// ....Compute element Matrix.....................................
public override double[,] ComputeMatrix()
{
double[] gaussCoord = { -1 / Math.Sqrt(3), 1 / Math.Sqrt(3) };
if (Modell.Material.TryGetValue(ElementMaterialId, out var abstractMaterial))
{
}
material = (Material)abstractMaterial;
ElementMaterial = material ?? throw new ArgumentNullException(nameof(material));
var conductivity = material.Leitfähigkeit[0];
MatrizenAlgebra.Clear(elementMatrix);
foreach (var coor1 in gaussCoord)
{
foreach (var coor2 in gaussCoord)
{
var z0 = coor1;
var z1 = coor2;
ComputeGeometry(z0, z1);
Sx = ComputeSx(z0, z1);
// Ke = C*Sx*SxT*determinant
MatrizenAlgebra.MultAddMatrixTransposed(elementMatrix, Determinant * conductivity, Sx, Sx);
}
}
return(elementMatrix);
}
// ... compute element matrix ..................................
public override double[,] ComputeMatrix()
{
ComputeGeometry();
var factor = ElementMaterial.MaterialWerte[0] * ElementCrossSection.QuerschnittsWerte[0] / length;
var sx = ComputeSx();
_stiffnessMatrix = MatrizenAlgebra.MultTransposedRect(factor, sx);
return(_stiffnessMatrix);
}
// ....Compute element matrix.....................................
public override double[,] ComputeMatrix()
{
ComputeGeometry();
ComputeStrainDisplacementTransformation();
ComputeMaterial();
// Ke = 0.5*thickness*determinant*BT*E*B
var temp = MatrizenAlgebra.MultTransposedMatrix(0.5 * ElementCrossSection.QuerschnittsWerte[0] * Determinant, b, e);
matrix = MatrizenAlgebra.Mult(temp, b);
return(matrix);
}
public override double[] ComputeElementState(double z0, double z1, double z2)
{
for (var i = 0; i < 8; i++)
{
elementTemperatures[i] = Nodes[i].NodalDof[0];
}
// midPointHeatState = E * Sx(transponiert) * Temperatures
var midpointHeatState = MatrizenAlgebra.MultTransposed(Sx, elementTemperatures);
midpointHeatState = MatrizenAlgebra.Mult(e, midpointHeatState);
return(midpointHeatState);
}
// --- Elementverhalten ----------------------------------
// ....Berechne Elementspannungen: sigma = E * B * Ue (Element Verformungen) ......
public override double[] ComputeZustandsvektor()
{
for (var i = 0; i < NodesPerElement; i++)
{
var nodalDof = i * 2;
elementVerformungen[nodalDof] = Nodes[i].NodalDof[0];
elementVerformungen[nodalDof + 1] = Nodes[i].NodalDof[1];
}
var temp = MatrizenAlgebra.Mult(e, b);
var elementSpannungen = MatrizenAlgebra.Mult(temp, elementVerformungen);
return(elementSpannungen);
}
public override double[] ComputeElementState(double z0, double z1)
{
_ = new double[2]; // in element
ComputeGeometry(z0, z1);
Sx = ComputeSx(z0, z1);
for (var i = 0; i < NodesPerElement; i++)
{
elementTemperatures[i] = Nodes[i].NodalDof[0];
}
var conductivity = material.Leitfähigkeit[0];
var midpointHeatState = MatrizenAlgebra.MultTransposed(-conductivity, Sx, elementTemperatures);
return(midpointHeatState);
}
// --- Elementverhalten ----------------------------------
// ....Berechne Elementspannungen: sigma = E * B * Ue (Elementverformungen) ......
public override double[] ComputeZustandsvektor()
{
for (var i = 0; i < 8; i++)
{
var nodalDof = i * ElementDof;
elementDeformations[nodalDof] = Nodes[i].NodalDof[0];
elementDeformations[nodalDof + 1] = Nodes[i].NodalDof[1];
elementDeformations[nodalDof + 2] = Nodes[i].NodalDof[2];
}
var temp = MatrizenAlgebra.Mult(e, b);
var elementStresses = MatrizenAlgebra.Mult(temp, elementDeformations);
return(elementStresses);
}
// ... compute end forces of frame element........................
public override double[] ComputeElementState()
{
var localMatrix = ComputeLocalMatrix();
var vector = ComputeZustandsvektor();
// contribution of the node deformations
ElementState = MatrizenAlgebra.Mult(localMatrix, vector);
ElementState[2] = -ElementState[2];
// contribution of the beam loads
foreach (var item in modell.PunktLasten)
{
if (!(item.Value is PunktLast punktLast))
{
continue;
}
if (punktLast.ElementId != ElementId)
{
continue;
}
vector = punktLast.ComputeLocalLoadVector();
vector[2] = -vector[2];
for (var i = 0; i < vector.Length; i++)
{
ElementState[i] -= vector[i];
}
}
foreach (var item in modell.ElementLasten)
{
if (!(item.Value is LinienLast linienLast))
{
continue;
}
if (linienLast.ElementId != ElementId)
{
continue;
}
vector = linienLast.ComputeLocalLoadVector();
vector[2] = -vector[2];
for (var i = 0; i < vector.Length; i++)
{
ElementState[i] -= vector[i];
}
}
return(ElementState);
}
protected double[,] ComputeSx(double z0, double z1, double z2)
{
var zx = new double[3, 3];
zx[0, 0] = (xz[1, 1] * xz[2, 2] - xz[1, 2] * xz[2, 1]) / Determinant;
zx[1, 0] = -(xz[1, 0] * xz[2, 2] - xz[1, 2] * xz[2, 0]) / Determinant;
zx[2, 0] = (xz[1, 0] * xz[2, 1] - xz[1, 1] * xz[2, 0]) / Determinant;
zx[0, 1] = -(xz[0, 1] * xz[2, 2] - xz[0, 2] * xz[2, 1]) / Determinant;
zx[1, 1] = (xz[0, 0] * xz[2, 2] - xz[0, 2] * xz[2, 0]) / Determinant;
zx[2, 1] = -(xz[0, 0] * xz[2, 1] - xz[0, 1] * xz[2, 0]) / Determinant;
zx[0, 2] = (xz[0, 1] * xz[1, 2] - xz[0, 2] * xz[1, 1]) / Determinant;
zx[1, 2] = -(xz[0, 0] * xz[1, 2] - xz[0, 2] * xz[1, 0]) / Determinant;
zx[2, 2] = (xz[0, 0] * xz[1, 1] - xz[0, 1] * xz[1, 0]) / Determinant;
Sx = MatrizenAlgebra.Mult(sz, zx);
return(Sx);
}
// ....Compute element Matrix.....................................
public override double[,] ComputeMatrix()
{
ComputeGeometry();
if (Modell.Material.TryGetValue(ElementMaterialId, out var abstractMaterial))
{
}
material = (Material)abstractMaterial;
ElementMaterial = material;
if (material == null)
{
return(elementMatrix);
}
var conductivity = material.Leitfähigkeit[0];
// Ke = area*c*Sx*SxT
elementMatrix = MatrizenAlgebra.RectMultMatrixTransposed(0.5 * Determinant * conductivity, Sx, Sx);
return(elementMatrix);
}
// ....Compute element matrix.....................................
public override double[,] ComputeMatrix()
{
if (Modell.Material.TryGetValue(ElementMaterialId, out var abstractMaterial))
{
}
Material = (Material)abstractMaterial;
double[] gCoord = { -1 / Math.Sqrt(5.0 / 3), 0, 1 / (Math.Sqrt(5.0 / 3)) };
double[] gWeight = { (5.0 / 9), (8.0 / 9), (5.0 / 9) }; // gaussian coordinates, weights
_ = new double[8, 3];
MatrizenAlgebra.Clear(elementMatrix);
// material matrix für ebene Verzerrung (plane strain)
var conduct = ((Material)Material)?.Leitfähigkeit;
if (conduct != null)
{
e[0, 0] = conduct[0];
e[1, 1] = conduct[1];
e[2, 2] = conduct[2];
}
for (var i = 0; i < gCoord.Length; i++)
{
var z0 = gCoord[i]; var g0 = gWeight[i];
for (var j = 0; j < gCoord.Length; j++)
{
var z1 = gCoord[j]; var g1 = gWeight[j];
for (var k = 0; k < gCoord.Length; k++)
{
var z2 = gCoord[k]; var g2 = gWeight[k];
ComputeGeometry(z0, z1, z2);
Sx = ComputeSx(z0, z1, z2);
// Ke = determinant*g0*g1*g2*Sx*E*SxT
var temp = MatrizenAlgebra.Mult(Sx, e);
MatrizenAlgebra.MultAddMatrixTransposed(elementMatrix, Determinant * g0 * g1 * g2, temp, Sx);
}
}
}
return(elementMatrix);
}
public override double[] ComputeElementState(double z0, double z1)
{
var elementWärmeStatus = new double[2]; // in element
ComputeGeometry();
if (Modell.Material.TryGetValue(ElementMaterialId, out var abstractMaterial))
{
}
material = (Material)abstractMaterial;
ElementMaterial = material;
if (Modell.Elemente.TryGetValue(ElementId, out element))
{
for (var i = 0; i < element.Nodes.Length; i++)
{
if (Modell.Knoten.TryGetValue(element.NodeIds[i], out knoten))
{
}
//Debug.Assert(node != null, nameof(node) + " != null");
if (knoten != null)
{
elementTemperatures[i] = knoten.NodalDof[0];
}
}
if (material == null)
{
return(elementWärmeStatus);
}
var conductivity = material.Leitfähigkeit[0];
elementWärmeStatus = MatrizenAlgebra.MultTransposed(-conductivity, Sx, elementTemperatures);
}
else
{
throw new ModellAusnahme("Element2D3: " + ElementId + " nicht im Modell gefunden");
}
return(elementWärmeStatus);
}
// ....Compute element matrix.....................................
public override double[,] ComputeMatrix()
{
MatrizenAlgebra.Clear(elementMatrix);
ComputeMaterial();
for (var i = 0; i < GCoord.Length; i++)
{
z0 = GCoord[i]; g0 = GWeight[i];
for (var j = 0; j < GCoord.Length; j++)
{
z1 = GCoord[j]; g1 = GWeight[j];
for (var k = 0; k < GCoord.Length; k++)
{
z2 = GCoord[k]; g2 = GWeight[k];
ComputeGeometry(z0, z1, z2);
Sx = ComputeSx(z0, z1, z2);
ComputeStrainDisplacementTransformation();
// Ke = determinant*g0*g1*g2*BT*E*B
var temp = MatrizenAlgebra.MultTransposedMatrix(Determinant * g0 * g1 * g2, b, e);
MatrizenAlgebra.MultAddMatrix(elementMatrix, temp, b);
}
}
}
return(elementMatrix);
}
//... solveEigenstates() ....................................................
public void SolveEigenstates()
{
// allocate the solution vectors
x = new double[numberOfStates][];
p = new double[numberOfStates][];
for (var i = 0; i < numberOfStates; i++)
{
x[i] = new double[dimension];
p[i] = new double[dimension];
}
eigenValue = new double[numberOfStates];
// reduce the number of eigenvalues to the maximum possible number
var counter = 0;
for (row = 0; row < dimension; row++)
{
if (status[row])
{
counter++;
}
}
if (numberOfStates > dimension - counter)
{
numberOfStates = dimension - counter;
}
var profileSolverStatus =
new ProfillöserStatus(a, w, y, status, profile);
// loop over the specified number of eigenstates
for (state = 0; state < numberOfStates; state++)
{
raleigh = 0;
s = 0;
// set start vector w0
for (row = 0; row < dimension; row++)
{
if (status[row])
{
w[row] = 0;
}
else
{
w[row] = 1;
}
}
// start iteration for next eigenstate
double m;
do
{
// increment iteration counter
s++;
// check if number of iterations is greater Smax
if (s > SMax)
{
throw new BerechnungAusnahme("Eigensolver: too many iterations " + s);
}
// B-orthogonalization of w(s-1) with respect to all smaller
// eigenvectors x[0] to x[state-1]
for (var i = 0; i < state; i++)
{
var c = 0.0;
// compute c(i) and subtract c(i)*p(i) from w
for (row = 0; row < dimension; row++)
{
if (!status[row])
{
c += w[row] * x[i][row];
}
}
for (row = 0; row < dimension; row++)
{
if (!status[row])
{
w[row] -= c * p[i][row];
}
}
}
// solve A * y(s) = w(s-1) for y(s)
profileSolverStatus.SetRhs(w);
profileSolverStatus.SolvePrimal();
// compute z(s) = B * y(s)
z = MatrizenAlgebra.Mult(b, y, status, profile);
// compute m2 = 1 / (y[s] * z[s])
double sum = 0;
for (row = 0; row < dimension; row++)
{
if (!status[row])
{
sum += y[row] * z[row];
}
}
m2 = 1 / sum;
// compute Rayleigh quotient r = m2 * y(s)(T) * w(s-1)
// and the difference ( r(s) - r(s-1) )
sum = 0;
for (row = 0; row < dimension; row++)
{
if (!status[row])
{
sum += y[row] * w[row];
}
}
sum *= m2;
deltaRaleigh = sum - raleigh;
raleigh = sum;
// compute w(s) = m(s) * z(s)
m = Math.Sqrt(Math.Abs(m2));
for (row = 0; row < dimension; row++)
{
if (!status[row])
{
w[row] = m * z[row];
}
}
// continue iteration as long as change in Rayleigh factor (r(s)-r(s-1)
// is greater than error bound
} while (Math.Abs(deltaRaleigh) > Math.Abs(RaleighFactor * raleigh));
// store computed eigenstate and vector p=w=Bx for B-orthogonalization
eigenValue[state] = raleigh;
for (row = 0; row < dimension; row++)
{
x[state][row] = m * y[row];
p[state][row] = w[row];
}
}
}
public void Perform()
{
double alfa, beta, gamma, theta;
if (method == 1)
{
alfa = 0; theta = 1; beta = parameter1; gamma = parameter2;
}
else if (method == 2)
{
beta = 1.0 / 6; gamma = 0.5; alfa = 0; theta = parameter1;
}
else if (method == 3)
{
theta = 1; alfa = parameter1; gamma = 0.5 - alfa; beta = 0.25 * (1 - alfa) * (1 - alfa);
}
else
{
throw new BerechnungAusnahme("TimeIntegration2NdOrderStatus: invalid method identifier entered");
}
var gammaDt = gamma * dt;
var betaDt2 = beta * dt * dt;
var gammaDtTheta = gamma * dt * theta;
var dt1MGamma = dt * (1 - gamma);
var dt2MBetaDt2 = dt * dt / 2 - beta * dt * dt;
var thetaDt = theta * dt;
var thetaDt1MGamma = theta * dt * (1 - gamma);
var theta2Dt2MBetaDt2 = theta * theta * dt2MBetaDt2;
var betaDt2Theta2AlfaP1 = beta * dt * dt * theta * theta * (1 + alfa);
var primal = new double[dimension];
var dual = new double[dimension];
var timeSteps = displacement.Length;
acceleration = new double[timeSteps][];
for (var i = 0; i < timeSteps; i++)
{
acceleration[i] = new double[dimension];
}
// calculate initial accelerations at unrestrained dof, for M[i]>0
var rhs = MatrizenAlgebra.Mult(k, displacement[0], status, profile);
for (var i = 0; i < dimension; i++)
{
// if (status[i]) continue; ODER wenn M[i]=0 continue --> RHS[i]=0
if (status[i] | m[i] == 0)
{
continue;
}
rhs[i] = (forcingFunction[0][i] - c[i] * velocity[0][i] - rhs[i]) / m[i];
acceleration[0][i] = rhs[i];
}
// constant coefficient matrix
var cm = new double[dimension][];
for (var row = 0; row < dimension; row++)
{
cm[row] = new double[row + 1 - profile[row]];
for (var col = 0; col <= (row - profile[row]); col++)
{
cm[row][col] = betaDt2Theta2AlfaP1 * k[row][col];
}
cm[row][row - profile[row]] += m[row] + gammaDtTheta * c[row];
}
var profileSolverStatus = new ProfillöserStatus(
cm, rhs, primal, dual, status, profile);
profileSolverStatus.Decompose();
for (var counter = 1; counter < timeSteps; counter++)
{
// calculate displacement(hat) and velocity(hat) at n+1
for (var i = 0; i < dimension; i++)
{
displacement[counter][i] = displacement[counter - 1][i]
+ thetaDt * velocity[0][i]
+ theta2Dt2MBetaDt2 * acceleration[counter - 1][i];
velocity[1][i] = velocity[0][i] + thetaDt1MGamma * acceleration[counter - 1][i];
}
// calculate new RHS
for (var i = 0; i < dimension; i++)
{
rhs[i] = (1 + alfa) * displacement[counter][i] - alfa * displacement[counter - 1][i];
}
rhs = MatrizenAlgebra.Mult(k, rhs, status, profile);
for (var i = 0; i < dimension; i++)
{
if (!status[i])
{
rhs[i] = (1 - theta) * forcingFunction[counter - 1][i]
+ theta * forcingFunction[counter][i]
- c[i] * velocity[1][i] - rhs[i];
}
}
// backsubstitution
profileSolverStatus.SetRhs(rhs);
profileSolverStatus.SolvePrimal();
// displacements, velocities and accelerations at next time step
for (var i = 0; i < dimension; i++)
{
if (status[i])
{
continue;
}
acceleration[counter][i] = acceleration[counter - 1][i]
+ (primal[i]
- acceleration[counter - 1][i]) / theta;
displacement[counter][i] = displacement[counter - 1][i]
+ dt * velocity[0][i]
+ dt2MBetaDt2 * acceleration[counter - 1][i]
+ betaDt2 * acceleration[counter][i];
velocity[0][i] = velocity[0][i]
+ dt1MGamma * acceleration[counter - 1][i]
+ gammaDt * acceleration[counter][i];
}
}
}
public void Perform()
{
var alfaDt = alfa * dt;
var dtAlfa = dt * (1 - alfa);
var primal = new double[dimension];
var timeSteps = Temperature.Length;
That = new double[dimension];
Tdot = new double[timeSteps][];
for (var i = 0; i < timeSteps; i++)
{
Tdot[i] = new double[dimension];
}
// calculate initial temperature gradients
var rhs = MatrizenAlgebra.Mult(k, Temperature[0], profile);
for (var i = 0; i < dimension; i++)
{
Tdot[0][i] = (ForcingFunction[0][i] - rhs[i]) / c[i];
}
// evaluate constant coefficient matrix
var cm = new double[dimension][];
for (var row = 0; row < dimension; row++)
{
cm[row] = new double[row + 1 - profile[row]];
for (var col = 0; col <= (row - profile[row]); col++)
{
cm[row][col] = alfaDt * k[row][col];
}
cm[row][row - profile[row]] += c[row];
}
var profileSolverStatus =
new ProfillöserStatus(cm, rhs, primal, status, profile);
profileSolverStatus.Decompose();
for (var counter = 1; counter < timeSteps; counter++)
{
// calculate temperature gradients at restrained nodes
for (var i = 0; i < dimension; i++)
{
if (status[i])
{
Tdot[counter][i] = (Temperature[counter][i] - Temperature[counter - 1][i]) / dt;
}
}
// calculate T(hat) for next step
for (var i = 0; i < dimension; i++)
{
if (status[i])
{
That[i] = Temperature[counter][i];
}
else
{
That[i] = Temperature[counter - 1][i] + dtAlfa * Tdot[counter - 1][i];
}
}
// modification of RHS
rhs = MatrizenAlgebra.Mult(k, That, status, profile);
var rhSfr = MatrizenAlgebra.MultUl(cm, Tdot[counter], status, profile);
for (var i = 0; i < dimension; i++)
{
rhs[i] = ForcingFunction[counter][i] - rhSfr[i] - rhs[i];
}
// backsubstitution
profileSolverStatus.SetRhs(rhs);
profileSolverStatus.SolvePrimal();
// temperatures and gradients at next time step for unrestrained nodes
for (var i = 0; i < dimension; i++)
{
if (status[i])
{
continue;
}
Tdot[counter][i] = primal[i];
Temperature[counter][i] = That[i] + alfaDt * primal[i];
}
}
}
