/// <summary>
/// Defines an 1D NURBS shape function for an edge element.
/// </summary>
/// <param name="element">An <see cref="Element"/> of type <see cref="NurbsElement1D"/>.</param>
/// <param name="controlPoints">A <see cref="List{T}"/> containing the control points of the element.</param>
/// <param name="edge">The one-dimensional boundary entity that contains the <paramref name="element"/>.</param>
public Nurbs1D(Element element, IList <ControlPoint> controlPoints, Edge edge)
{
GaussQuadrature gauss = new GaussQuadrature();
IList <GaussLegendrePoint3D> gaussPoints = gauss.CalculateElementGaussPoints(edge.Degree, element.Knots.ToArray());
var parametricGaussPointKsi = Vector.CreateZero(edge.Degree + 1);
for (int i = 0; i < edge.Degree + 1; i++)
{
parametricGaussPointKsi[i] = gaussPoints[i].Ksi;
}
var bsplinesKsi = new BSPLines1D(edge.Degree, edge.KnotValueVector, parametricGaussPointKsi);
bsplinesKsi.calculateBSPLinesAndDerivatives();
int supportKsi = edge.Degree + 1;
int numberOfElementControlPoints = supportKsi;
NurbsValues = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsDerivativeValuesKsi = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
for (int i = 0; i < supportKsi; i++)
{
double sumKsi = 0;
double sumdKsi = 0;
for (int j = 0; j < numberOfElementControlPoints; j++)
{
int index = controlPoints[j].ID;
sumKsi += bsplinesKsi.BSPLineValues[index, i] * controlPoints[j].WeightFactor;
sumdKsi += bsplinesKsi.BSPLineDerivativeValues[index, i] * controlPoints[j].WeightFactor;
}
for (int j = 0; j < numberOfElementControlPoints; j++)
{
int indexKsi = controlPoints[j].ID;
NurbsValues[j, i] = bsplinesKsi.BSPLineValues[indexKsi, i] * controlPoints[j].WeightFactor / sumKsi;
NurbsDerivativeValuesKsi[j, i] = controlPoints[j].WeightFactor * (bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] * sumKsi -
bsplinesKsi.BSPLineValues[indexKsi, i] * sumdKsi) / Math.Pow(sumKsi, 2);
}
}
}
/// <summary>
/// Two-dimensional T-spline shape functions from Bezier extraction for <see cref="TSplineKirchhoffLoveShellElementMaterial"/>.
/// </summary>
/// <param name="element">An <see cref="Element"/> of type <see cref="TSplineKirchhoffLoveShellElementMaterial"/>.</param>
/// <param name="controlPoints">A <see cref="List{T}"/> containing the control points of the element.</param>
public ShapeTSplines2DFromBezierExtraction(TSplineKirchhoffLoveShellElementMaterial element, ControlPoint[] controlPoints)
{
GaussQuadrature gauss = new GaussQuadrature();
IList <GaussLegendrePoint3D> gaussPoints = gauss.CalculateElementGaussPoints(element.DegreeKsi, element.DegreeHeta,
new List <Knot>
{
new Knot()
{
ID = 0, Ksi = -1, Heta = -1, Zeta = 0
},
new Knot()
{
ID = 1, Ksi = -1, Heta = 1, Zeta = 0
},
new Knot()
{
ID = 2, Ksi = 1, Heta = -1, Zeta = 0
},
new Knot()
{
ID = 3, Ksi = 1, Heta = 1, Zeta = 0
}
});
var parametricGaussPointKsi = Vector.CreateZero(element.DegreeKsi + 1);
for (int i = 0; i < element.DegreeKsi + 1; i++)
{
parametricGaussPointKsi[i] = gaussPoints[i * (element.DegreeHeta + 1)].Ksi;
}
var parametricGaussPointHeta = Vector.CreateZero(element.DegreeHeta + 1);
for (int i = 0; i < element.DegreeHeta + 1; i++)
{
parametricGaussPointHeta[i] = gaussPoints[i].Heta;
}
Vector knotValueVectorKsi = Vector.CreateZero((element.DegreeKsi + 1) * 2);
Vector knotValueVectorHeta = Vector.CreateZero((element.DegreeHeta + 1) * 2);
for (int i = 0; i < element.DegreeKsi + 1; i++)
{
knotValueVectorKsi[i] = -1;
knotValueVectorKsi[element.DegreeKsi + 1 + i] = 1;
}
for (int i = 0; i < element.DegreeHeta + 1; i++)
{
knotValueVectorHeta[i] = -1;
knotValueVectorHeta[element.DegreeHeta + 1 + i] = 1;
}
BSPLines1D bernsteinKsi = new BSPLines1D(element.DegreeKsi, knotValueVectorKsi, parametricGaussPointKsi);
BSPLines1D bernsteinHeta = new BSPLines1D(element.DegreeHeta, knotValueVectorHeta, parametricGaussPointHeta);
bernsteinKsi.calculateBSPLinesAndDerivatives();
bernsteinHeta.calculateBSPLinesAndDerivatives();
int supportKsi = element.DegreeKsi + 1;
int supportHeta = element.DegreeHeta + 1;
var bKsi = MatrixPart(supportKsi, bernsteinKsi.BSPLineValues);
var bdKsi = MatrixPart(supportKsi, bernsteinKsi.BSPLineDerivativeValues);
var bddKsi = MatrixPart(supportKsi, bernsteinKsi.BSPLineSecondDerivativeValues);
var bheta = MatrixPart(supportHeta, bernsteinHeta.BSPLineValues);
var bdheta = MatrixPart(supportHeta, bernsteinHeta.BSPLineDerivativeValues);
var bddheta = MatrixPart(supportHeta, bernsteinHeta.BSPLineSecondDerivativeValues);
var bernsteinShapeFunctions = KroneckerProduct(bKsi, bheta);
Matrix bernsteinShapeFunctionDerivativesKsi = KroneckerProduct(bheta, bdKsi);
Matrix bernsteinShapeFunctionDerivativesHeta = KroneckerProduct(bdheta, bKsi);
Matrix bernsteinShapeFunctionSecondDerivativesKsi = KroneckerProduct(bheta, bddKsi);
Matrix bernsteinShapeFunctionSecondDerivativesHeta = KroneckerProduct(bddheta, bKsi);
Matrix bernsteinShapeFunctionSecondDerivativesKsiHeta = KroneckerProduct(bdheta, bdKsi);
Matrix rationalTSplines = element.ExtractionOperator * bernsteinShapeFunctions;
Matrix rationalTSplineDerivativesKsi = element.ExtractionOperator * bernsteinShapeFunctionDerivativesKsi;
Matrix rationalTSplineDerivativesHeta = element.ExtractionOperator * bernsteinShapeFunctionDerivativesHeta;
Matrix rationalTSplineSecondDerivativesKsi = element.ExtractionOperator * bernsteinShapeFunctionSecondDerivativesKsi;
Matrix rationalTSplineSecondDerivativesHeta = element.ExtractionOperator * bernsteinShapeFunctionSecondDerivativesHeta;
Matrix rationalTSplineSecondDerivativesKsiHeta = element.ExtractionOperator * bernsteinShapeFunctionSecondDerivativesKsiHeta;
TSplineValues = Matrix.CreateZero(controlPoints.Length, supportKsi * supportHeta);
TSplineDerivativeValuesKsi = Matrix.CreateZero(controlPoints.Length, supportKsi * supportHeta);
TSplineDerivativeValuesHeta = Matrix.CreateZero(controlPoints.Length, supportKsi * supportHeta);
TSplineSecondDerivativesValueKsi = Matrix.CreateZero(controlPoints.Length, supportKsi * supportHeta);
TSplineSecondDerivativesValueHeta = Matrix.CreateZero(controlPoints.Length, supportKsi * supportHeta);
TSplineSecondDerivativesValueKsiHeta = Matrix.CreateZero(controlPoints.Length, supportKsi * supportHeta);
for (int i = 0; i < supportKsi; i++)
{
for (int j = 0; j < supportHeta; j++)
{
double sumKsiHeta = 0;
double sumdKsiHeta = 0;
double sumKsidHeta = 0;
double sumdKsidKsi = 0;
double sumdHetadHeta = 0;
double sumdKsidHeta = 0;
var index = i * supportHeta + j;
for (int k = 0; k < controlPoints.Length; k++)
{
sumKsiHeta += rationalTSplines[k, index] * controlPoints[k].WeightFactor;
sumdKsiHeta += rationalTSplineDerivativesKsi[k, index] * controlPoints[k].WeightFactor;
sumKsidHeta += rationalTSplineDerivativesHeta[k, index] * controlPoints[k].WeightFactor;
sumdKsidKsi += rationalTSplineSecondDerivativesKsi[k, index] * controlPoints[k].WeightFactor;
sumdHetadHeta += rationalTSplineSecondDerivativesHeta[k, index] * controlPoints[k].WeightFactor;
sumdKsidHeta += rationalTSplineSecondDerivativesKsiHeta[k, index] * controlPoints[k].WeightFactor;
}
for (int k = 0; k < controlPoints.Length; k++)
{
TSplineValues[k, index] = rationalTSplines[k, index] * controlPoints[k].WeightFactor / sumKsiHeta;
TSplineDerivativeValuesKsi[k, index] = (rationalTSplineDerivativesKsi[k, index] * sumKsiHeta -
rationalTSplines[k, index] * sumdKsiHeta) /
Math.Pow(sumKsiHeta, 2) * controlPoints[k].WeightFactor;
TSplineDerivativeValuesHeta[k, index] = (rationalTSplineDerivativesHeta[k, index] * sumKsiHeta -
rationalTSplines[k, index] * sumKsidHeta) /
Math.Pow(sumKsiHeta, 2) * controlPoints[k].WeightFactor;
TSplineSecondDerivativesValueKsi[k, index] = (rationalTSplineSecondDerivativesKsi[k, index] / sumKsiHeta -
2 * rationalTSplineDerivativesKsi[k, index] * sumdKsiHeta /
Math.Pow(sumKsiHeta, 2) -
rationalTSplines[k, index] * sumdKsidKsi / Math.Pow(sumKsiHeta, 2) +
2 * rationalTSplines[k, index] * Math.Pow(sumdKsiHeta, 2) /
Math.Pow(sumKsiHeta, 3)) * controlPoints[k].WeightFactor;
TSplineSecondDerivativesValueHeta[k, index] = (rationalTSplineSecondDerivativesHeta[k, index] / sumKsiHeta -
2 * rationalTSplineDerivativesHeta[k, index] * sumKsidHeta /
Math.Pow(sumKsiHeta, 2) -
rationalTSplines[k, index] * sumdHetadHeta /
Math.Pow(sumKsiHeta, 2) +
2 * rationalTSplines[k, index] * Math.Pow(sumKsidHeta, 2) /
Math.Pow(sumKsiHeta, 3)) * controlPoints[k].WeightFactor;
TSplineSecondDerivativesValueKsiHeta[k, index] = (rationalTSplineSecondDerivativesKsiHeta[k, index] / sumKsiHeta -
rationalTSplineDerivativesKsi[k, index] * sumKsidHeta /
Math.Pow(sumKsiHeta, 2) -
rationalTSplineDerivativesHeta[k, index] * sumdKsiHeta /
Math.Pow(sumKsiHeta, 2) -
rationalTSplines[k, index] * sumdKsidHeta /
Math.Pow(sumKsiHeta, 2) +
2 * rationalTSplines[k, index] * sumdKsiHeta * sumKsidHeta /
Math.Pow(sumKsiHeta, 3)) *
controlPoints[k].WeightFactor;
}
}
}
}
/// <summary>
/// Defines 3D NURBS shape functions given the control points.
/// </summary>
/// <param name="element">An <see cref="Element"/> of type <see cref="NurbsElement3D"/>.</param>
/// <param name="controlPoints">A <see cref="List{T}"/> containing the control points of the element.</param>
public Nurbs3D(Element element, ControlPoint[] controlPoints)
{
GaussQuadrature gauss = new GaussQuadrature();
IList <GaussLegendrePoint3D> gaussPoints = gauss.CalculateElementGaussPoints(element.Patch.DegreeKsi, element.Patch.DegreeHeta, element.Patch.DegreeZeta, element.Knots.ToArray());
var parametricGaussPointKsi = Vector.CreateZero(element.Patch.DegreeKsi + 1);
for (int i = 0; i < element.Patch.DegreeKsi + 1; i++)
{
parametricGaussPointKsi[i] = gaussPoints[i * (element.Patch.DegreeZeta + 1) * (element.Patch.DegreeHeta + 1)].Ksi;
}
var parametricGaussPointHeta = Vector.CreateZero(element.Patch.DegreeHeta + 1);
for (int i = 0; i < element.Patch.DegreeHeta + 1; i++)
{
parametricGaussPointHeta[i] = gaussPoints[i * (element.Patch.DegreeZeta + 1)].Heta;
}
var parametricGaussPointZeta = Vector.CreateZero(element.Patch.DegreeZeta + 1);
for (int i = 0; i < element.Patch.DegreeZeta + 1; i++)
{
parametricGaussPointZeta[i] = gaussPoints[i].Zeta;
}
BSPLines1D bsplinesKsi = new BSPLines1D(element.Patch.DegreeKsi, element.Patch.KnotValueVectorKsi, parametricGaussPointKsi);
BSPLines1D bsplinesHeta = new BSPLines1D(element.Patch.DegreeHeta, element.Patch.KnotValueVectorHeta, parametricGaussPointHeta);
BSPLines1D bsplinesZeta = new BSPLines1D(element.Patch.DegreeZeta, element.Patch.KnotValueVectorZeta, parametricGaussPointZeta);
bsplinesKsi.calculateBSPLinesAndDerivatives();
bsplinesHeta.calculateBSPLinesAndDerivatives();
bsplinesZeta.calculateBSPLinesAndDerivatives();
int supportKsi = element.Patch.DegreeKsi + 1;
int supportHeta = element.Patch.DegreeHeta + 1;
int supportZeta = element.Patch.DegreeZeta + 1;
int numberOfElementControlPoints = supportKsi * supportHeta * supportZeta;
NurbsValues = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsDerivativeValuesKsi = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsDerivativeValuesHeta = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsDerivativeValuesZeta = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
for (int i = 0; i < supportKsi; i++)
{
for (int j = 0; j < supportHeta; j++)
{
for (int k = 0; k < supportZeta; k++)
{
double sumKsiHetaZeta = 0;
double sumdKsiHetaZeta = 0;
double sumKsidHetaZeta = 0;
double sumKsiHetadZeta = 0;
for (int m = 0; m < numberOfElementControlPoints; m++)
{
int indexKsi = controlPoints[m].ID / (element.Patch.NumberOfControlPointsHeta * element.Patch.NumberOfControlPointsZeta);
int indexHeta = controlPoints[m].ID % (element.Patch.NumberOfControlPointsHeta * element.Patch.NumberOfControlPointsZeta) / element.Patch.NumberOfControlPointsZeta;
int indexZeta = controlPoints[m].ID % (element.Patch.NumberOfControlPointsHeta * element.Patch.NumberOfControlPointsZeta) % element.Patch.NumberOfControlPointsZeta;
sumKsiHetaZeta += bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
bsplinesZeta.BSPLineValues[indexZeta, k] *
controlPoints[m].WeightFactor;
sumdKsiHetaZeta += bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
bsplinesZeta.BSPLineValues[indexZeta, k] *
controlPoints[m].WeightFactor;
sumKsidHetaZeta += bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] *
bsplinesZeta.BSPLineValues[indexZeta, k] *
controlPoints[m].WeightFactor;
sumKsiHetadZeta += bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
bsplinesZeta.BSPLineDerivativeValues[indexZeta, k] *
controlPoints[m].WeightFactor;
}
for (int m = 0; m < numberOfElementControlPoints; m++)
{
int indexKsi = controlPoints[m].ID / (element.Patch.NumberOfControlPointsHeta * element.Patch.NumberOfControlPointsZeta);
int indexHeta = controlPoints[m].ID % (element.Patch.NumberOfControlPointsHeta * element.Patch.NumberOfControlPointsZeta) / element.Patch.NumberOfControlPointsZeta;
int indexZeta = controlPoints[m].ID % (element.Patch.NumberOfControlPointsHeta * element.Patch.NumberOfControlPointsZeta) % element.Patch.NumberOfControlPointsZeta;
NurbsValues[m, i *supportHeta *supportZeta + j * supportZeta + k] =
bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
bsplinesZeta.BSPLineValues[indexZeta, k] *
controlPoints[m].WeightFactor / sumKsiHetaZeta;
NurbsDerivativeValuesKsi[m, i *supportHeta *supportZeta + j * supportZeta + k] =
(bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] * sumKsiHetaZeta -
bsplinesKsi.BSPLineValues[indexKsi, i] * sumdKsiHetaZeta) *
bsplinesHeta.BSPLineValues[indexHeta, j] *
bsplinesZeta.BSPLineValues[indexZeta, k] *
controlPoints[m].WeightFactor / Math.Pow(sumKsiHetaZeta, 2);
NurbsDerivativeValuesHeta[m, i *supportHeta *supportZeta + j * supportZeta + k] =
bsplinesKsi.BSPLineValues[indexKsi, i] *
(bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] * sumKsiHetaZeta -
bsplinesHeta.BSPLineValues[indexHeta, j] * sumKsidHetaZeta) *
bsplinesZeta.BSPLineValues[indexZeta, k] *
controlPoints[m].WeightFactor / Math.Pow(sumKsiHetaZeta, 2);
NurbsDerivativeValuesZeta[m, i *supportHeta *supportZeta + j * supportZeta + k] =
bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
(bsplinesZeta.BSPLineDerivativeValues[indexZeta, k] * sumKsiHetaZeta -
bsplinesZeta.BSPLineValues[indexZeta, k] * sumKsiHetadZeta) *
controlPoints[m].WeightFactor / Math.Pow(sumKsiHetaZeta, 2);
}
}
}
}
}
/// <summary>
/// Defines n 2D NURBS shape function for a face element.
/// </summary>
/// <param name="element">An <see cref="Element"/> of type <see cref="NurbsElement2D"/>.</param>
/// <param name="controlPoints">A <see cref="List{T}"/> containing the control points of the element.</param>
/// <param name="face">The two-dimensional boundary entities where the <paramref name="element"/> shape functions will be evaluated.</param>
public Nurbs2D(Element element, ControlPoint[] controlPoints, Face face)
{
var degreeKsi = face.Degrees[0];
var degreeHeta = face.Degrees[1];
var knotValueVectorKsi = face.KnotValueVectors[0];
var knotValueVectorHeta = face.KnotValueVectors[1];
var numberOfControlPointsHeta = knotValueVectorHeta.Length - degreeHeta - 1;
GaussQuadrature gauss = new GaussQuadrature();
IList <GaussLegendrePoint3D> gaussPoints =
gauss.CalculateElementGaussPoints(degreeKsi, degreeHeta, element.Knots.ToArray());
var parametricGaussPointKsi = Vector.CreateZero(degreeKsi + 1);
for (int i = 0; i < degreeKsi + 1; i++)
{
parametricGaussPointKsi[i] = gaussPoints[i * (degreeHeta + 1)].Ksi;
}
var parametricGaussPointHeta = Vector.CreateZero(degreeHeta + 1);
for (int i = 0; i < degreeHeta + 1; i++)
{
parametricGaussPointHeta[i] = gaussPoints[i].Heta;
}
BSPLines1D bsplinesKsi = new BSPLines1D(degreeKsi, knotValueVectorKsi, parametricGaussPointKsi);
BSPLines1D bsplinesHeta = new BSPLines1D(degreeHeta, knotValueVectorHeta, parametricGaussPointHeta);
bsplinesKsi.calculateBSPLinesAndDerivatives();
bsplinesHeta.calculateBSPLinesAndDerivatives();
int supportKsi = degreeKsi + 1;
int supportHeta = degreeHeta + 1;
int numberOfElementControlPoints = supportKsi * supportHeta;
NurbsValues = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsDerivativeValuesKsi = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsDerivativeValuesHeta = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsSecondDerivativeValueKsi = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsSecondDerivativeValueHeta = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
NurbsSecondDerivativeValueKsiHeta = Matrix.CreateZero(numberOfElementControlPoints, gaussPoints.Count);
for (int i = 0; i < supportKsi; i++)
{
for (int j = 0; j < supportHeta; j++)
{
double sumKsiHeta = 0;
double sumdKsiHeta = 0;
double sumKsidHeta = 0;
double sumdKsidKsi = 0;
double sumdHetadHeta = 0;
double sumdKsidHeta = 0;
for (int k = 0; k < numberOfElementControlPoints; k++)
{
int indexKsi =
face.ControlPointsDictionary.First(cp => cp.Value.ID == controlPoints[k].ID).Key /
numberOfControlPointsHeta;
int indexHeta =
face.ControlPointsDictionary.First(cp => cp.Value.ID == controlPoints[k].ID).Key %
numberOfControlPointsHeta;
sumKsiHeta += bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
controlPoints[k].WeightFactor;
sumdKsiHeta += bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
controlPoints[k].WeightFactor;
sumKsidHeta += bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] *
controlPoints[k].WeightFactor;
sumdKsidKsi += bsplinesKsi.BSPLineSecondDerivativeValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
controlPoints[k].WeightFactor;
sumdHetadHeta += bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineSecondDerivativeValues[indexHeta, j] *
controlPoints[k].WeightFactor;
sumdKsidHeta += bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] *
bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] *
controlPoints[k].WeightFactor;
}
for (int k = 0; k < numberOfElementControlPoints; k++)
{
int indexKsi =
face.ControlPointsDictionary.First(cp => cp.Value.ID == controlPoints[k].ID).Key /
numberOfControlPointsHeta;
int indexHeta =
face.ControlPointsDictionary.First(cp => cp.Value.ID == controlPoints[k].ID).Key %
numberOfControlPointsHeta;
NurbsValues[k, i *supportHeta + j] =
bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
controlPoints[k].WeightFactor / sumKsiHeta;
NurbsDerivativeValuesKsi[k, i *supportHeta + j] =
bsplinesHeta.BSPLineValues[indexHeta, j] * controlPoints[k].WeightFactor *
(bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] * sumKsiHeta -
bsplinesKsi.BSPLineValues[indexKsi, i] * sumdKsiHeta) / Math.Pow(sumKsiHeta, 2);
NurbsDerivativeValuesHeta[k, i *supportHeta + j] =
bsplinesKsi.BSPLineValues[indexKsi, i] * controlPoints[k].WeightFactor *
(bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] * sumKsiHeta -
bsplinesHeta.BSPLineValues[indexHeta, j] * sumKsidHeta) / Math.Pow(sumKsiHeta, 2);
NurbsSecondDerivativeValueKsi[k, i *supportHeta + j] =
bsplinesHeta.BSPLineValues[indexHeta, j] * controlPoints[k].WeightFactor *
(bsplinesKsi.BSPLineSecondDerivativeValues[indexKsi, i] / sumKsiHeta -
2 * bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] * sumdKsiHeta /
Math.Pow(sumKsiHeta, 2) -
bsplinesKsi.BSPLineValues[indexKsi, i] * sumdKsidKsi / Math.Pow(sumKsiHeta, 2) +
2 * bsplinesKsi.BSPLineValues[indexKsi, i] * Math.Pow(sumdKsiHeta, 2) /
Math.Pow(sumKsiHeta, 3));
NurbsSecondDerivativeValueHeta[k, i *supportHeta + j] =
bsplinesKsi.BSPLineValues[indexKsi, i] * controlPoints[k].WeightFactor *
(bsplinesHeta.BSPLineSecondDerivativeValues[indexHeta, j] / sumKsiHeta -
2 * bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] * sumKsidHeta /
Math.Pow(sumKsiHeta, 2) -
bsplinesHeta.BSPLineValues[indexHeta, j] * sumdHetadHeta / Math.Pow(sumKsiHeta, 2) +
2 * bsplinesHeta.BSPLineValues[indexHeta, j] * Math.Pow(sumKsidHeta, 2) /
Math.Pow(sumKsiHeta, 3));
NurbsSecondDerivativeValueKsiHeta[k, i *supportHeta + j] =
controlPoints[k].WeightFactor *
(bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] *
bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] / sumKsiHeta -
bsplinesKsi.BSPLineDerivativeValues[indexKsi, i] *
bsplinesHeta.BSPLineValues[indexHeta, j] *
sumKsidHeta / Math.Pow(sumKsiHeta, 2) -
bsplinesKsi.BSPLineValues[indexKsi, i] *
bsplinesHeta.BSPLineDerivativeValues[indexHeta, j] *
sumdKsiHeta / Math.Pow(sumKsiHeta, 2) -
bsplinesKsi.BSPLineValues[indexKsi, i] * bsplinesHeta.BSPLineValues[indexHeta, j] *
sumdKsidHeta / Math.Pow(sumKsiHeta, 2) +
2 * bsplinesKsi.BSPLineValues[indexKsi, i] * bsplinesHeta.BSPLineValues[indexHeta, j] *
sumdKsiHeta * sumKsidHeta / Math.Pow(sumKsiHeta, 3));
}
}
}
}
