/// <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, correlates with <see cref="IGeometricalCellsData.RefElements"/>
/// - 2nd index: spatial direction of derivative
/// </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>
protected 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>
/// Returns the orthonormal approximation polynomials \f$ \phi_n \f$ up to a specific degree.
/// </summary>
/// <returns>
/// A polynomial list for each involved reference element
/// - index: reference element, correlates with <see cref="IGeometricalCellsData.RefElements"/>
/// </returns>
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>
/// 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, correlates with <see cref="IGeometricalCellsData.RefElements"/>
/// - 2nd index: spatial direction of first derivative
/// - 3rd index: spatial direction of second derivative
/// </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);
}