public AuxiliaryStatesTensors ComputeTensorsAt(CartesianPoint globalIntegrationPoint,
TipCoordinateSystem tipCoordinateSystem)
{
// Common calculations
PolarPoint2D polarCoordinates =
tipCoordinateSystem.TransformPointGlobalCartesianToLocalPolar(globalIntegrationPoint);
var commonValues = new CommonValues(polarCoordinates.R, polarCoordinates.Theta);
// Displacement field derivatives
Matrix2by2 displacementGradientMode1, displacementGradientMode2;
TipJacobians polarJacobians = tipCoordinateSystem.CalculateJacobiansAt(polarCoordinates);
ComputeDisplacementDerivatives(polarJacobians, commonValues,
out displacementGradientMode1, out displacementGradientMode2);
// Strains
Tensor2D strainTensorMode1 = ComputeStrainTensor(displacementGradientMode1);
Tensor2D strainTensorMode2 = ComputeStrainTensor(displacementGradientMode2);
// Stresses
Tensor2D stressTensorMode1, stressTensorMode2;
ComputeStressTensors(commonValues, out stressTensorMode1, out stressTensorMode2);
return(new AuxiliaryStatesTensors(displacementGradientMode1, displacementGradientMode2,
strainTensorMode1, strainTensorMode2, stressTensorMode1, stressTensorMode2));
}
public EvaluatedFunction2D EvaluateAllAt(PolarPoint2D point, TipJacobians jacobian)
{
double sqrtR = Math.Sqrt(point.R);
double cosThetaHalf = Math.Cos(point.Theta / 2.0);
double sinThetaHalf = Math.Sin(point.Theta / 2.0);
double value = sqrtR * cosThetaHalf;
var gradientPolar = Vector2.Create(0.5 / sqrtR * cosThetaHalf, -0.5 * sqrtR * sinThetaHalf);
Vector2 gradientGlobal = jacobian.TransformScalarFieldDerivativesLocalPolarToGlobalCartesian(gradientPolar);
return(new EvaluatedFunction2D(value, gradientGlobal));
}
private void ComputeDisplacementDerivatives(TipJacobians polarJacobians, CommonValues val,
out Matrix2by2 displacementGradientMode1, out Matrix2by2 displacementGradientMode2)
{
// Temporary values and derivatives of the differentiated quantities. See documentation for their derivation.
double E = material.HomogeneousYoungModulus;
double v = material.HomogeneousPoissonRatio;
double vEq = material.HomogeneousEquivalentPoissonRatio;
double k = (3.0 - vEq) / (1.0 + vEq);
double a = (1.0 + v) / (E * Math.Sqrt(2.0 * Math.PI));
double b = val.sqrtR;
double b_r = 0.5 / val.sqrtR;
// Mode 1
{
// Temporary values that differ between the 2 modes
double c1 = val.cosThetaOver2 * (k - val.cosTheta);
double c2 = val.sinThetaOver2 * (k - val.cosTheta);
double c1_theta = -0.5 * c2 + val.cosThetaOver2 * val.sinTheta;
double c2_theta = 0.5 * c1 + val.sinThetaOver2 * val.sinTheta;
// The vector field derivatives w.r.t. to the local polar coordinates.
// The vector components refer to the local cartesian system though.
var polarGradient = Matrix2by2.CreateFromArray(
new double[, ] {
{ a *b_r *c1, a *b *c1_theta }, { a *b_r *c2, a *b *c2_theta }
});
// The vector field derivatives w.r.t. to the local cartesian coordinates.
displacementGradientMode1 =
polarJacobians.TransformVectorFieldDerivativesLocalPolarToLocalCartesian(polarGradient);
}
// Mode 2
{
double paren1 = 2.0 + k + val.cosTheta;
double paren2 = 2.0 - k - val.cosTheta;
double c1 = val.sinThetaOver2 * paren1;
double c2 = val.cosThetaOver2 * paren2;
double c1_theta = 0.5 * val.cosThetaOver2 * paren1 - val.sinThetaOver2 * val.sinTheta;
double c2_theta = -0.5 * val.sinThetaOver2 * paren2 + val.cosThetaOver2 * val.sinTheta;
// The vector field derivatives w.r.t. to the local polar coordinates.
// The vector components refer to the local cartesian system though.
var polarGradient = Matrix2by2.CreateFromArray(
new double[, ] {
{ a *b_r *c1, a *b *c1_theta }, { a *b_r *c2, a *b *c2_theta }
});
// The vector field derivatives w.r.t. to the local cartesian coordinates.
displacementGradientMode2 =
polarJacobians.TransformVectorFieldDerivativesLocalPolarToLocalCartesian(polarGradient);
}
}
public EvaluatedFunction2D[] EvaluateAllAt(NaturalPoint point, XContinuumElement2D element,
EvalInterpolation2D interpolation)
{
PolarPoint2D polarPoint = TipSystem.TransformPointGlobalCartesianToLocalPolar(
interpolation.TransformPointNaturalToGlobalCartesian());
TipJacobians tipJacobians = TipSystem.CalculateJacobiansAt(polarPoint);
var enrichments = new EvaluatedFunction2D[enrichmentFunctions.Count];
for (int i = 0; i < enrichments.Length; ++i)
{
enrichments[i] = enrichmentFunctions[i].EvaluateAllAt(polarPoint, tipJacobians);
}
return(enrichments);
}