/* inline functions */
/* -------------------------------------------------------------------------- */
#if defined (AKANTU_INCLUDE_INLINE_IMPL)
# include "newmark-beta_inline_impl.cc"
#endif
/**
* 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}
*
*/
class CentralDifference : public NewmarkBeta {
public:
CentralDifference() : NewmarkBeta(0., .5) {};
};
//#include "integration_scheme/central_difference.hh"
/// undamped trapezoidal rule (implicit)
class TrapezoidalRule2 : public NewmarkBeta {
public:
TrapezoidalRule2() : NewmarkBeta(.5, .5) { };
};
__END_AKANTU__
#endif /* __AKANTU_NEWMARK_BETA_HH__ */
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