void evaluate(
const ConstGeometricalDataSlice<CoordinateType>& testGeomData,
const ConstGeometricalDataSlice<CoordinateType>& trialGeomData,
CollectionOf2dSlicesOfNdArrays<ValueType>& result) const {
const int coordCount = 3;
assert(testGeomData.dimWorld() == coordCount);
assert(result.size() == 1);
CoordinateType distanceSq = 0.;
CoordinateType nTest_diff = 0;
CoordinateType nTrial_diff = 0;
CoordinateType nTest_nTrial = 0.;
for (int coordIndex = 0; coordIndex < coordCount; ++coordIndex) {
CoordinateType diff = trialGeomData.global(coordIndex) -
testGeomData.global(coordIndex);
distanceSq += diff * diff;
nTest_diff += diff * testGeomData.normal(coordIndex);
nTrial_diff += diff * trialGeomData.normal(coordIndex);
nTest_nTrial += testGeomData.normal(coordIndex) *
trialGeomData.normal(coordIndex);
}
CoordinateType distance = sqrt(distanceSq);
CoordinateType commonFactor =
static_cast<CoordinateType>(-1. / (4. * M_PI)) /
(distance * distanceSq * distanceSq);
result[0](0, 0) = commonFactor * (
nTest_nTrial * distanceSq -
static_cast<CoordinateType>(3.) * nTest_diff * nTrial_diff);
}
void evaluate(
const ConstBasisDataSlice<ValueType>& basisData,
const ConstGeometricalDataSlice<CoordinateType>& geomData,
_1dSliceOf3dArray<ValueType>& result) const {
assert(basisData.componentCount() == 1);
const int dimWorld = geomData.dimWorld();
for (int dim = 0; dim < dimWorld; ++dim)
result(dim) = basisData.values(0) * geomData.normal(dim);
}
void evaluate(const ConstGeometricalDataSlice<CoordinateType> &testGeomData,
const ConstGeometricalDataSlice<CoordinateType> &trialGeomData,
CollectionOf2dSlicesOfNdArrays<ValueType> &result) const {
const int coordCount = 3;
assert(testGeomData.dimWorld() == coordCount);
assert(result.size() == 1);
ValueType sum = 0;
for (int coordIndex = 0; coordIndex < coordCount; ++coordIndex) {
ValueType diff =
testGeomData.global(coordIndex) - trialGeomData.global(coordIndex);
sum += diff * diff;
}
result[0](0, 0) = static_cast<CoordinateType>(1. / (4. * M_PI)) / sqrt(sum);
}
void evaluate(
const ConstBasisDataSlice<ValueType>& basisData,
const ConstGeometricalDataSlice<CoordinateType>& geomData,
_1dSliceOf3dArray<ValueType>& result) const {
assert(basisData.componentCount() == 1);
assert(geomData.dimWorld() == 3);
// result := gradient of the shape function extended outside
// the surface so that its normal derivative on the surf. is zero
result(0) = basisData.derivatives(0, 0) * geomData.jacobianInverseTransposed(0, 0) +
basisData.derivatives(0, 1) * geomData.jacobianInverseTransposed(0, 1);
result(1) = basisData.derivatives(0, 0) * geomData.jacobianInverseTransposed(1, 0) +
basisData.derivatives(0, 1) * geomData.jacobianInverseTransposed(1, 1);
result(2) = basisData.derivatives(0, 0) * geomData.jacobianInverseTransposed(2, 0) +
basisData.derivatives(0, 1) * geomData.jacobianInverseTransposed(2, 1);
}
void evaluate(
const ConstGeometricalDataSlice<CoordinateType>& testGeomData,
const ConstGeometricalDataSlice<CoordinateType>& trialGeomData,
CollectionOf2dSlicesOfNdArrays<ValueType>& result) const {
const int coordCount = 3;
assert(testGeomData.dimWorld() == coordCount);
assert(result.size() == 1);
/* Define coefficients for the regular part of the periodic Green's kernel for the 3D Laplace equation in a periodic box of size 1.
* We use here an expansion with harmonics up to degree 20, which yields an approximation correct up to precision 1e-10.
* The singular part contains the 27 Green's function for the neighbouring cells and a corrective term for a uniform neutralizing
* background charge.
*/
const int deg_max = 9;
const CoordinateType Harmonic_Coefficients[34] = {
0.01101709645861367,
0.01316794887913183,
5.954412827470426e-05,
-0.0002227937273979395,
0.0002062746586075765,
0.0001551394663489125,
0.0002363741367920927,
5.288864558642977e-06,
-1.065878569810829e-05,
-1.268649253304447e-05,
2.855411046592645e-05,
2.503036078346336e-05,
1.043475591993333e-05,
3.395059459839415e-05,
1.255601659344675e-06,
-1.846641841052362e-06,
-1.981606873099481e-06,
-2.406130170392715e-06,
1.25747200483749e-06,
6.475400668217489e-07,
7.957080434716371e-07,
9.28658402338717e-07,
1.321949011147113e-06,
9.74822898071474e-08,
-1.330176890868469e-07,
-4.957180138283109e-08,
-2.127820763047173e-07,
-1.370001191920438e-07,
2.762838013591894e-08,
-5.682089767011293e-09,
1.844902148801811e-08,
2.90202283948248e-08,
3.429155607637934e-08,
1.998351880226059e-08
};
const CoordinateType rescaling = static_cast<CoordinateType>( 3/(2.*M_PI) + 3/(sqrt(2)*M_PI) + 2/(sqrt(3)*M_PI));
CoordinateType ri2[3*coordCount];
CoordinateType r2 = 0;
CoordinateType diff[3];
ValueType res = 0;
const CoordinateType common = static_cast<CoordinateType>(1. / (4. * M_PI));
const CoordinateType ONE = static_cast<CoordinateType>(1.);
for (int coordIndex = 0; coordIndex < coordCount; ++coordIndex) {
diff[coordIndex] = boost::math::tools::fmod_workaround(testGeomData.global(coordIndex)
- trialGeomData.global(coordIndex)
+ static_cast<CoordinateType>(0.5), ONE);
if (diff[coordIndex] > 0)
diff[coordIndex] -= static_cast<CoordinateType>(0.5);
else
diff[coordIndex] += static_cast<CoordinateType>(0.5); // Periodization
ri2[ 3*coordIndex] = (diff[coordIndex]-ONE)*(diff[coordIndex]-ONE);
ri2[1+3*coordIndex] = diff[coordIndex]*diff[coordIndex];
ri2[2+3*coordIndex] = (diff[coordIndex]+ONE)*(diff[coordIndex]+ONE);
r2 += ri2[1+3*coordIndex];
}
// Loop over the 27 adjacent cells (!)
for (int i = 0; i < 3; ++i)
{
for (int j = 3; j < 6; ++j)
{
for (int k = 6; k < 9; ++k)
{ res += common / sqrt(ri2[i] + ri2[j] + ri2[k]); }
}
}
// corrective term for a neutral background charge
res += static_cast<CoordinateType>(1./6.) * r2 - rescaling;
// Approximation of the correction using spherical harmonics
int ind = 0;
// Transform to spherical coordinates
CoordinateType rdegree = r2;
CoordinateType rfactor = sqrt(r2), theta = acos(diff[2]/sqrt(r2)), phi = atan2(diff[1], diff[0]);
for (int degree = 4; degree < deg_max; degree += 2)
{
rdegree = rdegree * r2;
for (int order = 0; order < degree + 1; order += 4)
{
res += Harmonic_Coefficients[ind] * boost::math::spherical_harmonic_r(degree, order, theta, phi) * rdegree;
++ind;
}
}
result[0](0, 0) = res;
}