Implicit Euler time integrator of a second order ODE (equivently a system of coupled first order ODEs). More...

#include <ImplicitEuler.hpp>

Inheritance diagram for polyfem::time_integrator::ImplicitEuler:

Collaboration diagram for polyfem::time_integrator::ImplicitEuler:

Public Member Functions
	ImplicitEuler ()

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.

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

virtual	~ImplicitTimeIntegrator ()=default

virtual void	set_parameters (const json &params)
	Set the time integrator parameters from a json object.

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.

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.

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.

Detailed Description

Implicit Euler time integrator of a second order ODE (equivently a system of coupled first order ODEs).

\[ x^{t+1} = x^t + \Delta t v^{t+1}\newline v^{t+1} = v^t + \Delta t a^{t+1} \]

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

Definition at line 13 of file ImplicitEuler.hpp.

Constructor & Destructor Documentation

◆ ImplicitEuler()

polyfem::time_integrator::ImplicitEuler::ImplicitEuler ( )

inline

Definition at line 16 of file ImplicitEuler.hpp.

Member Function Documentation

◆ acceleration_scaling()

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

overridevirtual

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

\[ \Delta t^2 \]

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 28 of file ImplicitEuler.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::dt().

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◆ compute_acceleration()

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

overridevirtual

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

\[ a = \frac{v - v^t}{\Delta t} \]

Parameters

v	current velocity

Returns: value for \(a\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 23 of file ImplicitEuler.cpp.

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

Referenced by update_quantities().

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◆ compute_velocity()

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

overridevirtual

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

\[ v = \frac{x - x^t}{\Delta t} \]

Parameters

x	current solution vector

Returns: value for \(v\)

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 18 of file ImplicitEuler.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::dt(), x, and polyfem::time_integrator::ImplicitTimeIntegrator::x_prev().

Referenced by update_quantities().

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◆ dv_dx()

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

overridevirtual

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

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

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

\[ \frac{\partial v}{\partial x^{t-1}} = 0 \]

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 33 of file ImplicitEuler.cpp.

References polyfem::time_integrator::ImplicitTimeIntegrator::dt().

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◆ update_quantities()

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

overridevirtual

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

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

Parameters

x	new solution vector

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 5 of file ImplicitEuler.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::ImplicitEuler::x_tilde ( ) const

overridevirtual

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

\[ \tilde{x} = x^t + \Delta t v^t \]

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

Implements polyfem::time_integrator::ImplicitTimeIntegrator.

Definition at line 13 of file ImplicitEuler.cpp.

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

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

src/polyfem/time_integrator/ImplicitEuler.hpp
src/polyfem/time_integrator/ImplicitEuler.cpp

Public Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ ImplicitEuler()

Member Function Documentation

◆ acceleration_scaling()

◆ compute_acceleration()

◆ compute_velocity()

◆ dv_dx()

◆ update_quantities()

◆ x_tilde()