private Matrix CalculateConstitutiveMatrix(NurbsKirchhoffLoveShellElement element, Vector surfaceBasisVector1, Vector surfaceBasisVector2)
{
var auxMatrix1 = Matrix.CreateZero(2, 2);
auxMatrix1[0, 0] = surfaceBasisVector1.DotProduct(surfaceBasisVector1);
auxMatrix1[0, 1] = surfaceBasisVector1.DotProduct(surfaceBasisVector2);
auxMatrix1[1, 0] = surfaceBasisVector2.DotProduct(surfaceBasisVector1);
auxMatrix1[1, 1] = surfaceBasisVector2.DotProduct(surfaceBasisVector2);
(Matrix inverse, double det) = auxMatrix1.InvertAndDeterminant();
var material = ((IContinuumMaterial2D)element.Patch.Material);
var constitutiveMatrix = Matrix.CreateFromArray(new double[3, 3]
{
{
inverse[0, 0] * inverse[0, 0],
material.PoissonRatio * inverse[0, 0] * inverse[1, 1] + (1 - material.PoissonRatio) * inverse[1, 0] * inverse[1, 0],
inverse[0, 0] * inverse[1, 0]
},
{
material.PoissonRatio *inverse[0, 0] * inverse[1, 1] + (1 - material.PoissonRatio) * inverse[1, 0] * inverse[1, 0],
inverse[1, 1] * inverse[1, 1],
inverse[1, 1] * inverse[1, 0]
},
{
inverse[0, 0] * inverse[1, 0],
inverse[1, 1] * inverse[1, 0],
0.5 * (1 - material.PoissonRatio) * inverse[0, 0] * inverse[1, 1] + (1 + material.PoissonRatio) * inverse[1, 0] * inverse[1, 0]
},
});
return(constitutiveMatrix);
}
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);
}
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 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);
}