TimeAdvanceNewmark - Class to deal the $theta$ -method and TimeAdvanceNewmark scheme. More...

#include <TimeAdvanceNewmark.hpp>

Inheritance diagram for TimeAdvanceNewmark< feVectorType >:

Collaboration diagram for TimeAdvanceNewmark< feVectorType >:

Private Attributes
Real	M_theta
	Coefficient of TimeAdvanceNewmark: . More...

Real	M_gamma
	Coefficient of TimeAdvanceNewmark: $\gamma$ . More...

Public Types
typedef TimeAdvance< feVectorType >	super

typedef super::feVector_Type	feVector_Type

typedef super::feVectorContainer_Type	feVectorContainer_Type

typedef super::feVectorContainerPtr_Type	feVectorContainerPtr_Type

typedef feVectorContainerPtr_Type::iterator	feVectorContainerPtrIterate_Type

typedef super::feVectorSharedPtrContainer_Type	feVectorSharedPtrContainer_Type

Constructor & Destructor
	TimeAdvanceNewmark ()
	Empty Constructor. More...

virtual	~TimeAdvanceNewmark ()
	Destructor. More...

Methods
void	shiftRight (const feVector_Type &solution)
	Update the state vector. More...

void	RHSFirstDerivative (const Real &timeStep, feVectorType &rhsContribution) const
	Update the right hand side of the time derivative formula. More...

void	updateRHSSecondDerivative (const Real &timeStep=1)
	Update the right hand side of the time derivative formula. More...

void	showMe (std::ostream &output=std::cout) const
	Show the properties of temporal scheme. More...

Set Methods
void	setup (const UInt &, const UInt &)
	Initialize the parameters of time advance scheme. More...

void	setup (const std::vector< Real > &coefficients, const UInt &orderDerivative)
	Initialize the parameters of time advance scheme. More...

void	setInitialCondition (const feVector_Type &x0)
	Initialize the StateVector. More...

void	setInitialCondition (const feVector_Type &x0, const feVector_Type &v0)
	initialize the state vector More...

void	setInitialCondition (const feVector_Type &x0, const feVector_Type &v0, const feVector_Type &w0)
	initialize the state vector More...

void	setInitialCondition (const feVectorSharedPtrContainer_Type &x0)
	Initialize the state vector. More...

Get Methods
Real	coefficientExtrapolation (const UInt &i) const
	Return the -th coefficient of the unk's extrapolation. More...

Real	coefficientExtrapolationFirstDerivative (const UInt &i) const
	Return the -th coefficient of the velocity's extrapolation. More...

void	extrapolation (feVector_Type &extrapolation) const
	Compute the polynomial extrapolation of solution. More...

void	extrapolationFirstDerivative (feVector_Type &extrapolation) const
	Compute the polynomial extrapolation of velocity. More...

feVector_Type	firstDerivative () const
	Return the current velocity. More...

feVector_Type	secondDerivative () const
	Return the current acceleration. More...

Additional Inherited Members
Public Types inherited from TimeAdvance< feVectorType >
typedef feVectorType	feVector_Type

typedef ScalarVector	container_Type

typedef std::vector< feVector_Type >	feVectorContainer_Type

typedef std::vector< feVector_Type * >	feVectorContainerPtr_Type

typedef feVectorContainerPtr_Type::iterator	feVectorContainerPtrIterate_Type

typedef std::vector< std::shared_ptr< feVector_Type > >	feVectorSharedPtrContainer_Type

Public Member Functions inherited from TimeAdvance< feVectorType >
	TimeAdvance ()
	Empty Constructor. More...

virtual	~TimeAdvance ()
	Destructor. More...

Detailed Description

template<typename feVectorType = VectorEpetra>
class LifeV::TimeAdvanceNewmark< feVectorType >

TimeAdvanceNewmark - Class to deal the $theta$ -method and TimeAdvanceNewmark scheme.

Author: Matteo Pozzoli matte.nosp@m.o1.p.nosp@m.ozzol.nosp@m.i@ma.nosp@m.il.po.nosp@m.limi.nosp@m..it

This class can be used to approximate problems of the first order and the second order in time. In the first case the temporal scheme is a theta-method, while in the second case is a TimeAdvanceNewmark scheme.

This class defines the state vector $X^{n+1}$ , a suitable extrapolation of vector $X^*$ and opportune coefficients used to determinate $X^{n+1}$ and $X^*$ .

First order problem:

$M u' + A(u) = f$

the state vector is
$X^{n+1} = (U^{n+1},V^{n+1}, U^{n}, V^{n})$

. where is an approximation of and of $\dot{u}$ .

We consider a parameter $\theta$ , and we apply the following theta method:

$U^{n+1} = U^{n} + \Delta t \theta V^{n+1} + (1 − \theta) V^n,$

so the approximation of velocity at timestep is:

$V^{ n+1} = 1/ (\theta * \Delta t) (U^{n+1}-U^n)+( 1 - 1 / \theta) V^n;$

We can linearize non-linear term as $A(U^*) U^{n+1}$ , where the becomes:

$U^∗ = U^n + \Delta t V^n,$

The coefficients $\alpha$ , depend on , and we can use a explicit -method.

Second order Method:

$M \ddot{u} + D(u, \dot{u})+ A(u) = f$

the state vector is \form#202
where U is an approximation of \form#193 and \form#121  of \form#122 and \form#137 of \form#203 .

We introduce the parameters ( \form#189, \form#188)  and we apply the following TimeAdvanceNewmark method:

$U^{n+1} = U^{n} + \Delta t \theta V^{n+1} + (1 − \theta) V^n,$

so the approximation of velocity at time step \form#123 is

$V^{ n+1} = \beta / (\theta * \Delta t) (U^{n+1}-U^n)+( 1 - \gamma/ \theta) V^n+ (1- \gamma /(2\theta)) \Delta t W^n;$

and we determine \form#164 by

$W^{n+1} =1/(\theta \Delta t^2) (U^{n+1}-U^{n}) - 1 / \Delta t V^n + (1-1/(2\theta)) W^n$

.

We can  linearize non-linear term \form#196  as \form#197 and \form#206 as \form#207,
where \form#144 is given by

$U^∗ = U^n + \Delta t V^n$

 and \form#145 is

$V^* =V^n+ \Delta t W^n$

 The coefficients \form#210, \form#199, \form#211, \form#212 depend on \form#189 and \form#188.
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Definition at line 121 of file TimeAdvanceNewmark.hpp.

Member Typedef Documentation

◆ super

typedef TimeAdvance< feVectorType > super

Definition at line 130 of file TimeAdvanceNewmark.hpp.

◆ feVector_Type

typedef super::feVector_Type feVector_Type

Definition at line 132 of file TimeAdvanceNewmark.hpp.

◆ feVectorContainer_Type

typedef super::feVectorContainer_Type feVectorContainer_Type

Definition at line 134 of file TimeAdvanceNewmark.hpp.

◆ feVectorContainerPtr_Type

typedef super::feVectorContainerPtr_Type feVectorContainerPtr_Type

Definition at line 137 of file TimeAdvanceNewmark.hpp.

◆ feVectorContainerPtrIterate_Type

typedef feVectorContainerPtr_Type::iterator feVectorContainerPtrIterate_Type

Definition at line 140 of file TimeAdvanceNewmark.hpp.

◆ feVectorSharedPtrContainer_Type

typedef super::feVectorSharedPtrContainer_Type feVectorSharedPtrContainer_Type

Definition at line 143 of file TimeAdvanceNewmark.hpp.

Constructor & Destructor Documentation

◆ TimeAdvanceNewmark()

TimeAdvanceNewmark ( )

Empty Constructor.

Definition at line 303 of file TimeAdvanceNewmark.hpp.

◆ ~TimeAdvanceNewmark()

virtual ~TimeAdvanceNewmark ( )

inlinevirtual

Destructor.

Definition at line 154 of file TimeAdvanceNewmark.hpp.

Member Function Documentation

◆ shiftRight()

void shiftRight ( const feVector_Type & solution )

virtual

Update the state vector.

Update the vectors of the previous time steps by shifting on the right the old values.

Parameters

solution current (new) value of the state vector

Implements TimeAdvance< feVectorType >.

Definition at line 311 of file TimeAdvanceNewmark.hpp.

◆ RHSFirstDerivative()

void RHSFirstDerivative	(	const Real &	timeStep,
		feVectorType &	rhsContribution
	)		const

Update the right hand side $ f_V $ of the time derivative formula.

Return the right hand side $ f_V $ of the time derivative formula

Parameters

timeStep defined the time step need to compute the

Returns: rhsV

Definition at line 355 of file TimeAdvanceNewmark.hpp.

◆ updateRHSSecondDerivative()

void updateRHSSecondDerivative ( const Real & timeStep = 1 )

Update the right hand side $ f_W $ of the time derivative formula.

Set and Return the right hand side $ f_W $ of the time derivative formula

Parameters

timeStep defined the time step need to compute the

Definition at line 370 of file TimeAdvanceNewmark.hpp.

◆ showMe()

void showMe ( std::ostream & output = std::cout ) const

Show the properties of temporal scheme.

Definition at line 386 of file TimeAdvanceNewmark.hpp.

◆ setup() [1/2]

void setup	(	const UInt &	,
		const UInt &
	)

inline

Initialize the parameters of time advance scheme.

Parameters

order	define the order of BDF;
orderDerivatve	define the order of derivative;

Definition at line 195 of file TimeAdvanceNewmark.hpp.

◆ setup() [2/2]

void setup	(	const std::vector< Real > &	coefficients,
		const UInt &	orderDerivative
	)

Initialize the parameters of time advance scheme.

Parameters

coefficients	define the TimeAdvanceNewmark's coefficients (, );
orderDerivative	define the order of derivative;

Definition at line 418 of file TimeAdvanceNewmark.hpp.

◆ setInitialCondition() [1/4]

void setInitialCondition ( const feVector_Type & x0 )

Initialize the StateVector.

Initialize all the entries of the unknown vector to be derived with the vector x0 (duplicated). this class is virtual because used in bdf;

Parameters

x0	is the initial solution;

Definition at line 509 of file TimeAdvanceNewmark.hpp.

◆ setInitialCondition() [2/4]

void setInitialCondition	(	const feVector_Type &	x0,
		const feVector_Type &	v0
	)

initialize the state vector

Initialize all the entries of the unknown vector to be derived with the vector x0, v0 (duplicated). this class is virtual because used in $\theta$ -method scheme;

Parameters

x0	is the initial unk;
v0	is the initial velocity

initialize zero

Definition at line 539 of file TimeAdvanceNewmark.hpp.

◆ setInitialCondition() [3/4]

void setInitialCondition	(	const feVector_Type &	x0,
		const feVector_Type &	v0,
		const feVector_Type &	w0
	)

initialize the state vector

Initialize all the entries of the unknown vector to be derived with the vector x0, v0,w0 (duplicated). this class is virtual because used in Newamrk scheme;

Parameters

x0	is the initial solution;
v0	is the initial velocity
w0	is the initial acceleration

initialize zero

Definition at line 566 of file TimeAdvanceNewmark.hpp.

◆ setInitialCondition() [4/4]

void setInitialCondition ( const feVectorSharedPtrContainer_Type & x0 )

Initialize the state vector.

Initialize all the entries of the unknown vector to be derived with a set of vectors x0 note: this is taken as a copy (not a reference), since x0 is resized inside the method.

initialize zero

Definition at line 594 of file TimeAdvanceNewmark.hpp.

◆ coefficientExtrapolation()

Real coefficientExtrapolation ( const UInt & i ) const

Return the $i$ -th coefficient of the unk's extrapolation.

Parameters

i	index of extrapolation coefficient

Returns: beta

Definition at line 634 of file TimeAdvanceNewmark.hpp.

◆ coefficientExtrapolationFirstDerivative()

Real coefficientExtrapolationFirstDerivative ( const UInt & i ) const

Return the $i$ -th coefficient of the velocity's extrapolation.

Parameters

Definition at line 654 of file TimeAdvanceNewmark.hpp.

◆ extrapolation()

void extrapolation ( feVector_Type & extrapolation ) const

Compute the polynomial extrapolation of solution.

Compute the polynomial extrapolation approximation of order $n-1$ of $u^{n+1}$ defined by the n stored state vectors

Definition at line 672 of file TimeAdvanceNewmark.hpp.

◆ extrapolationFirstDerivative()

void extrapolationFirstDerivative ( feVector_Type & extrapolation ) const

Compute the polynomial extrapolation of velocity.

Compute the polynomial extrapolation approximation of order $n-1$ of $u^{n+1}$ defined by the n stored state vectors

Definition at line 685 of file TimeAdvanceNewmark.hpp.

◆ firstDerivative()

feVector_Type firstDerivative ( ) const

inline

Return the current velocity.

Definition at line 276 of file TimeAdvanceNewmark.hpp.

◆ secondDerivative()

feVector_Type secondDerivative ( ) const

inline

Return the current acceleration.

Definition at line 282 of file TimeAdvanceNewmark.hpp.

Field Documentation

◆ M_theta

Real M_theta

private

Coefficient of TimeAdvanceNewmark: $theta$ .

Definition at line 292 of file TimeAdvanceNewmark.hpp.

◆ M_gamma

Real M_gamma

private

Coefficient of TimeAdvanceNewmark: $\gamma$ .

Definition at line 295 of file TimeAdvanceNewmark.hpp.

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

TimeAdvanceNewmark.hpp

Private Attributes

Public Types

Constructor & Destructor

Methods

Set Methods

Get Methods

Additional Inherited Members

Detailed Description

template<typename feVectorType = VectorEpetra> class LifeV::TimeAdvanceNewmark< feVectorType >

Member Typedef Documentation

◆ super

◆ feVector_Type

◆ feVectorContainer_Type

◆ feVectorContainerPtr_Type

◆ feVectorContainerPtrIterate_Type

◆ feVectorSharedPtrContainer_Type

Constructor & Destructor Documentation

◆ TimeAdvanceNewmark()

◆ ~TimeAdvanceNewmark()

Member Function Documentation

◆ shiftRight()

◆ RHSFirstDerivative()

◆ updateRHSSecondDerivative()

◆ showMe()

◆ setup() [1/2]

◆ setup() [2/2]

◆ setInitialCondition() [1/4]

◆ setInitialCondition() [2/4]

◆ setInitialCondition() [3/4]

◆ setInitialCondition() [4/4]

◆ coefficientExtrapolation()

◆ coefficientExtrapolationFirstDerivative()

◆ extrapolation()

◆ extrapolationFirstDerivative()

◆ firstDerivative()

◆ secondDerivative()

Field Documentation

◆ M_theta

◆ M_gamma

template<typename feVectorType = VectorEpetra>
class LifeV::TimeAdvanceNewmark< feVectorType >