/// <summary>
/// Creates a node-set and a corresponding set of nodal polynomials;
/// </summary>
public void SelectNodalPolynomials(int px, out NodeSet Nodes, out PolynomialList NodalBasis, out int[] Type, out int[] EntityIndex, Func <Polynomial, bool> ModalBasisSelector = null, int[] NodeTypeFilter = null)
{
MultidimensionalArray a, b;
SelectNodalPolynomials(px, out Nodes, out NodalBasis, out Type, out EntityIndex, out a, out b, ModalBasisSelector, NodeTypeFilter);
Debug.Assert(ContainsZero(NodalBasis), "interpolation poly not found, 4");
}
/// <summary>
/// Returns the 1st derivatives of the orthonormal approximation polynomials,
/// i.e. \f$ \nabla_{\vec{\xi}} \phi_n \f$ up to a specific degree.
/// </summary>
/// <returns>
/// 1st index: reference element;
/// 2nd index: spatial direction
/// </returns>
public PolynomialList[,] GetOrthonormalPolynomials1stDeriv(int Degree)
{
PolynomialList[,] R;
if (!this.m_Polynomial1stDerivLists.TryGetValue(Degree, out R))
{
var Krefs = this.m_Owner.iGeomCells.RefElements;
int D = this.m_Owner.SpatialDimension;
R = new PolynomialList[Krefs.Length, D];
PolynomialList[] Polys = this.GetOrthonormalPolynomials(Degree);
int[] DerivExp = new int[D];
for (int d = 0; d < D; d++)
{
DerivExp[d] = 1;
for (int iKref = 0; iKref < Krefs.Length; iKref++)
{
int N = Polys[iKref].Count;
Polynomial[] tmp = new Polynomial[N];
for (int n = 0; n < N; n++)
{
tmp[n] = Polys[iKref][n].Derive(DerivExp);
}
R[iKref, d] = new PolynomialList(tmp);
}
DerivExp[d] = 0;
}
this.m_Polynomial1stDerivLists.Add(Degree, R);
}
return(R);
}
/// <summary>
/// Used by the <see cref="BasisValues"/>-cache.
/// </summary>
MultidimensionalArray EvaluateBasis(NodeSet NS, int MinDegree)
{
int iKref = NS.GetVolumeRefElementIndex(this.m_Owner);
PolynomialList Polys = this.GetOrthonormalPolynomials(MinDegree)[iKref];
MultidimensionalArray R = MultidimensionalArray.Create(NS.NoOfNodes, Polys.Count);
Polys.Evaluate(NS, R);
return(R);
}
/// <summary>
/// Computes the matrix matrix
/// V_ij = sum_k{p_i(nodes(k) * p_j(nodes(k)) * weights(k)}
/// for a given quadrature rule (where p_i and p_j are the orthonormal
/// polynomials supplied by the quadrature rule) and reports an error
/// if this matrix differs from the identity matrix
/// </summary>
/// <param name="order">The <b>requested</b> order</param>
/// <param name="rule">The quadrature rule</param>
/// <param name="polynomials">The orthonormal polynomials</param>
private void TestMassMatrix(int order, QuadRule rule, PolynomialList polynomials)
{
MultidimensionalArray[] values = new MultidimensionalArray[polynomials.Count];
for (int i = 0; i < polynomials.Count; i++)
{
values[i] = MultidimensionalArray.Create(rule.NoOfNodes);
polynomials[i].Evaluate(values[i], rule.Nodes);
}
//For every exactly integrable polynomial i
for (int i = 0; i < polynomials.Count; i++)
{
// Check if quadrature rule is suitable
if (polynomials[i].AbsoluteDegree > order)
{
continue;
}
//For every exactly integrable co-polynomial j
for (int j = 0; j <= i; j++)
{
// Check if quadrature rule is suitable
if (polynomials[i].AbsoluteDegree + polynomials[j].AbsoluteDegree > order)
{
continue;
}
double result = 0;
for (int k = 0; k < rule.NoOfNodes; k++)
{
result += values[i][k] * values[j][k] * rule.Weights[k];
}
//Elements should be zero except of diagonal elements
double expectedResult = 0.0;
if (i == j)
{
expectedResult = 1.0;
}
//Store maximum absolute error for the report we want to create
double error = Math.Abs(result - expectedResult);
//Console.WriteLine("MassMatrixError: {0}", error);
Assert.That(
error <= 1e-9,
String.Format(
"MassMatrix[" + i + "," + j + "] should be {0} but is off by {1:e} for order {2}",
expectedResult,
error,
order));
}
}
}
/// <summary>
/// Change-of-basis, in cell 0
/// </summary>
/// <param name="jCell"></param>
/// <param name="pl"></param>
public MultidimensionalArray GetChangeofBasisMatrix(int jCell, PolynomialList pl)
{
var m_Context = this.GridDat;
int N = this.Length;
int M = pl.Count;
int J = this.GridDat.iLogicalCells.NoOfLocalUpdatedCells;
if (jCell < 0 || jCell >= J)
{
throw new ArgumentOutOfRangeException("cell index out of range");
}
MultidimensionalArray Mtx = MultidimensionalArray.Create(N, M);
var cellMask = new CellMask(m_Context, new[] { new Chunk()
{
i0 = jCell, Len = 1
} }, MaskType.Geometrical);
// we project the basis function from 'jCell1' onto 'jCell0'
CellQuadrature.GetQuadrature(new int[2] {
N, M
}, m_Context,
(new CellQuadratureScheme(true, cellMask)).Compile(m_Context, this.Degree + pl.MaxAbsoluteDegree), // integrate over target cell
delegate(int i0, int Length, QuadRule QR, MultidimensionalArray _EvalResult) {
NodeSet nodes_Cell0 = QR.Nodes;
Debug.Assert(Length == 1);
//NodesGlobal.Allocate(1, nodes_Cell0.GetLength(0), nodes_Cell0.GetLength(1));
//m_Context.TransformLocal2Global(nodes_Cell0, jCell0, 1, NodesGlobal, 0);
//var nodes_Cell1 = new NodeSet(GridDat.iGeomCells.GetRefElement(jCell1), nodes_Cell0.GetLength(0), nodes_Cell0.GetLength(1));
//m_Context.TransformGlobal2Local(NodesGlobal.ExtractSubArrayShallow(0, -1, -1), nodes_Cell1, jCell1, null);
//nodes_Cell1.LockForever();
var phi_0 = this.CellEval(nodes_Cell0, jCell, 1).ExtractSubArrayShallow(0, -1, -1);
MultidimensionalArray R = MultidimensionalArray.Create(QR.NoOfNodes, pl.Count);
pl.Evaluate(nodes_Cell0, R);
var EvalResult = _EvalResult.ExtractSubArrayShallow(0, -1, -1, -1);
EvalResult.Multiply(1.0, R, phi_0, 0.0, "knm", "km", "kn");
},
delegate(int i0, int Length, MultidimensionalArray ResultsOfIntegration) {
Debug.Assert(Length == 1);
var res = ResultsOfIntegration.ExtractSubArrayShallow(0, -1, -1);
Mtx.Clear();
Mtx.Acc(1.0, res);
}).Execute();
return(Mtx);
}
static MultidimensionalArray Transform(RefElement Kref, Cell Cl, NodeSet Nodes)
{
int D = Kref.SpatialDimension;
PolynomialList polys = Kref.GetInterpolationPolynomials(Cl.Type);
MultidimensionalArray polyVals = polys.Values.GetValues(Nodes);
MultidimensionalArray GlobalVerticesOut = MultidimensionalArray.Create(Nodes.NoOfNodes, D);
for (int d = 0; d < D; d++)
{
GlobalVerticesOut.ExtractSubArrayShallow(-1, d)
.Multiply(1.0, polyVals, Cl.TransformationParams.ExtractSubArrayShallow(-1, d), 0.0, "k", "kn", "n");
}
return(GlobalVerticesOut);
}
/// <summary>
/// Returns the orthonormal approximation polynomials \f$ \phi_n \f$ up to a specific degree.
/// </summary>
public PolynomialList[] GetOrthonormalPolynomials(int Degree)
{
PolynomialList[] R;
if (!this.m_PolynomialLists.TryGetValue(Degree, out R))
{
var Krefs = this.m_Owner.iGeomCells.RefElements;
R = new PolynomialList[Krefs.Length];
for (int iKref = 0; iKref < Krefs.Length; iKref++)
{
R[iKref] = Krefs[iKref].GetOrthonormalPolynomials(Degree);
}
this.m_PolynomialLists.Add(Degree, R);
}
return(R);
}
/// <summary>
/// Projects every basis polynomial for the quadrature rule of the
/// given order onto a single cell grid supplied by the sub-class and
/// approximates the LInfinity error of the projection.
/// </summary>
/// <remarks>
/// The DG projection uses a quadrature rule of 2*n+1 for a basis of
/// order n. Thus, make sure <paramref name="order"/> that a quadrature
/// rule of order 2*order+1 exists before calling this method</remarks>
/// <param name="order">The <b>requested</b> order</param>
/// <param name="polynomials">The orthonormal polynomials</param>
/// <param name="rule">The tested quadrature rule</param>
private void TestProjectionError(int order, PolynomialList polynomials, QuadRule rule)
{
Basis basis = new Basis(context, order);
for (int i = 0; i < polynomials.Count; i++)
{
if (2 * polynomials[i].AbsoluteDegree + 1 > order)
{
// Not integrable _exactly_ with this rule
continue;
}
DGField field = new SinglePhaseField(basis);
ScalarFunction function = delegate(MultidimensionalArray input, MultidimensionalArray output) {
polynomials[i].Evaluate(output, new NodeSet(rule.RefElement, input));
};
field.ProjectField(function);
double LInfError = 0;
for (int j = 0; j < rule.NoOfNodes; j++)
{
MultidimensionalArray result = MultidimensionalArray.Create(1, 1);
NodeSet point = new NodeSet(context.iGeomCells.GetRefElement(0), 1, rule.SpatialDim);
for (int k = 0; k < rule.SpatialDim; k++)
{
point[0, k] = rule.Nodes[j, k];
}
point.LockForever();
//Evaluate pointwise because we don't want the accumulated
//result
field.Evaluate(0, 1, point, result, 0.0);
MultidimensionalArray exactResult = MultidimensionalArray.Create(1);
polynomials[i].Evaluate(exactResult, point);
LInfError = Math.Max(Math.Abs(result[0, 0] - exactResult[0]), LInfError);
}
//Console.WriteLine("ProjectionError: {0}", LInfError);
Assert.IsTrue(
LInfError <= 1e-13,
String.Format(
"Projection error too high for polynomial {0}. Error: {1:e}",
i,
LInfError));
}
}
private void DoMaths()
{
PolynomialList.Clear();
double eps = 0;
int nodesCount, degree;
if (!int.TryParse(NodesCountBind, out nodesCount) || !int.TryParse(PolynomialDegreeBind, out degree) ||
(!ApproxByExplicitDegree && !double.TryParse(ErrorMarginBind, out eps)))
{
MessageBox.Show(Locale["#MethArgParsing"], Locale["#ParsingErr"], MessageBoxButton.OK,
MessageBoxImage.Error);
}
else
{
try
{
AccuratePlot =
NumCore.NumCore.GetAccuratePlotDataPoints(SelectedFunction, DrawInterval).ToList();
Polynomial approx;
var timer = new Stopwatch();
timer.Start();
ApproximationCriterium criterium;
if (ApproxByExplicitDegree)
{
criterium = new ApproximationByPolynomialLevel(degree, UseCotes);
}
else
{
criterium = new ApproximationByAccuracy(eps, UseCotes);
}
var points =
NumCore.NumCore.GetApproximatedPlotDataPoints(SelectedFunction, ApproxInterval, nodesCount,
criterium, out approx);
timer.Stop();
ApproxPlot = points.Select(x => new DataPoint(x.X, x.Y)).ToList();
ApproxTime = timer.ElapsedTicks.ToString();
Error = NumCore.NumCore.GetError(SelectedFunction, approx).ToString();
Polynomial = GetPolynom(approx);
}
catch (Exception e)
{
MessageBox.Show(e.Message + Locale["#BadIntervalCont"], Locale["#ParsingErr"],
MessageBoxButton.OK,
MessageBoxImage.Error);
}
}
}
private string GetPolynom(Polynomial approx)
{
var result = "";
var coefs = approx.Coefficients;
var prec = coefs.Count - 1;
var pr = prec;
foreach (var coef in coefs)
{
var key = "x" + (pr / 10 > 0 ? PolynomCoefs[pr / 10] + PolynomCoefs[pr % 10] : PolynomCoefs[pr]);
PolynomialList.Add(new KeyValuePair <string, double>(key, coef));
pr--;
}
for (var i = prec / 10; i >= 0; i--)
{
for (var j = prec % 10; j >= 0; j--)
{
var coef = coefs[10 * i + j];
if (i == 0 && j == 0)
{
return(result + (coef > 0 ? $"+{coef:N2}" : $"-{Math.Abs(coef):N2}"));
}
result += coef > 0 ? $"+{coef:N2}x" : $"-{Math.Abs(coef):N2}x";
if (i > 0)
{
for (var k = 1; k <= i; k++)
{
result += PolynomCoefs[k];
prec--;
}
}
result += PolynomCoefs[j];
}
}
return(result);
}
/// <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>
/// Updates <see cref="lambdaBasis"/>, <see cref="baseRule"/> and
/// <see cref="basisValuesEdge"/> every time a different order is
/// requested in <see cref="GetQuadRuleSet"/>
/// </summary>
/// <param name="order">
/// The new order of the moment-fitting basis
/// </param>
private void SwitchOrder(int order)
{
int iKref = this.LevelSetData.GridDat.Cells.RefElements.IndexOf(this.RefElement, (A, B) => object.ReferenceEquals(A, B));
int noOfFaces = LevelSetData.GridDat.Grid.RefElements[iKref].NoOfFaces;
int D = RefElement.FaceRefElement.SpatialDimension;
Polynomial[] basePolynomials = RefElement.FaceRefElement.GetOrthonormalPolynomials(order).ToArray();
Polynomial[] antiderivativePolynomials =
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];
// Make sure divergence is Phi again
pNew.Coeff[j] /= D;
}
antiderivativePolynomials[i * D + d] = pNew;
}
}
lambdaBasis = new PolynomialList(antiderivativePolynomials);
int minNoOfPoints = GetNumberOfLambdas();
int minOrder = 1;
double safetyFactor = 1.6;
while (RefElement.FaceRefElement.GetQuadratureRule(minOrder).NoOfNodes < safetyFactor * minNoOfPoints)
{
minOrder += 1;
}
QuadRule singleEdgeRule = RefElement.FaceRefElement.GetQuadratureRule(minOrder);
baseRule = new CellBoundaryFromEdgeRuleFactory <CellBoundaryQuadRule>(
LevelSetData.GridDat,
LevelSetData.GridDat.Grid.RefElements[0],
new FixedRuleFactory <QuadRule>(singleEdgeRule)).
GetQuadRuleSet(new CellMask(LevelSetData.GridDat, Chunk.GetSingleElementChunk(0)), -1).
First().Rule;
//Basis singleEdgeBasis = new Basis(tracker.GridDat, order, RefElement.FaceRefElement);
PolynomialList singleEdgeBasis = new PolynomialList(RefElement.FaceRefElement.GetOrthonormalPolynomials(order));
//MultidimensionalArray edgeNodes = singleEdgeRule.Nodes.CloneAs();
//basisValuesEdge = MultidimensionalArray.Create(
// singleEdgeRule.NoOfNodes, singleEdgeBasis.Count);
//MultidimensionalArray monomials = Polynomial.GetMonomials(
// edgeNodes, RefElement.FaceRefElement.SpatialDimension, singleEdgeBasis.Degree);
//for (int j = 0; j < singleEdgeBasis.MinimalLength; j++) {
// singleEdgeBasis.Polynomials[iKref, j].Evaluate(
// basisValuesEdge.ExtractSubArrayShallow(-1, j),
// edgeNodes,
// monomials);
//}
this.basisValuesEdge = singleEdgeBasis.Values.GetValues(singleEdgeRule.Nodes);
}
/// <summary>
/// Creates a node-set and a corresponding set of nodal polynomials;
/// </summary>
public void SelectNodalPolynomials(int px, out NodeSet Nodes, out PolynomialList _NodalBasis, out int[] Type, out int[] EntityIndex, out MultidimensionalArray Nodal2Modal, out MultidimensionalArray Modal2Nodal, Func <Polynomial, bool> ModalBasisSelector = null, int[] NodeTypeFilter = null)
{
int D = this.SpatialDimension;
if (ModalBasisSelector == null)
{
ModalBasisSelector = delegate(Polynomial p) {
Debug.Assert(p.Coeff.Length == p.Exponents.GetLength(0));
Debug.Assert(p.Exponents.GetLength(1) == D);
for (int l = 0; l < p.Coeff.Length; l++)
{
for (int d = 0; d < D; d++)
{
if (p.Exponents[l, d] > (px - 1))
{
return(false);
}
}
}
return(true);
}
}
;
// Find node set
// =============
GetNodeSet(px, out Nodes, out Type, out EntityIndex, NodeTypeFilter);
int NoOfNodes = Type.Length;
#if DEBUG
double[,] DIST = new double[NoOfNodes, NoOfNodes];
for (int j1 = 0; j1 < NoOfNodes; j1++)
{
for (int j2 = 0; j2 < NoOfNodes; j2++)
{
if (j2 == j1)
{
continue;
}
var node_j1 = Nodes.GetRow(j1);
var node_j2 = Nodes.GetRow(j2);
DIST[j1, j2] = GenericBlas.L2Dist(node_j1, node_j2);
if (DIST[j1, j2] <= 1.0e-8)
{
throw new ApplicationException("internal error.");
}
}
}
#endif
// Find modal basis of approximation space
// =======================================
var ortho_polys = this.OrthonormalPolynomials;
List <Polynomial> Basis = new List <Polynomial>();
int deg;
if (px > 1)
{
for (deg = 0; deg <= (px - 1) * D; deg++)
{
var r = ortho_polys.Where(p => ((p.AbsoluteDegree == deg) && ModalBasisSelector(p)));
Basis.AddRange(r);
if (Basis.Count >= NoOfNodes)
{
break;
}
}
}
else
{
var r = ortho_polys.Where(pol => pol.AbsoluteDegree <= 1);
Basis.AddRange(r);
deg = 1;
}
if (Basis.Count != NoOfNodes)
{
throw new ApplicationException("Basis selection failed.");
}
// check for basis selection
// ==========================
// case Triangle, Tetra: the complete P_{px-1}(x,y) should be selected
// case Quad, Cube: P_{px-1}(x)*P_{px-1)(y) subset of P_{px-1}(x,y) will be selected
int NoOfPolys = NoOfNodes;
int[] idx_Basis = new int[NoOfPolys];
for (int i = 0; i < NoOfPolys; i++)
{
idx_Basis[i] = Array.IndexOf(ortho_polys, Basis[i]);
Debug.Assert(idx_Basis[i] >= 0);
}
// Construct nodal Polynomials
// ===========================
MultidimensionalArray PolyAtNodes = MultidimensionalArray.Create(NoOfPolys, NoOfNodes);
for (int i = 0; i < NoOfNodes; i++)
{
Basis[i].Evaluate(PolyAtNodes.ExtractSubArrayShallow(-1, i), Nodes);
}
//FullMatrix MtxPolyAtNodes = new FullMatrix(NoOfPolys, NoOfNodes);
//MtxPolyAtNodes.Set(PolyAtNodes);
Debug.Assert(!PolyAtNodes.ContainsNanOrInf(), "Nodal Poly generation, illegal value");
var Sol = PolyAtNodes.GetInverse();
Debug.Assert(!Sol.ContainsNanOrInf(), "Nodal Poly generation, solution, illegal value");
Modal2Nodal = MultidimensionalArray.Create(ortho_polys.Where(p => p.AbsoluteDegree <= deg).Count(), NoOfPolys);
Nodal2Modal = MultidimensionalArray.Create(Modal2Nodal.NoOfCols, Modal2Nodal.NoOfRows);
MultidimensionalArray _Modal2Nodal = MultidimensionalArray.Create(NoOfPolys, NoOfPolys);
MultidimensionalArray _Nodal2Modal = MultidimensionalArray.Create(NoOfPolys, NoOfPolys);
var b = new double[NoOfPolys];
var rhs = new double[NoOfNodes];
Polynomial[] NodalBasis = new Polynomial[NoOfNodes];
for (int k = 0; k < NoOfNodes; k++)
{
Array.Clear(rhs, 0, rhs.Length);
rhs[k] = 1.0;
Sol.GEMV(1.0, rhs, 0.0, b);
for (int i = 0; i < NoOfPolys; i++)
{
if (Math.Abs(b[i]) > 1.0e-12)
{
if (NodalBasis[k] == null)
{
NodalBasis[k] = b[i] * Basis[i];
}
else
{
NodalBasis[k] = NodalBasis[k] + b[i] * Basis[i];
}
}
else
{
//Console.WriteLine("tresh {0}, {1}", Math.Abs(b[i]), b[i]);
}
_Modal2Nodal[i, k] = b[i];
}
//Console.WriteLine("k: {0}, NoOfPolys {1}, isnull {2} ", k, NoOfPolys, NodalBasis[k] == null);
}
_NodalBasis = new PolynomialList(NodalBasis);
Debug.Assert(ContainsZero(_NodalBasis), "interpolation poly not found, 5");
_Modal2Nodal.InvertTo(_Nodal2Modal);
for (int k = 0; k < NoOfNodes; k++)
{
for (int i = 0; i < NoOfPolys; i++)
{
Modal2Nodal[idx_Basis[i], k] = _Modal2Nodal[i, k];
Nodal2Modal[i, idx_Basis[k]] = _Nodal2Modal[i, k];
}
}
}
}
/// <summary>
/// Orthonormalization for curved elements
/// </summary>
protected override MultidimensionalArray Compute_OrthonormalizationTrafo(int j0, int Len, int Degree)
{
// init
// ====
int iKref = this.m_Owner.iGeomCells.GetRefElementIndex(j0);
PolynomialList Polys = this.GetOrthonormalPolynomials(Degree)[iKref];
int N = Polys.Count;
CellType cellType = this.m_Owner.iGeomCells.GetCellType(j0);
#if DEBUG
// checking
for (int j = 1; j < Len; j++)
{
int jCell = j + j0;
if (this.m_Owner.iGeomCells.GetCellType(jCell) != cellType)
{
throw new NotSupportedException("All cells in chunk must have same type.");
}
}
#endif
// storage for result
MultidimensionalArray NonlinOrtho = MultidimensionalArray.Create(Len, N, N);
if (m_Owner.iGeomCells.IsCellAffineLinear(j0))
{
// affine-linear branch
// ++++++++++++++++++++
MultidimensionalArray scl = this.Scaling;
for (int j = 0; j < Len; j++)
{
int jCell = j + j0;
double scl_j = scl[jCell];
for (int n = 0; n < N; n++)
{
NonlinOrtho[j, n, n] = scl_j;
}
}
}
else
{
// nonlinear cells branch
// ++++++++++++++++++++++
// init
// ====
var Kref = m_Owner.iGeomCells.GetRefElement(j0);
int deg; // polynomial degree of integrand: Degree of Jacobi determinat + 2* degree of basis polynomials in ref.-space.
{
int D = m_Owner.SpatialDimension;
deg = Kref.GetInterpolationDegree(cellType);
if (deg > 1)
{
deg -= 1;
}
deg *= D;
deg += 2 * Degree;
}
var qr = Kref.GetQuadratureRule((int)deg);
int K = qr.NoOfNodes;
// evaluate basis polys in ref space
// =================================
MultidimensionalArray BasisValues = this.EvaluateBasis(qr.Nodes, Degree);
Debug.Assert(BasisValues.GetLength(0) == K);
if (BasisValues.GetLength(0) > N)
{
BasisValues = BasisValues.ExtractSubArrayShallow(new int[] { 0, 0 }, new int[] { K - 1, N - 1 });
}
// compute \f$ A_{k n m} = \phi_{k n} \phi_{k m} w_{k} \f$
// (cell-INdependentd part of mass matrix computation)
// ==================================================================
MultidimensionalArray A = MultidimensionalArray.Create(K, N, N);
A.Multiply(1.0, qr.Weights, BasisValues, BasisValues, 0.0, "knm", "k", "kn", "km");
// Determine mass matrix \f$ M \f$ in all cells by quadrature
// \f$ M_{n m} = \sum_{k} J_k A_{k n m} \f$
// =====================================================
MultidimensionalArray J = m_Owner.JacobianDeterminat.GetValue_Cell(qr.Nodes, j0, Len);
// store mass-matrix in 'NonlinOrtho' to save mem alloc
NonlinOrtho.Multiply(1.0, A, J, 0.0, "jmn", "knm", "jk");
// Compute change-of-basis for all cells
// (invese of cholesky)
// =====================================
for (int j = 0; j < Len; j++)
{
MultidimensionalArray Mj = NonlinOrtho.ExtractSubArrayShallow(j, -1, -1); // mass-matrix of basis on refrence element in cell j+j0
#if DEBUG
MultidimensionalArray MjClone = Mj.CloneAs();
#endif
//Mj.InvertSymmetrical();
Mj.SymmetricLDLInversion(Mj, null);
// clear lower triangular part
// Debug.Assert(Mj.NoOfCols == N);
for (int n = 1; n < N; n++)
{
for (int m = 0; m < n; m++)
{
Mj[n, m] = 0.0;
}
}
#if DEBUG
MultidimensionalArray B = NonlinOrtho.ExtractSubArrayShallow(j, -1, -1);
MultidimensionalArray Bt = B.Transpose();
double MjNorm = MjClone.InfNorm();
double Bnorm = B.InfNorm();
MultidimensionalArray check = IMatrixExtensions.GEMM(Bt, MjClone, B);
check.AccEye(-1.0);
double checkNorm = check.InfNorm();
double RelErr = checkNorm / Math.Max(MjNorm, Bnorm);
Debug.Assert(RelErr < 1.0e-5, "Fatal error in numerical orthonomalization on nonlinear cell.");
#endif
}
}
// return
// =========
return(NonlinOrtho);
}
/// <summary>
/// Not implemented.
/// </summary>
protected override void GetInterpolationNodes_NonLin(CellType Type, out NodeSet InterpolationNodes, out PolynomialList InterpolationPolynomials, out int[] NodeType, out int[] EntityIndex)
{
throw new NotImplementedException();
}
/// <summary>
/// Returns the 2nd derivatives of the orthonormal approximation polynomials,
/// i.e. \f$ \nabla_{\vec{\xi}} \phi_n \f$ up to a specific degree.
/// </summary>
/// <returns>
/// 1st index: reference element;
/// 2nd index: spatial direction
/// 2nd index: spatial direction
/// </returns>
public PolynomialList[,,] GetOrthonormalPolynomials2ndDeriv(int Degree)
{
PolynomialList[,,] R;
if (!this.m_Polynomial2ndDerivLists.TryGetValue(Degree, out R))
{
var Krefs = this.m_Owner.iGeomCells.RefElements;
int D = this.m_Owner.SpatialDimension;
R = new PolynomialList[Krefs.Length, D, D];
PolynomialList[] Polys = this.GetOrthonormalPolynomials(Degree);
int[] DerivExp = new int[D];
for (int d1 = 0; d1 < D; d1++)
{
DerivExp[d1]++;
for (int d2 = 0; d2 < D; d2++)
{
DerivExp[d2]++;
for (int iKref = 0; iKref < Krefs.Length; iKref++)
{
int N = Polys[iKref].Count;
Polynomial[] tmp = new Polynomial[N];
for (int n = 0; n < N; n++)
{
tmp[n] = Polys[iKref][n].Derive(DerivExp);
}
R[iKref, d1, d2] = new PolynomialList(tmp);
}
DerivExp[d2]--;
}
DerivExp[d1]--;
}
#if DEBUG
for (int d1 = 0; d1 < D; d1++)
{
for (int d2 = d1 + 1; d2 < D; d2++)
{
for (int iKref = 0; iKref < Krefs.Length; iKref++)
{
int N = Polys[iKref].Count;
for (int n = 0; n < N; n++)
{
Polynomial P_d1d2 = R[iKref, d1, d2][n];
Polynomial P_d2d1 = R[iKref, d2, d1][n];
Debug.Assert(P_d1d2.Equals(P_d2d1), "Hessian seems unsymmetric.");
}
}
}
}
#endif
this.m_Polynomial2ndDerivLists.Add(Degree, R);
}
return(R);
}