asl::FDPoroElasticity Class Reference

Numerical method which computes homogenious isotropic poro-elasticity equation. More...

#include <aslFDPoroElasticity.h>

Inheritance diagram for asl::FDPoroElasticity:

Public Member Functions
	FDPoroElasticity ()
	FDPoroElasticity (Data d, Data pl, Param bM, Param sM, Param k, const VectorTemplate *vT)
	~FDPoroElasticity ()
virtual void	init ()
	Builds the necesery internal data and kernels. More...
virtual void	execute ()
	Executes the numerical procedure. More...
Data	getPressureData () const
Data	getLiquidPressureData () const
void	setNSubsteps (unsigned int n)
	defaul value 10 More...
	FDPoroElasticity ()
	FDPoroElasticity (Data d, Data pl, Param bM, Param sM, Param k, const VectorTemplate *vT)
	~FDPoroElasticity ()
virtual void	init ()
	Builds the necesery internal data and kernels. More...
virtual void	execute ()
	Executes the numerical procedure. More...
Data	getPressureData () const
Data	getLiquidPressureData () const
void	setNSubsteps (unsigned int n)
	defaul value 10 More...
Public Member Functions inherited from asl::ElasticityCommonA
	ElasticityCommonA ()
	ElasticityCommonA (Data d, Param bM, Param sM, const VectorTemplate *vT)
	~ElasticityCommonA ()
void	setVectorTemplate (const VectorTemplate *vT)
VectorTemplate	getVectorTemplate ()
void	setForce (Param f)
Data	getDisplacementData () const
Data	getDisplacementInternalData () const
const Param	getBulkModulus () const
const Param	getShearModulus () const
	ElasticityCommonA ()
	ElasticityCommonA (Data d, Param bM, Param sM, const VectorTemplate *vT)
	~ElasticityCommonA ()
void	setVectorTemplate (const VectorTemplate *vT)
VectorTemplate	getVectorTemplate ()
void	setForce (Param f)
Data	getDisplacementData () const
Data	getDisplacementInternalData () const
const Param	getBulkModulus () const
const Param	getShearModulus () const
Public Member Functions inherited from asl::NumMethod
virtual	~NumMethod ()
virtual	~NumMethod ()

Additional Inherited Members
Public Types inherited from asl::ElasticityCommonA
typedef SPDataWithGhostNodesACLData	Data
typedef acl::VectorOfElements	Param
typedef SPDataWithGhostNodesACLData	Data
typedef acl::VectorOfElements	Param
Public Attributes inherited from asl::ElasticityCommonA
const VectorTemplate *	vectorTemplate
Protected Attributes inherited from asl::ElasticityCommonA
std::unique_ptr< acl::Kernel >	kernel
Data	displacementData
Data	displacementInternalData
Param	bulkModulus
Param	shearModulus
Param	force

Detailed Description

Numerical method which computes homogenious isotropic poro-elasticity equation.

The classic poroelastic equations were originally developed by Biot (1941) to represent biphasic soil consolidation. The equations can be written in the following form:

\[ (K+\mu/3)\nabla_j \nabla_k u_k+ \mu \Delta u_j - a \nabla_j p = - (\rho_s-\rho_f)g_j \]

\[ a\nabla_i\dot u_i+ \frac{1}{S} \dot p = \nabla_i k \nabla_i p, \]

where

• \(\vec u\) is the displacement vector,
• \( p \) the interstitial pressure,
• \( a \) the ratio of fluid volume extracted to volume change of the tissue under compression,
• \(\rho_s\) the solid fraction density,
• \(\rho_f\) the fluid density,
• \(\vec g \) the gravitational vector,
• \( 1/S \) the amount of fluid which can be forced into the tissue under constant volume,
• \( k \) the hydraulic conductivity,
• \(K\) is the bulk modulus,
• \(\mu\) is the shear modulus,.

In order to solve the first equation we will introduce a fictisionus time \( \tau \):

\[ \partial_\tau u = (K+\mu/3)\nabla_j \nabla_k u_k+ \mu \Delta u_j - a \nabla_j p + (\rho_s-\rho_f)g_j \]

\[ a\partial_t\nabla_i u_i+ \frac{1}{S} \partial_t p = \nabla_i k \nabla_i p, \]

This implementation is aided to improove stability for incompressible materials. Let us reformulate the first eqation by the following way:

\[ \partial_\tau u = \mu \Delta u_j - \nabla_j \tilde p - a\nabla_j p + (\rho_s-\rho_f)g_j, \]

\[ \tilde p = -(K+\mu/3) \nabla_k u_k. \]

then the second eqation will turn to:

\[ \partial_t p = \frac{aS}{K+\mu/3}\partial_t\tilde p + S\nabla_i k \nabla_i p, \]

The equation for \(\tilde p\) leads to an instability for low values of the Poisson ration (for nearly incompressible materials). Therefore the last equation we would like to replace by a relaxation one:

\[ \partial_\tau \tilde p = - w\left( (K+\mu/3)\nabla_k u_k + \tilde p\right), \]

where \( w \) is a relaxation parameter.

Let's return to single time variable. Two time steps are related as follows \( \delta_t = N \delta_\tau \). Than \( \partial_\tau = N\partial_t \). The resulting set of equations is:

\[ \partial_\tau u = \mu \Delta u_j - \nabla_j \tilde p - a\nabla_j p + (\rho_s-\rho_f)g_j, \]

\[ \partial_\tau \tilde p = - w\left( (K+\mu/3)\nabla_k u_k + \tilde p\right), \]

\[ \partial_\tau p = \frac{aS}{K+\mu/3}\partial_\tau\tilde p + \frac{Sk}{N}\Delta p, \]

Definition at line 82 of file aslFDPoroElasticity.h.

Constructor & Destructor Documentation

asl::FDPoroElasticity::FDPoroElasticity

(

)

asl::FDPoroElasticity::FDPoroElasticity	(	Data	d,
		Data	pl,
		Param	bM,
		Param	sM,
		Param	k,
		const VectorTemplate *	vT
	)

Parameters

d	is a displacement field
pl	is a pressure of liquid field
bM	is the bulk modulus
sM	is the shear modulus
k	is hydraulic conductivity
vT	is a vector template

asl::FDPoroElasticity::~FDPoroElasticity

(

)

asl::FDPoroElasticity::FDPoroElasticity

(

)

asl::FDPoroElasticity::FDPoroElasticity	(	Data	d,
		Data	pl,
		Param	bM,
		Param	sM,
		Param	k,
		const VectorTemplate *	vT
	)

Parameters

d	is a displacement field
pl	is a pressure of liquid field
bM	is the bulk modulus
sM	is the shear modulus
k	is hydraulic conductivity
vT	is a vector template

asl::FDPoroElasticity::~FDPoroElasticity

(

)

Member Function Documentation

virtual void asl::FDPoroElasticity::execute ( )

virtual

Executes the numerical procedure.

Implements asl::ElasticityCommonA.

virtual void asl::FDPoroElasticity::execute ( )

virtual

Executes the numerical procedure.

Implements asl::ElasticityCommonA.

ElasticityCommonA::Data asl::FDPoroElasticity::getLiquidPressureData ( ) const

inline

Definition at line 159 of file aslFDPoroElasticity.h.

Data asl::FDPoroElasticity::getLiquidPressureData ( ) const

inline

Data asl::FDPoroElasticity::getPressureData ( ) const

inline

ElasticityCommonA::Data asl::FDPoroElasticity::getPressureData ( ) const

inline

Definition at line 154 of file aslFDPoroElasticity.h.

virtual void asl::FDPoroElasticity::init ( )

virtual

Builds the necesery internal data and kernels.

Implements asl::ElasticityCommonA.

virtual void asl::FDPoroElasticity::init ( )

virtual

Builds the necesery internal data and kernels.

Implements asl::ElasticityCommonA.

void asl::FDPoroElasticity::setNSubsteps

(

unsigned int

n

)

defaul value 10

void asl::FDPoroElasticity::setNSubsteps

(

unsigned int

n

)

defaul value 10

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

• /build/asl-cIsmox/asl-0.1.6/debian/tmp/usr/include/ASL/num/aslFDPoroElasticity.h