Backward Differential Formulas. More...

#include <BDF.hpp>

Inheritance diagram for polyfem::time_integrator::BDF:

Collaboration diagram for polyfem::time_integrator::BDF:

Public Member Functions
	BDF (const int order=1, const DynamicOrder dynamic_order=DynamicOrder::Second)

void	set_parameters (const json &params) override
	Set the number of steps 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	beta_dt () const
	Compute \(\beta\Delta t\).

Eigen::VectorXd	weighted_sum_x_prevs () const
	Compute the weighted sum of the previous solutions.

Eigen::VectorXd	weighted_sum_v_prevs () const
	Compute the weighted sum of the previous velocities.

Public Member Functions inherited from polyfem::time_integrator::ImplicitTimeIntegrator
	ImplicitTimeIntegrator (const DynamicOrder dynamic_order)

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.

Static Public Member Functions
static const std::vector< double > &	alphas (const int i)
	Retrieve the alphas used for BDF with `i` steps.

static double	betas (const int i)
	Retrieve the value of beta used for BDF with `i` steps.

Static Public Member Functions inherited from polyfem::time_integrator::ImplicitTimeIntegrator
static std::shared_ptr< ImplicitTimeIntegrator >	construct_time_integrator (const json &params, DynamicOrder dynamic_order=DynamicOrder::Second)
	Factory method for constructing an implicit time integrator.

static std::shared_ptr< BDF >	construct_bdf_integrator (const json &params, DynamicOrder dynamic_order=DynamicOrder::Second)
	Construct a BDF integrator for algorithms using BDF-specific operations.

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

Protected Member Functions
int	max_steps () const override
	Get the maximum number of steps to use for integration.

Protected Member Functions inherited from polyfem::time_integrator::ImplicitTimeIntegrator
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.

Protected Attributes
int	max_steps_ = 1
	The maximum number of steps to use for integration.

Protected Attributes inherited from polyfem::time_integrator::ImplicitTimeIntegrator
const DynamicOrder	dynamic_order_

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
Public Types inherited from polyfem::time_integrator::ImplicitTimeIntegrator
enum class	DynamicOrder { First , Second }

Detailed Description

Backward Differential Formulas.

\[ x^{t+1} = \left(\sum_{i=0}^{n-1} \alpha_{i} x^{t-i}\right)+ \Delta t \beta v^{t+1}\newline v^{t+1} = \left(\sum_{i=0}^{n-1} \alpha_{i} v^{t-i}\right)+ \Delta t \beta a^{t+1} \]

See also: https://en.wikipedia.org/wiki/Backward_differentiation_formula

Definition at line 13 of file BDF.hpp.

Constructor & Destructor Documentation

◆ BDF()

polyfem::time_integrator::BDF::BDF	(	const int	order = `1`,
		const DynamicOrder	dynamic_order = `DynamicOrder::Second`
	)

Definition at line 7 of file BDF.cpp.

References polyfem::log_and_throw_error(), and max_steps_.

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Member Function Documentation

◆ acceleration_scaling()

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

overridevirtual

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

\[ \beta^2 \Delta t^2 \]

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 117 of file BDF.cpp.

References beta_dt(), betas(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), polyfem::time_integrator::ImplicitTimeIntegrator::dynamic_order_, polyfem::time_integrator::ImplicitTimeIntegrator::First, and polyfem::time_integrator::ImplicitTimeIntegrator::steps().

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◆ alphas()

const std::vector< double > & polyfem::time_integrator::BDF::alphas ( const int i )

static

Retrieve the alphas used for BDF with i steps.

Parameters

i	number of steps

See also: https://en.wikipedia.org/wiki/Backward_differentiation_formula#General_formula

Definition at line 22 of file BDF.cpp.

Referenced by dv_dx(), weighted_sum_v_prevs(), and weighted_sum_x_prevs().

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◆ beta_dt()

double polyfem::time_integrator::BDF::beta_dt ( ) const

Compute \(\beta\Delta t\).

Definition at line 135 of file BDF.cpp.

References betas(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), and polyfem::time_integrator::ImplicitTimeIntegrator::steps().

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

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◆ betas()

double polyfem::time_integrator::BDF::betas ( const int i )

static

Retrieve the value of beta used for BDF with i steps.

Parameters

i	number of steps

See also: https://en.wikipedia.org/wiki/Backward_differentiation_formula#General_formula

Definition at line 36 of file BDF.cpp.

Referenced by acceleration_scaling(), beta_dt(), and x_tilde().

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◆ compute_acceleration()

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

overridevirtual

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

\[ a = \frac{v - \sum_{i=0}^{n-1} \alpha_i v^{t-i}}{\beta \Delta t} \]

Parameters

v	current velocity

Returns: value for \(a\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 109 of file BDF.cpp.

References beta_dt(), polyfem::time_integrator::ImplicitTimeIntegrator::dynamic_order_, polyfem::time_integrator::ImplicitTimeIntegrator::First, and weighted_sum_v_prevs().

Referenced by update_quantities().

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◆ compute_velocity()

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

overridevirtual

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

\[ v = \frac{x - \sum_{i=0}^{n-1} \alpha_i x^{t-i}}{\beta \Delta t} \]

Parameters

x	current solution vector

Returns: value for \(v\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 104 of file BDF.cpp.

References beta_dt(), weighted_sum_x_prevs(), and x.

Referenced by update_quantities().

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◆ dv_dx()

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

overridevirtual

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

\[ \frac{\partial v}{\partial x} = \frac{1}{\beta \Delta t} \]

\[ \frac{\partial v}{\partial x^{t-i}} = \frac{-\alpha_i}{\beta \Delta t} \]

\[ \frac{\partial v}{\partial x^{t-n}} = 0 \]

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 126 of file BDF.cpp.

References alphas(), beta_dt(), and polyfem::time_integrator::ImplicitTimeIntegrator::steps().

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◆ max_steps()

int polyfem::time_integrator::BDF::max_steps ( ) const

inlineoverrideprotectedvirtual

Get the maximum number of steps to use for integration.

Reimplemented from polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 95 of file BDF.hpp.

References max_steps_.

Referenced by update_quantities().

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◆ set_parameters()

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

overridevirtual

Set the number of steps parameters from a json object.

Parameters

params json containing {"steps": 1}

Reimplemented from polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 15 of file BDF.cpp.

References polyfem::log_and_throw_error(), and max_steps_.

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◆ update_quantities()

void polyfem::time_integrator::BDF::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 76 of file BDF.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::a_prevs_, compute_acceleration(), compute_velocity(), max_steps(), polyfem::time_integrator::ImplicitTimeIntegrator::steps(), polyfem::time_integrator::ImplicitTimeIntegrator::v_prevs_, x, and polyfem::time_integrator::ImplicitTimeIntegrator::x_prevs_.

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◆ weighted_sum_v_prevs()

Eigen::VectorXd polyfem::time_integrator::BDF::weighted_sum_v_prevs ( ) const

Compute the weighted sum of the previous velocities.

\[ \sum_{i=0}^{n-1} \alpha_i v^{t-i} \]

Definition at line 63 of file BDF.cpp.

References alphas(), polyfem::time_integrator::ImplicitTimeIntegrator::steps(), polyfem::time_integrator::ImplicitTimeIntegrator::v_prev(), and polyfem::time_integrator::ImplicitTimeIntegrator::v_prevs_.

Referenced by polyfem::DiffCache::cache_transient(), compute_acceleration(), and x_tilde().

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◆ weighted_sum_x_prevs()

Eigen::VectorXd polyfem::time_integrator::BDF::weighted_sum_x_prevs ( ) const

Compute the weighted sum of the previous solutions.

\[ \sum_{i=0}^{n-1} \alpha_i x^{t-i} \]

Definition at line 50 of file BDF.cpp.

References alphas(), polyfem::time_integrator::ImplicitTimeIntegrator::steps(), polyfem::time_integrator::ImplicitTimeIntegrator::x_prev(), and polyfem::time_integrator::ImplicitTimeIntegrator::x_prevs_.

Referenced by compute_velocity(), and x_tilde().

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◆ x_tilde()

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

overridevirtual

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

\[ \tilde{x} = \left(\sum_{i=0}^{n-1} \alpha_i x^{t-i}\right) + \beta \Delta t \left(\sum_{i=0}^{n-1} \alpha_i v^{t-i}\right) \]

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

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 96 of file BDF.cpp.

References betas(), polyfem::time_integrator::ImplicitTimeIntegrator::dt(), polyfem::time_integrator::ImplicitTimeIntegrator::dynamic_order_, polyfem::time_integrator::ImplicitTimeIntegrator::First, polyfem::time_integrator::ImplicitTimeIntegrator::steps(), weighted_sum_v_prevs(), and weighted_sum_x_prevs().

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Member Data Documentation

◆ max_steps_

int polyfem::time_integrator::BDF::max_steps_ = 1

protected

The maximum number of steps to use for integration.

Definition at line 98 of file BDF.hpp.

Referenced by BDF(), max_steps(), and set_parameters().

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

src/polyfem/time_integrator/BDF.hpp
src/polyfem/time_integrator/BDF.cpp

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ BDF()

Member Function Documentation

◆ acceleration_scaling()

◆ alphas()

◆ beta_dt()

◆ betas()

◆ compute_acceleration()

◆ compute_velocity()

◆ dv_dx()

◆ max_steps()

◆ set_parameters()

◆ update_quantities()

◆ weighted_sum_v_prevs()

◆ weighted_sum_x_prevs()

◆ x_tilde()

Member Data Documentation

◆ max_steps_