From c47b4d03bf5d3a1b66fcb49f36174f8904f60d5a Mon Sep 17 00:00:00 2001
From: Oliver Sander <oliver.sander@tu-dresden.de>
Date: Fri, 16 Oct 2015 17:01:32 +0200
Subject: [PATCH] Implement Projected Finite Elements with values in SO(3)
---
dune/gfe/localprojectedfefunction.hh | 282 ++++++++++++++++++++++++++-
1 file changed, 281 insertions(+), 1 deletion(-)
diff --git a/dune/gfe/localprojectedfefunction.hh b/dune/gfe/localprojectedfefunction.hh
index a653f310..189deb1f 100644
--- a/dune/gfe/localprojectedfefunction.hh
+++ b/dune/gfe/localprojectedfefunction.hh
@@ -7,8 +7,28 @@
#include <dune/geometry/type.hh>
+#include <dune/gfe/rotation.hh>
+#include <dune/gfe/svd.hh>
+#include <dune/gfe/linearalgebra.hh>
+
namespace Dune {
+template< class K, int m, int n, int p >
+Dune::FieldMatrix< K, m, p > operator* ( const Dune::FieldMatrix< K, m, n > &A, const Dune::FieldMatrix< K, n, p > &B)
+{
+ typedef typename Dune::FieldMatrix< K, m, p > :: size_type size_type;
+ Dune::FieldMatrix< K, m, p > ret;
+
+ for( size_type i = 0; i < m; ++i ) {
+
+ for( size_type j = 0; j < p; ++j ) {
+ ret[ i ][ j ] = K( 0 );
+ for( size_type k = 0; k < n; ++k )
+ ret[ i ][ j ] += A[ i ][ k ] * B[ k ][ j ];
+ }
+ }
+ return ret;
+}
namespace GFE {
/** \brief Interpolate in an embedding Euclidean space, and project back onto the Riemannian manifold
@@ -18,9 +38,12 @@ namespace Dune {
* \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>
+ template <int dim, class ctype, class LocalFiniteElement, class TS>
class LocalProjectedFEFunction
{
+ public:
+ using TargetSpace=TS;
+ private:
typedef typename TargetSpace::ctype RT;
typedef typename TargetSpace::EmbeddedTangentVector EmbeddedTangentVector;
@@ -151,6 +174,263 @@ namespace Dune {
return result;
}
+ /** \brief Interpolate in an embedding Euclidean space, and project back onto the Riemannian manifold -- specialization for SO(3)
+ *
+ * \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
+ */
+ template <int dim, class ctype, class LocalFiniteElement, class field_type>
+ class LocalProjectedFEFunction<dim,ctype,LocalFiniteElement,Rotation<field_type,3> >
+ {
+ public:
+ typedef Rotation<field_type,3> TargetSpace;
+ private:
+ typedef typename TargetSpace::ctype RT;
+
+ typedef typename TargetSpace::EmbeddedTangentVector EmbeddedTangentVector;
+ static const int embeddedDim = EmbeddedTangentVector::dimension;
+
+ static const int spaceDim = TargetSpace::TangentVector::dimension;
+
+ static FieldMatrix<field_type,3,3> transpose(const FieldMatrix<field_type,3,3>& matrix)
+ {
+ FieldMatrix<field_type,3,3> result;
+
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ result[i][j] = matrix[j][i];
+
+ return result;
+ }
+
+ static FieldMatrix<field_type,3,3> polarFactor(const FieldMatrix<field_type,3,3>& matrix)
+ {
+ FieldVector<field_type,3> W;
+ FieldMatrix<field_type,3,3> V, VT;
+ auto polar = matrix;
+
+ // returns a decomposition U W VT, where U is returned in the first argument
+ svdcmp<field_type,3,3>(polar, W, V);
+
+ // \todo Merge this transposition with the following multiplication
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ VT[i][j] = V[j][i];
+
+ polar.rightmultiply(VT);
+ return polar;
+ }
+
+ /**
+ * \param A The argument of the projection
+ * \param value The image of the projection
+ */
+ static std::array<std::array<FieldMatrix<field_type,3,3>, 3>, 3> derivativeOfProjection(const FieldMatrix<field_type,3,3> A,
+ Rotation<field_type,3> value)
+ {
+ std::array<std::array<FieldMatrix<field_type,3,3>, 3>, 3> result;
+
+ /////////////////////////////////////////////////////////////////////////////////
+ // Step I: Compute the singular value decomposition of A
+ /////////////////////////////////////////////////////////////////////////////////
+
+ FieldVector<field_type,3> sigma;
+ FieldMatrix<field_type,3,3> V, VT;
+ FieldMatrix<field_type,3,3> U = A;
+
+ // returns a decomposition U diag(sigma) VT, where U is returned in the first argument
+ svdcmp<field_type,3,3>(U, sigma, V);
+
+ // Compute VT from V (svdcmp returns the latter)
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ VT[i][j] = V[j][i];
+
+ /////////////////////////////////////////////////////////////////////////////////
+ // Step II: Compute the derivative direction X
+ /////////////////////////////////////////////////////////////////////////////////
+
+ for (int dir0=0; dir0<3; dir0++)
+ {
+ for (int dir1=0; dir1<3; dir1++)
+ {
+ FieldMatrix<field_type,3,3> X(0);
+ X[dir0][dir1] = 1.0;
+
+ /////////////////////////////////////////////////////////////////////////////////
+ // Step ???: Markus' magic formula
+ /////////////////////////////////////////////////////////////////////////////////
+
+ FieldMatrix<field_type,3,3> UTXV = transpose(U)*X*V;
+
+ FieldMatrix<field_type,3,3> center;
+ center[0][0] = 0.0;
+ center[0][1] = (UTXV[0][1] - UTXV[1][0]) / (sigma[0] + sigma[1]);
+ center[0][2] = (UTXV[0][2] - UTXV[2][0]) / (sigma[0] + sigma[2]);
+
+ center[1][0] = (UTXV[1][0] - UTXV[0][1]) / (sigma[1] + sigma[0]);
+ center[1][1] = 0.0;
+ center[1][2] = (UTXV[1][2] - UTXV[2][1]) / (sigma[1] + sigma[2]);
+
+ center[2][0] = (UTXV[2][0] - UTXV[0][2]) / (sigma[2] + sigma[0]);
+ center[2][1] = (UTXV[2][1] - UTXV[1][2]) / (sigma[2] + sigma[1]);
+ center[2][2] = 0.0;
+
+ // Compute U center VT
+ FieldMatrix<field_type,3,3> PAX = U*center*VT;
+
+ // Copy into result array
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ result[i][j][dir0][dir1] = PAX[i][j];
+
+ }
+ }
+
+ return result;
+ }
+
+
+ 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
+ */
+ LocalProjectedFEFunction(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_.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
+ {
+ Rotation<field_type,3> result;
+
+ // 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);
+
+ // Interpolate in R^{3x3}
+ FieldMatrix<field_type,3,3> interpolatedMatrix(0);
+ for (size_t i=0; i<coefficients_.size(); i++)
+ {
+ FieldMatrix<field_type,3,3> coefficientAsMatrix;
+ coefficients_[i].matrix(coefficientAsMatrix);
+ interpolatedMatrix.axpy(w[i][0], coefficientAsMatrix);
+ }
+
+ // Project back onto SO(3)
+ result.set(polarFactor(interpolatedMatrix));
+
+ return result;
+ }
+
+ /** \brief Evaluate the derivative of the function */
+ DerivativeType 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);
+ }
+
+ /** \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 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
+ {
+ // 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);
+
+ std::vector<Dune::FieldMatrix<ctype,1,dim> > wDer;
+ localFiniteElement_.localBasis().evaluateJacobian(local,wDer);
+
+ // Compute matrix representations for all coefficients (we only have them in quaternion representation)
+ std::vector<Dune::FieldMatrix<field_type,3,3> > coefficientsAsMatrix(coefficients_.size());
+ for (size_t i=0; i<coefficients_.size(); i++)
+ coefficients_[i].matrix(coefficientsAsMatrix[i]);
+
+ // Interpolate in R^{3x3}
+ FieldMatrix<field_type,3,3> interpolatedMatrix(0);
+ for (size_t i=0; i<coefficients_.size(); i++)
+ interpolatedMatrix.axpy(w[i][0], coefficientsAsMatrix[i]);
+
+ auto polarFactor = this->polarFactor(interpolatedMatrix);
+
+ Tensor3<RT,dim,3,3> derivative(0);
+
+ for (size_t dir=0; dir<dim; dir++)
+ for (size_t i=0; i<3; i++)
+ for (size_t j=0; j<3; j++)
+ for (size_t k=0; k<coefficients_.size(); k++)
+ derivative[dir][i][j] += wDer[k][0][dir] * coefficientsAsMatrix[k][i][j];
+
+ auto derivativeOfProjection = this->derivativeOfProjection(interpolatedMatrix,q);
+
+ Tensor3<field_type,dim,3,3> intermediateResult(0);
+
+ for (size_t dir=0; dir<dim; dir++)
+ for (size_t i=0; i<3; i++)
+ for (size_t j=0; j<3; j++)
+ for (size_t k=0; k<3; k++)
+ for (size_t l=0; l<3; l++)
+ intermediateResult[dir][i][j] += derivativeOfProjection[i][j][k][l]*derivative[dir][k][l];
+
+ // One more application of the chain rule: we need to go from orthogonal matrices to quaternions
+ Tensor3<field_type,4,3,3> derivativeOfMatrixToQuaternion = Rotation<field_type,3>::derivativeOfMatrixToQuaternion(polarFactor);
+
+ DerivativeType result(0);
+
+ for (size_t dir0=0; dir0<4; dir0++)
+ for (size_t dir1=0; dir1<dim; dir1++)
+ for (size_t i=0; i<3; i++)
+ for (size_t j=0; j<3; j++)
+ result[dir0][dir1] += derivativeOfMatrixToQuaternion[dir0][i][j] * intermediateResult[dir1][i][j];
+
+ return result;
+ }
+
+ /** \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_;
+
+ };
+
}
}
--
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