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Oliver Sander authored[[Imported from SVN: r8051]]
Oliver Sander authored[[Imported from SVN: r8051]]
localgeodesicfefunction.hh 31.10 KiB
#ifndef LOCAL_GEODESIC_FE_FUNCTION_HH
#define LOCAL_GEODESIC_FE_FUNCTION_HH
#include <vector>
#include <dune/common/fvector.hh>
#include <dune/gfe/averagedistanceassembler.hh>
#include <dune/gfe/targetspacertrsolver.hh>
#include <dune/gfe/rigidbodymotion.hh>
#include <dune/gfe/tensor3.hh>
#include <dune/gfe/tensorssd.hh>
#include <dune/gfe/linearalgebra.hh>
// forward declaration
template <class LocalFiniteElement, class TargetSpace>
class LocalGfeTestFunctionBasis;
/** \brief A function defined by simplicial geodesic interpolation
from the reference element to a Riemannian manifold.
\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 LocalGeodesicFEFunction
{
typedef typename TargetSpace::EmbeddedTangentVector EmbeddedTangentVector;
static const int embeddedDim = EmbeddedTangentVector::dimension;
static const int spaceDim = TargetSpace::TangentVector::dimension;
friend class LocalGfeTestFunctionBasis<LocalFiniteElement,TargetSpace>;
public:
/** \brief Constructor
* \param localFiniteElement A Lagrangian finite element that provides the interpolation points
* \param coefficients Values of the function at the Lagrange points
*/
LocalGeodesicFEFunction(const LocalFiniteElement& localFiniteElement,
const std::vector<TargetSpace>& coefficients)
: localFiniteElement_(localFiniteElement),
coefficients_(coefficients)
{}
/** \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 */
Dune::FieldMatrix<ctype, EmbeddedTangentVector::dimension, dim> evaluateDerivative(const Dune::FieldVector<ctype, dim>& local) const;
/** \brief For debugging: Evaluate the derivative of the function using a finite-difference approximation*/
Dune::FieldMatrix<ctype, EmbeddedTangentVector::dimension, dim> evaluateDerivativeFD(const Dune::FieldVector<ctype, dim>& local) const;
/** \brief Evaluate the derivative of the function value with respect to a coefficient */
void evaluateDerivativeOfValueWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Dune::FieldMatrix<double,embeddedDim,embeddedDim>& derivative) const;
/** \brief Evaluate the derivative of the function value with respect to a coefficient */
void evaluateFDDerivativeOfValueWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Dune::FieldMatrix<double,embeddedDim,embeddedDim>& derivative) const;
/** \brief Evaluate the derivative of the gradient of the function with respect to a coefficient */
void evaluateDerivativeOfGradientWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Tensor3<double,embeddedDim,embeddedDim,dim>& result) const;
/** \brief Evaluate the derivative of the gradient of the function with respect to a coefficient */
void evaluateFDDerivativeOfGradientWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Tensor3<double,embeddedDim,embeddedDim,dim>& result) const;
/** \brief Get the i'th base coefficient. */
TargetSpace coefficient(int i) const
{
return coefficients_[i];
}
private:
static Dune::FieldMatrix<double,embeddedDim,embeddedDim> pseudoInverse(const Dune::FieldMatrix<double,embeddedDim,embeddedDim>& dFdq,
const TargetSpace& q)
{
const int shortDim = TargetSpace::TangentVector::dimension;
// the orthonormal frame
Dune::FieldMatrix<ctype,shortDim,embeddedDim> O = q.orthonormalFrame();
// compute A = O dFDq O^T
Dune::FieldMatrix<ctype,shortDim,shortDim> A;
for (int i=0; i<shortDim; i++)
for (int j=0; j<shortDim; j++) {
A[i][j] = 0;
for (int k=0; k<embeddedDim; k++)
for (int l=0; l<embeddedDim; l++)
A[i][j] += O[i][k]*dFdq[k][l]*O[j][l];
}
A.invert();
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> result;
for (int i=0; i<embeddedDim; i++)
for (int j=0; j<embeddedDim; j++) {
result[i][j] = 0;
for (int k=0; k<shortDim; k++)
for (int l=0; l<shortDim; l++)
result[i][j] += O[k][i]*A[k][l]*O[l][j];
}
return result;
}
/** \brief Compute derivate of F(w,q) (the derivative of the weighted distance fctl) wrt to w */
Dune::Matrix<Dune::FieldMatrix<ctype,1,1> > computeDFdw(const TargetSpace& q) const
{
Dune::Matrix<Dune::FieldMatrix<ctype,1,1> > dFdw(embeddedDim,localFiniteElement_.localBasis().size());
for (size_t i=0; i<localFiniteElement_.localBasis().size(); i++) {
Dune::FieldVector<ctype,embeddedDim> tmp = TargetSpace::derivativeOfDistanceSquaredWRTSecondArgument(coefficients_[i], q);
for (int j=0; j<embeddedDim; j++)
dFdw[j][i] = tmp[j];
}
return dFdw;
}
Tensor3<ctype,embeddedDim,embeddedDim,embeddedDim> computeDqDqF(const std::vector<Dune::FieldVector<ctype,1> >& w, const TargetSpace& q) const
{
Tensor3<ctype,embeddedDim,embeddedDim,embeddedDim> result;
result = 0;
for (size_t i=0; i<w.size(); i++)
result.axpy(w[i][0], TargetSpace::thirdDerivativeOfDistanceSquaredWRTSecondArgument(coefficients_[i],q));
return result;
}
/** \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 LocalGeodesicFEFunction<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);
// The energy functional whose mimimizer is the value of the geodesic interpolation
AverageDistanceAssembler<TargetSpace> assembler(coefficients_, w);
TargetSpaceRiemannianTRSolver<TargetSpace> solver;
solver.setup(&assembler,
coefficients_[0], // initial iterate
1e-14, // tolerance
100, // maxTrustRegionSteps
2, // initial trust region radius
100, // inner iterations
1e-14 // inner tolerance
);
solver.solve();
return solver.getSol();
}
template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
Dune::FieldMatrix<ctype, TargetSpace::EmbeddedTangentVector::dimension, dim> LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
evaluateDerivative(const Dune::FieldVector<ctype, dim>& local) const
{
Dune::FieldMatrix<ctype, embeddedDim, dim> result;
#if 0 // this is probably faster than the general implementation, but we leave it out for testing purposes
if (dim==1) {
EmbeddedTangentVector tmp = TargetSpace::interpolateDerivative(coefficients_[0], coefficients_[1], local[0]);
for (int i=0; i<embeddedDim; i++)
result[i][0] = tmp[i];
}
#endif
// ////////////////////////////////////////////////////////////////////////
// The derivative is evaluated using the implicit function theorem.
// Hence we need to solve a small system of linear equations.
// ////////////////////////////////////////////////////////////////////////
// the function value at the point where we are evaluating the derivative TargetSpace q = evaluate(local);
// the matrix that turns coordinates on the reference simplex into coordinates on the standard simplex
std::vector<Dune::FieldMatrix<ctype,1,dim> > BNested(coefficients_.size());
localFiniteElement_.localBasis().evaluateJacobian(local, BNested);
Dune::Matrix<Dune::FieldMatrix<double,1,1> > B(coefficients_.size(), dim);
for (size_t i=0; i<coefficients_.size(); i++)
for (size_t j=0; j<dim; j++)
B[i][j] = BNested[i][0][j];
// compute negative derivative of F(w,q) (the derivative of the weighted distance fctl) wrt to w
Dune::Matrix<Dune::FieldMatrix<ctype,1,1> > dFdw = computeDFdw(q);
dFdw *= -1;
// multiply the two previous matrices: the result is the right hand side
Dune::Matrix<Dune::FieldMatrix<ctype,1,1> > RHS = dFdw * B;
// the actual system matrix
std::vector<Dune::FieldVector<ctype,1> > w;
localFiniteElement_.localBasis().evaluateFunction(local, w);
AverageDistanceAssembler<TargetSpace> assembler(coefficients_, w);
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> dFdq(0);
assembler.assembleEmbeddedHessian(q,dFdq);
// ////////////////////////////////////
// solve the system
//
// We want to solve
// dFdq * x = rhs
// However as dFdq is defined in the embedding space it has a positive rank.
// Hence we use the orthonormal frame at q to get rid of its kernel.
// That means we instead solve
// O dFdq O^T O x = O rhs
// ////////////////////////////////////
const int shortDim = TargetSpace::TangentVector::dimension;
// the orthonormal frame
Dune::FieldMatrix<ctype,shortDim,embeddedDim> O = q.orthonormalFrame();
// compute A = O dFDq O^T
Dune::FieldMatrix<ctype,shortDim,shortDim> A;
for (int i=0; i<shortDim; i++)
for (int j=0; j<shortDim; j++) {
A[i][j] = 0;
for (int k=0; k<embeddedDim; k++)
for (int l=0; l<embeddedDim; l++)
A[i][j] += O[i][k]*dFdq[k][l]*O[j][l];
}
for (int i=0; i<dim; i++) {
Dune::FieldVector<ctype,embeddedDim> rhs;
for (int j=0; j<embeddedDim; j++)
rhs[j] = RHS[j][i];
Dune::FieldVector<ctype,shortDim> shortRhs;
O.mv(rhs,shortRhs);
Dune::FieldVector<ctype,shortDim> shortX;
A.solve(shortX,shortRhs);
Dune::FieldVector<ctype,embeddedDim> x;
O.mtv(shortX,x);
for (int j=0; j<embeddedDim; j++)
result[j][i] = x[j];
}
return result;
}
template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
Dune::FieldMatrix<ctype, TargetSpace::EmbeddedTangentVector::dimension, dim> LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
evaluateDerivativeFD(const Dune::FieldVector<ctype, dim>& local) const
{
double eps = 1e-6;
Dune::FieldMatrix<ctype, EmbeddedTangentVector::dimension, dim> result;
for (int i=0; i<dim; i++) {
Dune::FieldVector<ctype, dim> forward = local;
Dune::FieldVector<ctype, dim> backward = local;
forward[i] += eps;
backward[i] -= eps;
EmbeddedTangentVector fdDer = evaluate(forward).globalCoordinates() - evaluate(backward).globalCoordinates();
fdDer /= 2*eps;
for (int j=0; j<EmbeddedTangentVector::dimension; j++)
result[j][i] = fdDer[j];
}
return result;
}
template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
void LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
evaluateDerivativeOfValueWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Dune::FieldMatrix<double,embeddedDim,embeddedDim>& result) const
{
// the function value at the point where we are evaluating the derivative
TargetSpace q = evaluate(local);
// dFdq
std::vector<Dune::FieldVector<ctype,1> > w;
localFiniteElement_.localBasis().evaluateFunction(local,w);
AverageDistanceAssembler<TargetSpace> assembler(coefficients_, w);
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> dFdq(0);
assembler.assembleEmbeddedHessian(q,dFdq);
const int shortDim = TargetSpace::TangentVector::dimension;
// the orthonormal frame
Dune::FieldMatrix<ctype,shortDim,embeddedDim> O = q.orthonormalFrame();
// compute A = O dFDq O^T
Dune::FieldMatrix<ctype,shortDim,shortDim> A;
for (int i=0; i<shortDim; i++)
for (int j=0; j<shortDim; j++) {
A[i][j] = 0;
for (int k=0; k<embeddedDim; k++)
for (int l=0; l<embeddedDim; l++)
A[i][j] += O[i][k]*dFdq[k][l]*O[j][l];
}
//
Dune::FieldMatrix<double,embeddedDim,embeddedDim> rhs = TargetSpace::secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(coefficients_[coefficient], q);
rhs *= -w[coefficient];
for (int i=0; i<embeddedDim; i++) {
Dune::FieldVector<ctype,shortDim> shortRhs;
O.mv(rhs[i],shortRhs);
Dune::FieldVector<ctype,shortDim> shortX;
A.solve(shortX,shortRhs);
O.mtv(shortX,result[i]);
}
}
template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
void LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
evaluateFDDerivativeOfValueWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Dune::FieldMatrix<double,embeddedDim,embeddedDim>& result) const
{
double eps = 1e-6;
// the function value at the point where we are evaluating the derivative
const Dune::FieldMatrix<double,spaceDim,embeddedDim> B = coefficients_[coefficient].orthonormalFrame();
Dune::FieldMatrix<double,spaceDim,embeddedDim> interimResult;
std::vector<TargetSpace> cornersPlus = coefficients_;
std::vector<TargetSpace> cornersMinus = coefficients_;
for (int j=0; j<spaceDim; j++) {
typename TargetSpace::EmbeddedTangentVector forwardVariation = B[j];
forwardVariation *= eps;
typename TargetSpace::EmbeddedTangentVector backwardVariation = B[j];
backwardVariation *= -eps;
cornersPlus [coefficient] = TargetSpace::exp(coefficients_[coefficient], forwardVariation);
cornersMinus[coefficient] = TargetSpace::exp(coefficients_[coefficient], backwardVariation);
LocalGeodesicFEFunction<dim,double,LocalFiniteElement,TargetSpace> fPlus(localFiniteElement_,cornersPlus);
LocalGeodesicFEFunction<dim,double,LocalFiniteElement,TargetSpace> fMinus(localFiniteElement_,cornersMinus);
TargetSpace hPlus = fPlus.evaluate(local);
TargetSpace hMinus = fMinus.evaluate(local);
interimResult[j] = hPlus.globalCoordinates();
interimResult[j] -= hMinus.globalCoordinates();
interimResult[j] /= 2*eps;
}
for (int i=0; i<embeddedDim; i++)
for (int j=0; j<embeddedDim; j++) {
result[i][j] = 0;
for (int k=0; k<spaceDim; k++)
result[i][j] += B[k][i]*interimResult[k][j];
}
}
template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
void LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
evaluateDerivativeOfGradientWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Tensor3<double,embeddedDim,embeddedDim,dim>& result) const
{
// the function value at the point where we are evaluating the derivative
TargetSpace q = evaluate(local);
// the matrix that turns coordinates on the reference simplex into coordinates on the standard simplex
std::vector<Dune::FieldMatrix<ctype,1,dim> > BNested(coefficients_.size());
localFiniteElement_.localBasis().evaluateJacobian(local, BNested);
Dune::Matrix<Dune::FieldMatrix<double,1,1> > B(coefficients_.size(), dim);
for (size_t i=0; i<coefficients_.size(); i++)
for (size_t j=0; j<dim; j++)
B[i][j] = BNested[i][0][j];
// the actual system matrix
std::vector<Dune::FieldVector<ctype,1> > w;
localFiniteElement_.localBasis().evaluateFunction(local,w);
AverageDistanceAssembler<TargetSpace> assembler(coefficients_, w);
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> dFdq(0);
assembler.assembleEmbeddedHessian(q,dFdq);
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> mixedDerivative = TargetSpace::secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(coefficients_[coefficient], q);
TensorSSD<double,embeddedDim,embeddedDim> dvDwF(coefficients_.size());
dvDwF = 0;
for (int i=0; i<embeddedDim; i++)
for (int j=0; j<embeddedDim; j++)
dvDwF(i, j, coefficient) = mixedDerivative[i][j];
// dFDq is not invertible, if the target space is embedded into a higher-dimensional
// Euclidean space. Therefore we use its pseudo inverse. I don't think that is the
// best way, though.
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> dFdqPseudoInv = pseudoInverse(dFdq,q);
//
Tensor3<double,embeddedDim,embeddedDim,embeddedDim> dvDqF
= TargetSpace::thirdDerivativeOfDistanceSquaredWRTFirst1AndSecond2Argument(coefficients_[coefficient], q);
dvDqF = w[coefficient] * dvDqF;
// Put it all together
Dune::FieldMatrix<double,embeddedDim,embeddedDim> dvq;
evaluateDerivativeOfValueWRTCoefficient(local,coefficient,dvq);
Dune::FieldMatrix<ctype, embeddedDim, dim> derivative = evaluateDerivative(local);
Tensor3<double, embeddedDim,embeddedDim,embeddedDim> dqdqF;
dqdqF = computeDqDqF(w,q);
TensorSSD<double, embeddedDim,embeddedDim> dqdwF(coefficients_.size());
for (size_t k=0; k<coefficients_.size(); k++) {
Dune::FieldMatrix<ctype,embeddedDim,embeddedDim> hesse = TargetSpace::secondDerivativeOfDistanceSquaredWRTSecondArgument(coefficients_[k], q);
for (int i=0; i<embeddedDim; i++)
for (int j=0; j<embeddedDim; j++)
dqdwF(i, j, k) = hesse[i][j];
}
TensorSSD<double, embeddedDim,embeddedDim> dqdwF_times_dvq(coefficients_.size());
for (int i=0; i<embeddedDim; i++)
for (int j=0; j<embeddedDim; j++)
for (size_t k=0; k<coefficients_.size(); k++) {
dqdwF_times_dvq(i, j, k) = 0;
for (int l=0; l<embeddedDim; l++)
dqdwF_times_dvq(i, j, k) += dqdwF(l, j, k) * dvq[l][i];
}
Tensor3<double, embeddedDim,embeddedDim,dim> foo;
foo = -1 * dvDqF*derivative - (dvq*dqdqF)*derivative;
TensorSSD<double,embeddedDim,embeddedDim> bar(dim);
bar = dvDwF * B + dqdwF_times_dvq*B;
for (int i=0; i<embeddedDim; i++) for (int j=0; j<embeddedDim; j++)
for (int k=0; k<dim; k++)
foo[i][j][k] -= bar(i, j, k);
result = 0;
for (int i=0; i<embeddedDim; i++)
for (int j=0; j<embeddedDim; j++)
for (int k=0; k<dim; k++)
for (int l=0; l<embeddedDim; l++)
result[i][j][k] += dFdqPseudoInv[j][l] * foo[i][l][k];
}
template <int dim, class ctype, class LocalFiniteElement, class TargetSpace>
void LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,TargetSpace>::
evaluateFDDerivativeOfGradientWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Tensor3<double,embeddedDim,embeddedDim,dim>& result) const
{
double eps = 1e-6;
static const int embeddedDim = TargetSpace::EmbeddedTangentVector::dimension;
for (int j=0; j<embeddedDim; j++) {
std::vector<TargetSpace> cornersPlus = coefficients_;
std::vector<TargetSpace> cornersMinus = coefficients_;
typename TargetSpace::CoordinateType aPlus = coefficients_[coefficient].globalCoordinates();
typename TargetSpace::CoordinateType aMinus = coefficients_[coefficient].globalCoordinates();
aPlus[j] += eps;
aMinus[j] -= eps;
cornersPlus[coefficient] = TargetSpace(aPlus);
cornersMinus[coefficient] = TargetSpace(aMinus);
LocalGeodesicFEFunction<dim,double,LocalFiniteElement,TargetSpace> fPlus(localFiniteElement_,cornersPlus);
LocalGeodesicFEFunction<dim,double,LocalFiniteElement,TargetSpace> fMinus(localFiniteElement_,cornersMinus);
Dune::FieldMatrix<double,embeddedDim,dim> hPlus = fPlus.evaluateDerivative(local);
Dune::FieldMatrix<double,embeddedDim,dim> hMinus = fMinus.evaluateDerivative(local);
result[j] = hPlus;
result[j] -= hMinus;
result[j] /= 2*eps;
TargetSpace q = evaluate(local);
Dune::FieldVector<double,embeddedDim> foo;
for (int l=0; l<dim; l++) {
for (int k=0; k<embeddedDim; k++)
foo[k] = result[j][k][l];
foo = q.projectOntoTangentSpace(foo);
for (int k=0; k<embeddedDim; k++)
result[j][k][l] = foo[k];
}
}
}
/** \brief A function defined by simplicial geodesic interpolation
from the reference element to a RigidBodyMotion.
This is a specialization for speeding up the code.
We use that a RigidBodyMotion is a product manifold.
\tparam dim Dimension of the reference element
\tparam ctype Type used for coordinates on the reference element*/
template <int dim, class ctype, class LocalFiniteElement>
class LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,RigidBodyMotion<3> >
{
typedef RigidBodyMotion<3> TargetSpace;
typedef typename TargetSpace::EmbeddedTangentVector EmbeddedTangentVector;
static const int embeddedDim = EmbeddedTangentVector::dimension;
static const int spaceDim = TargetSpace::TangentVector::dimension;
public:
/** \brief Constructor */
LocalGeodesicFEFunction(const LocalFiniteElement& localFiniteElement,
const std::vector<TargetSpace>& coefficients)
: localFiniteElement_(localFiniteElement),
translationCoefficients_(coefficients.size())
{
assert(localFiniteElement.localBasis().size() == coefficients.size());
for (size_t i=0; i<coefficients.size(); i++)
translationCoefficients_[i] = coefficients[i].r;
std::vector<Rotation<3,ctype> > orientationCoefficients(coefficients.size());
for (size_t i=0; i<coefficients.size(); i++)
orientationCoefficients[i] = coefficients[i].q;
orientationFEFunction_ = std::auto_ptr<LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,Rotation<3,double> > > (new LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,Rotation<3,double> >(localFiniteElement,orientationCoefficients));
}
/** \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
{
TargetSpace 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);
result.r = 0;
for (size_t i=0; i<w.size(); i++)
result.r.axpy(w[i][0], translationCoefficients_[i]);
result.q = orientationFEFunction_->evaluate(local);
return result;
}
/** \brief Evaluate the derivative of the function */
Dune::FieldMatrix<ctype, embeddedDim, dim> evaluateDerivative(const Dune::FieldVector<ctype, dim>& local) const
{
Dune::FieldMatrix<ctype, embeddedDim, dim> result(0);
// get translation part
std::vector<Dune::FieldMatrix<ctype,1,dim> > sfDer(translationCoefficients_.size());
localFiniteElement_.localBasis().evaluateJacobian(local, sfDer);
for (size_t i=0; i<translationCoefficients_.size(); i++)
for (int j=0; j<3; j++)
result[j].axpy(translationCoefficients_[i][j], sfDer[i][0]);
// get orientation part
Dune::FieldMatrix<ctype,4,dim> qResult = orientationFEFunction_->evaluateDerivative(local);
for (int i=0; i<4; i++)
for (int j=0; j<dim; j++)
result[3+i][j] = qResult[i][j];
return result;
}
/** \brief For debugging: Evaluate the derivative of the function using a finite-difference approximation*/
Dune::FieldMatrix<ctype, embeddedDim, dim> evaluateDerivativeFD(const Dune::FieldVector<ctype, dim>& local) const
{
Dune::FieldMatrix<ctype, embeddedDim, dim> result(0);
// get translation part
std::vector<Dune::FieldMatrix<ctype,1,dim> > sfDer(translationCoefficients_.size());
localFiniteElement_.localBasis().evaluateJacobian(local, sfDer);
for (size_t i=0; i<translationCoefficients_.size(); i++)
for (int j=0; j<3; j++)
result[j].axpy(translationCoefficients_[i][j], sfDer[i][0]);
// get orientation part
Dune::FieldMatrix<ctype,4,dim> qResult = orientationFEFunction_->evaluateDerivativeFD(local);
for (int i=0; i<4; i++)
for (int j=0; j<dim; j++)
result[3+i][j] = qResult[i][j];
return result;
}
/** \brief Evaluate the derivative of the function value with respect to a coefficient */
void evaluateDerivativeOfValueWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Dune::FieldMatrix<double,embeddedDim,embeddedDim>& derivative) const
{
derivative = 0;
// Translation part
std::vector<Dune::FieldVector<ctype,1> > w;
localFiniteElement_.localBasis().evaluateFunction(local,w);
for (int i=0; i<3; i++)
derivative[i][i] = w[coefficient];
// Rotation part
Dune::FieldMatrix<ctype,4,4> qDerivative;
orientationFEFunction_->evaluateDerivativeOfValueWRTCoefficient(local,coefficient,qDerivative);
for (int i=0; i<4; i++)
for (int j=0; j<4; j++)
derivative[3+i][3+j] = qDerivative[i][j];
}
/** \brief Evaluate the derivative of the function value with respect to a coefficient */
void evaluateFDDerivativeOfValueWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Dune::FieldMatrix<double,embeddedDim,embeddedDim>& derivative) const
{
derivative = 0;
// Translation part
std::vector<Dune::FieldVector<ctype,1> > w;
localFiniteElement_.localBasis().evaluateFunction(local,w);
for (int i=0; i<3; i++)
derivative[i][i] = w[coefficient];
// Rotation part Dune::FieldMatrix<ctype,4,4> qDerivative;
orientationFEFunction_->evaluateFDDerivativeOfValueWRTCoefficient(local,coefficient,qDerivative);
for (int i=0; i<4; i++)
for (int j=0; j<4; j++)
derivative[3+i][3+j] = qDerivative[i][j];
}
/** \brief Evaluate the derivative of the gradient of the function with respect to a coefficient */
void evaluateDerivativeOfGradientWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Tensor3<double,embeddedDim,embeddedDim,dim>& derivative) const
{
derivative = 0;
// Translation part
std::vector<Dune::FieldMatrix<ctype,1,dim> > w;
localFiniteElement_.localBasis().evaluateJacobian(local,w);
for (int i=0; i<3; i++)
derivative[i][i] = w[coefficient][0];
// Rotation part
Tensor3<ctype,4,4,dim> qDerivative;
orientationFEFunction_->evaluateDerivativeOfGradientWRTCoefficient(local,coefficient,qDerivative);
for (int i=0; i<4; i++)
for (int j=0; j<4; j++)
for (int k=0; k<dim; k++)
derivative[3+i][3+j][k] = qDerivative[i][j][k];
}
/** \brief Evaluate the derivative of the gradient of the function with respect to a coefficient */
void evaluateFDDerivativeOfGradientWRTCoefficient(const Dune::FieldVector<ctype, dim>& local,
int coefficient,
Tensor3<double,embeddedDim,embeddedDim,dim>& derivative) const
{
derivative = 0;
// Translation part
std::vector<Dune::FieldMatrix<ctype,1,dim> > w;
localFiniteElement_.localBasis().evaluateJacobian(local,w);
for (int i=0; i<3; i++)
derivative[i][i] = w[coefficient][0];
// Rotation part
Tensor3<ctype,4,4,dim> qDerivative;
orientationFEFunction_->evaluateFDDerivativeOfGradientWRTCoefficient(local,coefficient,qDerivative);
for (int i=0; i<4; i++)
for (int j=0; j<4; j++)
for (int k=0; k<dim; k++)
derivative[3+i][3+j][k] = qDerivative[i][j][k];
}
TargetSpace coefficient(int i) const {
return TargetSpace(translationCoefficients_[i],orientationFEFunction_->coefficient(i));
}
private:
/** \brief The scalar local finite element, which provides the weighting factors
\todo We really only need the local basis
*/
const LocalFiniteElement& localFiniteElement_;
// The two factors of a RigidBodyMotion
//LocalGeodesicFEFunction<dim,ctype,RealTuple<3> > translationFEFunction_;
std::vector<Dune::FieldVector<double,3> > translationCoefficients_;
std::auto_ptr<LocalGeodesicFEFunction<dim,ctype,LocalFiniteElement,Rotation<3,double> > > orientationFEFunction_;
};
#endif