From f053f105ed6d2703ff8898e1896aa4292f926a17 Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Mon, 11 Apr 2011 08:17:38 +0000
Subject: [PATCH] add method for the third derivative of the squared distance
wrt the second argument
[[Imported from SVN: r7134]]
---
dune/gfe/unitvector.hh | 35 +++++++++++++++++++++++++++++++++++
1 file changed, 35 insertions(+)
diff --git a/dune/gfe/unitvector.hh b/dune/gfe/unitvector.hh
index 65b98373..dedc20d1 100644
--- a/dune/gfe/unitvector.hh
+++ b/dune/gfe/unitvector.hh
@@ -223,6 +223,41 @@ public:
}
+ /** \brief Compute the mixed third derivative \partial d^3 / \partial da db^2
+
+ Unlike the distance itself the squared distance is differentiable at zero
+ */
+ static Tensor3<double,N,N,N> thirdDerivativeOfDistanceSquaredWRTSecondArgument(const UnitVector& p, const UnitVector& q) {
+
+ Tensor3<double,N,N,N> result;
+
+ double sp = p.data_ * q.data_;
+
+ // The projection matrix onto the tangent space at p and q
+ Dune::FieldMatrix<double,N,N> Pq;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];
+
+ Dune::FieldVector<double,N> pProjected = q.projectOntoTangentSpace(p.globalCoordinates());
+
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ for (int k=0; k<N; k++) {
+
+ result[i][j][k] = thirdDerivativeOfArcCosSquared(sp) * pProjected[i] * pProjected[j] * pProjected[k]
+ - secondDerivativeOfArcCosSquared(sp) * ((i==j)*sp + p.globalCoordinates()[i]*q.globalCoordinates()[j])*pProjected[k]
+ - secondDerivativeOfArcCosSquared(sp) * ((i==k)*sp + p.globalCoordinates()[i]*q.globalCoordinates()[k])*pProjected[j]
+ - secondDerivativeOfArcCosSquared(sp) * pProjected[i] * Pq[j][k] * sp
+ + derivativeOfArcCosSquared(sp) * ((i==j)*q.globalCoordinates()[k] + (i==k)*q.globalCoordinates()[j]) * sp
+ - derivativeOfArcCosSquared(sp) * p.globalCoordinates()[i] * Pq[j][k];
+ }
+
+ result = Pq * result;
+
+ return result;
+ }
+
/** \brief Compute the mixed third derivative \partial d^3 / \partial da db^2
Unlike the distance itself the squared distance is differentiable at zero
--
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