PorousFlowCubic Namespace Reference

Utility to produce the value and derivative of a cubic function at a point. More...

Functions
Real	cubic (Real x, Real x0, Real y0, Real y0p, Real x1, Real y1, Real y1p)
	Cubic function f(x) that satisfies f(x0) = y0 f'(x0) = y0p f(x1) = y1 f'(x1) = y1p. More...
Real	dcubic (Real x, Real x0, Real y0, Real y0p, Real x1, Real y1, Real y1p)
	Derivative of cubic function, f(x), with respect to x. More...

Detailed Description

Utility to produce the value and derivative of a cubic function at a point.

The cubic is defined by values and derivatives at two points.

Function Documentation

cubic()

Real PorousFlowCubic::cubic	(	Real	x,
		Real	x0,
		Real	y0,
		Real	y0p,
		Real	x1,
		Real	y1,
		Real	y1p
	)

Cubic function f(x) that satisfies f(x0) = y0 f'(x0) = y0p f(x1) = y1 f'(x1) = y1p.

Parameters

x	the argument

Returns

value of the cubic function at x

Definition at line 16 of file PorousFlowCubic.C.

Referenced by PorousFlowVanGenuchten::relativePermeabilityHys(), PorousFlowVanGenuchten::relativePermeabilityNWHys(), and TEST().

{
  mooseAssert(x0 != x1, "PorousFlowCubic: x0 cannot equal x1");
  const Real d = x1 - x0;
  const Real d2 = Utility::pow<2>(d);
  const Real mean = 0.5 * (x1 + x0);
  const Real sq3 = 0.5 * std::sqrt(3.0) * d;
  const Real term1 = y0p * (x - x0) * Utility::pow<2>(x - x1) /
                     d2; // term1(x0) = term1(x1) = term1'(x1) = 0, term1'(x0) = y0p
  const Real term2 = y1p * (x - x1) * Utility::pow<2>(x - x0) /
                     d2; // term2(x0) = term2(x1) = term2'(x0) = 0, term2'(x1) = y1p
  const Real term3 = (x - mean - sq3) * (x - mean) * (x - mean + sq3);
  // term3' = (x - mean) * (x - mean + sq3) + (x - mean - sq3) * (x - mean + sq3) + (x - mean - sq3)
  // * (x - mean)
  //        = 3 (x - mean)^2 - sq3^2
  // note term3' = 0 when x = mean +/- sq3/sqrt(3) = 0.5 * (x1 + x0) +/- 0.5 * (x1 - x0) = {x1, x0}
  const Real term3_x0 = (x0 - mean - sq3) * (x0 - mean) * (x0 - mean + sq3);
  const Real term3_x1 = (x1 - mean - sq3) * (x1 - mean) * (x1 - mean + sq3);
  return (y0 * (term3 - term3_x1) + y1 * (term3_x0 - term3)) / (term3_x0 - term3_x1) + term1 +
         term2;
}

dcubic()

Real PorousFlowCubic::dcubic	(	Real	x,
		Real	x0,
		Real	y0,
		Real	y0p,
		Real	x1,
		Real	y1,
		Real	y1p
	)

Derivative of cubic function, f(x), with respect to x.

f(x) satisfies f(x0) = y0 f'(x0) = y0p f(x1) = y1 f'(x1) = y1p

Parameters

x	the argument

Returns

the derivative of the cubic function at x

Definition at line 39 of file PorousFlowCubic.C.

Referenced by PorousFlowVanGenuchten::drelativePermeabilityHys(), PorousFlowVanGenuchten::drelativePermeabilityNWHys(), and TEST().

{
  mooseAssert(x0 != x1, "PorousFlowCubic: x0 cannot equal x1");
  const Real d = x1 - x0;
  const Real d2 = Utility::pow<2>(d);
  const Real mean = 0.5 * (x1 + x0);
  const Real sq3 = 0.5 * std::sqrt(3.0) * d;
  const Real term1_prime = y0p * (Utility::pow<2>(x - x1) + (x - x0) * 2 * (x - x1)) / d2;
  const Real term2_prime = y1p * (Utility::pow<2>(x - x0) + (x - x1) * 2 * (x - x0)) / d2;
  const Real term3_prime =
      3.0 * Utility::pow<2>(mean) - 6 * mean * x - 0.75 * d2 + 3.0 * Utility::pow<2>(x);
  const Real term3_x0 = (x0 - mean - sq3) * (x0 - mean) * (x0 - mean + sq3);
  const Real term3_x1 = (x1 - mean - sq3) * (x1 - mean) * (x1 - mean + sq3);
  return (y0 - y1) * term3_prime / (term3_x0 - term3_x1) + term1_prime + term2_prime;
}