/// <summary>
/// Integrates the shear force along intersection of element with defined plane.
/// </summary>
/// <param name="element">The element.</param>
/// <param name="plane">The plane.</param>
/// <returns>Integrated force</returns>
public static Force IntegrateNormalForce(this TriangleFlatShell element, Plane plane, LoadCombination cmb)
{
//step 1: does intersect?
//step 2: get a tensor (no matter where, this is constant stress)
//step 3: find rotation amount for tensors
//step 4: rotate tensors
//step 5: integrate forces
if (!DoesIntersects(element, plane))
return Force.Zero;
var intersection = GetIntersection(element, plane);
//step 2
var tensor = element.GetInternalForce(0, 0, cmb);
//step 3
var t = element.GetTransformationMatrix().Transpose(); //transpose of t
var nLocal = (Vector)(t * plane.Normal.ToMatrix()).ToPoint();//project of N to local element coord system
var theta = Math.Atan2(nLocal.Y, nLocal.X) - Math.PI / 2;
//step 4
var rotatedTensor = MembraneStressTensor.Rotate(tensor.MembraneTensor, theta);
// shear direction
var shearLocalDirection = new Vector(Math.Cos(theta), Math.Sin(theta), 0);//direction of shear force in local element coord system, which is vector I rotated by amount theta
var shearGlobalDirection = (Vector)(t.Transpose() * shearLocalDirection.ToMatrix()).ToPoint();
var fShearAmount = rotatedTensor.Txy * element.Thickness * intersection.Length;//shear force
var shearForce = fShearAmount * shearGlobalDirection.GetUnit();
return new Force(shearForce, Vector.Zero);
}
/// <summary>
/// Determines whether the defined plane does intersect with defined triangle shell element or not.
/// </summary>
/// <param name="element">The element.</param>
/// <param name="plane">The plane.</param>
/// <returns>true, if intersects, false otherwise</returns>
public static bool DoesIntersects(this TriangleFlatShell element, Plane plane)
{
//http://geomalgorithms.com/a06-_intersect-2.html section "Intersection of a Triangle with a Plane"
//http://stackoverflow.com/questions/15688232/check-which-side-of-a-plane-points-are-on
//if this intersects with plane, then points must not be in same side of plane
var d = -Vector.Dot(plane.Normal, (Vector)plane.P);
var ps = element.Nodes.Select(i => i.Location).ToArray();
var signs =
ps.Select(i => Vector.Dot((Vector)i, plane.Normal) + d).Select(i => i < 0.0 ? -1 : (i > 0 ? 1 : 0)).ToArray();
if (signs.Any(i => i.Equals(0)))
{
return(false);
}
if (signs.Distinct().Count() == 1)
{
//all points on same side of plane
return(false);
}
return(true);
}
/// <summary>
/// Rotates the define local stress tensor into direction of plane.
/// </summary>
/// <param name="shell">The shell.</param>
/// <param name="localStressTensor">The local stress tensor.</param>
/// <param name="targetPlane">The target plane.</param>
/// <returns>
/// Rotated tensor
/// </returns>
/// <exception cref="System.NotImplementedException"></exception>
public static MembraneStressTensor RotateTensor(this TriangleFlatShell shell, MembraneStressTensor localStressTensor, Plane targetPlane)
{
var t = shell.GetTransformationMatrix().Transpose();
var plane = targetPlane;
var nLocal = (Vector)(t * plane.Normal.ToMatrix()).ToPoint();//project of N to local element coord system
var theta = Math.Atan2(nLocal.Y, nLocal.X) - Math.PI / 2;
var buf = MembraneStressTensor.Rotate(localStressTensor, theta);
return buf;
}
/// <summary>
/// Gets the intersection of triangle and plane as a line segment
/// </summary>
/// <param name="element">The element.</param>
/// <param name="plane">The plane.</param>
/// <returns>intersection</returns>
public static LineSegment GetIntersection(this TriangleFlatShell element, Plane plane)
{
if (!DoesIntersects(element, plane))
throw new InvalidOperationException();
var points = GetIntersectionPoints(element, plane, 5);
var buf = new LineSegment();
buf.P1 = points.First();
buf.P2 = points.Last();
return buf;
}
public static FlatShellStressTensor[] GeTensorsAtIntersection(this TriangleFlatShell element, Plane plane, int n)
{
var buf = new List<FlatShellStressTensor>();
var globalPoints = GetIntersectionPoints(element, plane, n);
if (!globalPoints.Any())
return buf.ToArray();
var t = element.GetTransformationMatrix().Transpose(); //transpose of t
var locals = globalPoints.Select(i => (t*i.ToMatrix()).ToPoint()).ToArray();
var lpts = element.GetLocalPoints();
var tensors =
locals.Select(i => element.GetInternalForce(i.X, i.Y, LoadCombination.DefaultLoadCombination)).ToArray();
return tensors;
}
/// <summary>
/// Gets the N points, equally spaced and lying of intersection line between current triangular flat shell and defined <see cref="plane" />.
/// </summary>
/// <param name="element">The element.</param>
/// <param name="plane">The plane.</param>
/// <param name="n">The n.</param>
/// <returns>
/// point list
/// </returns>
/// <remarks>
/// Assume a flat plane in 3D, if the flat plane intersects with this triangular flat shell, then a line segment is common part of the flat shell and plane.
/// this will equally divide that line section into n parts, then return the coordination of them
/// </remarks>
public static Point[] GetIntersectionPoints(this TriangleFlatShell element, Plane plane, int n)
{
var buf = new List<Point>();
//http://geomalgorithms.com/a06-_intersect-2.html section "Intersection of a Triangle with a Plane"
//http://stackoverflow.com/questions/15688232/check-which-side-of-a-plane-points-are-on
//if this intersects with plane, then points must not be in same side of plane
var d = -Vector.Dot(plane.Normal, (Vector)plane.P);
var ps = element.Nodes.Select(i => i.Location).ToArray();
var signs =
ps.Select(i => Vector.Dot((Vector)i, plane.Normal) + d).Select(i => i < 0.0 ? -1 : (i > 0 ? 1 : 0)).ToArray();
if (signs.Any(i => i.Equals(0)))
return buf.ToArray();
if (signs.Distinct().Count() == 1)
//all points on same side of plane
return buf.ToArray();
//find each pair that are on opposite side of plane
for (var i = 0; i < 3; i++)
{
for (var j = i; j < 3; j++)
{
if (i == j)
continue;
if (signs[i] != signs[j])
{
//points i & j are on opposite sides of plane
{
//eqs from here: hxxps://en.wikipedia.org/wiki/Line%E2%80%93plane_intersection
var l = ps[i] - ps[j];
var l0 = ps[j];
var N = plane.Normal;
var p0 = plane.P;
var D = (p0 - l0).Dot(N) / (l).Dot(N);
var p = l0 + D * l;//intersection point
buf.Add(p);
}
}
}
}
var sp = buf[0];
var ep = buf[1];
var buff = new List<Point>();
n--;
for (var i = 0; i < n+1; i++)
{
var t = sp + (ep - sp)*(1.0*i/n);
buff.Add(t);
}
return buff.ToArray();
}
public static Tuple<Force,Point> GetTotalForce(this TriangleFlatShell element, Plane plane, int n)
{
//step 1: find intersection points
//step 2: find tensors at intersection points
//step 3: find rotation amount for tensors
//step 4: rotate tensors
//step 5: integrate forces
//step 1
var globalPoints = GetIntersectionPoints(element, plane, n);
if (!globalPoints.Any())
return new Tuple<Force, Point>();
var t = element.GetTransformationMatrix().Transpose(); //transpose of t
var locals = globalPoints.Select(i => (t * i.ToMatrix()).ToPoint()).ToArray();
//step 2
var tensors = GeTensorsAtIntersection(element, plane, n);
var memTensors = tensors.Select(i => i.MembraneTensor).ToList();
var bendTensors = tensors.Select(i => i.BendingTensor).ToList();
//step 3
var nLocal = (Vector)(t * plane.Normal.ToMatrix()).ToPoint();//project of N to local element coord system
var theta = Math.Atan2(nLocal.Y, nLocal.X) - Math.PI/2;
//step 4
var rTensors = tensors.Select(i => MembraneStressTensor.Rotate(i.MembraneTensor, theta)).ToList();
var avg = rTensors.Aggregate((i, j) => i + j);
avg = MembraneStressTensor.Multiply(avg, 1.0 / tensors.Length);
var length = (globalPoints.First() - globalPoints.Last()).Length;
var fShearAmount = avg.Txy * element.Thickness * length;//shear force
var fCompAmount = avg.Sy * element.Thickness * length;//compressive force
// shear dirction
var shearLocalDirection = new Vector(Math.Cos(theta), Math.Sin(theta), 0);//direction of shear force in local element coord system, which is vector I rotated by amount theta
var shearGlobalDirection = (Vector) (t.Transpose()*shearLocalDirection.ToMatrix()).ToPoint();
//compressive direction
var compLocalDirection = new Vector(-Math.Sin(theta), Math.Cos(theta), 0);//direction of shear force in local element coord system, which is vector I rotated by amount theta
var compGlobalDirection = (Vector)(t.Transpose() * compLocalDirection.ToMatrix()).ToPoint();
var shearForce = fShearAmount*shearGlobalDirection.GetUnit();
var compForce = fCompAmount*compGlobalDirection.GetUnit();
var totForce = shearForce + compForce;
var bufFrc = new Force(totForce, Vector.Zero);
var loc = globalPoints.Aggregate((i, j) => (Point)((Vector)i + (Vector)j));
loc.X /= globalPoints.Length;
loc.Y /= globalPoints.Length;
loc.Z /= globalPoints.Length;
return new Tuple<Force, Point>(bufFrc, loc);
//return bufFrc;
}