/// <summary>
/// Constructs suitable quadrature rules cells in
/// <paramref name="mask"/>.
/// </summary>
/// <param name="mask">
/// Cells for which quadrature rules shall be created
/// </param>
/// <param name="order">
/// Desired order of the moment-fitting system. Assuming that
/// <see cref="surfaceRuleFactory"/> integrates the basis polynomials
/// exactly over the zero iso-contour (which it usually
/// doesn't!), the resulting quadrature rules will be exact up to this
/// order.
/// </param>
/// <returns>A set of quadrature rules</returns>
/// <remarks>
/// Since the selected level set is generally discontinuous across cell
/// boundaries, this method does not make use of the fact that
/// neighboring cells share edges. That is, the optimization will be
/// performed twice for each inner edge in <paramref name="mask"/>.
/// </remarks>
public IEnumerable <IChunkRulePair <QuadRule> > GetQuadRuleSet(ExecutionMask mask, int order)
{
using (var tr = new FuncTrace()) {
CellMask cellMask = mask as CellMask;
if (cellMask == null)
{
throw new ArgumentException("Mask must be a volume mask", "mask");
}
// Note: This is a parallel call, so do this early to avoid parallel confusion
localCellIndex2SubgridIndex = new SubGrid(cellMask).LocalCellIndex2SubgridIndex;
int maxLambdaDegree = order + 1;
int noOfLambdas = GetNumberOfLambdas(maxLambdaDegree);
int noOfEdges = LevelSetData.GridDat.Grid.RefElements[0].NoOfFaces;
int D = RefElement.SpatialDimension;
// Get the basis polynomials and integrate them analytically
Polynomial[] basePolynomials = RefElement.GetOrthonormalPolynomials(order).ToArray();
Polynomial[] polynomials = new Polynomial[basePolynomials.Length * D];
for (int i = 0; i < basePolynomials.Length; i++)
{
Polynomial p = basePolynomials[i];
for (int d = 0; d < D; d++)
{
Polynomial pNew = p.CloneAs();
for (int j = 0; j < p.Coeff.Length; j++)
{
pNew.Exponents[j, d]++;
pNew.Coeff[j] /= pNew.Exponents[j, d];
pNew.Coeff[j] /= D; // Make sure divergence is Phi again
}
polynomials[i * D + d] = pNew;
}
}
// basePolynomials[i] == div(polynomials[i*D], ... , polynomials[i*D + D - 1])
lambdaBasis = new PolynomialList(polynomials);
if (RestrictNodes)
{
trafos = new AffineTrafo[mask.NoOfItemsLocally];
foreach (Chunk chunk in mask)
{
foreach (var cell in chunk.Elements.AsSmartEnumerable())
{
CellMask singleElementMask = new CellMask(
LevelSetData.GridDat, Chunk.GetSingleElementChunk(cell.Value));
LineAndPointQuadratureFactory.LineQRF lineFactory = this.edgeRuleFactory as LineAndPointQuadratureFactory.LineQRF;
if (lineFactory == null)
{
throw new Exception();
}
var lineRule = lineFactory.GetQuadRuleSet(singleElementMask, order).Single().Rule;
var pointRule = lineFactory.m_Owner.GetPointFactory().GetQuadRuleSet(singleElementMask, order).Single().Rule;
// Also add point rule points since line rule points
// are constructed from Gauss rules that do not include
// the end points
BoundingBox box = new BoundingBox(lineRule.Nodes);
box.AddPoints(pointRule.Nodes);
int noOfRoots = pointRule.Nodes.GetLength(0);
if (noOfRoots <= 1)
{
// Cell is considered cut because the level set
// is very close, but actually isn't. Note that
// we can NOT omit the cell (as in the surface
// case) as it will be missing in the list of
// uncut cells, i.e. this cell would be ignored
// completely
trafos[localCellIndex2SubgridIndex[cell.Value]] =
AffineTrafo.Identity(RefElement.SpatialDimension);
continue;
}
else if (noOfRoots == 2)
{
// Go a bit into the direction of the normal
// from the center between the nodes in order
// not to miss regions with strong curvature
double[] center = box.Min.CloneAs();
center.AccV(1.0, box.Max);
center.ScaleV(0.5);
NodeSet centerNode = new NodeSet(RefElement, center);
centerNode.LockForever();
MultidimensionalArray normal = LevelSetData.GetLevelSetReferenceNormals(centerNode, cell.Value, 1);
MultidimensionalArray dist = LevelSetData.GetLevSetValues(centerNode, cell.Value, 1);
double scaling = Math.Sqrt(LevelSetData.GridDat.Cells.JacobiDet[cell.Value]);
double[] newPoint = new double[D];
for (int d = 0; d < D; d++)
{
newPoint[d] = center[d] - normal[0, 0, d] * dist[0, 0] / scaling;
}
box.AddPoint(newPoint);
// Make sure points stay in box
for (int d = 0; d < D; d++)
{
box.Min[d] = Math.Max(box.Min[d], -1);
box.Max[d] = Math.Min(box.Max[d], 1);
}
}
MultidimensionalArray preImage = RefElement.Vertices.ExtractSubArrayShallow(
new int[] { 0, 0 }, new int[] { D, D - 1 });
MultidimensionalArray image = MultidimensionalArray.Create(D + 1, D);
image[0, 0] = box.Min[0]; // Top left
image[0, 1] = box.Max[1];
image[1, 0] = box.Max[0]; // Top right
image[1, 1] = box.Max[1];
image[2, 0] = box.Min[0]; // Bottom left;
image[2, 1] = box.Min[1];
AffineTrafo trafo = AffineTrafo.FromPoints(preImage, image);
trafos[localCellIndex2SubgridIndex[cell.Value]] = trafo;
}
}
}
LambdaCellBoundaryQuadrature cellBoundaryQuadrature =
new LambdaCellBoundaryQuadrature(this, edgeRuleFactory, cellMask);
cellBoundaryQuadrature.Execute();
LambdaLevelSetSurfaceQuadrature surfaceQuadrature =
new LambdaLevelSetSurfaceQuadrature(this, surfaceRuleFactory, cellMask);
surfaceQuadrature.Execute();
// Must happen _after_ all parallel calls (e.g., definition of
// the sub-grid or quadrature) in order to avoid problems in
// parallel runs
if (mask.NoOfItemsLocally == 0)
{
var empty = new ChunkRulePair <QuadRule> [0];
return(empty);
}
if (cachedRules.ContainsKey(order))
{
order = cachedRules.Keys.Where(cachedOrder => cachedOrder >= order).Min();
CellMask cachedMask = new CellMask(mask.GridData, cachedRules[order].Select(p => p.Chunk).ToArray());
if (cachedMask.Equals(mask))
{
return(cachedRules[order]);
}
else
{
throw new NotImplementedException(
"Case not yet covered yet in combination with caching; deactivate caching to get rid of this message");
}
}
double[,] quadResults = cellBoundaryQuadrature.Results;
foreach (Chunk chunk in mask)
{
for (int i = 0; i < chunk.Len; i++)
{
int iSubGrid = localCellIndex2SubgridIndex[chunk.i0 + i];
switch (jumpType)
{
case JumpTypes.Heaviside:
for (int k = 0; k < noOfLambdas; k++)
{
quadResults[iSubGrid, k] -= surfaceQuadrature.Results[iSubGrid, k];
}
break;
case JumpTypes.OneMinusHeaviside:
for (int k = 0; k < noOfLambdas; k++)
{
quadResults[iSubGrid, k] += surfaceQuadrature.Results[iSubGrid, k];
}
break;
case JumpTypes.Sign:
for (int k = 0; k < noOfLambdas; k++)
{
quadResults[iSubGrid, k] -= 2.0 * surfaceQuadrature.Results[iSubGrid, k];
}
break;
default:
throw new NotImplementedException();
}
}
}
BitArray voidCellsArray = new BitArray(LevelSetData.GridDat.Cells.NoOfLocalUpdatedCells);
BitArray fullCellsArray = new BitArray(LevelSetData.GridDat.Cells.NoOfLocalUpdatedCells);
foreach (Chunk chunk in cellMask)
{
foreach (var cell in chunk.Elements)
{
double rhsL2Norm = 0.0;
for (int k = 0; k < noOfLambdas; k++)
{
double entry = quadResults[localCellIndex2SubgridIndex[cell], k];
rhsL2Norm += entry * entry;
}
if (rhsL2Norm < 1e-14)
{
// All integrals are zero => cell not really cut
// (level set is tangent) and fully in void region
voidCellsArray[cell] = true;
continue;
}
double l2NormFirstIntegral = quadResults[localCellIndex2SubgridIndex[cell], 0];
l2NormFirstIntegral *= l2NormFirstIntegral;
double rhsL2NormWithoutFirst = rhsL2Norm - l2NormFirstIntegral;
// Beware: This check is only sensible if basis is orthonormal on RefElement!
if (rhsL2NormWithoutFirst < 1e-14 &&
Math.Abs(l2NormFirstIntegral - RefElement.Volume) < 1e-14)
{
// All integrals are zero except integral over first integrand
// If basis is orthonormal, this implies that cell is uncut and
// fully in non-void region since then
// \int_K \Phi_i dV = \int_A \Phi_i dV = \delta_{0,i}
// However, we have to compare RefElement.Volume since
// integration is performed in reference coordinates!
fullCellsArray[cell] = true;
}
}
}
var result = new List <ChunkRulePair <QuadRule> >(cellMask.NoOfItemsLocally);
CellMask emptyCells = new CellMask(LevelSetData.GridDat, voidCellsArray);
foreach (Chunk chunk in emptyCells)
{
foreach (int cell in chunk.Elements)
{
QuadRule emptyRule = QuadRule.CreateEmpty(RefElement, 1, RefElement.SpatialDimension);
emptyRule.Nodes.LockForever();
result.Add(new ChunkRulePair <QuadRule>(
Chunk.GetSingleElementChunk(cell), emptyRule));
}
}
CellMask fullCells = new CellMask(LevelSetData.GridDat, fullCellsArray);
foreach (Chunk chunk in fullCells)
{
foreach (int cell in chunk.Elements)
{
QuadRule fullRule = RefElement.GetQuadratureRule(order);
result.Add(new ChunkRulePair <QuadRule>(
Chunk.GetSingleElementChunk(cell), fullRule));
}
}
CellMask realCutCells = cellMask.Except(emptyCells).Except(fullCells);
if (RestrictNodes)
{
foreach (Chunk chunk in realCutCells)
{
foreach (int cell in chunk.Elements)
{
CellMask singleElementMask = new CellMask(
LevelSetData.GridDat, Chunk.GetSingleElementChunk(cell));
AffineTrafo trafo = trafos[localCellIndex2SubgridIndex[cell]];
Debug.Assert(Math.Abs(trafo.Matrix.Determinant()) > 1e-10);
NodeSet nodes = GetNodes(noOfLambdas).CloneAs();
NodeSet mappedNodes = new NodeSet(RefElement, trafo.Transform(nodes));
mappedNodes.LockForever();
// Remove nodes in negative part
MultidimensionalArray levelSetValues = LevelSetData.GetLevSetValues(mappedNodes, cell, 1);
List <int> nodesToBeCopied = new List <int>(mappedNodes.GetLength(0));
for (int n = 0; n < nodes.GetLength(0); n++)
{
if (levelSetValues[0, n] >= 0.0)
{
nodesToBeCopied.Add(n);
}
}
NodeSet reducedNodes = new NodeSet(
this.RefElement, nodesToBeCopied.Count, D);
for (int n = 0; n < nodesToBeCopied.Count; n++)
{
for (int d = 0; d < D; d++)
{
reducedNodes[n, d] = mappedNodes[nodesToBeCopied[n], d];
}
}
reducedNodes.LockForever();
QuadRule optimizedRule = GetOptimizedRule(
cell,
trafo,
reducedNodes,
quadResults,
order);
result.Add(new ChunkRulePair <QuadRule>(
singleElementMask.Single(), optimizedRule));
}
}
}
else
{
// Use same nodes in all cells
QuadRule[] optimizedRules = GetOptimizedRules(
realCutCells, GetNodes(noOfLambdas), quadResults, order);
int ruleIndex = 0;
foreach (Chunk chunk in realCutCells)
{
foreach (var cell in chunk.Elements)
{
result.Add(new ChunkRulePair <QuadRule>(
Chunk.GetSingleElementChunk(cell), optimizedRules[ruleIndex]));
ruleIndex++;
}
}
}
cachedRules[order] = result.OrderBy(p => p.Chunk.i0).ToArray();
return(cachedRules[order]);
}
}
/// <summary>
/// Uses a moment-fitting basis of order <paramref name="order"/> for
/// determining optimal weights of a quadrature rule for a sub-region
/// of each edge of each cell in the given <paramref name="mask"/>.
/// </summary>
/// <param name="mask">
/// Cells for which rules shall be created
/// </param>
/// <param name="order">
/// Desired order of the moment-fitting system. Assuming that
/// <see cref="edgeSurfaceRuleFactory"/> integrates the basis
/// polynomials exactly over the zero iso-contour (which it usually
/// doesn't), the resulting quadrature rules will be exact up to this
/// order.
/// </param>
/// <returns>
/// A set of quadrature rules
/// </returns>
private CellBoundaryQuadRule[] GetOptimizedRules(CellMask mask, int order)
{
using (var tr = new FuncTrace()) {
int maxLambdaDegree = order + 1;
int noOfLambdas = GetNumberOfLambdas();
int noOfFaces = LevelSetData.GridDat.Grid.RefElements[0].NoOfFaces;
int D = LevelSetData.GridDat.SpatialDimension;
int noOfNodesPerEdge = baseRule.NumbersOfNodesPerFace[0];
CellBoundaryQuadRule[] optimizedRules = new CellBoundaryQuadRule[mask.NoOfItemsLocally];
Debug.Assert(
baseRule.NumbersOfNodesPerFace.Distinct().Count() == 1,
"Assumption violated: Number of nodes varies from edge to edge.");
Debug.Assert(noOfLambdas < noOfNodesPerEdge, "Not enough integration points");
LambdaEdgeBoundaryQuadrature cellBoundaryQuadrature =
new LambdaEdgeBoundaryQuadrature(this, CoFaceQuadRuleFactory, maxLambdaDegree, mask);
cellBoundaryQuadrature.Execute();
double[, ,] boundaryResults = cellBoundaryQuadrature.Results;
LambdaLevelSetSurfaceQuadrature surfaceQuadrature =
new LambdaLevelSetSurfaceQuadrature(this, edgeSurfaceRuleFactory, maxLambdaDegree, mask);
surfaceQuadrature.Execute();
double[, ,] surfaceResults = surfaceQuadrature.Results;
int noOfRhs = 0;
int[,] rhsIndexMap = new int[mask.NoOfItemsLocally, noOfFaces];
foreach (Chunk chunk in mask)
{
MultidimensionalArray levelSetValues = LevelSetData.GetLevSetValues(signTestRule.Nodes, chunk.i0, chunk.Len);
for (int i = 0; i < chunk.Len; i++)
{
int cell = i + chunk.i0;
int iSubGrid = localCellIndex2SubgridIndex[cell];
optimizedRules[iSubGrid] = new CellBoundaryQuadRule()
{
Nodes = baseRule.Nodes,
Weights = MultidimensionalArray.Create(baseRule.NoOfNodes),
NumbersOfNodesPerFace = baseRule.NumbersOfNodesPerFace.CloneAs()
};
int noOfProcessedNodes = 0;
for (int e = 0; e < noOfFaces; e++)
{
int noOfNodesOnEdge = baseRule.NumbersOfNodesPerFace[e];
bool faceIsCut = false;
for (int k = 0; k < noOfLambdas; k++)
{
faceIsCut |= surfaceResults[iSubGrid, e, k].Abs() > 1e-9;
}
//edgeIsCut = tracker.EdgeIsCut[cell, e];
if (!faceIsCut)
{
// Sign is checked in multiple points to avoid
// some weird edge cases. Sign with most
// occurrences on the test nodes wins
int numNeg = 0;
int numPos = 0;
int offset = signTestRule.NumbersOfNodesPerFace.Take(e).Sum();
for (int j = 0; j < signTestRule.NumbersOfNodesPerFace[e]; j++)
{
double val = levelSetValues[i, offset + j];
if (val < 0.0)
{
numNeg++;
}
else if (val > 0.0)
{
numPos++;
}
}
int sign = numPos - numNeg;
if (sign == 0)
{
throw new Exception(String.Format(
"Could not determine sign of face {0} of cell {1}", e, cell));
}
switch (jumpType)
{
case JumpTypes.Heaviside:
if (sign > 0)
{
CopyWeights(baseRule, optimizedRules[iSubGrid], e, 1.0);
}
break;
case JumpTypes.OneMinusHeaviside:
if (sign < 0)
{
CopyWeights(baseRule, optimizedRules[iSubGrid], e, -1.0);
}
break;
case JumpTypes.Sign:
CopyWeights(baseRule, optimizedRules[iSubGrid], e, levelSetValues[i, e].Sign());
break;
default:
throw new NotImplementedException();
}
rhsIndexMap[iSubGrid, e] = -1;
noOfProcessedNodes += noOfNodesOnEdge;
continue;
}
rhsIndexMap[iSubGrid, e] = noOfRhs;
noOfRhs++;
noOfProcessedNodes += noOfNodesOnEdge;
}
}
}
// Leading dimension of B (rhs); required by DGELSY
int LDB = Math.Max(noOfLambdas, noOfNodesPerEdge);
double[] rhs = new double[LDB * noOfRhs];
foreach (Chunk chunk in mask)
{
for (int i = 0; i < chunk.Len; i++)
{
int cell = i + chunk.i0;
int iSubGrid = localCellIndex2SubgridIndex[cell];
for (int e = 0; e < noOfFaces; e++)
{
int rhsIndex = rhsIndexMap[iSubGrid, e];
if (rhsIndex < 0)
{
continue;
}
switch (jumpType)
{
case JumpTypes.Heaviside:
for (int k = 0; k < noOfLambdas; k++)
{
rhs[rhsIndex * LDB + k] = boundaryResults[iSubGrid, e, k] - surfaceResults[iSubGrid, e, k];
}
break;
case JumpTypes.OneMinusHeaviside:
for (int k = 0; k < noOfLambdas; k++)
{
rhs[rhsIndex * LDB + k] = boundaryResults[iSubGrid, e, k] + surfaceResults[iSubGrid, e, k];
}
break;
case JumpTypes.Sign:
for (int k = 0; k < noOfLambdas; k++)
{
rhs[rhsIndex * LDB + k] = boundaryResults[iSubGrid, e, k] - 2.0 * surfaceResults[iSubGrid, e, k];
}
break;
default:
throw new NotImplementedException();
}
}
}
}
if (rhs.Length > 0)
{
LAPACK.F77_LAPACK.DGELSY(noOfLambdas, noOfNodesPerEdge, basisValuesEdge.Storage, rhs, noOfRhs, 1e-12);
}
foreach (Chunk chunk in mask)
{
for (int i = 0; i < chunk.Len; i++)
{
int cell = i + chunk.i0;
int iSubGrid = localCellIndex2SubgridIndex[cell];
int noOfProcessedNodes = 0;
for (int e = 0; e < noOfFaces; e++)
{
int noOfNodesOnEdge = baseRule.NumbersOfNodesPerFace[e];
int rhsIndex = rhsIndexMap[iSubGrid, e];
if (rhsIndex < 0)
{
noOfProcessedNodes += noOfNodesOnEdge;
continue;
}
for (int j = 0; j < noOfNodesOnEdge; j++)
{
optimizedRules[iSubGrid].Weights[noOfProcessedNodes + j] = rhs[rhsIndex * LDB + j];
}
noOfProcessedNodes += noOfNodesOnEdge;
}
double max = optimizedRules[iSubGrid].Weights.Max(d => d.Abs());
if (max > 2.0 * RefElement.Volume)
{
tr.Info(String.Format(
"Warning: Abnormally large integration weight detected"
+ " for level set edge volume integral in cell {0}"
+ " (|w| = {1}). This may indicate a loss of"
+ " integration accuracy.",
cell,
max));
}
}
}
return(optimizedRules);
}
}