/// <summary>
/// Calculates displacements of knots for post-processing with Paraview.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsElement3D"/>.</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 = (NurbsElement3D)element;
var elementControlPoints = nurbsElement.ControlPoints.ToArray();
var elementKnots = nurbsElement.Knots.ToArray();
var knotParametricCoordinatesKsi = Vector.CreateFromArray(new double[] { elementKnots[0].Ksi, elementKnots[4].Ksi });
var knotParametricCoordinatesHeta = Vector.CreateFromArray(new double[] { elementKnots[0].Heta, elementKnots[2].Heta });
var knotParametricCoordinatesΖeta = Vector.CreateFromArray(new double[] { elementKnots[0].Zeta, elementKnots[1].Zeta });
Nurbs3D nurbs = new Nurbs3D(nurbsElement, elementControlPoints, knotParametricCoordinatesKsi, knotParametricCoordinatesHeta, knotParametricCoordinatesΖeta);
var knotDisplacements = new double[8, 3];
var paraviewKnotRenumbering = new int[] { 0, 4, 2, 6, 1, 5, 3, 7 };
for (int j = 0; j < elementKnots.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];
knotDisplacements[paraviewKnotRenumbering[j], 2] += nurbs.NurbsValues[i, j] * localDisplacements[i, 2];
}
}
return(knotDisplacements);
}
/// <summary>
/// Calculates the stiffness matrix of the element.
/// </summary>
/// <param name="element">An element of type <see cref="NurbsElement3D"/>.</param>
/// <returns>An <see cref="IMatrix"/> containing the stiffness matrix of an <see cref="NurbsElement3D"/>.</returns>
public IMatrix StiffnessMatrix(IElement element)
{
var nurbsElement = (NurbsElement3D)element;
var elementControlPoints = nurbsElement.ControlPoints.ToArray();
IList <GaussLegendrePoint3D> gaussPoints = CreateElementGaussPoints(nurbsElement);
Matrix stiffnessMatrixElement = Matrix.CreateZero(nurbsElement.ControlPointsDictionary.Count * 3, nurbsElement.ControlPointsDictionary.Count * 3);
Nurbs3D nurbs = new Nurbs3D(nurbsElement, elementControlPoints);
for (int j = 0; j < gaussPoints.Count; j++)
{
Matrix jacobianMatrix = CalculateJacobian(elementControlPoints, nurbs, j);
double jacdet = CalculateJacobianDeterminant(jacobianMatrix);
Matrix inverseJacobian = CalculateInverseJacobian(jacobianMatrix, jacdet);
Matrix B1 = CalculateDeformationMatrix1(inverseJacobian);
Matrix B2 = CalculateDeformationMatrix2(elementControlPoints, nurbs, j);
Matrix B = B1 * B2;
IMatrixView E = ((IContinuumMaterial3D)nurbsElement.Patch.Material).ConstitutiveMatrix;
Matrix stiffnessMatrixGaussPoint = B.ThisTransposeTimesOtherTimesThis(E) * jacdet * gaussPoints[j].WeightFactor;
for (int m = 0; m < elementControlPoints.Length * 3; m++)
{
for (int n = 0; n < elementControlPoints.Length * 3; n++)
{
stiffnessMatrixElement[m, n] += stiffnessMatrixGaussPoint[m, n];
}
}
}
return(stiffnessMatrixElement);
}
private static Matrix CalculateJacobian(IList <ControlPoint> elementControlPoints, Nurbs3D nurbs, int j)
{
Matrix jacobianMatrix = Matrix.CreateZero(3, 3);
for (int k = 0; k < elementControlPoints.Count; 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;
jacobianMatrix[2, 0] += nurbs.NurbsDerivativeValuesZeta[k, j] * elementControlPoints[k].X;
jacobianMatrix[2, 1] += nurbs.NurbsDerivativeValuesZeta[k, j] * elementControlPoints[k].Y;
jacobianMatrix[2, 2] += nurbs.NurbsDerivativeValuesZeta[k, j] * elementControlPoints[k].Z;
}
return(jacobianMatrix);
}
private static Matrix CalculateDeformationMatrix2(IList <ControlPoint> elementControlPoints, Nurbs3D nurbs, int j)
{
Matrix B2 = Matrix.CreateZero(9, 3 * elementControlPoints.Count);
for (int column = 0; column < 3 * elementControlPoints.Count; column += 3)
{
B2[0, column] += nurbs.NurbsDerivativeValuesKsi[column / 3, j];
B2[1, column] += nurbs.NurbsDerivativeValuesHeta[column / 3, j];
B2[2, column] += nurbs.NurbsDerivativeValuesZeta[column / 3, j];
B2[3, column + 1] += nurbs.NurbsDerivativeValuesKsi[column / 3, j];
B2[4, column + 1] += nurbs.NurbsDerivativeValuesHeta[column / 3, j];
B2[5, column + 1] += nurbs.NurbsDerivativeValuesZeta[column / 3, j];
B2[6, column + 2] += nurbs.NurbsDerivativeValuesKsi[column / 3, j];
B2[7, column + 2] += nurbs.NurbsDerivativeValuesHeta[column / 3, j];
B2[8, column + 2] += nurbs.NurbsDerivativeValuesZeta[column / 3, j];
}
return(B2);
}
