Implicit Newmark-beta method. More...

#include <ImplicitNewmark.hpp>

Inheritance diagram for polyfem::time_integrator::ImplicitNewmark:

Collaboration diagram for polyfem::time_integrator::ImplicitNewmark:

Public Member Functions
	ImplicitNewmark ()

void	set_parameters (const json &params) override
	Set the `gamma` and `beta` parameters from a json object.

void	update_quantities (const Eigen::VectorXd &x) override
	Update the time integration quantities (i.e., \(x\), \(v\), and \(a\)).

Eigen::VectorXd	x_tilde () const override
	Compute the predicted solution to be used in the inertia term \((x-\tilde{x})^TM(x-\tilde{x})\).

Eigen::VectorXd	compute_velocity (const Eigen::VectorXd &x) const override
	Compute the current velocity given the current solution and using the stored previous solution(s).

Eigen::VectorXd	compute_acceleration (const Eigen::VectorXd &v) const override
	Compute the current acceleration given the current velocity and using the stored previous velocity(s).

double	acceleration_scaling () const override
	Compute the acceleration scaling used to scale forces when integrating a second order ODE.

double	dv_dx (const unsigned prev_ti=0) const override
	Compute the derivative of the velocity with respect to the solution.

double	da_dx (const unsigned prev_ti=0) const

double	beta () const
	\(\beta\) parameter for blending accelerations in the solution update.

double	gamma () const
	\(\gamma\) parameter for blending accelerations in the velocity update.

Public Member Functions inherited from polyfem::time_integrator::ImplicitTimeIntegrator
	ImplicitTimeIntegrator ()

virtual	~ImplicitTimeIntegrator ()=default

virtual void	init (const Eigen::MatrixXd &x_prevs, const Eigen::MatrixXd &v_prevs, const Eigen::MatrixXd &a_prevs, double dt)
	Initialize the time integrator with the previous values for \(x\), \(v\), and \(a\).

const double &	dt () const
	Access the time step size.

virtual void	save_state (const std::string &state_path) const
	Save the values of \(x\), \(v\), and \(a\).

const Eigen::VectorXd &	x_prev () const
	Get the most recent previous solution value.

const Eigen::VectorXd &	v_prev () const
	Get the most recent previous velocity value.

const Eigen::VectorXd &	a_prev () const
	Get the most recent previous acceleration value.

const std::deque< Eigen::VectorXd > &	x_prevs () const
	Get the (relevant) history of previous solution value.

const std::deque< Eigen::VectorXd > &	v_prevs () const
	Get the (relevant) history of previous velocity value.

const std::deque< Eigen::VectorXd > &	a_prevs () const
	Get the (relevant) history of previous acceleration value.

int	steps () const
	Get the current number of steps to use for integration.

Protected Attributes
double	beta_ = 0.25
	\(\beta\) parameter for blending accelerations in the solution update.

double	gamma_ = 0.5
	\(\gamma\) parameter for blending accelerations in the velocity update.

Protected Attributes inherited from polyfem::time_integrator::ImplicitTimeIntegrator
double	dt_ = 1
	Time step size.

std::deque< Eigen::VectorXd >	x_prevs_
	Store the necessary previous values of the solution for single or multi-step integration.

std::deque< Eigen::VectorXd >	v_prevs_
	Store the necessary previous values of the velocity for single or multi-step integration.

std::deque< Eigen::VectorXd >	a_prevs_
	Store the necessary previous values of the acceleration for single or multi-step integration.

Additional Inherited Members
Static Public Member Functions inherited from polyfem::time_integrator::ImplicitTimeIntegrator
static std::shared_ptr< ImplicitTimeIntegrator >	construct_time_integrator (const json &params)
	Factory method for constructing implicit time integrators from the name of the integrator.

static const std::vector< std::string > &	get_time_integrator_names ()
	Get a vector of the names of possible ImplicitTimeIntegrators.

Protected Member Functions inherited from polyfem::time_integrator::ImplicitTimeIntegrator
virtual int	max_steps () const
	Get the maximum number of steps to use for integration.

void	set_x_prev (const Eigen::VectorXd &x_prev)
	Convenience functions for setting the most recent previous solution.

void	set_v_prev (const Eigen::VectorXd &v_prev)
	Convenience functions for setting the most recent previous velocity.

void	set_a_prev (const Eigen::VectorXd &a_prev)
	Convenience functions for setting the most recent previous acceleration.

Detailed Description

Implicit Newmark-beta method.

\[ x^{t+1} = x^t + \Delta t v^t + \frac{\Delta t^2}{2}((1-2\beta)a^t + 2 \beta a^{t+1})\newline v^{t+1} = v^t + (1-\gamma)\Delta ta^t + \gamma \Delta ta^{t+1} \]

See also: https://en.wikipedia.org/wiki/Newmark-beta_method

Definition at line 13 of file ImplicitNewmark.hpp.

Constructor & Destructor Documentation

◆ ImplicitNewmark()

polyfem::time_integrator::ImplicitNewmark::ImplicitNewmark ( )

inline

Definition at line 16 of file ImplicitNewmark.hpp.

Member Function Documentation

◆ acceleration_scaling()

double polyfem::time_integrator::ImplicitNewmark::acceleration_scaling ( ) const

overridevirtual

Compute the acceleration scaling used to scale forces when integrating a second order ODE.

\[ \beta \Delta t^2 \]

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 35 of file ImplicitNewmark.cpp.

References beta(), and polyfem::time_integrator::ImplicitTimeIntegrator::dt().

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◆ beta()

double polyfem::time_integrator::ImplicitNewmark::beta ( ) const

inline

\(\beta\) parameter for blending accelerations in the solution update.

Definition at line 73 of file ImplicitNewmark.hpp.

References beta_.

Referenced by acceleration_scaling(), compute_velocity(), dv_dx(), and x_tilde().

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◆ compute_acceleration()

Eigen::VectorXd polyfem::time_integrator::ImplicitNewmark::compute_acceleration ( const Eigen::VectorXd & v ) const

overridevirtual

Compute the current acceleration given the current velocity and using the stored previous velocity(s).

\[ a = \frac{v - v^t - (1 - \gamma) \Delta t a^t)}{\gamma \Delta t} \]

Parameters

v	current velocity

Returns: value for \(a\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 30 of file ImplicitNewmark.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::a_prev(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), gamma(), and polyfem::time_integrator::ImplicitTimeIntegrator::v_prev().

Referenced by update_quantities().

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◆ compute_velocity()

Eigen::VectorXd polyfem::time_integrator::ImplicitNewmark::compute_velocity ( const Eigen::VectorXd & x ) const

overridevirtual

Compute the current velocity given the current solution and using the stored previous solution(s).

\[ a^{t+1} = \frac{x - (x^t + \Delta t v^t + \Delta t^2 (0.5 - \beta) a^t)}{\beta \Delta t^2}\newline v = v^t + \Delta t (1 - \gamma) a^t + \Delta t \gamma a^{t+1} = (\gamma / \beta) / \Delta t (x - x^t) + (1 - \gamma / \beta) v^t + (1 - \gamma / \beta / 2) \Delta t a^t \]

Parameters

x	current solution vector

Returns: value for \(v\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 24 of file ImplicitNewmark.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::a_prev(), beta(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), gamma(), polyfem::time_integrator::ImplicitTimeIntegrator::v_prev(), x, and polyfem::time_integrator::ImplicitTimeIntegrator::x_prev().

Referenced by update_quantities().

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◆ da_dx()

double polyfem::time_integrator::ImplicitNewmark::da_dx ( const unsigned prev_ti = 0 ) const

Definition at line 52 of file ImplicitNewmark.cpp.

References da_dx(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), dv_dx(), and gamma().

Referenced by da_dx(), and dv_dx().

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◆ dv_dx()

double polyfem::time_integrator::ImplicitNewmark::dv_dx ( const unsigned prev_ti = 0 ) const

overridevirtual

Compute the derivative of the velocity with respect to the solution.

\[ \frac{\partial v}{\partial x} = \frac{\gamma}{\beta\Delta t} \]

\[ \frac{\partial v}{\partial x^t} = ? \]

\[ \frac{\partial v}{\partial x^{t-1}} = ? \]

Parameters

prev_ti index of the previous solution to use (0 -> current; 1 -> previous; 2 -> second previous; etc.)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 40 of file ImplicitNewmark.cpp.

References beta(), da_dx(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), dv_dx(), and gamma().

Referenced by da_dx(), and dv_dx().

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◆ gamma()

double polyfem::time_integrator::ImplicitNewmark::gamma ( ) const

inline

\(\gamma\) parameter for blending accelerations in the velocity update.

Definition at line 75 of file ImplicitNewmark.hpp.

References gamma_.

Referenced by compute_acceleration(), compute_velocity(), da_dx(), and dv_dx().

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◆ set_parameters()

void polyfem::time_integrator::ImplicitNewmark::set_parameters ( const json & params )

overridevirtual

Set the gamma and beta parameters from a json object.

Parameters

params json containing {"gamma": 0.5, "beta": 0.25}

Reimplemented from polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 5 of file ImplicitNewmark.cpp.

References beta_, and gamma_.

◆ update_quantities()

void polyfem::time_integrator::ImplicitNewmark::update_quantities ( const Eigen::VectorXd & x )

overridevirtual

Update the time integration quantities (i.e., \(x\), \(v\), and \(a\)).

Parameters

x	new solution vector

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 11 of file ImplicitNewmark.cpp.

References compute_acceleration(), compute_velocity(), polyfem::time_integrator::ImplicitTimeIntegrator::set_a_prev(), polyfem::time_integrator::ImplicitTimeIntegrator::set_v_prev(), polyfem::time_integrator::ImplicitTimeIntegrator::set_x_prev(), and x.

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◆ x_tilde()

Eigen::VectorXd polyfem::time_integrator::ImplicitNewmark::x_tilde ( ) const

overridevirtual

Compute the predicted solution to be used in the inertia term \((x-\tilde{x})^TM(x-\tilde{x})\).

\[ \tilde{x} = x^t + \Delta t (v^t + (0.5 - \beta) \Delta t a^t) \]

Returns: value for \(\tilde{x}\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 19 of file ImplicitNewmark.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::a_prev(), beta(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), polyfem::time_integrator::ImplicitTimeIntegrator::v_prev(), and polyfem::time_integrator::ImplicitTimeIntegrator::x_prev().

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Member Data Documentation

◆ beta_

double polyfem::time_integrator::ImplicitNewmark::beta_ = 0.25

protected

\(\beta\) parameter for blending accelerations in the solution update.

Definition at line 79 of file ImplicitNewmark.hpp.

Referenced by beta(), and set_parameters().

◆ gamma_

double polyfem::time_integrator::ImplicitNewmark::gamma_ = 0.5

protected

\(\gamma\) parameter for blending accelerations in the velocity update.

Definition at line 81 of file ImplicitNewmark.hpp.

Referenced by gamma(), and set_parameters().

The documentation for this class was generated from the following files:

src/polyfem/time_integrator/ImplicitNewmark.hpp
src/polyfem/time_integrator/ImplicitNewmark.cpp

Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ ImplicitNewmark()

Member Function Documentation

◆ acceleration_scaling()

◆ beta()

◆ compute_acceleration()

◆ compute_velocity()

◆ da_dx()

◆ dv_dx()

◆ gamma()

◆ set_parameters()

◆ update_quantities()

◆ x_tilde()

Member Data Documentation

◆ beta_

◆ gamma_