RichardsSUPGstandard Class Reference

standard SUPG relationships valid for the Richards equation. More...

#include <RichardsSUPGstandard.h>

Inheritance diagram for RichardsSUPGstandard:

Public Member Functions
	RichardsSUPGstandard (const InputParameters &parameters)
RealVectorValue	velSUPG (RealTensorValue perm, RealVectorValue gradp, Real density, RealVectorValue gravity) const
	SUPG velocity = -perm*(gradp - density*gravity) This points in direction of information propagation. More...
RealTensorValue	dvelSUPG_dgradp (RealTensorValue perm) const
	derivative of SUPG velocity wrt gradient of porepressure More...
RealVectorValue	dvelSUPG_dp (RealTensorValue perm, Real density_prime, RealVectorValue gravity) const
	derivative of SUPG velocity wrt porepressure (keeping gradp fixed) More...
RealVectorValue	bb (RealVectorValue vel, int dimen, RealVectorValue xi_prime, RealVectorValue eta_prime, RealVectorValue zeta_prime) const
	|bb| ~ 2*velocity/element_length More...
RealVectorValue	dbb2_dgradp (RealVectorValue vel, RealTensorValue dvel_dgradp, RealVectorValue xi_prime, RealVectorValue eta_prime, RealVectorValue zeta_prime) const
	derivative of bb*bb wrt gradient of porepressure More...
Real	dbb2_dp (RealVectorValue vel, RealVectorValue dvel_dp, RealVectorValue xi_prime, RealVectorValue eta_prime, RealVectorValue zeta_prime) const
	derivative of bb*bb wrt porepressure More...
Real	tauSUPG (RealVectorValue vel, Real traceperm, RealVectorValue b) const
	The SUPG tau parameter. More...
RealVectorValue	dtauSUPG_dgradp (RealVectorValue vel, RealTensorValue dvel_dgradp, Real traceperm, RealVectorValue b, RealVectorValue db2_dgradp) const
	derivative of tau wrt gradient of porepressure More...
Real	dtauSUPG_dp (RealVectorValue vel, RealVectorValue dvel_dp, Real traceperm, RealVectorValue b, Real db2_dp) const
	derivative of tau wrt porepressure (keeping gradp fixed) More...
bool	SUPG_trivial () const
	returns false in this case since everything is trivial More...
void	initialize ()
void	execute ()
void	finalize ()

Protected Attributes
Real	_p_SUPG
	the SUPG pressure parameter This dictates how strong the SUPG is _p_SUPG large means only a little SUPG _p_SUPG small means close to fully-upwind More...

Private Member Functions
Real	cosh_relation (Real alpha) const
	cosh(alpha)/sinh(alpha) - 1/alpha, modified at extreme values of alpha to prevent overflows More...
Real	cosh_relation_prime (Real alpha) const
	derivative of cosh_relation wrt alpha More...

Detailed Description

standard SUPG relationships valid for the Richards equation.

here i use a formula for "tau" presented in Appendix A of TJR Hughes, M Mallet and A Mizukami `‘A new finite element formulation for computational fluid dynamics:: II. Behond SUPG’' Computer Methods in Applied Mechanics and Engineering 54 (1986) 341–355

Definition at line 28 of file RichardsSUPGstandard.h.

Constructor & Destructor Documentation

RichardsSUPGstandard()

RichardsSUPGstandard::RichardsSUPGstandard

(

const InputParameters &

parameters

)

Definition at line 37 of file RichardsSUPGstandard.C.

  : RichardsSUPG(parameters), _p_SUPG(getParam<Real>("p_SUPG"))
{
}

Member Function Documentation

bb()

RealVectorValue RichardsSUPGstandard::bb ( RealVectorValue  vel, int  dimen, RealVectorValue  xi_prime, RealVectorValue  eta_prime, RealVectorValue  zeta_prime  ) const

virtual

|bb| ~ 2*velocity/element_length

Parameters

vel	SUPG velocity
dimen	dimension of problem
xi_prime	spatial gradient of the isoparametric coordinate xi
eta_prime	spatial gradient of the isoparametric coordinate eta
zeta_prime	spatial gradient of the isoparametric coordinate zeta

Implements RichardsSUPG.

Definition at line 87 of file RichardsSUPGstandard.C.

{
  RealVectorValue b;
  b(0) = xi_prime * vel;
  if (dimen >= 2)
    b(1) = eta_prime * vel;
  if (dimen == 3)
    b(2) = zeta_prime * vel;
  return b;
}

cosh_relation()

Real RichardsSUPGstandard::cosh_relation ( Real  alpha ) const

private

cosh(alpha)/sinh(alpha) - 1/alpha, modified at extreme values of alpha to prevent overflows

Definition at line 66 of file RichardsSUPGstandard.C.

Referenced by dtauSUPG_dgradp(), dtauSUPG_dp(), and tauSUPG().

{
  if (alpha >= 5.0 || alpha <= -5.0)
    return ((alpha > 0.0) ? 1.0 : -1.0) - 1.0 / alpha; // prevents overflows
  else if (alpha == 0)
    return 0.0;
  return 1.0 / std::tanh(alpha) - 1.0 / alpha;
}

cosh_relation_prime()

Real RichardsSUPGstandard::cosh_relation_prime ( Real  alpha ) const

private

derivative of cosh_relation wrt alpha

Definition at line 76 of file RichardsSUPGstandard.C.

Referenced by dtauSUPG_dgradp(), and dtauSUPG_dp().

{
  if (alpha >= 5.0 || alpha <= -5.0)
    return 1.0 / (alpha * alpha); // prevents overflows
  else if (alpha == 0)
    return 1.0 / 3.0;
  return 1.0 - Utility::pow<2>(std::cosh(alpha) / std::sinh(alpha)) + 1.0 / (alpha * alpha);
}

dbb2_dgradp()

RealVectorValue RichardsSUPGstandard::dbb2_dgradp ( RealVectorValue  vel, RealTensorValue  dvel_dgradp, RealVectorValue  xi_prime, RealVectorValue  eta_prime, RealVectorValue  zeta_prime  ) const

virtual

derivative of bb*bb wrt gradient of porepressure

Parameters

vel	SUPG velocity
dvel_dgradp	derivative of velocity wrt gradient of porepressure
xi_prime	spatial gradient of the isoparametric coordinate xi
eta_prime	spatial gradient of the isoparametric coordinate eta
zeta_prime	spatial gradient of the isoparametric coordinate zeta

Implements RichardsSUPG.

Definition at line 104 of file RichardsSUPGstandard.C.

{
  // if dvel_dgradp is symmetric so transpose is probably a waste of time
  return 2.0 * ((xi_prime * vel) * (dvel_dgradp.transpose() * xi_prime) +
                (eta_prime * vel) * (dvel_dgradp.transpose() * eta_prime) +
                (zeta_prime * vel) * (dvel_dgradp.transpose() * zeta_prime));
}

dbb2_dp()

Real RichardsSUPGstandard::dbb2_dp ( RealVectorValue  vel, RealVectorValue  dvel_dp, RealVectorValue  xi_prime, RealVectorValue  eta_prime, RealVectorValue  zeta_prime  ) const

virtual

derivative of bb*bb wrt porepressure

Parameters

vel	SUPG velocity
dvel_dp	derivative of velocity wrt porepressure
xi_prime	spatial gradient of the isoparametric coordinate xi
eta_prime	spatial gradient of the isoparametric coordinate eta
zeta_prime	spatial gradient of the isoparametric coordinate zeta

Implements RichardsSUPG.

Definition at line 118 of file RichardsSUPGstandard.C.

{
  return 2.0 *
         ((xi_prime * vel) * (dvel_dp * xi_prime) + (eta_prime * vel) * (dvel_dp * eta_prime) +
          (zeta_prime * vel) * (dvel_dp * zeta_prime));
}

dtauSUPG_dgradp()

RealVectorValue RichardsSUPGstandard::dtauSUPG_dgradp ( RealVectorValue  vel, RealTensorValue  dvel_dgradp, Real  traceperm, RealVectorValue  b, RealVectorValue  db2_dgradp  ) const

virtual

derivative of tau wrt gradient of porepressure

Parameters

vel	SUPG velocity
dvel_dgradp	derivative of the SUPG velocity wrt gradient of porepressure
traceperm	trace of the permeability tensor for the material
b	the b parameter: |b| ~ 2*SUPGvelocity/element_length
db2_dgradp	derivative of b*b wrt gradient of porepressure

Implements RichardsSUPG.

Definition at line 150 of file RichardsSUPGstandard.C.

{
  Real norm_vel = vel.norm();
  if (norm_vel == 0)
    return RealVectorValue();

  RealVectorValue norm_vel_dgradp(dvel_dgradp * vel / norm_vel);

  Real norm_b = b.norm();
  if (norm_b == 0)
    return RealVectorValue();
  RealVectorValue norm_b_dgradp = db2_dgradp / 2.0 / norm_b;

  Real h = 2 * norm_vel / norm_b; // h is a measure of the element length in the "a" direction
  RealVectorValue h_dgradp(2 * norm_vel_dgradp / norm_b -
                           2.0 * norm_vel * norm_b_dgradp / norm_b / norm_b);

  Real alpha = 0.5 * norm_vel * h / traceperm / _p_SUPG; // this is the Peclet number
  RealVectorValue alpha_dgradp =
      0.5 * (norm_vel_dgradp * h + norm_vel * h_dgradp) / traceperm / _p_SUPG;

  Real xi_tilde = RichardsSUPGstandard::cosh_relation(alpha);
  Real xi_tilde_prime = RichardsSUPGstandard::cosh_relation_prime(alpha);
  RealVectorValue xi_tilde_dgradp = xi_tilde_prime * alpha_dgradp;

  RealVectorValue tau_dgradp =
      xi_tilde_dgradp / norm_b - xi_tilde * norm_b_dgradp / (norm_b * norm_b);

  return tau_dgradp;
}

dtauSUPG_dp()

Real RichardsSUPGstandard::dtauSUPG_dp ( RealVectorValue  vel, RealVectorValue  dvel_dp, Real  traceperm, RealVectorValue  b, Real  db2_dp  ) const

virtual

derivative of tau wrt porepressure (keeping gradp fixed)

Parameters

vel	SUPG velocity
dvel_dp	derivative of the SUPG velocity wrt porepressure
traceperm	trace of the permeability tensor for the material
b	the b parameter: |b| ~ 2*SUPGvelocity/element_length
db2_dp	derivative of b*b wrt porepressure

Implements RichardsSUPG.

Definition at line 186 of file RichardsSUPGstandard.C.

{
  Real norm_vel = vel.norm();
  if (norm_vel == 0.0)
    return 0.0; // this deriv is not necessarily correct, but i can't see a better thing to do

  Real norm_vel_dp = dvel_dp * vel / norm_vel;

  Real norm_b = b.norm();
  if (norm_b == 0)
    return 0.0; // this deriv is not necessarily correct, but i can't see a better thing to do
  Real norm_b_dp = db2_dp / (2.0 * norm_b);

  Real h = 2.0 * norm_vel / norm_b; // h is a measure of the element length in the "a" direction
  Real h_dp = 2.0 * norm_vel_dp / norm_b - 2.0 * norm_vel * norm_b_dp / (norm_b * norm_b);

  Real alpha = 0.5 * norm_vel * h / (traceperm * _p_SUPG); // this is the Peclet number
  Real alpha_dp = 0.5 * (norm_vel_dp * h + norm_vel * h_dp) / (traceperm * _p_SUPG);

  Real xi_tilde = RichardsSUPGstandard::cosh_relation(alpha);
  Real xi_tilde_prime = RichardsSUPGstandard::cosh_relation_prime(alpha);
  Real xi_tilde_dp = xi_tilde_prime * alpha_dp;

  // Real tau = xi_tilde/norm_b;
  const Real tau_dp = xi_tilde_dp / norm_b - xi_tilde * norm_b_dp / (norm_b * norm_b);

  return tau_dp;
}

dvelSUPG_dgradp()

RealTensorValue RichardsSUPGstandard::dvelSUPG_dgradp ( RealTensorValue  perm ) const

virtual

derivative of SUPG velocity wrt gradient of porepressure

Parameters

perm	permeability tensor

Implements RichardsSUPG.

Definition at line 52 of file RichardsSUPGstandard.C.

{
  return -perm;
}

dvelSUPG_dp()

RealVectorValue RichardsSUPGstandard::dvelSUPG_dp ( RealTensorValue  perm, Real  density_prime, RealVectorValue  gravity  ) const

virtual

derivative of SUPG velocity wrt porepressure (keeping gradp fixed)

Parameters

perm	permeability tensor
density_prime	derivative of fluid density wrt porepressure
gravity	gravitational acceleration vector

Implements RichardsSUPG.

Definition at line 58 of file RichardsSUPGstandard.C.

{
  return perm * density_prime * gravity;
}

execute()

void RichardsSUPG::execute ( )

inherited

Definition at line 31 of file RichardsSUPG.C.

{
}

finalize()

void RichardsSUPG::finalize ( )

inherited

Definition at line 36 of file RichardsSUPG.C.

{
}

initialize()

void RichardsSUPG::initialize ( )

inherited

Definition at line 26 of file RichardsSUPG.C.

{
}

SUPG_trivial()

bool RichardsSUPGstandard::SUPG_trivial ( ) const

virtual

returns false in this case since everything is trivial

Implements RichardsSUPG.

Definition at line 220 of file RichardsSUPGstandard.C.

{
  return false;
}

tauSUPG()

Real RichardsSUPGstandard::tauSUPG ( RealVectorValue  vel, Real  traceperm, RealVectorValue  b  ) const

virtual

The SUPG tau parameter.

This dictates how strong the SUPG is

Parameters

vel	SUPG velocity
traceperm	trace of the permeability tensor for the material
b	the b parameter: |b| ~ 2*SUPGvelocity/element_length

Implements RichardsSUPG.

Definition at line 130 of file RichardsSUPGstandard.C.

{
  // vel = velocity, b = bb
  Real norm_v = vel.norm();
  Real norm_b = b.norm(); // Hughes et al investigate infinity-norm and 2-norm.  i just use 2-norm
                          // here.   norm_b ~ 2|a|/ele_length_in_direction_of_a

  if (norm_b == 0)
    return 0.0; // Only way for norm_b=0 is for zero ele size, or vel=0.  Either way we don't have
                // to upwind.

  Real h = 2.0 * norm_v / norm_b; // h is a measure of the element length in the "a" direction
  Real alpha = 0.5 * norm_v * h / (traceperm * _p_SUPG); // this is the Peclet number

  const Real xi_tilde = RichardsSUPGstandard::cosh_relation(alpha);

  return xi_tilde / norm_b;
}

velSUPG()

RealVectorValue RichardsSUPGstandard::velSUPG ( RealTensorValue  perm, RealVectorValue  gradp, Real  density, RealVectorValue  gravity  ) const

virtual

SUPG velocity = -perm*(gradp - density*gravity) This points in direction of information propagation.

Parameters

perm	permeability tensor
gradp	gradient of porepressure
density	fluid density
gravity	gravitational acceleration vector

Implements RichardsSUPG.

Definition at line 43 of file RichardsSUPGstandard.C.

{
  return -perm * (gradp - density * gravity); // points in direction of info propagation
}

Member Data Documentation

_p_SUPG

Real RichardsSUPGstandard::_p_SUPG

protected

the SUPG pressure parameter This dictates how strong the SUPG is _p_SUPG large means only a little SUPG _p_SUPG small means close to fully-upwind

Definition at line 148 of file RichardsSUPGstandard.h.

Referenced by dtauSUPG_dgradp(), dtauSUPG_dp(), and tauSUPG().

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

• richards/include/userobjects/RichardsSUPGstandard.h
• richards/src/userobjects/RichardsSUPGstandard.C