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spline: some fixes to make Spline::intersect() work correctly

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most of these issues are fallout from the conversion of the spline
class from a moments (second derivative) based approach to hermite
base functions:

- the second and third derivatives where incorrect, and the third
  derivative was not available in the public interface.
- The unit test did not check the derivatives
- The coefficients for the monomial basis were off by the factors
  stemming from the derivatives
- The intersectIntervall_() method used std::max() instead of
  std::min() at one place and still added the base offset for the x
  values as indicated by Stoer
This commit is contained in:
Andreas Lauser
2013-11-07 15:06:59 +01:00
parent 86ca995629
commit c9067c4bda
2 changed files with 80 additions and 15 deletions
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opm/core/utility/Spline.hpp
+36 -14
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@@ -858,6 +858,27 @@ public:
return evalDerivative2_(x, segmentIdx_(x));
}
/*!
* \brief Evaluate the spline's third derivative at a given position.
*
* \param x The value on the abscissa where the spline's
* derivative ought to be evaluated
*
* \param extrapolate If this parameter is set to true, the spline
* will be extended beyond its range by
* straight lines, if false calling extrapolate
* for \f$ x \not [x_{min}, x_{max}]\f$ will
* cause a failed assertation.
*/
Scalar evalThirdDerivative(Scalar x, bool extrapolate=false) const
{
assert(extrapolate || applies(x));
if (extrapolate)
return 0.0;
return evalDerivative3_(x, segmentIdx_(x));
}
/*!
* \brief Find the intersections of the spline with a cubic
* polynomial in the whole intervall, throws
@@ -1493,7 +1514,7 @@ protected:
Scalar alpha = 1 / delta;
return
alpha
alpha*alpha
*(h00_prime2_(t) * y_(i)
+ h10_prime2_(t) * slope_(i)*delta
+ h01_prime2_(t) * y_(i + 1)
@@ -1505,15 +1526,16 @@ protected:
Scalar evalDerivative3_(Scalar x, int i) const
{
// See http://en.wikipedia.org/wiki/Cubic_Hermite_spline
Scalar t = (x - x_(i))/h_(i + 1);
Scalar alpha = 1 / h_(i + 1);
Scalar delta = h_(i + 1);
Scalar t = (x - x_(i))/delta;
Scalar alpha = 1 / delta;
return
alpha
alpha*alpha*alpha
*(h00_prime3_(t)*y_(i)
+ h10_prime3_(t)*slope_(i)
+ h10_prime3_(t)*slope_(i)*delta
+ h01_prime3_(t)*y_(i + 1)
+ h11_prime3_(t)*slope_(i + 1));
+ h11_prime3_(t)*slope_(i + 1)*delta);
}
// hermite basis functions
@@ -1578,8 +1600,8 @@ protected:
// -1: spline is monotonously decreasing in the specified interval
int monotonic_(int i, Scalar x0, Scalar x1) const
{
Scalar a = 3*a_(i);
Scalar b = 2*b_(i);
Scalar a = a_(i);
Scalar b = b_(i);
Scalar c = c_(i);
if (std::abs(a) < 1e-20 && std::abs(b) < 1e-20 && std::abs(c) < 1e-20)
@@ -1631,12 +1653,11 @@ protected:
c_(segIdx) - c,
d_(segIdx) - d);
x0 = std::max(x_(segIdx), x0);
x1 = std::max(x_(segIdx+1), x1);
x1 = std::min(x_(segIdx+1), x1);
// filter the intersections outside of the specified intervall
int k = 0;
for (int j = 0; j < n; ++j) {
sol[j] += x_(segIdx); // add the offset of the intervall. For details see Stoer
if (x0 <= sol[j] && sol[j] <= x1) {
sol[k] = sol[j];
++k;
@@ -1691,19 +1712,20 @@ protected:
Scalar slope_(int i) const
{ return slopeVec_[i]; }
// returns the coefficient in front of the x^0 term. In Stoer this
// returns the coefficient in front of the x^3 term. In Stoer this
// is delta.
Scalar a_(int i) const
{ return evalDerivative3_(/*x=*/0, i); }
{ return evalDerivative3_(/*x=*/0, i)/6.0; }
// returns the coefficient in front of the x^2 term In Stoer this
// is gamma.
Scalar b_(int i) const
{ return evalDerivative2_(/*x=*/0, i); }
{ return evalDerivative2_(/*x=*/0, i)/2.0; }
// returns the coefficient in front of the x^1 term. In Stoer this
// is beta.
Scalar c_(int i) const { return evalDerivative_(/*x=*/0, i); }
Scalar c_(int i) const
{ return evalDerivative_(/*x=*/0, i); }
// returns the coefficient in front of the x^0 term. In Stoer this
// is alpha.