From 2e788ffc25f8e2f8e20fd6da4877b4694f1f404d Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Wed, 8 Feb 2012 18:01:03 +0000
Subject: [PATCH] implement the remaining curvature and bending terms for the
energy gradient. These are not working yet, though.
[[Imported from SVN: r8408]]
---
dune/gfe/cosseratenergystiffness.hh | 185 +++++++++++++++++++++++++---
1 file changed, 171 insertions(+), 14 deletions(-)
diff --git a/dune/gfe/cosseratenergystiffness.hh b/dune/gfe/cosseratenergystiffness.hh
index 5701d963..466fcb08 100644
--- a/dune/gfe/cosseratenergystiffness.hh
+++ b/dune/gfe/cosseratenergystiffness.hh
@@ -133,6 +133,53 @@ public: // for testing
}
+ /** \brief Compute the derivative of the gradient of the rotation with respect to the corner coefficients,
+ * but in matrix coordinates
+ *
+ \param value Value of the gfe function at a certain point
+ \param derivative First derivative of the gfe function wrt the coefficients at that point, in quaternion coordinates
+ */
+ static void compute_dDR_dv(const RigidBodyMotion<double,3>& value,
+ const Dune::FieldMatrix<double,7,gridDim>& derOfValueWRTx,
+ const Dune::FieldMatrix<double,7,7>& derOfValueWRTCoefficient,
+ const Tensor3<double,7,7,gridDim>& derOfGradientWRTCoefficient,
+ Dune::array<Tensor3<double,3,3,4>, 3>& dDR_dv)
+ {
+ // The LocalGFEFunction class gives us the derivatives of the orientation variable,
+ // but as a map into quaternion space. To obtain matrix coordinates we use the
+ // chain rule, which means that we have to multiply the given derivative with
+ // the derivative of the embedding of the unit quaternion into the space of 3x3 matrices.
+ // This second derivative is almost given by the method getFirstDerivativesOfDirectors.
+ // However, since the directors of a given unit quaternion are the _columns_ of the
+ // corresponding orthogonal matrix, we need to invert the i and j indices
+ //
+ // So, if I am not mistaken, DR[i][j][k] contains \partial R_ij / \partial k
+ Tensor3<double,3 , 3, 4> dd_dq;
+ value.q.getFirstDerivativesOfDirectors(dd_dq);
+
+ /** \todo This is a constant -- don't recompute it every time */
+ Dune::array<Tensor3<double,3,4,4>, 3> dd_dq_dq;
+ Rotation<double,3>::getSecondDerivativesOfDirectors(dd_dq_dq);
+
+ for (size_t i=0; i<4; i++)
+ dDR_dv[i] = 0;
+
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ for (int k=0; k<gridDim; k++)
+ for (int v_i=0; v_i<4; v_i++)
+ for (int l=0; l<4; l++)
+ dDR_dv[i][j][k][v_i] += dd_dq_dq[j][i][l][k] * derOfValueWRTCoefficient[l+3][v_i] * derOfValueWRTx[l+3][k];
+
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ for (int k=0; k<gridDim; k++)
+ for (int v_i=0; v_i<4; v_i++)
+ for (int l=0; l<4; l++)
+ dDR_dv[i][j][k][v_i] += dd_dq[j][i][l] * derOfGradientWRTCoefficient[l+3][v_i][k];
+
+ }
+
public:
/** \brief Constructor with a set of material parameters
@@ -258,6 +305,18 @@ public:
const Tensor3<double,7,7,gridDim>& derOfGradientWRTCoefficient,
const Dune::FieldMatrix<double,3,3>& U) const;
+ void curvatureEnergyGradient(typename TargetSpace::EmbeddedTangentVector& embeddedLocalGradient,
+ const Dune::FieldMatrix<double,3,3>& R,
+ const Tensor3<double,3,3,3>& DR,
+ const Dune::array<Tensor3<double,3,3,4>, 3>& dDR_dv) const;
+
+ void bendingEnergyGradient(typename TargetSpace::EmbeddedTangentVector& embeddedLocalGradient,
+ const Dune::FieldMatrix<double,3,3>& R,
+ const Tensor3<double,3,3,4>& dR_dv,
+ const Tensor3<double,3,3,3>& DR,
+ const Dune::array<Tensor3<double,3,3,4>, 3>& dDR_dv) const;
+
+
/** \brief The shell thickness */
double thickness_;
@@ -412,7 +471,7 @@ energy(const Entity& element,
else
neumannFunction_->evaluate(it->geometry().global(quad[pt].position()), neumannValue);
- // Constant Neumann force in z-direction
+ // Only translational dofs are affected by the Neumann force
energy += thickness_ * (neumannValue * value.r) * quad[pt].weight() * integrationElement;
}
@@ -521,6 +580,87 @@ longQuadraticMembraneEnergyGradient(typename TargetSpace::EmbeddedTangentVector&
}
+
+template <class GridView, class LocalFiniteElement, int dim>
+void CosseratEnergyLocalStiffness<GridView, LocalFiniteElement, dim>::
+curvatureEnergyGradient(typename TargetSpace::EmbeddedTangentVector& embeddedLocalGradient,
+ const Dune::FieldMatrix<double,3,3>& R,
+ const Tensor3<double,3,3,3>& DR,
+ const Dune::array<Tensor3<double,3,3,4>, 3>& dDR_dv) const
+{
+ embeddedLocalGradient = 0;
+
+ for (size_t v_i=0; v_i<4; v_i++) {
+
+ for (size_t i=0; i<3; i++)
+ for (size_t j=0; j<3; j++)
+ for (size_t k=0; k<3; k++)
+ embeddedLocalGradient[v_i+3] += DR[i][j][k] * dDR_dv[i][j][k][v_i];
+
+ }
+
+ embeddedLocalGradient *= mu_ * q_ * std::pow(L_c_, q_) * std::pow(DR.frobenius_norm(), q_ - 2);
+}
+
+
+template <class GridView, class LocalFiniteElement, int dim>
+void CosseratEnergyLocalStiffness<GridView, LocalFiniteElement, dim>::
+bendingEnergyGradient(typename TargetSpace::EmbeddedTangentVector& embeddedLocalGradient,
+ const Dune::FieldMatrix<double,3,3>& R,
+ const Tensor3<double,3,3,4>& dR_dv,
+ const Tensor3<double,3,3,3>& DR,
+ const Dune::array<Tensor3<double,3,3,4>, 3>& dDR_dv) const
+{
+ embeddedLocalGradient = 0;
+
+ // left-multiply the derivative of the third director (in DR[][2][]) with R^T
+ Dune::FieldMatrix<double,3,3> RT_DR3;
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++) {
+ RT_DR3[i][j] = 0;
+ for (int k=0; k<3; k++)
+ RT_DR3[i][j] += R[k][i] * DR[k][2][j];
+ }
+
+ // -----------------------------------------------------------------------------
+
+ Tensor3<double,3,3,4> d_RT_DR3(0);
+ for (size_t v_i=0; v_i<4; v_i++)
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ for (int k=0; k<3; k++)
+ d_RT_DR3[i][j][v_i] += dR_dv[k][i][v_i] * DR[k][2][j] + R[k][i] * dDR_dv[k][2][j][v_i];
+
+ ////////////////////////////////////////////////////////////////////////////////
+ // "Shear--Stretch Energy"
+ ////////////////////////////////////////////////////////////////////////////////
+
+ for (size_t v_i=0; v_i<4; v_i++)
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ embeddedLocalGradient[v_i+3] += mu_ * 0.5 * (RT_DR3[i][j]+RT_DR3[j][i]) * (d_RT_DR3[i][j][v_i] + d_RT_DR3[j][i][v_i]);
+
+
+ ////////////////////////////////////////////////////////////////////////////////
+ // "First-order Drill Energy"
+ ////////////////////////////////////////////////////////////////////////////////
+
+ for (size_t v_i=0; v_i<4; v_i++)
+ for (int i=0; i<3; i++)
+ for (int j=0; j<3; j++)
+ embeddedLocalGradient[v_i+3] += mu_c_ * 0.5 * (RT_DR3[i][j]-RT_DR3[j][i]) * (d_RT_DR3[i][j][v_i] - d_RT_DR3[j][i][v_i]);
+
+ ////////////////////////////////////////////////////////////////////////////////
+ // "Elongational Stretch Energy"
+ ////////////////////////////////////////////////////////////////////////////////
+
+ double factor = 2 * mu_*lambda_ / (2*mu_ + lambda_) * (RT_DR3[0][0] + RT_DR3[1][1] + RT_DR3[2][2]);
+ for (size_t v_i=0; v_i<4; v_i++)
+ for (int i=0; i<3; i++)
+ embeddedLocalGradient[v_i+3] += factor * d_RT_DR3[i][i][v_i];
+
+}
+
#if 0 // out-commented, because not completely implemented yet
template <class GridView, class LocalFiniteElement, int dim>
void CosseratEnergyLocalStiffness<GridView, LocalFiniteElement, dim>::
@@ -631,14 +771,23 @@ assembleGradient(const Entity& element,
derOfGradientWRTCoefficient[ii][jj][kk] += referenceDerivativeDerivative[ii][jj][ll] * jacobianInverseTransposed[kk][ll];
}
-
+ // ---------------------------------------------------------------------
+ Dune::array<Tensor3<double,3,3,4>, 3> dDR_dv;
+ compute_dDR_dv(value, derivative, derOfValueWRTCoefficient, derOfGradientWRTCoefficient, dDR_dv);
+
// Add the local energy density
if (gridDim==2) {
typename TargetSpace::EmbeddedTangentVector tmp(0);
longQuadraticMembraneEnergyGradient(tmp,R,dR_dv,derivative,derOfGradientWRTCoefficient,U);
+ //embeddedLocalGradient[i].axpy(weight * thickness_, tmp);
+
+ tmp = 0;
+ curvatureEnergyGradient(tmp,R,DR,dDR_dv);
embeddedLocalGradient[i].axpy(weight * thickness_, tmp);
- //energy += weight * thickness_ * curvatureEnergyGradient(DR);
- //energy += weight * std::pow(thickness_,3) / 12.0 * bendingEnergyGradient(R,DR);
+
+ tmp = 0;
+ bendingEnergyGradient(tmp,R,dR_dv,DR,dDR_dv);
+ //embeddedLocalGradient[i].axpy(weight * std::pow(thickness_,3) / 12.0, tmp);
} else if (gridDim==3) {
assert(gridDim==2); // 3d not implemented yet
// energy += weight * quadraticMembraneEnergyGradient(U);
@@ -646,13 +795,16 @@ assembleGradient(const Entity& element,
} else
DUNE_THROW(Dune::NotImplemented, "CosseratEnergyStiffness for 1d grids");
+ }
+
+ }
//////////////////////////////////////////////////////////////////////////////
// Assemble boundary contributions
//////////////////////////////////////////////////////////////////////////////
assert(neumannFunction_);
-#if 0
+
for (typename Entity::LeafIntersectionIterator it = element.ileafbegin(); it != element.ileafend(); ++it) {
if (not neumannBoundary_ or not neumannBoundary_->contains(*it))
@@ -668,9 +820,6 @@ assembleGradient(const Entity& element,
const double integrationElement = it->geometry().integrationElement(quad[pt].position());
- // The value of the local function
- RigidBodyMotion<double,dim> value = localGeodesicFEFunction.evaluate(quadPos);
-
// Value of the Neumann data at the current position
Dune::FieldVector<double,3> neumannValue;
@@ -679,16 +828,24 @@ assembleGradient(const Entity& element,
else
neumannFunction_->evaluate(it->geometry().global(quad[pt].position()), neumannValue);
- // Constant Neumann force in z-direction
- energy += thickness_ * (neumannValue * value.r) * quad[pt].weight() * integrationElement;
+ // loop over all the element's degrees of freedom and compute the gradient wrt it
+ for (size_t i=0; i<localSolution.size(); i++) {
+
+ // --------------------------------------------------------------------
+ Dune::FieldMatrix<double,7,7> derOfValueWRTCoefficient;
+ localGeodesicFEFunction.evaluateDerivativeOfValueWRTCoefficient(quadPos, i, derOfValueWRTCoefficient);
+
+ // Only translational dofs are affected by the Neumann force
+ for (size_t v_i=0; v_i<3; v_i++)
+ for (size_t j=0; j<3; j++)
+ embeddedLocalGradient[i][v_i] += thickness_ * (neumannValue[j] * derOfValueWRTCoefficient[j][v_i]) * quad[pt].weight() * integrationElement;
+ }
}
}
-#endif
- }
-
- }
+
+
// transform to coordinates on the tangent space
localGradient.resize(embeddedLocalGradient.size());
--
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