/// <summary>
/// Creates an n by n + 1 (where n is the length of the provided array
/// of source panels) matrix representing the linear equations to be
/// solved for the strengths of the source panels.
/// </summary>
/// <param name="sourcePanels">
/// The array of source panels.
/// </param>
/// <param name="vPanels">
/// A vector representing the velocity vector of the panels (the
/// negative far-field flow velocity).
/// </param>
/// <returns>The equation matrix.</returns>
public static double[,] CreateEquationMatrix(
UnitStrengthRectangularSourcePanel[] sourcePanels, Vector3 vPanels)
{
int rows = sourcePanels.Length;
int columns = rows + 1;
double[,] equationMatrix = new double[rows, columns];
for (int i = 0; i < rows; i++)
{
Vector3 surfaceNormal =
Vector3.Cross(sourcePanels[i].U, sourcePanels[i].V);
for (int j = 0; j < columns - 1; j++)
{
Vector3 unscaledVelocity = Velocity.ComputeSteadyState(
sourcePanels[i].Position, sourcePanels[j]);
equationMatrix[i, j] =
Vector3.Dot(unscaledVelocity, surfaceNormal);
}
equationMatrix[i, columns - 1] =
Vector3.Dot(vPanels, surfaceNormal);
}
return(equationMatrix);
}
public void ComputeSteadyState_WithDrInTheUVPlane_ReturnsTheCorrectValue()
{
// Arrange
float aOver2 = 1;
float bOver2 = 2;
Vector3 dr = new(3, 4, 0);
Vector3 u = Vector3.UnitX;
Vector3 v = Vector3.UnitY;
float d11 = (dr - aOver2 * u - bOver2 * v).Length();
float d12 = (dr - aOver2 * u + bOver2 * v).Length();
float d21 = (dr + aOver2 * u - bOver2 * v).Length();
float d22 = (dr + aOver2 * u + bOver2 * v).Length();
float uDotDr = Vector3.Dot(u, dr);
float vDotDr = Vector3.Dot(v, dr);
float ln1 = MathF.Log(MathF.Abs(
(-bOver2 - vDotDr + d11) * (bOver2 - vDotDr + d22) /
((-bOver2 - vDotDr + d21) * (bOver2 - vDotDr + d12))));
float ln2 = MathF.Log(MathF.Abs(
(-aOver2 - uDotDr + d11) * (aOver2 - uDotDr + d22) /
((aOver2 - uDotDr + d21) * (-aOver2 - uDotDr + d12))));
Vector3 correctResult = 1 / (4 * MathF.PI) * (ln1 * u + ln2 * v);
// Act
Vector3 result = Velocity.ComputeSteadyState(aOver2, bOver2, dr, u, v);
// Assert
Assert.AreEqual(correctResult, result);
}
public void SolveForSourcePanelStrengths_WithUnitCubeInput_SatisfiesTheTangentialFlowConstraint()
{
// Arrange
UnitStrengthRectangularSourcePanel[] sourcePanels = new UnitStrengthRectangularSourcePanel[6]
{
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0.5F, 0, 0), U = Vector3.UnitY, V = Vector3.UnitZ
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(-0.5F, 0, 0), U = Vector3.UnitY, V = Vector3.UnitZ
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, 0.5F, 0), U = Vector3.UnitZ, V = Vector3.UnitX
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, -0.5F, 0), U = Vector3.UnitZ, V = Vector3.UnitX
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, 0, 0.5F), U = Vector3.UnitX, V = Vector3.UnitY
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, 0, -0.5F), U = Vector3.UnitX, V = Vector3.UnitY
}
};
Vector3 vPanels = new Vector3(10, 0, 0);
// Act
double[] strengths = Solver.SolveForSourcePanelStrengths(sourcePanels, vPanels);
bool flowIsTangentialAtEachControlPoint = true;
for (int i = 0; i < sourcePanels.Length; i++)
{
Vector3 surfaceNormal = Vector3.Cross(sourcePanels[i].U, sourcePanels[i].V);
Vector3 velocity = vPanels;
for (int j = 0; j < sourcePanels.Length; j++)
{
velocity += (float)strengths[j] * Velocity.ComputeSteadyState(sourcePanels[i].Position, sourcePanels[j]);
}
if (Vector3.Dot(velocity, surfaceNormal) != 0)
{
flowIsTangentialAtEachControlPoint = false;
}
}
// Assert
Assert.IsTrue(flowIsTangentialAtEachControlPoint);
}
public void ComputeSteadyState_WithDrEqualToTheZeroVector_ReturnsTheZeroVector()
{
// Arrange
float aOver2 = 1;
float bOver2 = 2;
Vector3 dr = Vector3.Zero;
Vector3 u = Vector3.UnitX;
Vector3 v = Vector3.UnitY;
// Act
Vector3 result = Velocity.ComputeSteadyState(aOver2, bOver2, dr, u, v);
// Assert
Assert.AreEqual(Vector3.Zero, result);
}
public void ComputeSteadyState_WithDrOutsideTheUVPlane_ReturnsTheCorrectValue()
{
// Arrange
float aOver2 = 1;
float bOver2 = 2;
Vector3 dr = new(3, 4, 5);
Vector3 u = Vector3.UnitX;
Vector3 v = Vector3.UnitY;
float d11 = (dr - aOver2 * u - bOver2 * v).Length();
float d12 = (dr - aOver2 * u + bOver2 * v).Length();
float d21 = (dr + aOver2 * u - bOver2 * v).Length();
float d22 = (dr + aOver2 * u + bOver2 * v).Length();
float uDotDr = Vector3.Dot(u, dr);
float vDotDr = Vector3.Dot(v, dr);
float ln1 = MathF.Log(MathF.Abs(
(-bOver2 - vDotDr + d11) * (bOver2 - vDotDr + d22) /
((-bOver2 - vDotDr + d21) * (bOver2 - vDotDr + d12))));
float ln2 = MathF.Log(MathF.Abs(
(-aOver2 - uDotDr + d11) * (aOver2 - uDotDr + d22) /
((aOver2 - uDotDr + d21) * (-aOver2 - uDotDr + d12))));
Vector3 c = dr - uDotDr * u - vDotDr * v;
float sqrtK = c.Length();
float atan11 = MathF.Atan2(
(-aOver2 - uDotDr) * (-bOver2 - vDotDr), sqrtK * d11);
float atan12 = MathF.Atan2(
(-aOver2 - uDotDr) * (bOver2 - vDotDr), sqrtK * d12);
float atan21 = MathF.Atan2(
(aOver2 - uDotDr) * (-bOver2 - vDotDr), sqrtK * d21);
float atan22 = MathF.Atan2(
(aOver2 - uDotDr) * (bOver2 - vDotDr), sqrtK * d22);
float atanSum = atan22 - atan21 - atan12 + atan11;
Vector3 correctResult = 1 / (4 * MathF.PI) * (ln1 * u + ln2 * v +
(atanSum / sqrtK) * c);
// Act
Vector3 result = Velocity.ComputeSteadyState(aOver2, bOver2, dr, u, v);
// Assert
Assert.AreEqual(correctResult, result);
}
public void CreateEquationMatrix_WithUnitCubeInput_CreatesTheCorrectLHS()
{
// Arrange
UnitStrengthRectangularSourcePanel[] sourcePanels = new UnitStrengthRectangularSourcePanel[6]
{
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0.5F, 0, 0), U = Vector3.UnitY, V = Vector3.UnitZ
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(-0.5F, 0, 0), U = Vector3.UnitY, V = Vector3.UnitZ
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, 0.5F, 0), U = Vector3.UnitZ, V = Vector3.UnitX
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, -0.5F, 0), U = Vector3.UnitZ, V = Vector3.UnitX
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, 0, 0.5F), U = Vector3.UnitX, V = Vector3.UnitY
},
new UnitStrengthRectangularSourcePanel
{
AOver2 = 0.5F, BOver2 = 0.5F, Position = new Vector3(0, 0, -0.5F), U = Vector3.UnitX, V = Vector3.UnitY
}
};
Vector3 vPanels = new Vector3(10, 0, 0);
double[,] correctLHS = new double[6, 6]
{
{
Velocity.ComputeSteadyState(sourcePanels[0].Position, sourcePanels[0]).X,
Velocity.ComputeSteadyState(sourcePanels[0].Position, sourcePanels[1]).X,
Velocity.ComputeSteadyState(sourcePanels[0].Position, sourcePanels[2]).X,
Velocity.ComputeSteadyState(sourcePanels[0].Position, sourcePanels[3]).X,
Velocity.ComputeSteadyState(sourcePanels[0].Position, sourcePanels[4]).X,
Velocity.ComputeSteadyState(sourcePanels[0].Position, sourcePanels[5]).X
},
{
Velocity.ComputeSteadyState(sourcePanels[1].Position, sourcePanels[0]).X,
Velocity.ComputeSteadyState(sourcePanels[1].Position, sourcePanels[1]).X,
Velocity.ComputeSteadyState(sourcePanels[1].Position, sourcePanels[2]).X,
Velocity.ComputeSteadyState(sourcePanels[1].Position, sourcePanels[3]).X,
Velocity.ComputeSteadyState(sourcePanels[1].Position, sourcePanels[4]).X,
Velocity.ComputeSteadyState(sourcePanels[1].Position, sourcePanels[5]).X
},
{
Velocity.ComputeSteadyState(sourcePanels[2].Position, sourcePanels[0]).Y,
Velocity.ComputeSteadyState(sourcePanels[2].Position, sourcePanels[1]).Y,
Velocity.ComputeSteadyState(sourcePanels[2].Position, sourcePanels[2]).Y,
Velocity.ComputeSteadyState(sourcePanels[2].Position, sourcePanels[3]).Y,
Velocity.ComputeSteadyState(sourcePanels[2].Position, sourcePanels[4]).Y,
Velocity.ComputeSteadyState(sourcePanels[2].Position, sourcePanels[5]).Y
},
{
Velocity.ComputeSteadyState(sourcePanels[3].Position, sourcePanels[0]).Y,
Velocity.ComputeSteadyState(sourcePanels[3].Position, sourcePanels[1]).Y,
Velocity.ComputeSteadyState(sourcePanels[3].Position, sourcePanels[2]).Y,
Velocity.ComputeSteadyState(sourcePanels[3].Position, sourcePanels[3]).Y,
Velocity.ComputeSteadyState(sourcePanels[3].Position, sourcePanels[4]).Y,
Velocity.ComputeSteadyState(sourcePanels[3].Position, sourcePanels[5]).Y
},
{
Velocity.ComputeSteadyState(sourcePanels[4].Position, sourcePanels[0]).Z,
Velocity.ComputeSteadyState(sourcePanels[4].Position, sourcePanels[1]).Z,
Velocity.ComputeSteadyState(sourcePanels[4].Position, sourcePanels[2]).Z,
Velocity.ComputeSteadyState(sourcePanels[4].Position, sourcePanels[3]).Z,
Velocity.ComputeSteadyState(sourcePanels[4].Position, sourcePanels[4]).Z,
Velocity.ComputeSteadyState(sourcePanels[4].Position, sourcePanels[5]).Z
},
{
Velocity.ComputeSteadyState(sourcePanels[5].Position, sourcePanels[0]).Z,
Velocity.ComputeSteadyState(sourcePanels[5].Position, sourcePanels[1]).Z,
Velocity.ComputeSteadyState(sourcePanels[5].Position, sourcePanels[2]).Z,
Velocity.ComputeSteadyState(sourcePanels[5].Position, sourcePanels[3]).Z,
Velocity.ComputeSteadyState(sourcePanels[5].Position, sourcePanels[4]).Z,
Velocity.ComputeSteadyState(sourcePanels[5].Position, sourcePanels[5]).Z
}
};
// Act
double[,] result = Solver.CreateEquationMatrix(sourcePanels, vPanels);
bool lhsIsCorrect = true;
for (int i = 0; i < 6; i++)
{
for (int j = 0; j < 6; j++)
{
if (result[i, j] != correctLHS[i, j])
{
lhsIsCorrect = false;
}
}
}
// Assert
Assert.IsTrue(lhsIsCorrect);
}
public void ComputeSteadyState()
{
Vector3 _ = Velocity.ComputeSteadyState(_r, _sourcePanel);
}