protected override double[] FindRoots(LinearPSI <Cube> psi, MultidimensionalArray X, int heightDirection, double[] bounds, int cell)
{
MultidimensionalArray XonPsi = psi.ProjectOnto(X);
XonPsi = XonPsi.ExtractSubArrayShallow(0, -1);
double[] start = XonPsi.To1DArray();
double[] end = XonPsi.To1DArray();
start[heightDirection] = -1;
end[heightDirection] = 1;
HMF.LineSegment line = new HMF.LineSegment(3, RefElement, start, end);
LevelSet levelSet = lsData.LevelSet as LevelSet;
line.ProjectBasisPolynomials(levelSet.Basis);
double[] roots = rootFinder.GetRoots(line, levelSet, cell, this.iKref);
return(roots);
}
/// <summary>
/// Gathers the line segments bounding <see cref="RefElement"/> in
/// reference coordinates.
/// </summary>
/// <returns>
/// An array of line segments
/// <list type="bullet">
/// <item>1st index: Edge index</item>
/// <item>2nd index: Edge of edge index</item>
/// </list>
/// </returns>
/// <remarks>
/// Each line segment will appear twice in the result, since it is
/// stored for each edge separately. However, this method ensures that
/// corresponding segments are represented by the same object (reference
/// equality) which ensures that this does not affect performance.
/// </remarks>
private LineSegment[,] GetReferenceLineSegments()
{
int D = lsData.GridDat.SpatialDimension;
int noOfEdges = lsData.GridDat.Grid.RefElements[0].NoOfFaces;
int noOfEdgesOfEdge = RefElement.FaceRefElement.NoOfFaces;
MultidimensionalArray edgeOfEdgeVertices = MultidimensionalArray.Create(2, 1);
edgeOfEdgeVertices[0, 0] = -1.0;
edgeOfEdgeVertices[1, 0] = 1.0;
MultidimensionalArray edgeVertices = MultidimensionalArray.Create(
edgeOfEdgeVertices.GetLength(0), RefElement.FaceRefElement.SpatialDimension);
MultidimensionalArray volumeVertices = MultidimensionalArray.Create(
edgeVertices.GetLength(0), D);
// Remember encountered segments so that $lineSegments contains no
// duplicates and roots can be cached efficiently
Dictionary <LineSegment, LineSegment> seenSegments = new Dictionary <LineSegment, LineSegment>(
noOfEdges * noOfEdgesOfEdge / 2);
LineSegment[,] lineSegments = new LineSegment[noOfEdges, noOfEdgesOfEdge];
LevelSet levelSetField = lsData.LevelSet as LevelSet;
for (int ee = 0; ee < noOfEdgesOfEdge; ee++)
{
RefElement.FaceRefElement.TransformFaceCoordinates(ee, edgeOfEdgeVertices, edgeVertices);
for (int e = 0; e < noOfEdges; e++)
{
lsData.GridDat.Grid.RefElements[0].TransformFaceCoordinates(
e, edgeVertices, volumeVertices);
//double[] start = new double[D];
//double[] end = new double[D];
//for (int d = 0; d < D; d++) {
// start[d] = volumeVertices[0, d];
// end[d] = volumeVertices[1, d];
//}
var start = volumeVertices.GetRowPt(0);
var end = volumeVertices.GetRowPt(1);
LineSegment newSegment = new LineSegment(D, this.RefElement, start, end, rootFindingAlgorithm: RootFindingAlgorithm);
// Assert that the segment does not already exist
LineSegment segment;
if (!seenSegments.TryGetValue(newSegment, out segment))
{
segment = newSegment;
seenSegments.Add(segment, segment);
if (levelSetField != null)
{
segment.ProjectBasisPolynomials(levelSetField.Basis);
}
//tracker.Subscribe(segment);
}
lineSegments[e, ee] = segment;
}
}
return(lineSegments);
}
/// <summary>
/// Returns a set of <see cref="CellEdgeBoundaryQuadRule"/>s that
/// enables the integration over sub-segments of the edges of the edges
/// of a (three-dimensional) domain. This is obviously only useful if
/// the integrand has a discontinuity that is aligned with the zero
/// iso-contour of the level set function.
/// </summary>
/// <param name="mask"></param>
/// <param name="order"></param>
/// <returns></returns>
public IEnumerable <IChunkRulePair <CellEdgeBoundaryQuadRule> > GetQuadRuleSet(ExecutionMask mask, int order)
{
if (mask == null)
{
mask = CellMask.GetFullMask(this.lsData.GridDat, MaskType.Geometrical);
}
if (mask is CellMask == false)
{
throw new ArgumentException("Edge mask required", "mask");
}
if (mask.MaskType != MaskType.Geometrical)
{
throw new ArgumentException("Expecting a geometrical mask.");
}
if (lastOrder != order)
{
cache.Clear();
}
QuadRule baseRule = lineSimplex.GetQuadratureRule(order);
int D = lsData.GridDat.SpatialDimension;
int noOfEdges = lsData.GridDat.Grid.RefElements[0].NoOfFaces;
int noOfEdgesOfEdge = RefElement.FaceRefElement.NoOfFaces;
var result = new List <ChunkRulePair <CellEdgeBoundaryQuadRule> >(mask.NoOfItemsLocally);
foreach (Chunk chunk in mask)
{
for (int i = 0; i < chunk.Len; i++)
{
int cell = i + chunk.i0;
if (cache.ContainsKey(cell))
{
result.Add(new ChunkRulePair <CellEdgeBoundaryQuadRule>(
Chunk.GetSingleElementChunk(cell),
cache[cell]));
continue;
}
List <Vector> nodes = new List <Vector>();
List <double> weights = new List <double>();
if (lsData.GridDat.Cells.Cells2Edges[cell].Length != noOfEdges)
{
throw new NotImplementedException("Not implemented for hanging nodes");
}
int[] noOfNodesPerEdge = new int[noOfEdges];
int[,] noOfNodesPerEdgeOfEdge = new int[noOfEdges, noOfEdgesOfEdge];
for (int e = 0; e < noOfEdges; e++)
{
int edge = Math.Abs(lsData.GridDat.Cells.Cells2Edges[cell][e]) - 1;
double edgeDet = lsData.GridDat.Edges.SqrtGramian[edge];
for (int ee = 0; ee < noOfEdgesOfEdge; ee++)
{
LineSegment refSegment = referenceLineSegments[e, ee];
double edgeOfEdgeDet = RefElement.FaceRefElement.FaceTrafoGramianSqrt[ee];
double[] roots = refSegment.GetRoots(lsData.LevelSet, cell, 0);
LineSegment[] subSegments = refSegment.Split(roots);
for (int k = 0; k < subSegments.Length; k++)
{
// Evaluate sub segment at center to determine sign
NodeSet _point = new NodeSet(this.RefElement, subSegments[k].GetPointOnSegment(0.0));
double scaling = edgeOfEdgeDet * subSegments[k].Length / refSegment.Length;
if (jumpType != JumpTypes.Implicit)
{
//using (tracker.GridDat.NSC.CreateLock(
// MultidimensionalArray.CreateWrapper(point, 1, D), 0, -1.0)) {
MultidimensionalArray levelSetValue = lsData.GetLevSetValues(_point, cell, 1);
switch (jumpType)
{
case JumpTypes.Heaviside:
if (levelSetValue[0, 0] <= 0.0)
{
continue;
}
break;
case JumpTypes.OneMinusHeaviside:
if (levelSetValue[0, 0] > 0.0)
{
continue;
}
break;
case JumpTypes.Sign:
scaling *= levelSetValue[0, 0].Sign();
break;
default:
throw new NotImplementedException();
}
}
for (int m = 0; m < baseRule.NoOfNodes; m++)
{
// Base rule _always_ is a line rule, thus Nodes[*, _0_]
var point = subSegments[k].GetPointOnSegment(baseRule.Nodes[m, 0]);
weights.Add(baseRule.Weights[m] * scaling);
nodes.Add(point);
noOfNodesPerEdge[e]++;
noOfNodesPerEdgeOfEdge[e, ee]++;
}
}
}
}
if (weights.Count == 0)
{
CellEdgeBoundaryQuadRule emptyRule =
CellEdgeBoundaryQuadRule.CreateEmpty(1, RefElement);
emptyRule.Nodes.LockForever();
cache.Add(cell, emptyRule);
result.Add(new ChunkRulePair <CellEdgeBoundaryQuadRule>(
Chunk.GetSingleElementChunk(cell), emptyRule));
continue;
}
NodeSet localNodes = new NodeSet(this.RefElement, nodes.Count, D);
for (int j = 0; j < nodes.Count; j++)
{
for (int d = 0; d < D; d++)
{
localNodes[j, d] = nodes[j][d];
}
}
localNodes.LockForever();
CellEdgeBoundaryQuadRule subdividedRule = new CellEdgeBoundaryQuadRule()
{
OrderOfPrecision = order,
Weights = MultidimensionalArray.Create(weights.Count),
Nodes = localNodes,
NumbersOfNodesPerFace = noOfNodesPerEdge,
NumbersOfNodesPerFaceOfFace = noOfNodesPerEdgeOfEdge
};
subdividedRule.Weights.SetSubVector(weights, -1);
cache.Add(cell, subdividedRule);
result.Add(new ChunkRulePair <CellEdgeBoundaryQuadRule>(
Chunk.GetSingleElementChunk(cell), subdividedRule));
}
}
return(result);
}
/// <summary>
/// Uses a safe-guarded Newton method to find all roots on
/// <paramref name="segment"/>
/// </summary>
/// <param name="segment"></param>
/// <param name="levelSet"></param>
/// <param name="cell"></param>
/// <param name="iKref"></param>
/// <returns></returns>
public double[] GetRoots(LineSegment segment, ILevelSet levelSet, int cell, int iKref)
{
LevelSet levelSetField = levelSet as LevelSet;
if (levelSetField == null)
{
throw new NotImplementedException("Method currently only works for polynomial level sets");
}
int maxNoOfCoefficientsPerDimension = levelSetField.Basis.Degree + 1;
int noOfPolynomials = segment.ProjectedPolynomialCoefficients.GetLength(1);
double[] coefficients = new double[maxNoOfCoefficientsPerDimension];
for (int i = 0; i < noOfPolynomials; i++)
{
double dgCoefficient = levelSetField.Coordinates[cell, i];
for (int j = 0; j < maxNoOfCoefficientsPerDimension; j++)
{
coefficients[j] += dgCoefficient * segment.ProjectedPolynomialCoefficients[iKref, i, j];
}
}
//if(coefficients.L2NormPow2() < this.Tolerance) {
// // ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// // special case :
// // the zero-level-set is probably parallel to this line segment
// // ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// return new double[] { -1.0, 1.0 };
//}
double[] roots;
unsafe
{
fixed(double *pCoeff = &coefficients[coefficients.Length - 1])
{
double xLow = -1.0;
double xHigh = 1.0;
int NO_OF_BRACKETS = 8;
List <double> lowerBounds = new List <double>();
List <double> upperBounds = new List <double>();
double dx2 = 2.0 / NO_OF_BRACKETS;
double x = xLow;
double fOld = Eval(x, pCoeff, coefficients.Length);
for (int i = 0; i < NO_OF_BRACKETS; i++)
{
x += dx2;
double fNew = Eval(x, pCoeff, coefficients.Length);
if (i == 0 && fOld.Abs() < Tolerance)
{
lowerBounds.Add(x - dx2);
upperBounds.Add(x);
fOld = fNew;
continue;
}
else
{
if (fNew.Abs() < Tolerance)
{
lowerBounds.Add(x - dx2);
upperBounds.Add(x);
x += dx2;
fOld = Eval(x, pCoeff, coefficients.Length);
continue;
}
}
if (fNew * fOld <= 0.0)
{
lowerBounds.Add(x - dx2);
upperBounds.Add(x);
}
fOld = fNew;
}
// Actual Newton-Raphson
int MAX_ITERATIONS = 50;
roots = new double[lowerBounds.Count];
for (int j = 0; j < lowerBounds.Count; j++)
{
xLow = lowerBounds[j];
xHigh = upperBounds[j];
double fLeft = Eval(xLow, pCoeff, coefficients.Length);
double fRight = Eval(xHigh, pCoeff, coefficients.Length);
if (fLeft.Abs() < Tolerance)
{
roots[j] = xLow;
break;
}
if (fRight.Abs() < Tolerance)
{
roots[j] = xHigh;
break;
}
if (fLeft.Sign() == fRight.Sign())
{
throw new Exception();
}
if (fLeft > 0.0)
{
xLow = upperBounds[j];
xHigh = lowerBounds[j];
}
double root = 0.5 * (xLow + xHigh);
double f;
double df;
Eval(root, pCoeff, coefficients.Length, out f, out df);
double dxOld = (xHigh - xLow).Abs();
double dx = dxOld;
int i = 0;
while (true)
{
if (i > MAX_ITERATIONS)
{
throw new Exception("Max iterations exceeded");
}
double a = ((root - xHigh) * df - f) * ((root - xLow) * df - f);
if (a > 0.0 || 2.0 * Math.Abs(f) > Math.Abs(dxOld * df))
{
// Newton out of range or too slow -> Bisect
dxOld = dx;
dx = 0.5 * (xHigh - xLow);
root = xLow + dx;
}
else
{
// Take Newton step
dxOld = dx;
dx = -f / df;
//// Convergence acceleration according to Yao2014
//double fDelta = Eval(root + dx, pCoeff, coefficients.Length);
//dx = -(f + fDelta) / df;
root += dx;
}
Eval(root, pCoeff, coefficients.Length, out f, out df);
if (Math.Abs(f) <= Tolerance)
{
roots[j] = root;
break;
}
if (f < 0.0)
{
xLow = root;
}
else
{
xHigh = root;
}
i++;
}
}
}
}
return(roots);
}
/// <summary>
/// Finds the roots using <see cref="GSL.gsl_poly_complex_solve"/>
/// </summary>
/// <param name="segment"></param>
/// <param name="levelSet"></param>
/// <param name="cell"></param>
/// <param name="iKref"></param>
/// <returns></returns>
public double[] GetRoots(LineSegment segment, ILevelSet levelSet, int cell, int iKref)
{
LevelSet levelSetField = levelSet as LevelSet;
if (levelSetField == null)
{
throw new NotImplementedException("Method currently only works for polynomial level sets");
}
int maxNoOfCoefficientsPerDimension = levelSetField.Basis.Degree + 1;
int noOfPolynomials = segment.ProjectedPolynomialCoefficients.GetLength(1);
double[] coefficients = new double[maxNoOfCoefficientsPerDimension];
for (int i = 0; i < noOfPolynomials; i++)
{
double dgCoefficient = levelSetField.Coordinates[cell, i];
for (int j = 0; j < maxNoOfCoefficientsPerDimension; j++)
{
coefficients[j] += dgCoefficient * segment.ProjectedPolynomialCoefficients[iKref, i, j];
}
}
// Make sure "leading" coefficient (i.e., the last element of the
// list of coefficients) is not too small since this will make gsl
// crash (or lead to bogus results)
int newLength = coefficients.Length;
while (newLength > 0 && Math.Abs(coefficients[newLength - 1]) < EPSILON)
{
newLength--;
}
if (newLength != coefficients.Length)
{
coefficients = coefficients.Take(newLength).ToArray();
}
// Make sure polynomial is not constant since this will make gsl crash
if (newLength < 2)
{
return(new double[0]);
}
int maxNoOfRoots = coefficients.Length - 1;
double[] roots = new double[2 * maxNoOfRoots];
GCHandle coefficientHandle = GCHandle.Alloc(coefficients, GCHandleType.Pinned);
GCHandle rootsHandle = GCHandle.Alloc(roots, GCHandleType.Pinned);
IntPtr workSpace = GSL.gsl_poly_complex_workspace_alloc(coefficients.Length);
GSL.gsl_poly_complex_solve(
coefficientHandle.AddrOfPinnedObject(),
coefficients.Length,
workSpace,
rootsHandle.AddrOfPinnedObject());
GSL.gsl_poly_complex_workspace_free(workSpace);
coefficientHandle.Free();
rootsHandle.Free();
List <double> realRoots = new List <double>(levelSetField.Basis.Degree);
for (int i = 0; i < maxNoOfRoots; i++)
{
double realPart = roots[2 * i];
double imaginaryPart = roots[2 * i + 1];
// Exclude |$realPart| == 1.0 since it doesn't matter
if (imaginaryPart == 0.0 && Math.Abs(realPart) <= 1.0 - EPSILON)
{
realRoots.Add(realPart);
}
}
return(realRoots.OrderBy(d => d).ToArray());
}