public Matrix BuildLumpedMassMatrix()
{
int numDofs = 2 * Nodes.Count;
var lumpedMass = Matrix.CreateZero(numDofs, numDofs);
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForConsistentMass);
// Contribution of each Gauss point to the element's area
//TODO: Perhaps I could calculate the volume of the element without going through each Gauss point. Probably the
// nodes are needed instead of the GPs. For linear elements I can find the area geometrically (as polygons).
//TODO: this should have been cached when integrating other quantities (e.g. stiffness)
double area = 0;
for (int gp = 0; gp < QuadratureForConsistentMass.IntegrationPoints.Count; ++gp)
{
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
area += jacobian.DirectDeterminant * QuadratureForConsistentMass.IntegrationPoints[gp].Weight;
}
// Divide the total mass uniformly for each node
double nodalMass = Thickness * area * dynamicProperties.Density / Nodes.Count;
for (int i = 0; i < numDofs; ++i)
{
lumpedMass[i, i] = nodalMass;
}
return(lumpedMass);
}
public Matrix BuildDiffusionConductivityMatrix()
{
int numDofs = Nodes.Count;
var conductivity = Matrix.CreateZero(numDofs, numDofs);
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForStiffness);
for (int gp = 0; gp < QuadratureForStiffness.IntegrationPoints.Count; ++gp)
{
// Calculate the necessary quantities for the integration
//Matrix constitutive = (Matrix)(materialsAtGaussPoints[gp].ConstitutiveMatrix); // ugly cast will be removed along with the legacy Matrix classes
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
Matrix shapeGradientsCartesian =
jacobian.TransformNaturalDerivativesToCartesian(shapeGradientsNatural[gp]);
Matrix deformation = BuildDeformationMatrix(shapeGradientsCartesian);
// Contribution of this gauss point to the element stiffness matrix
Matrix partialK = deformation.Transpose() * deformation;
//Matrix partialΚ = deformation.Transpose() * (constitutive * deformation);
//partialK.Scale(materialsAtGaussPoints[gaussPoint].ThermalConductivity);
double dA = jacobian.DirectDeterminant * QuadratureForStiffness.IntegrationPoints[gp].Weight; //TODO: this is used by all methods that integrate. I should cache it.
conductivity.AxpyIntoThis(partialK, dA * material.ThermalConductivity);
}
conductivity.ScaleIntoThis(Thickness);
return(conductivity);
}
public Matrix BuildStabilizingConductivityMatrix()
{
int numDofs = Nodes.Count;
var convection = Matrix.CreateZero(numDofs, numDofs);
IReadOnlyList <double[]> shapeFunctions =
Interpolation.EvaluateFunctionsAtGaussPoints(QuadratureForConsistentMass);
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForConsistentMass);
for (int gp = 0; gp < QuadratureForConsistentMass.IntegrationPoints.Count; ++gp)
{
Vector shapeFunctionMatrix = BuildShapeFunctionMatrix(shapeFunctions[gp]);
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
Matrix shapeGradientsCartesian =
jacobian.TransformNaturalDerivativesToCartesian(shapeGradientsNatural[gp]);
Matrix deformation = BuildDeformationMatrix(shapeGradientsCartesian);
Vector deformationX = deformation.GetRow(0);
Vector deformationY = deformation.GetRow(1);
//Matrix partial = shapeFunctionMatrix.TensorProduct(shapeFunctionMatrix);
Matrix partial = deformationX.TensorProduct(deformationY);
double dA = jacobian.DirectDeterminant * QuadratureForConsistentMass.IntegrationPoints[gp].Weight;
convection.AxpyIntoThis(partial, dA);
}
//WARNING: the following needs to change for non uniform density. Perhaps the integration order too.
convection.ScaleIntoThis(-Math.Pow(material.ThermalConvection, 2));
return(convection);
}
public double CalculateArea()
{
//TODO: Linear elements can use the more efficient rules for volume of polygons. Therefore this method should be
// delegated to the interpolation.
//TODO: A different integration rule should be used for integrating constant functions. For linear elements there
// is only 1 Gauss point (most probably), therefore the computational cost could be the same as using the
// polygonal formulas.
double area = 0.0;
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForStiffness);
for (int gp = 0; gp < QuadratureForStiffness.IntegrationPoints.Count; ++gp)
{
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
area += jacobian.DirectDeterminant * QuadratureForStiffness.IntegrationPoints[gp].Weight; //TODO: this is used by all methods that integrate. I should cache it.
}
return(area);
}
public EvalInterpolation2D(IReadOnlyList <Node> elementNodes, double[] shapeFunctions, Matrix shapeGradientsNatural,
IsoparametricJacobian2D jacobian)
{
int numNodes = elementNodes.Count;
#if DEBUG
if ((shapeFunctions.Length != numNodes) || (shapeGradientsNatural.NumRows != numNodes))
{
throw new ArgumentException($"There are {numNodes} nodes, but {ShapeFunctions.Length} shape functions"
+ $" and {shapeGradientsNatural.NumRows} natural shape derivatives.");
}
#endif
this.elementNodes = elementNodes;
this.ShapeFunctions = shapeFunctions;
this.ShapeGradientsNatural = shapeGradientsNatural;
this.Jacobian = jacobian;
this.ShapeGradientsCartesian = jacobian.TransformNaturalDerivativesToCartesian(shapeGradientsNatural);
}
public Matrix BuildCapacityMatrix()
{
int numDofs = Nodes.Count;
var capacity = Matrix.CreateZero(numDofs, numDofs);
IReadOnlyList <double[]> shapeFunctions =
Interpolation.EvaluateFunctionsAtGaussPoints(QuadratureForConsistentMass);
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForConsistentMass);
for (int gp = 0; gp < QuadratureForConsistentMass.IntegrationPoints.Count; ++gp)
{
Vector shapeFunctionMatrix = BuildShapeFunctionMatrix(shapeFunctions[gp]);
Matrix partial = shapeFunctionMatrix.TensorProduct(shapeFunctionMatrix);
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
double dA = jacobian.DirectDeterminant * QuadratureForConsistentMass.IntegrationPoints[gp].Weight;
capacity.AxpyIntoThis(partial, dA);
}
//WARNING: the following needs to change for non uniform density. Perhaps the integration order too.
capacity.ScaleIntoThis(Thickness * material.Density * material.SpecialHeatCoeff);
return(capacity);
}
/// <summary>
/// Calculate strains (exx, eyy, 2exy) and stresses (sxx, syy, sxy) at integration points, store them in the materials
/// and return them (e.g. for postprocessing). The order of the tensors is the same as the order of the integration
/// points defined by <see cref="QuadratureForStiffness"/>.
/// </summary>
/// <param name="localDisplacements"></param>
/// <returns></returns>
public (IReadOnlyList <double[]> strains, IReadOnlyList <double[]> stresses) UpdateStrainsStressesAtGaussPoints(
double[] localDisplacements)
{
int numGPs = QuadratureForStiffness.IntegrationPoints.Count;
var strains = new double[numGPs][];
var stresses = new double[numGPs][];
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForStiffness);
for (int gp = 0; gp < numGPs; ++gp)
{
IMatrixView constitutive = materialsAtGaussPoints[gp].ConstitutiveMatrix;
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
Matrix shapeGrandientsCartesian =
jacobian.TransformNaturalDerivativesToCartesian(shapeGradientsNatural[gp]);
Matrix deformation = BuildDeformationMatrix(shapeGrandientsCartesian);
strains[gp] = deformation.Multiply(localDisplacements);
stresses[gp] = constitutive.Multiply(strains[gp]);
}
return(strains, stresses);
}
public Matrix BuildConsistentMassMatrix()
{
int numDofs = 2 * Nodes.Count;
var mass = Matrix.CreateZero(numDofs, numDofs);
IReadOnlyList <double[]> shapeFunctions =
Interpolation.EvaluateFunctionsAtGaussPoints(QuadratureForConsistentMass);
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForConsistentMass);
for (int gp = 0; gp < QuadratureForConsistentMass.IntegrationPoints.Count; ++gp)
{
Matrix shapeFunctionMatrix = BuildShapeFunctionMatrix(shapeFunctions[gp]);
//Matrix partial = shapeFunctionMatrix.Transpose() * shapeFunctionMatrix;
Matrix partial = shapeFunctionMatrix.MultiplyRight(shapeFunctionMatrix, true, false);
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
double dA = jacobian.DirectDeterminant * QuadratureForConsistentMass.IntegrationPoints[gp].Weight;
mass.AxpyIntoThis(partial, dA);
}
//WARNING: the following needs to change for non uniform density. Perhaps the integration order too.
mass.ScaleIntoThis(Thickness * dynamicProperties.Density);
return(mass);
}
public Matrix BuildStiffnessMatrix()
{
int numDofs = 2 * Nodes.Count;
var stiffness = Matrix.CreateZero(numDofs, numDofs);
IReadOnlyList <Matrix> shapeGradientsNatural =
Interpolation.EvaluateNaturalGradientsAtGaussPoints(QuadratureForStiffness);
for (int gp = 0; gp < QuadratureForStiffness.IntegrationPoints.Count; ++gp)
{
// Calculate the necessary quantities for the integration
IMatrixView constitutive = materialsAtGaussPoints[gp].ConstitutiveMatrix;
var jacobian = new IsoparametricJacobian2D(Nodes, shapeGradientsNatural[gp]);
Matrix shapeGradientsCartesian =
jacobian.TransformNaturalDerivativesToCartesian(shapeGradientsNatural[gp]);
Matrix deformation = BuildDeformationMatrix(shapeGradientsCartesian);
// Contribution of this gauss point to the element stiffness matrix
Matrix partial = deformation.ThisTransposeTimesOtherTimesThis(constitutive);
double dA = jacobian.DirectDeterminant * QuadratureForStiffness.IntegrationPoints[gp].Weight; //TODO: this is used by all methods that integrate. I should cache it.
stiffness.AxpyIntoThis(partial, dA);
}
stiffness.ScaleIntoThis(Thickness);
return(stiffness);
}