private Matrix CalculateMembraneDeformationMatrix(Nurbs2D nurbs, int j, Vector surfaceBasisVector1,
Vector surfaceBasisVector2, NurbsKirchhoffLoveShellElement element)
{
var elementControlPoints = element.ControlPoints.ToArray();
Matrix dRIa = Matrix.CreateZero(3, elementControlPoints.Length);
for (int i = 0; i < elementControlPoints.Length; i++)
{
for (int m = 0; m < 3; m++)
{
dRIa[m, i] = nurbs.NurbsDerivativeValuesHeta[i, j] * surfaceBasisVector1[m] +
nurbs.NurbsDerivativeValuesKsi[i, j] * surfaceBasisVector2[m];
}
}
Matrix Bmembrane = Matrix.CreateZero(3, elementControlPoints.Length * 3);
for (int column = 0; column < elementControlPoints.Length * 3; column += 3)
{
Bmembrane[0, column] = nurbs.NurbsDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[0];
Bmembrane[0, column + 1] = nurbs.NurbsDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[1];
Bmembrane[0, column + 2] = nurbs.NurbsDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[2];
Bmembrane[1, column] = nurbs.NurbsDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[0];
Bmembrane[1, column + 1] = nurbs.NurbsDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[1];
Bmembrane[1, column + 2] = nurbs.NurbsDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[2];
Bmembrane[2, column] = dRIa[0, column / 3];
Bmembrane[2, column + 1] = dRIa[1, column / 3];
Bmembrane[2, column + 2] = dRIa[2, column / 3];
}
return(Bmembrane);
}
private static void CalculateNeumannLoad2D(Element element, NeumannBoundaryCondition neumann, Dictionary <int, double> neumannLoad,
Nurbs2D nurbs, int j, double jacdet, IList <GaussLegendrePoint3D> gaussPoints, double xGaussPoint, double yGaussPoint, double zGaussPoint,
Vector surfaceBasisVector3)
{
var elementControlPoints = element.ControlPoints.ToArray();
for (int k = 0; k < elementControlPoints.Length; k++)
{
int dofIDX =
element.Model.GlobalDofOrdering.GlobalFreeDofs[elementControlPoints[k], StructuralDof.TranslationX];
int dofIDY =
element.Model.GlobalDofOrdering.GlobalFreeDofs[elementControlPoints[k], StructuralDof.TranslationY];
int dofIDZ =
element.Model.GlobalDofOrdering.GlobalFreeDofs[elementControlPoints[k], StructuralDof.TranslationZ];
if (neumannLoad.ContainsKey(dofIDX))
{
neumannLoad[dofIDX] += nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor *
neumann.Value(xGaussPoint, yGaussPoint, zGaussPoint)[0] *
surfaceBasisVector3[0];
}
else
{
neumannLoad.Add(
dofIDX,
nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor *
neumann.Value(xGaussPoint, yGaussPoint, zGaussPoint)[0] * surfaceBasisVector3[0]);
}
if (neumannLoad.ContainsKey(dofIDY))
{
neumannLoad[dofIDY] += nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor *
neumann.Value(xGaussPoint, yGaussPoint, zGaussPoint)[1] *
surfaceBasisVector3[1];
}
else
{
neumannLoad.Add(
dofIDY,
nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor * neumann.Value(xGaussPoint, yGaussPoint, zGaussPoint)[1] * surfaceBasisVector3[1]);
}
if (neumannLoad.ContainsKey(dofIDZ))
{
neumannLoad[dofIDZ] += nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor * neumann.Value(xGaussPoint, yGaussPoint, zGaussPoint)[2] * surfaceBasisVector3[2];
}
else
{
neumannLoad.Add(dofIDZ,
nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor * neumann.Value(xGaussPoint, yGaussPoint, zGaussPoint)[2] * surfaceBasisVector3[2]);
}
}
}
/// <summary>
/// Calculates the stiffness matrix of the element.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsKirchhoffLoveShellElement"/>.</param>
/// <returns>An <see cref="IMatrix"/> containing the stiffness matrix of an <see cref="NurbsKirchhoffLoveShellElement"/>.</returns>
public IMatrix StiffnessMatrix(IElement element)
{
var shellElement = (NurbsKirchhoffLoveShellElement)element;
var elementControlPoints = shellElement.ControlPoints.ToArray();
IList <GaussLegendrePoint3D> gaussPoints = CreateElementGaussPoints(shellElement);
Matrix stiffnessMatrixElement = Matrix.CreateZero(shellElement.ControlPointsDictionary.Count * 3, shellElement.ControlPointsDictionary.Count * 3);
Nurbs2D nurbs = new Nurbs2D(shellElement, elementControlPoints);
for (int j = 0; j < gaussPoints.Count; j++)
{
var jacobianMatrix = CalculateJacobian(shellElement, nurbs, j);
var hessianMatrix = CalculateHessian(shellElement, nurbs, j);
var surfaceBasisVector1 = CalculateSurfaceBasisVector1(jacobianMatrix, 0);
var surfaceBasisVector2 = CalculateSurfaceBasisVector1(jacobianMatrix, 1);
var surfaceBasisVector3 = surfaceBasisVector1.CrossProduct(surfaceBasisVector2);
var J1 = surfaceBasisVector3.Norm2();
surfaceBasisVector3.ScaleIntoThis(1 / J1);
var surfaceBasisVectorDerivative1 = CalculateSurfaceBasisVector1(hessianMatrix, 0);
var surfaceBasisVectorDerivative2 = CalculateSurfaceBasisVector1(hessianMatrix, 1);
var surfaceBasisVectorDerivative12 = CalculateSurfaceBasisVector1(hessianMatrix, 2);
Matrix constitutiveMatrix = CalculateConstitutiveMatrix(shellElement, surfaceBasisVector1, surfaceBasisVector2);
var Bmembrane = CalculateMembraneDeformationMatrix(nurbs, j, surfaceBasisVector1, surfaceBasisVector2, shellElement);
var Bbending = CalculateBendingDeformationMatrix(surfaceBasisVector3, nurbs, j, surfaceBasisVector2, surfaceBasisVectorDerivative1, surfaceBasisVector1, J1, surfaceBasisVectorDerivative2, surfaceBasisVectorDerivative12, shellElement);
double membraneStiffness = ((IIsotropicContinuumMaterial2D)shellElement.Patch.Material).YoungModulus * shellElement.Patch.Thickness /
(1 - Math.Pow(((IIsotropicContinuumMaterial2D)shellElement.Patch.Material).PoissonRatio, 2));
var Kmembrane = Bmembrane.Transpose() * constitutiveMatrix * Bmembrane * membraneStiffness * J1 *
gaussPoints[j].WeightFactor;
double bendingStiffness = ((IIsotropicContinuumMaterial2D)shellElement.Patch.Material).YoungModulus * Math.Pow(shellElement.Patch.Thickness, 3) /
12 / (1 - Math.Pow(((IIsotropicContinuumMaterial2D)shellElement.Patch.Material).PoissonRatio, 2));
var Kbending = Bbending.Transpose() * constitutiveMatrix * Bbending * bendingStiffness * J1 *
gaussPoints[j].WeightFactor;
stiffnessMatrixElement.AddIntoThis(Kmembrane);
stiffnessMatrixElement.AddIntoThis(Kbending);
}
return(stiffnessMatrixElement);
}
/// <summary>
/// Calculates displacements of knots for post-processing with Paraview.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsElement2D"/>.</param>
/// <param name="localDisplacements">A <see cref="Matrix"/> containing the displacements for the degrees of freedom of the element.</param>
/// <returns>A <see cref="double"/> array calculating the displacement of the element Knots'.
/// The rows of the matrix denote the knot numbering while the columns the displacements for each degree of freedom.</returns>
public double[,] CalculateDisplacementsForPostProcessing(Element element, Matrix localDisplacements)
{
Contract.Requires(element != null, "The element cannot be null");
var nurbsElement = (NurbsElement2D)element;
var elementControlPoints = nurbsElement.ControlPoints.ToArray();
var elemenetKnots = nurbsElement.Knots.ToArray();
var knotParametricCoordinatesKsi = Vector.CreateFromArray(new double[] { elemenetKnots[0].Ksi, elemenetKnots[2].Ksi });
var knotParametricCoordinatesHeta = Vector.CreateFromArray(new double[] { elemenetKnots[0].Heta, elemenetKnots[1].Heta });
Nurbs2D nurbs = new Nurbs2D(nurbsElement, elementControlPoints, knotParametricCoordinatesKsi, knotParametricCoordinatesHeta);
var knotDisplacements = new double[4, 2];
var paraviewKnotRenumbering = new int[] { 0, 3, 1, 2 };
for (int j = 0; j < elemenetKnots.Length; j++)
{
for (int i = 0; i < elementControlPoints.Length; i++)
{
knotDisplacements[paraviewKnotRenumbering[j], 0] += nurbs.NurbsValues[i, j] * localDisplacements[i, 0];
knotDisplacements[paraviewKnotRenumbering[j], 1] += nurbs.NurbsValues[i, j] * localDisplacements[i, 1];
}
}
return(knotDisplacements);
}
private static Matrix JacobianMatrixForLoadCalculation(Element element, Nurbs2D nurbs, int j, out double xGaussPoint,
out double yGaussPoint, out double zGaussPoint)
{
var elementControlPoints = element.ControlPoints.ToArray();
Matrix jacobianMatrix = Matrix.CreateZero(2, 3);
xGaussPoint = 0;
yGaussPoint = 0;
zGaussPoint = 0;
for (int k = 0; k < elementControlPoints.Length; k++)
{
xGaussPoint += nurbs.NurbsValues[k, j] * elementControlPoints[k].X;
yGaussPoint += nurbs.NurbsValues[k, j] * elementControlPoints[k].Y;
zGaussPoint += nurbs.NurbsValues[k, j] * elementControlPoints[k].Z;
jacobianMatrix[0, 0] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].X;
jacobianMatrix[0, 1] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Y;
jacobianMatrix[0, 2] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Z;
jacobianMatrix[1, 0] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].X;
jacobianMatrix[1, 1] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Y;
jacobianMatrix[1, 2] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Z;
}
return(jacobianMatrix);
}
/// <summary>
/// This method calculates the Neumann boundary condition when applied to a two-dimensional NURBS element.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsElement2D"/></param>
/// <param name="face">An two dimensional boundary entity. For more info see <see cref="Face"/>.</param>
/// <param name="neumann"><inheritdoc cref="NeumannBoundaryCondition"/>.</param>
/// <returns>A <see cref="Dictionary{TKey,TValue}"/> where integer values denote the degree of freedom that has a value double load value due to the enforcement of the <see cref="NeumannBoundaryCondition"/>.</returns>
public Dictionary <int, double> CalculateLoadingCondition(Element element, Face face, NeumannBoundaryCondition neumann)
{
Contract.Requires(element != null, "The element cannot be null");
Contract.Requires(face != null, "The face cannot be null");
Contract.Requires(neumann != null, "The Neumann Boundary condition cannot be null");
IList <GaussLegendrePoint3D> gaussPoints =
CreateElementGaussPoints(element, face.Degrees[0], face.Degrees[1]);
Dictionary <int, double> neumannLoad = new Dictionary <int, double>();
var elementControlPoints = element.ControlPoints.ToArray();
Nurbs2D nurbs = new Nurbs2D(element, elementControlPoints, face);
for (int j = 0; j < gaussPoints.Count; j++)
{
var jacobianMatrix = JacobianMatrixForLoadCalculation(element, nurbs, j, out var xGaussPoint, out var yGaussPoint, out var zGaussPoint);
Vector surfaceBasisVector1 = Vector.CreateZero(3);
surfaceBasisVector1[0] = jacobianMatrix[0, 0];
surfaceBasisVector1[1] = jacobianMatrix[0, 1];
surfaceBasisVector1[2] = jacobianMatrix[0, 2];
Vector surfaceBasisVector2 = Vector.CreateZero(3);
surfaceBasisVector2[0] = jacobianMatrix[1, 0];
surfaceBasisVector2[1] = jacobianMatrix[1, 1];
surfaceBasisVector2[2] = jacobianMatrix[1, 2];
Vector surfaceBasisVector3 = surfaceBasisVector1.CrossProduct(surfaceBasisVector2);
double jacdet = jacobianMatrix[0, 0] * jacobianMatrix[1, 1]
- jacobianMatrix[1, 0] * jacobianMatrix[0, 1];
CalculateNeumannLoad2D(element, neumann, neumannLoad, nurbs, j, jacdet, gaussPoints, xGaussPoint, yGaussPoint, zGaussPoint, surfaceBasisVector3);
}
return(neumannLoad);
}
/// <summary>
/// This method calculates the Pressure boundary condition when applied to a two-dimensional NURBS element.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsElement2D"/>.</param>
/// <param name="face">An two dimensional boundary entity. For more info see <see cref="Face"/>.</param>
/// <param name="pressure"><inheritdoc cref="NeumannBoundaryCondition"/>.</param>
/// <returns>A <see cref="Dictionary{TKey,TValue}"/> where integer values denote the degree of freedom that has a value double load value due to the enforcement of the <see cref="PressureBoundaryCondition"/>.</returns>
public Dictionary <int, double> CalculateLoadingCondition(Element element, Face face, PressureBoundaryCondition pressure)
{
Contract.Requires(element != null, "The element cannot be null");
Contract.Requires(face != null, "The face cannot be null");
Contract.Requires(pressure != null, "The pressure boundary condition cannot be null");
var dofs = new IDofType[] { StructuralDof.TranslationX, StructuralDof.TranslationY, StructuralDof.TranslationZ };
IList <GaussLegendrePoint3D> gaussPoints =
CreateElementGaussPoints(element, face.Degrees[0], face.Degrees[1]);
Dictionary <int, double> pressureLoad = new Dictionary <int, double>();
var elementControlPoints = element.ControlPoints.ToArray();
Nurbs2D nurbs = new Nurbs2D(element, elementControlPoints, face);
for (int j = 0; j < gaussPoints.Count; j++)
{
Matrix jacobianMatrix = Matrix.CreateZero(2, 3);
double xGaussPoint = 0;
double yGaussPoint = 0;
double zGaussPoint = 0;
for (int k = 0; k < elementControlPoints.Length; k++)
{
xGaussPoint += nurbs.NurbsValues[k, j] * elementControlPoints[k].X;
yGaussPoint += nurbs.NurbsValues[k, j] * elementControlPoints[k].Y;
zGaussPoint += nurbs.NurbsValues[k, j] * elementControlPoints[k].Z;
jacobianMatrix[0, 0] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].X;
jacobianMatrix[0, 1] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Y;
jacobianMatrix[0, 2] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Z;
jacobianMatrix[1, 0] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].X;
jacobianMatrix[1, 1] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Y;
jacobianMatrix[1, 2] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Z;
}
Vector surfaceBasisVector1 = Vector.CreateZero(3);
surfaceBasisVector1[0] = jacobianMatrix[0, 0];
surfaceBasisVector1[1] = jacobianMatrix[0, 1];
surfaceBasisVector1[2] = jacobianMatrix[0, 2];
Vector surfaceBasisVector2 = Vector.CreateZero(3);
surfaceBasisVector2[0] = jacobianMatrix[1, 0];
surfaceBasisVector2[1] = jacobianMatrix[1, 1];
surfaceBasisVector2[2] = jacobianMatrix[1, 2];
Vector surfaceBasisVector3 = surfaceBasisVector1.CrossProduct(surfaceBasisVector2);
double jacdet = (jacobianMatrix[0, 0] * jacobianMatrix[1, 1])
- (jacobianMatrix[1, 0] * jacobianMatrix[0, 1]);
for (int k = 0; k < elementControlPoints.Length; k++)
{
for (int m = 0; m < 3; m++)
{
int dofID = element.Model.GlobalDofOrdering.GlobalFreeDofs[elementControlPoints[k], dofs[m]];
if (pressureLoad.ContainsKey(dofID))
{
pressureLoad[dofID] += nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor *
pressure.Value * surfaceBasisVector3[m];
}
else
{
pressureLoad.Add(dofID,
nurbs.NurbsValues[k, j] * jacdet * gaussPoints[j].WeightFactor * pressure.Value * surfaceBasisVector3[m]);
}
}
}
}
return(pressureLoad);
}
/// <summary>
/// Calculates the stiffness matrix of the element.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsElement2D"/>.</param>
/// <returns>An <see cref="IMatrix"/> containing the stiffness matrix of an <see cref="NurbsElement2D"/>.</returns>
public IMatrix StiffnessMatrix(IElement element)
{
Contract.Requires(element != null, "The element cannot be null");
var nurbsElement = (NurbsElement2D)element;
var gaussPoints = CreateElementGaussPoints(nurbsElement);
var stiffnessMatrixElement = Matrix.CreateZero(
nurbsElement.ControlPointsDictionary.Count * 2,
nurbsElement.ControlPointsDictionary.Count * 2);
var elementControlPoints = nurbsElement.ControlPoints.ToArray();
var nurbs = new Nurbs2D(nurbsElement, elementControlPoints);
for (var j = 0; j < gaussPoints.Count; j++)
{
var jacobianMatrix = Matrix.CreateZero(2, 2);
for (int k = 0; k < elementControlPoints.Length; k++)
{
jacobianMatrix[0, 0] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].X;
jacobianMatrix[0, 1] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Y;
jacobianMatrix[1, 0] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].X;
jacobianMatrix[1, 1] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Y;
}
double jacdet = (jacobianMatrix[0, 0] * jacobianMatrix[1, 1])
- (jacobianMatrix[1, 0] * jacobianMatrix[0, 1]);
Matrix B1 = Matrix.CreateZero(3, 4);
B1[0, 0] += jacobianMatrix[1, 1] / jacdet;
B1[0, 1] += -jacobianMatrix[0, 1] / jacdet;
B1[1, 2] += -jacobianMatrix[1, 0] / jacdet;
B1[1, 3] += jacobianMatrix[0, 0] / jacdet;
B1[2, 0] += -jacobianMatrix[1, 0] / jacdet;
B1[2, 1] += jacobianMatrix[0, 0] / jacdet;
B1[2, 2] += jacobianMatrix[1, 1] / jacdet;
B1[2, 3] += -jacobianMatrix[0, 1] / jacdet;
Matrix B2 = Matrix.CreateZero(4, 2 * elementControlPoints.Length);
for (int column = 0; column < 2 * elementControlPoints.Length; column += 2)
{
B2[0, column] += nurbs.NurbsDerivativeValuesKsi[column / 2, j];
B2[1, column] += nurbs.NurbsDerivativeValuesHeta[column / 2, j];
B2[2, column + 1] += nurbs.NurbsDerivativeValuesKsi[column / 2, j];
B2[3, column + 1] += nurbs.NurbsDerivativeValuesHeta[column / 2, j];
}
Matrix B = B1 * B2;
IMatrixView elasticityMatrix = ((IContinuumMaterial2D)nurbsElement.Patch.Material).ConstitutiveMatrix;
Matrix stiffnessMatrixGaussPoint = B.ThisTransposeTimesOtherTimesThis(elasticityMatrix);
stiffnessMatrixGaussPoint *= jacdet * gaussPoints[j].WeightFactor * nurbsElement.Patch.Thickness;
for (int m = 0; m < elementControlPoints.Length * 2; m++)
{
for (int n = 0; n < elementControlPoints.Length * 2; n++)
{
stiffnessMatrixElement[m, n] += stiffnessMatrixGaussPoint[m, n];
}
}
}
return(stiffnessMatrixElement);
}
private Matrix CalculateBendingDeformationMatrix(Vector surfaceBasisVector3, Nurbs2D nurbs, int j,
Vector surfaceBasisVector2, Vector surfaceBasisVectorDerivative1, Vector surfaceBasisVector1, double J1,
Vector surfaceBasisVectorDerivative2, Vector surfaceBasisVectorDerivative12, NurbsKirchhoffLoveShellElement element)
{
var elementControlPoints = element.ControlPoints.ToArray();
Matrix Bbending = Matrix.CreateZero(3, elementControlPoints.Length * 3);
for (int column = 0; column < elementControlPoints.Length * 3; column += 3)
{
#region BI1
var BI1 = surfaceBasisVector3.CrossProduct(surfaceBasisVector3);
BI1.ScaleIntoThis(nurbs.NurbsDerivativeValuesHeta[column / 3, j]);
var auxVector = surfaceBasisVector2.CrossProduct(surfaceBasisVector3);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesKsi[column / 3, j]);
BI1.AddIntoThis(auxVector);
BI1.ScaleIntoThis(surfaceBasisVector3.DotProduct(surfaceBasisVectorDerivative1));
auxVector = surfaceBasisVector1.CrossProduct(surfaceBasisVectorDerivative1);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesHeta[column / 3, j]);
BI1.AddIntoThis(auxVector);
BI1.ScaleIntoThis(1 / J1);
auxVector[0] = surfaceBasisVector3[0];
auxVector[1] = surfaceBasisVector3[1];
auxVector[2] = surfaceBasisVector3[2];
auxVector.ScaleIntoThis(-nurbs.NurbsSecondDerivativeValueKsi[column / 3, j]);
BI1.AddIntoThis(auxVector);
#endregion BI1
#region BI2
Vector BI2 = surfaceBasisVector3.CrossProduct(surfaceBasisVector3);
BI2.ScaleIntoThis(nurbs.NurbsDerivativeValuesHeta[column / 3, j]);
auxVector = surfaceBasisVector2.CrossProduct(surfaceBasisVector3);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesKsi[column / 3, j]);
BI2.AddIntoThis(auxVector);
BI2.ScaleIntoThis(surfaceBasisVector3.DotProduct(surfaceBasisVectorDerivative2));
auxVector = surfaceBasisVector1.CrossProduct(surfaceBasisVectorDerivative2);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesHeta[column / 3, j]);
BI2.AddIntoThis(auxVector);
auxVector = surfaceBasisVectorDerivative2.CrossProduct(surfaceBasisVector2);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesKsi[column / 3, j]);
BI2.AddIntoThis(auxVector);
BI2.ScaleIntoThis(1 / J1);
auxVector[0] = surfaceBasisVector3[0];
auxVector[1] = surfaceBasisVector3[1];
auxVector[2] = surfaceBasisVector3[2];
auxVector.ScaleIntoThis(-nurbs.NurbsSecondDerivativeValueHeta[column / 3, j]);
BI2.AddIntoThis(auxVector);
#endregion BI2
#region BI3
Vector BI3 = surfaceBasisVector3.CrossProduct(surfaceBasisVector3);
BI3.ScaleIntoThis(nurbs.NurbsDerivativeValuesHeta[column / 3, j]);
auxVector = surfaceBasisVector2.CrossProduct(surfaceBasisVector3);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesKsi[column / 3, j]);
BI3.AddIntoThis(auxVector);
BI3.ScaleIntoThis(surfaceBasisVector3.DotProduct(surfaceBasisVectorDerivative12));
auxVector = surfaceBasisVector1.CrossProduct(surfaceBasisVectorDerivative12);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesHeta[column / 3, j]);
BI3.AddIntoThis(auxVector);
auxVector = surfaceBasisVectorDerivative2.CrossProduct(surfaceBasisVector2);
auxVector.ScaleIntoThis(nurbs.NurbsDerivativeValuesKsi[column / 3, j]);
BI3.AddIntoThis(auxVector);
BI3.ScaleIntoThis(1 / J1);
auxVector[0] = surfaceBasisVector3[0];
auxVector[1] = surfaceBasisVector3[1];
auxVector[2] = surfaceBasisVector3[2];
auxVector.ScaleIntoThis(-nurbs.NurbsSecondDerivativeValueKsiHeta[column / 3, j]);
BI3.AddIntoThis(auxVector);
#endregion BI3
Bbending[0, column] = BI1[0];
Bbending[0, column + 1] = BI1[1];
Bbending[0, column + 2] = BI1[2];
Bbending[1, column] = BI2[0];
Bbending[1, column + 1] = BI2[1];
Bbending[1, column + 2] = BI2[2];
Bbending[2, column] = 2 * BI3[0];
Bbending[2, column + 1] = 2 * BI3[1];
Bbending[2, column + 2] = 2 * BI3[2];
}
return(Bbending);
}
private static Matrix CalculateJacobian(NurbsKirchhoffLoveShellElement shellElement, Nurbs2D nurbs, int j)
{
var elementControlPoints = shellElement.ControlPoints.ToArray();
Matrix jacobianMatrix = Matrix.CreateZero(2, 3);
for (int k = 0; k < elementControlPoints.Length; k++)
{
jacobianMatrix[0, 0] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].X;
jacobianMatrix[0, 1] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Y;
jacobianMatrix[0, 2] += nurbs.NurbsDerivativeValuesKsi[k, j] * elementControlPoints[k].Z;
jacobianMatrix[1, 0] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].X;
jacobianMatrix[1, 1] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Y;
jacobianMatrix[1, 2] += nurbs.NurbsDerivativeValuesHeta[k, j] * elementControlPoints[k].Z;
}
return(jacobianMatrix);
}
private static Matrix CalculateHessian(NurbsKirchhoffLoveShellElement shellElement, Nurbs2D nurbs, int j)
{
var elementControlPoints = shellElement.ControlPoints.ToArray();
Matrix hessianMatrix = Matrix.CreateZero(3, 3);
for (int k = 0; k < elementControlPoints.Length; k++)
{
hessianMatrix[0, 0] += nurbs.NurbsSecondDerivativeValueKsi[k, j] * elementControlPoints[k].X;
hessianMatrix[0, 1] += nurbs.NurbsSecondDerivativeValueKsi[k, j] * elementControlPoints[k].Y;
hessianMatrix[0, 2] += nurbs.NurbsSecondDerivativeValueKsi[k, j] * elementControlPoints[k].Z;
hessianMatrix[1, 0] += nurbs.NurbsSecondDerivativeValueHeta[k, j] * elementControlPoints[k].X;
hessianMatrix[1, 1] += nurbs.NurbsSecondDerivativeValueHeta[k, j] * elementControlPoints[k].Y;
hessianMatrix[1, 2] += nurbs.NurbsSecondDerivativeValueHeta[k, j] * elementControlPoints[k].Z;
hessianMatrix[2, 0] += nurbs.NurbsSecondDerivativeValueKsiHeta[k, j] * elementControlPoints[k].X;
hessianMatrix[2, 1] += nurbs.NurbsSecondDerivativeValueKsiHeta[k, j] * elementControlPoints[k].Y;
hessianMatrix[2, 2] += nurbs.NurbsSecondDerivativeValueKsiHeta[k, j] * elementControlPoints[k].Z;
}
return(hessianMatrix);
}