Interpolation functions that works by retracting onto a tangent space

This is the interpolation method used by Ingo Münch and Wolfgang Müller for
Cosserat materials.  My implementation is only able to do all that I needed
for the illustrations in the GAMM Rundbrief article.  I'm sure there'll be
more fixes needed if you actually want to do finite elements with this.

[[Imported from SVN: r9940]]

dune/gfe/localtangentfefunction.hh

+#ifndef DUNE_GFE_LOCALTANGENTFEFUNCTION_HH
+#define DUNE_GFE_LOCALTANGENTFEFUNCTION_HH
+
+#include <vector>
+
+#include <dune/common/fvector.hh>
+
+#include <dune/geometry/type.hh>
+
+namespace Dune {
+
+  namespace GFE {
+
+    /** \brief Interpolate on a tangent space
+     *
+     * \tparam dim Dimension of the reference element
+     * \tparam ctype Type used for coordinates on the reference element
+     * \tparam LocalFiniteElement A Lagrangian finite element whose shape functions define the interpolation weights
+     * \tparam TargetSpace The manifold that the function takes its values in
+     */
+    template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
+    class LocalTangentFEFunction
+    {
+      typedef typename TargetSpace::ctype RT;
+
+      typedef typename TargetSpace::EmbeddedTangentVector EmbeddedTangentVector;
+      static const int embeddedDim = EmbeddedTangentVector::dimension;
+
+      static const int spaceDim = TargetSpace::TangentVector::dimension;
+
+    public:
+
+      /** \brief The type used for derivatives */
+      typedef Dune::FieldMatrix<RT, embeddedDim, dim> DerivativeType;
+
+      /** \brief Constructor
+       * \param localFiniteElement A Lagrangian finite element that provides the interpolation points
+       * \param coefficients Values of the function at the Lagrange points
+       */
+      LocalTangentFEFunction(const LocalFiniteElement& localFiniteElement,
+                               const std::vector<TargetSpace>& coefficients)
+      : localFiniteElement_(localFiniteElement),
+      coefficients_(coefficients)
+      {
+        assert(localFiniteElement_.localBasis().size() == coefficients_.size());
+      }
+
+      /** \brief The number of Lagrange points */
+      unsigned int size() const
+      {
+        return localFiniteElement_.localBasis().size();
+      }
+
+      /** \brief The type of the reference element */
+      Dune::GeometryType type() const
+      {
+        return localFiniteElement_.type();
+      }
+
+      /** \brief Evaluate the function */
+      TargetSpace evaluate(const Dune::FieldVector<ctype, dim>& local) const;
+
+      /** \brief Evaluate the derivative of the function */
+      DerivativeType evaluateDerivative(const Dune::FieldVector<ctype, dim>& local) const;
+
+      /** \brief Evaluate the derivative of the function, if you happen to know the function value (much faster!)
+       *        \param local Local coordinates in the reference element where to evaluate the derivative
+       *        \param q Value of the local gfe function at 'local'.  If you provide something wrong here the result will be wrong, too!
+       */
+      DerivativeType evaluateDerivative(const Dune::FieldVector<ctype, dim>& local,
+                                        const TargetSpace& q) const;
+
+      /** \brief Get the i'th base coefficient. */
+      TargetSpace coefficient(int i) const
+      {
+        return coefficients_[i];
+      }
+    private:
+
+      /** \brief The scalar local finite element, which provides the weighting factors
+       *        \todo We really only need the local basis
+       */
+      const LocalFiniteElement& localFiniteElement_;
+
+      /** \brief The coefficient vector */
+      std::vector<TargetSpace> coefficients_;
+
+    };
+
+    template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
+    TargetSpace LocalTangentFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
+    evaluate(const Dune::FieldVector<ctype, dim>& local) const
+    {
+      // Evaluate the weighting factors---these are the Lagrangian shape function values at 'local'
+      std::vector<Dune::FieldVector<ctype,1> > w;
+      localFiniteElement_.localBasis().evaluateFunction(local,w);
+
+      // This is the base point of the tangent space that we are interpolating on
+      // HACK: This point is reasonable only for spheres
+      typename TargetSpace::CoordinateType baseEmb(0);
+      baseEmb[0] = 1.0;
+      TargetSpace base(baseEmb);
+
+      // Map all points onto the tangent space at 'base'
+      std::vector<typename TargetSpace::EmbeddedTangentVector> tangentCoefficients(coefficients_.size());
+      for (size_t i=0; i<coefficients_.size(); i++)
+        tangentCoefficients[i] = TargetSpace::log(base,coefficients_[i]);
+
+      // Interpolate on the tangent space
+      typename TargetSpace::EmbeddedTangentVector c(0);
+      for (size_t i=0; i<tangentCoefficients.size(); i++)
+        c.axpy(w[i][0], tangentCoefficients[i]);
+
+      return TargetSpace::exp(base,c);
+    }
+
+    template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
+    typename LocalTangentFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::DerivativeType
+    LocalTangentFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
+    evaluateDerivative(const Dune::FieldVector<ctype, dim>& local) const
+    {
+      // the function value at the point where we are evaluating the derivative
+      TargetSpace q = evaluate(local);
+
+      // Actually compute the derivative
+      return evaluateDerivative(local,q);
+    }
+
+    template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
+    typename LocalTangentFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::DerivativeType
+    LocalTangentFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
+    evaluateDerivative(const Dune::FieldVector<ctype, dim>& local, const TargetSpace& q) const
+    {
+      DUNE_THROW(NotImplemented, "evaluateDerivative not implemented yet!");
+    }
+
+  }
+
+}
+#endif