private double[,] CalculateMembraneDeformationMatrix(ShapeTSplines2DFromBezierExtraction tsplines, int j, double[] surfaceBasisVector1,
double[] surfaceBasisVector2, ControlPoint[] elementControlPoints)
{
var dRIa = new double[3, elementControlPoints.Length * 3];
for (var i = 0; i < elementControlPoints.Length; i++)
{
for (var m = 0; m < 3; m++)
{
dRIa[m, i] = tsplines.TSplineDerivativeValuesHeta[i, j] * surfaceBasisVector1[m] +
tsplines.TSplineDerivativeValuesKsi[i, j] * surfaceBasisVector2[m];
}
}
var Bmembrane = new double[3, elementControlPoints.Length * 3];
for (var column = 0; column < elementControlPoints.Length * 3; column += 3)
{
Bmembrane[0, column] = tsplines.TSplineDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[0];
Bmembrane[0, column + 1] = tsplines.TSplineDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[1];
Bmembrane[0, column + 2] = tsplines.TSplineDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[2];
Bmembrane[1, column] = tsplines.TSplineDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[0];
Bmembrane[1, column + 1] = tsplines.TSplineDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[1];
Bmembrane[1, column + 2] = tsplines.TSplineDerivativeValuesHeta[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);
}
/// <summary>
/// Calculates knots physical coordinates for post-processing with Paraview.
/// </summary>
/// <param name="element">An <see cref="Element"/> of type <see cref="TSplineKirchhoffLoveShellElementMaterial"/>.</param>
/// <returns>A <see cref="double"/> array calculating the coordinates 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[,] CalculatePointsForPostProcessing(TSplineKirchhoffLoveShellElementMaterial element)
{
var localCoordinates = new double[4, 2]
{
{ -1, -1 },
{ -1, 1 },
{ 1, -1 },
{ 1, 1 }
};
var knotParametricCoordinatesKsi = Vector.CreateFromArray(new double[] { -1, 1 });
var knotParametricCoordinatesHeta = Vector.CreateFromArray(new double[] { -1, 1 });
var elementControlPoints = element.ControlPoints.ToArray();
var tsplines = new ShapeTSplines2DFromBezierExtraction(element, elementControlPoints, knotParametricCoordinatesKsi, knotParametricCoordinatesHeta);
var knotDisplacements = new double[4, 3];
var paraviewKnotRenumbering = new int[] { 0, 3, 1, 2 };
for (int j = 0; j < localCoordinates.GetLength(0); j++)
{
for (int i = 0; i < elementControlPoints.Length; i++)
{
knotDisplacements[paraviewKnotRenumbering[j], 0] += tsplines.TSplineValues[i, j] * elementControlPoints[i].X;
knotDisplacements[paraviewKnotRenumbering[j], 1] += tsplines.TSplineValues[i, j] * elementControlPoints[i].Y;
knotDisplacements[paraviewKnotRenumbering[j], 2] += tsplines.TSplineValues[i, j] * elementControlPoints[i].Z;
}
}
return(knotDisplacements);
}
private Matrix CalculateMembraneDeformationMatrix(ShapeTSplines2DFromBezierExtraction tsplines, int j, Vector surfaceBasisVector1,
Vector surfaceBasisVector2, ControlPoint[] elementControlPoints)
{
Matrix dRIa = Matrix.CreateZero(3, elementControlPoints.Length * 3);
for (int i = 0; i < elementControlPoints.Length; i++)
{
for (int m = 0; m < 3; m++)
{
dRIa[m, i] = tsplines.TSplineDerivativeValuesHeta[i, j] * surfaceBasisVector1[m] +
tsplines.TSplineDerivativeValuesKsi[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] = tsplines.TSplineDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[0];
Bmembrane[0, column + 1] = tsplines.TSplineDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[1];
Bmembrane[0, column + 2] = tsplines.TSplineDerivativeValuesKsi[column / 3, j] * surfaceBasisVector1[2];
Bmembrane[1, column] = tsplines.TSplineDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[0];
Bmembrane[1, column + 1] = tsplines.TSplineDerivativeValuesHeta[column / 3, j] * surfaceBasisVector2[1];
Bmembrane[1, column + 2] = tsplines.TSplineDerivativeValuesHeta[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);
}
/// <summary>
/// This method calculates the internal forces of the element.
/// </summary>
/// <param name="element">An element of type <see cref="TSplineKirchhoffLoveShellElementMaterial"/>.</param>
/// <param name="localDisplacements">A <see cref="double"/> array containing the displacements for the degrees of freedom of the element.</param>
/// <param name="localdDisplacements">A <see cref="double"/> array containing the displacements change for the degrees of freedom of the element.</param>
/// <returns>A <see cref="double"/> array containing the forces all degrees of freedom.</returns>
public double[] CalculateForces(IElement element, double[] localDisplacements, double[] localdDisplacements)
{
var shellElement = (TSplineKirchhoffLoveShellElementMaterial)element;
var gaussPoints = CreateElementGaussPoints(shellElement);
var ElementNodalForces = new double[shellElement.ControlPointsDictionary.Count * 3];
var tsplines = new ShapeTSplines2DFromBezierExtraction(shellElement, shellElement.ControlPoints.ToArray());
var elementControlPoints = shellElement.ControlPoints.ToArray();
for (var j = 0; j < gaussPoints.Count; j++)
{
var jacobianMatrix = CalculateJacobian(elementControlPoints, tsplines, j);
var hessianMatrix = CalculateHessian(elementControlPoints, tsplines, j);
var surfaceBasisVector1 = CalculateSurfaceBasisVector1(jacobianMatrix, 0);
var surfaceBasisVector2 = CalculateSurfaceBasisVector1(jacobianMatrix, 1);
var surfaceBasisVector3 = new[]
{
surfaceBasisVector1[1] * surfaceBasisVector2[2] - surfaceBasisVector1[2] * surfaceBasisVector2[1],
surfaceBasisVector1[2] * surfaceBasisVector2[0] - surfaceBasisVector1[0] * surfaceBasisVector2[2],
surfaceBasisVector1[0] * surfaceBasisVector2[1] - surfaceBasisVector1[1] * surfaceBasisVector2[0]
};
var norm = surfaceBasisVector3.Sum(t => t * t);
var J1 = Math.Sqrt(norm);
for (int i = 0; i < surfaceBasisVector3.Length; i++)
{
surfaceBasisVector3[i] = surfaceBasisVector3[i] / J1;
}
var surfaceBasisVectorDerivative1 = CalculateSurfaceBasisVector1(hessianMatrix, 0);
var surfaceBasisVectorDerivative2 = CalculateSurfaceBasisVector1(hessianMatrix, 1);
var surfaceBasisVectorDerivative12 = CalculateSurfaceBasisVector1(hessianMatrix, 2);
var Bmembrane = CalculateMembraneDeformationMatrix(tsplines, j, surfaceBasisVector1,
surfaceBasisVector2, elementControlPoints);
var Bbending = CalculateBendingDeformationMatrix(surfaceBasisVector3, tsplines, j, surfaceBasisVector2,
surfaceBasisVectorDerivative1, surfaceBasisVector1, J1, surfaceBasisVectorDerivative2,
surfaceBasisVectorDerivative12, elementControlPoints);
var(MembraneForces, BendingMoments) =
IntegratedStressesOverThickness(gaussPoints, j);
var wfactor = J1 * gaussPoints[j].WeightFactor;
for (var i = 0; i < Bmembrane.GetLength(1); i++)
{
for (var k = 0; k < Bmembrane.GetLength(0); k++)
{
ElementNodalForces[i] += Bmembrane[k, i] * MembraneForces[k] * wfactor +
Bbending[k, i] * BendingMoments[k] * wfactor;
}
}
}
return(ElementNodalForces);
}
/// <summary>
/// Calculates the stiffness matrix of the element.
/// </summary>
/// <param name="element">An element of type <see cref="TSplineKirchhoffLoveShellElement"/>.</param>
/// <returns>An <see cref="IMatrix"/> containing the stiffness matrix of an <see cref="TSplineKirchhoffLoveShellElement"/>.</returns>
public IMatrix StiffnessMatrix(IElement element)
{
var shellElement = (TSplineKirchhoffLoveShellElement)element;
var gaussPoints = CreateElementGaussPoints(shellElement);
var stiffnessMatrixElement = Matrix.CreateZero(shellElement.ControlPointsDictionary.Count * 3,
shellElement.ControlPointsDictionary.Count * 3);
var elementControlPoints = shellElement.ControlPoints.ToArray();
var tsplines = new ShapeTSplines2DFromBezierExtraction(shellElement, elementControlPoints);
for (var j = 0; j < gaussPoints.Count; j++)
{
var jacobianMatrix = CalculateJacobian(elementControlPoints, tsplines, j);
var hessianMatrix = CalculateHessian(elementControlPoints, tsplines, 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);
var constitutiveMatrix = CalculateConstitutiveMatrix(shellElement, surfaceBasisVector1, surfaceBasisVector2);
var Bmembrane = CalculateMembraneDeformationMatrix(tsplines, j, surfaceBasisVector1,
surfaceBasisVector2, elementControlPoints);
var Bbending = CalculateBendingDeformationMatrix(surfaceBasisVector3, tsplines, j, surfaceBasisVector2,
surfaceBasisVectorDerivative1, surfaceBasisVector1, J1, surfaceBasisVectorDerivative2,
surfaceBasisVectorDerivative12, elementControlPoints);
var 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;
var 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);
}
private static Matrix CalculateJacobian(ControlPoint[] elementControlPoints, ShapeTSplines2DFromBezierExtraction tsplines, int j)
{
var jacobianMatrix = Matrix.CreateZero(2, 3);
for (var k = 0; k < elementControlPoints.Length; k++)
{
jacobianMatrix[0, 0] += tsplines.TSplineDerivativeValuesKsi[k, j] * elementControlPoints[k].X;
jacobianMatrix[0, 1] += tsplines.TSplineDerivativeValuesKsi[k, j] * elementControlPoints[k].Y;
jacobianMatrix[0, 2] += tsplines.TSplineDerivativeValuesKsi[k, j] * elementControlPoints[k].Z;
jacobianMatrix[1, 0] += tsplines.TSplineDerivativeValuesHeta[k, j] * elementControlPoints[k].X;
jacobianMatrix[1, 1] += tsplines.TSplineDerivativeValuesHeta[k, j] * elementControlPoints[k].Y;
jacobianMatrix[1, 2] += tsplines.TSplineDerivativeValuesHeta[k, j] * elementControlPoints[k].Z;
}
return(jacobianMatrix);
}
private static Matrix CalculateHessian(ControlPoint[] elementControlPoints, ShapeTSplines2DFromBezierExtraction tsplines, int j)
{
Matrix hessianMatrix = Matrix.CreateZero(3, 3);
for (int k = 0; k < elementControlPoints.Length; k++)
{
hessianMatrix[0, 0] += tsplines.TSplineSecondDerivativesValueKsi[k, j] * elementControlPoints[k].X;
hessianMatrix[0, 1] += tsplines.TSplineSecondDerivativesValueKsi[k, j] * elementControlPoints[k].Y;
hessianMatrix[0, 2] += tsplines.TSplineSecondDerivativesValueKsi[k, j] * elementControlPoints[k].Z;
hessianMatrix[1, 0] += tsplines.TSplineSecondDerivativesValueHeta[k, j] * elementControlPoints[k].X;
hessianMatrix[1, 1] += tsplines.TSplineSecondDerivativesValueHeta[k, j] * elementControlPoints[k].Y;
hessianMatrix[1, 2] += tsplines.TSplineSecondDerivativesValueHeta[k, j] * elementControlPoints[k].Z;
hessianMatrix[2, 0] += tsplines.TSplineSecondDerivativesValueKsiHeta[k, j] * elementControlPoints[k].X;
hessianMatrix[2, 1] += tsplines.TSplineSecondDerivativesValueKsiHeta[k, j] * elementControlPoints[k].Y;
hessianMatrix[2, 2] += tsplines.TSplineSecondDerivativesValueKsiHeta[k, j] * elementControlPoints[k].Z;
}
return(hessianMatrix);
}
private double[,] CalculateBendingDeformationMatrix(double[] surfaceBasisVector3, ShapeTSplines2DFromBezierExtraction tsplines, int j,
double[] surfaceBasisVector2, double[] surfaceBasisVectorDerivative1, double[] surfaceBasisVector1, double J1,
double[] surfaceBasisVectorDerivative2, double[] surfaceBasisVectorDerivative12, ControlPoint[] elementControlPoints)
{
var Bbending = new double[3, elementControlPoints.Length * 3];
var s1 = Vector.CreateFromArray(surfaceBasisVector1);
var s2 = Vector.CreateFromArray(surfaceBasisVector2);
var s3 = Vector.CreateFromArray(surfaceBasisVector3);
var s11 = Vector.CreateFromArray(surfaceBasisVectorDerivative1);
var s22 = Vector.CreateFromArray(surfaceBasisVectorDerivative2);
var s12 = Vector.CreateFromArray(surfaceBasisVectorDerivative12);
for (int column = 0; column < elementControlPoints.Length * 3; column += 3)
{
#region BI1
var BI1 = s3.CrossProduct(s3);
BI1.ScaleIntoThis(tsplines.TSplineDerivativeValuesHeta[column / 3, j]);
var auxVector = s2.CrossProduct(s3);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesKsi[column / 3, j]);
BI1.AddIntoThis(auxVector);
BI1.ScaleIntoThis(s3.DotProduct(s11));
auxVector = s1.CrossProduct(s11);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesHeta[column / 3, j]);
BI1.AddIntoThis(auxVector);
BI1.ScaleIntoThis(1 / J1);
auxVector[0] = surfaceBasisVector3[0];
auxVector[1] = surfaceBasisVector3[1];
auxVector[2] = surfaceBasisVector3[2];
auxVector.ScaleIntoThis(-tsplines.TSplineSecondDerivativesValueKsi[column / 3, j]);
BI1.AddIntoThis(auxVector);
#endregion BI1
#region BI2
IVector BI2 = s3.CrossProduct(s3);
BI2.ScaleIntoThis(tsplines.TSplineDerivativeValuesHeta[column / 3, j]);
auxVector = s2.CrossProduct(s3);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesKsi[column / 3, j]);
BI2.AddIntoThis(auxVector);
BI2.ScaleIntoThis(s3.DotProduct(s22));
auxVector = s1.CrossProduct(s22);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesHeta[column / 3, j]);
BI2.AddIntoThis(auxVector);
auxVector = s22.CrossProduct(s2);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesKsi[column / 3, j]);
BI2.AddIntoThis(auxVector);
BI2.ScaleIntoThis(1 / J1);
auxVector[0] = surfaceBasisVector3[0];
auxVector[1] = surfaceBasisVector3[1];
auxVector[2] = surfaceBasisVector3[2];
auxVector.ScaleIntoThis(-tsplines.TSplineSecondDerivativesValueHeta[column / 3, j]);
BI2.AddIntoThis(auxVector);
#endregion BI2
#region BI3
Vector BI3 = s3.CrossProduct(s3);
BI3.ScaleIntoThis(tsplines.TSplineDerivativeValuesHeta[column / 3, j]);
auxVector = s2.CrossProduct(s3);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesKsi[column / 3, j]);
BI3.AddIntoThis(auxVector);
BI3.ScaleIntoThis(s3.DotProduct(s12));
auxVector = s1.CrossProduct(s12);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesHeta[column / 3, j]);
BI3.AddIntoThis(auxVector);
auxVector = s22.CrossProduct(s2);
auxVector.ScaleIntoThis(tsplines.TSplineDerivativeValuesKsi[column / 3, j]);
BI3.AddIntoThis(auxVector);
BI3.ScaleIntoThis(1 / J1);
auxVector[0] = surfaceBasisVector3[0];
auxVector[1] = surfaceBasisVector3[1];
auxVector[2] = surfaceBasisVector3[2];
auxVector.ScaleIntoThis(-tsplines.TSplineSecondDerivativesValueKsiHeta[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);
}
/// <summary>
/// Calculates the stiffness matrix of the element.
/// </summary>
/// <param name="element">An element of type <see cref="TSplineKirchhoffLoveShellElementMaterial"/>.</param>
/// <returns>An <see cref="IMatrix"/> containing the stiffness matrix of an <see cref="TSplineKirchhoffLoveShellElementMaterial"/>.</returns>
public IMatrix StiffnessMatrix(IElement element)
{
var shellElement = (TSplineKirchhoffLoveShellElementMaterial)element;
var gaussPoints = _materialsAtThicknessGp.Keys.ToArray();
var elementControlPoints = shellElement.ControlPoints.ToArray();
ShapeTSplines2DFromBezierExtraction tsplines =
new ShapeTSplines2DFromBezierExtraction(shellElement, elementControlPoints);
var bRows = 3;
var bCols = elementControlPoints.Length * 3;
var stiffnessMatrix = new double[bCols, bCols];
var BmTranspose = new double[bCols, bRows];
var BbTranspose = new double[bCols, bRows];
var BmTransposeMultStiffness = new double[bCols, bRows];
var BbTransposeMultStiffness = new double[bCols, bRows];
var BmbTransposeMultStiffness = new double[bCols, bRows];
var BbmTransposeMultStiffness = new double[bCols, bRows];
for (int j = 0; j < gaussPoints.Length; j++)
{
var jacobianMatrix = CalculateJacobian(elementControlPoints, tsplines, j);
var hessianMatrix = CalculateHessian(elementControlPoints, tsplines, j);
var surfaceBasisVector1 = CalculateSurfaceBasisVector1(jacobianMatrix, 0);
var surfaceBasisVector2 = CalculateSurfaceBasisVector1(jacobianMatrix, 1);
var surfaceBasisVector3 = new[]
{
surfaceBasisVector1[1] * surfaceBasisVector2[2] - surfaceBasisVector1[2] * surfaceBasisVector2[1],
surfaceBasisVector1[2] * surfaceBasisVector2[0] - surfaceBasisVector1[0] * surfaceBasisVector2[2],
surfaceBasisVector1[0] * surfaceBasisVector2[1] - surfaceBasisVector1[1] * surfaceBasisVector2[0]
};
foreach (var integrationPointMaterial in _materialsAtThicknessGp[gaussPoints[j]].Values)
{
integrationPointMaterial.TangentVectorV1 = surfaceBasisVector1;
integrationPointMaterial.TangentVectorV2 = surfaceBasisVector2;
integrationPointMaterial.NormalVectorV3 = surfaceBasisVector3;
}
double norm = 0;
for (int i = 0; i < surfaceBasisVector3.Length; i++)
{
norm += surfaceBasisVector3[i] * surfaceBasisVector3[i];
}
var J1 = Math.Sqrt(norm);
for (int i = 0; i < surfaceBasisVector3.Length; i++)
{
surfaceBasisVector3[i] = surfaceBasisVector3[i] / J1;
}
var surfaceBasisVectorDerivative1 = CalculateSurfaceBasisVector1(hessianMatrix, 0);
var surfaceBasisVectorDerivative2 = CalculateSurfaceBasisVector1(hessianMatrix, 1);
var surfaceBasisVectorDerivative12 = CalculateSurfaceBasisVector1(hessianMatrix, 2);
var Bmembrane = CalculateMembraneDeformationMatrix(tsplines, j, surfaceBasisVector1,
surfaceBasisVector2, elementControlPoints);
var Bbending = CalculateBendingDeformationMatrix(surfaceBasisVector3, tsplines, j, surfaceBasisVector2,
surfaceBasisVectorDerivative1, surfaceBasisVector1, J1, surfaceBasisVectorDerivative2,
surfaceBasisVectorDerivative12, elementControlPoints);
var(MembraneConstitutiveMatrix, BendingConstitutiveMatrix, CouplingConstitutiveMatrix) =
IntegratedConstitutiveOverThickness(gaussPoints, j);
double wFactor = J1 * gaussPoints[j].WeightFactor;
double tempb = 0;
double tempm = 0;
Array.Clear(BmTranspose, 0, bRows * bCols);
Array.Clear(BbTranspose, 0, bRows * bCols);
for (int i = 0; i < bRows; i++)
{
for (int k = 0; k < bCols; k++)
{
BmTranspose[k, i] = Bmembrane[i, k] * wFactor;
BbTranspose[k, i] = Bbending[i, k] * wFactor;
}
}
double tempcm = 0;
double tempcb = 0;
double tempcc = 0;
Array.Clear(BmTransposeMultStiffness, 0, bRows * bCols);
Array.Clear(BbTransposeMultStiffness, 0, bRows * bCols);
Array.Clear(BmbTransposeMultStiffness, 0, bRows * bCols);
Array.Clear(BbmTransposeMultStiffness, 0, bRows * bCols);
for (int i = 0; i < bCols; i++)
{
for (int k = 0; k < bRows; k++)
{
tempm = BmTranspose[i, k];
tempb = BbTranspose[i, k];
for (int m = 0; m < bRows; m++)
{
tempcm = MembraneConstitutiveMatrix[k, m];
tempcb = BendingConstitutiveMatrix[k, m];
tempcc = CouplingConstitutiveMatrix[k, m];
BmTransposeMultStiffness[i, m] += tempm * tempcm;
BbTransposeMultStiffness[i, m] += tempb * tempcb;
BmbTransposeMultStiffness[i, m] += tempm * tempcc;
BbmTransposeMultStiffness[i, m] += tempb * tempcc;
}
}
}
double tempmb = 0;
double tempbm = 0;
double mem = 0;
double ben = 0;
for (int i = 0; i < bCols; i++)
{
for (int k = 0; k < bRows; k++)
{
tempm = BmTransposeMultStiffness[i, k];
tempb = BbTransposeMultStiffness[i, k];
tempmb = BmbTransposeMultStiffness[i, k];
tempbm = BbmTransposeMultStiffness[i, k];
for (int m = 0; m < bCols; m++)
{
mem = Bmembrane[k, m];
ben = Bbending[k, m];
stiffnessMatrix[i, m] += tempm * mem + tempb * ben + tempmb * ben + tempbm * mem;
}
}
}
}
return(Matrix.CreateFromArray(stiffnessMatrix));
}
/// <summary>
/// This method calculates the stresses of the element.
/// </summary>
/// <param name="element">An element of type <see cref="TSplineKirchhoffLoveShellElementMaterial"/></param>
/// <param name="localDisplacements">A <see cref="double"/> array containing the displacements for the degrees of freedom of the element.</param>
/// <param name="localdDisplacements">A <see cref="double"/> array containing the displacements change for the degrees of freedom of the element.</param>
/// <returns>A <see cref="Tuple{T1,T2}"/> of the stresses and strains of the element.</returns>
public Tuple <double[], double[]> CalculateStresses(IElement element, double[] localDisplacements, double[] localdDisplacements)
{
var shellElement = (TSplineKirchhoffLoveShellElementMaterial)element;
var elementControlPoints = shellElement.ControlPoints.ToArray();
var tsplines = new ShapeTSplines2DFromBezierExtraction(shellElement, shellElement.ControlPoints.ToArray());
for (var j = 0; j < _materialsAtThicknessGp.Keys.Count; j++)
{
var jacobianMatrix = CalculateJacobian(elementControlPoints, tsplines, j);
var hessianMatrix = CalculateHessian(elementControlPoints, tsplines, j);
var surfaceBasisVector1 = CalculateSurfaceBasisVector1(jacobianMatrix, 0);
var surfaceBasisVector2 = CalculateSurfaceBasisVector1(jacobianMatrix, 1);
var surfaceBasisVector3 = new[]
{
surfaceBasisVector1[1] * surfaceBasisVector2[2] - surfaceBasisVector1[2] * surfaceBasisVector2[1],
surfaceBasisVector1[2] * surfaceBasisVector2[0] - surfaceBasisVector1[0] * surfaceBasisVector2[2],
surfaceBasisVector1[0] * surfaceBasisVector2[1] - surfaceBasisVector1[1] * surfaceBasisVector2[0]
};
var norm = surfaceBasisVector3.Sum(t => t * t);
var J1 = Math.Sqrt(norm);
var surfaceBasisVectorDerivative1 = CalculateSurfaceBasisVector1(hessianMatrix, 0);
var surfaceBasisVectorDerivative2 = CalculateSurfaceBasisVector1(hessianMatrix, 1);
var surfaceBasisVectorDerivative12 = CalculateSurfaceBasisVector1(hessianMatrix, 2);
var Bmembrane = CalculateMembraneDeformationMatrix(tsplines, j, surfaceBasisVector1, surfaceBasisVector2, elementControlPoints);
var Bbending = CalculateBendingDeformationMatrix(surfaceBasisVector3, tsplines, j, surfaceBasisVector2,
surfaceBasisVectorDerivative1, surfaceBasisVector1, J1, surfaceBasisVectorDerivative2,
surfaceBasisVectorDerivative12, elementControlPoints);
var membraneStrain = new double[Bmembrane.GetLength(0)];
var bendingStrain = new double[Bmembrane.GetLength(0)];
for (var i = 0; i < Bmembrane.GetLength(1); i++)
{
for (var k = 0; k < Bmembrane.GetLength(0); k++)
{
membraneStrain[i] += Bmembrane[k, i] * localDisplacements[k];
bendingStrain[i] += Bbending[k, i] * localDisplacements[k];
}
}
foreach (var keyValuePair in _materialsAtThicknessGp[_materialsAtThicknessGp.Keys.ToList()[j]])
{
var thicknessPoint = keyValuePair.Key;
var material = keyValuePair.Value;
var gpStrain = new double[bendingStrain.Length];
var z = -thicknessPoint.Zeta;
for (var i = 0; i < bendingStrain.Length; i++)
{
gpStrain[i] += membraneStrain[i] + bendingStrain[i] * z;
}
material.UpdateMaterial(gpStrain);
}
}
return(new Tuple <double[], double[]>(new double[0], new double[0]));
}