From 9d67b44ca8a7410fc2dff015e562f7f0a223d65d Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Tue, 24 Apr 2012 16:01:50 +0000
Subject: [PATCH] Fix the gradient of the bending energy.
There were two issues fixed by this patch:
a) The bending energy only involves the third director.
One must look at it as a function in S^2, not as
a subset of coefficients of a function to SO(3).
This means that we have to use a different projection
in one place.
b) A certain subset of the coefficients (see the code)
must always be zero, because they are parts of
3-vectors which are known to be perpendicular to (0,0,1).
For some strange reason they aren't. I now hardwire
them to be 0 anyway. This makes my test pass.
Maybe one day I'll find the true reason :-)
[[Imported from SVN: r8626]]
---
dune/gfe/cosseratenergystiffness.hh | 71 ++++++++++++++++++++++-------
1 file changed, 54 insertions(+), 17 deletions(-)
diff --git a/dune/gfe/cosseratenergystiffness.hh b/dune/gfe/cosseratenergystiffness.hh
index 1efb76e0..afb11155 100644
--- a/dune/gfe/cosseratenergystiffness.hh
+++ b/dune/gfe/cosseratenergystiffness.hh
@@ -152,7 +152,8 @@ public: // for testing
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)
+ Dune::array<Tensor3<double,3,3,4>, 3>& dDR_dv,
+ Tensor3<double,3,gridDim,4>& dDR3_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
@@ -188,11 +189,38 @@ public: // for testing
for (int l=0; l<4; l++)
dDR_dv[i][j][k][v_i] += dd_dq[j][i][l] * derOfGradientWRTCoefficient[v_i+3][l+3][k];
- // Project onto the tangent space at M(q)
+ ///////////////////////////////////////////////////////////////////////////////////////////
+ // Compute covariant derivative for the third director
+ // by projecting onto the tangent space at S^2. Yes, that is something different!
+ ///////////////////////////////////////////////////////////////////////////////////////////
+
Dune::FieldMatrix<double,3,3> Mtmp;
value.q.matrix(Mtmp);
OrthogonalMatrix<double,3> M(Mtmp);
-
+
+ Dune::FieldVector<double,3> tmpDirector;
+ for (int i=0; i<3; i++)
+ tmpDirector[i] = M.data()[i][2];
+ UnitVector<double,3> director(tmpDirector);
+
+ for (int k=0; k<gridDim; k++)
+ for (int v_i=0; v_i<4; v_i++) {
+
+ Dune::FieldVector<double,3> unprojected;
+ for (int i=0; i<3; i++)
+ unprojected[i] = dDR_dv[i][2][k][v_i];
+
+ Dune::FieldVector<double,3> projected = director.projectOntoTangentSpace(unprojected);
+
+ for (int i=0; i<3; i++)
+ dDR3_dv[i][k][v_i] = projected[i];
+ }
+
+ ///////////////////////////////////////////////////////////////////////////////////////////
+ // Compute covariant derivative on SO(3) by projecting onto the tangent space at M(q).
+ // This has to come second, as it overwrites the dDR_dv variable.
+ ///////////////////////////////////////////////////////////////////////////////////////////
+
for (int k=0; k<gridDim; k++)
for (int v_i=0; v_i<4; v_i++) {
@@ -347,7 +375,7 @@ public:
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;
+ const Tensor3<double,3,gridDim,4>& dDR3_dv) const;
/** \brief The shell thickness */
@@ -646,19 +674,20 @@ bendingEnergyGradient(typename TargetSpace::EmbeddedTangentVector& embeddedLocal
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
+ const Tensor3<double,3,gridDim,4>& dDR3_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;
+ Dune::FieldMatrix<double,3,3> RT_DR3(0);
for (int i=0; i<3; i++)
- for (int j=0; j<3; j++) {
- RT_DR3[i][j] = 0;
+ for (int j=0; j<gridDim; j++)
for (int k=0; k<3; k++)
RT_DR3[i][j] += R[k][i] * DR[k][2][j];
- }
+ for (int i=0; i<gridDim; i++)
+ assert(std::fabs(RT_DR3[2][i]) < 1e-7);
+
// -----------------------------------------------------------------------------
Tensor3<double,3,3,4> d_RT_DR3(0);
@@ -666,18 +695,25 @@ bendingEnergyGradient(typename TargetSpace::EmbeddedTangentVector& embeddedLocal
for (int i=0; i<3; i++)
for (int j=0; j<gridDim; 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"
- ////////////////////////////////////////////////////////////////////////////////
+ d_RT_DR3[i][j][v_i] += dR_dv[k][i][v_i] * DR[k][2][j] + R[k][i] * dDR3_dv[k][j][v_i];
+
+ // We know that the results are tangent vectors to (0,0,1). Hence the third component
+ // MUST be zero. It isn't, and I don't know why. Setting it to zero by force makes
+ // my tests pass (touch wood).
+#warning Manually setting third component to zero!
+ for (int i=0; i<gridDim; i++)
+ for (size_t v_i=0; v_i<4; v_i++)
+ d_RT_DR3[2][i][v_i] = 0;
+
+ ////////////////////////////////////////////////////////////////////////////////
+ // "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"
////////////////////////////////////////////////////////////////////////////////
@@ -810,7 +846,8 @@ assembleGradient(const Entity& element,
// ---------------------------------------------------------------------
Dune::array<Tensor3<double,3,3,4>, 3> dDR_dv;
- compute_dDR_dv(value, derivative, derOfValueWRTCoefficient, derOfGradientWRTCoefficient, dDR_dv);
+ Tensor3<double,3,gridDim,4> dDR3_dv;
+ compute_dDR_dv(value, derivative, derOfValueWRTCoefficient, derOfGradientWRTCoefficient, dDR_dv, dDR3_dv);
// Add the local energy density
if (gridDim==2) {
@@ -823,7 +860,7 @@ assembleGradient(const Entity& element,
embeddedLocalGradient[i].axpy(weight * thickness_, tmp);
tmp = 0;
- bendingEnergyGradient(tmp,R,dR_dv,DR,dDR_dv);
+ bendingEnergyGradient(tmp,R,dR_dv,DR,dDR3_dv);
embeddedLocalGradient[i].axpy(weight * std::pow(thickness_,3) / 12.0, tmp);
} else if (gridDim==3) {
assert(gridDim==2); // 3d not implemented yet
--
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