From a6c5fa8e23111d4b26f41ca01cfc40555d91ce4d Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Tue, 31 Aug 2010 12:42:24 +0000
Subject: [PATCH] compute the mixed second derivative of the squared distance
function
[[Imported from SVN: r6293]]
---
dune/gfe/unitvector.hh | 37 +++++++++++++++++++++++++++++++++++++
1 file changed, 37 insertions(+)
diff --git a/dune/gfe/unitvector.hh b/dune/gfe/unitvector.hh
index 66b2fad9..03e9b36a 100644
--- a/dune/gfe/unitvector.hh
+++ b/dune/gfe/unitvector.hh
@@ -149,6 +149,43 @@ public:
return result;
}
+ /** \brief Compute the mixed second derivate \partial d^2 / \partial da db
+
+ Unlike the distance itself the squared distance is differentiable at zero
+ */
+ static Dune::FieldMatrix<double,dim,dim> secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(const UnitVector& a, const UnitVector& b) {
+
+ Dune::FieldMatrix<double,dim,dim> result;
+
+ double sp = a.data_ * b.data_;
+
+ // Compute vector A (see notes)
+ Dune::FieldMatrix<double,1,dim> row;
+ row[0] = b.globalCoordinates();
+ row *= secondDerivativeOfArcCosSquared(sp);
+
+ Dune::FieldMatrix<double,dim,1> column = b.projectOntoTangentSpace(a.globalCoordinates());
+
+ Dune::FieldMatrix<double,dim,dim> A;
+ // A = row * column
+ Dune::FMatrixHelp::multMatrix(column,row,A);
+
+ // Compute matrix B (see notes)
+ Dune::FieldMatrix<double,dim,dim> B;
+ for (int i=0; i<dim; i++)
+ for (int j=0; j<dim; j++)
+ B[i][j] = (i==j) - b.data_[i]*b.data_[j];
+
+ // Bring it all together
+ result = A;
+ result.axpy(derivativeOfArcCosSquared(sp), B);
+
+ for (int i=0; i<dim; i++)
+ result[i] = a.projectOntoTangentSpace(result[i]);
+
+ return result;
+ }
+
/** \brief Project tangent vector of R^n onto the tangent space */
EmbeddedTangentVector projectOntoTangentSpace(const EmbeddedTangentVector& v) const {
EmbeddedTangentVector result = v;
--
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