void
RankTwoTensorTest::d2thirdInvariantTest()
{
// Here i do a finite-difference calculation of the third
// derivative and compare with the closed-solution form
Real ep = 1E-5; // small finite-difference parameter
RankTwoTensor d1; // first derivative of thirdInvariant - from RankTwoTensor - do a finite-difference of this
RankFourTensor d2; // third derivative of third Invariant - from RankTwoTensor
RankTwoTensor mep; // matrix with shifted entries
RankTwoTensor d1ep; // first derivative of thirdInvariant of mep
mep = _m3;
d1 = _m3.dthirdInvariant();
d2 = _m3.d2thirdInvariant();
for (unsigned i = 0; i < 3; i++)
for (unsigned j = 0; j < 3; j++)
{
for (unsigned int k = 0; k < 3; k++)
for (unsigned int l = 0; l < 3; l++)
{
mep(k, l) += ep;
d1ep = mep.dthirdInvariant();
CPPUNIT_ASSERT_DOUBLES_EQUAL((d1ep(i, j) - d1(i, j))/ep, d2(i, j, k, l), ep);
mep(k, l) -= ep;
}
}
mep = _unsymmetric1;
d1 = _unsymmetric1.dthirdInvariant();
d2 = _unsymmetric1.d2thirdInvariant();
for (unsigned i = 0; i < 3; i++)
for (unsigned j = 0; j < 3; j++)
{
for (unsigned int k = 0; k < 3; k++)
for (unsigned int l = 0; l < 3; l++)
{
mep(k, l) += ep;
d1ep = mep.dthirdInvariant();
CPPUNIT_ASSERT_DOUBLES_EQUAL((d1ep(i, j) - d1(i, j))/ep, d2(i, j, k, l), ep);
mep(k, l) -= ep;
// note that because d1 and d2 explicitly symmeterise the matrix
// the derivative may or may not explicitly symmeterise
mep(k, l) += 0.5*ep;
mep(l, k) += 0.5*ep;
d1ep = mep.dthirdInvariant();
CPPUNIT_ASSERT_DOUBLES_EQUAL((d1ep(i, j) - d1(i, j))/ep, d2(i, j, k, l), ep);
mep(k, l) -= 0.5*ep;
mep(l, k) -= 0.5*ep;
}
}
}
