/// <inheritdoc />
public override Matrix GetGlobalStifnessMatrix()
{
//step 1 : get points in local system
//step 2 : get local stiffness matrix
//step 3 : expand local stiffness matrix
//step 4 : get global stiffness matrix
//step 1
var ls = GetLocalPoints();
var xs = new [] { ls[0].X, ls[1].X, ls[2].X };
var ys = new [] { ls[0].Y, ls[1].Y, ls[2].Y };
//step 2
var kl = CstElement.GetStiffnessMatrix(xs, ys, this._thickness, this.ElasticModulus, this.PoissonRatio,
this._formulationType);
//step 3
var currentOrder = new []
{
new FluentElementPermuteManager.ElementLocalDof(0, DoF.Dx),
new FluentElementPermuteManager.ElementLocalDof(0, DoF.Dy),
new FluentElementPermuteManager.ElementLocalDof(1, DoF.Dx),
new FluentElementPermuteManager.ElementLocalDof(1, DoF.Dy),
new FluentElementPermuteManager.ElementLocalDof(2, DoF.Dx),
new FluentElementPermuteManager.ElementLocalDof(2, DoF.Dy),
};
var kle = FluentElementPermuteManager.FullyExpand(kl, currentOrder, 3);
var lambda = GetTransformationMatrix();
var tr = Matrix.DiagonallyRepeat(lambda.Transpose(), 6); // eq. 5-16 page 78 (87 of file)
//step 4 : get global stiffness matrix
var buf = tr.Transpose() * kle * tr; //eq. 5-15 p77
return(buf);
}
private Matrix GetLocalMembraneStiffnessMatrix()
{
//cst
//step 1 : get points in local system
//step 2 : get local stiffness matrix
//step 3 : expand local stiffness matrix
//step 4 : get global stiffness matrix
//step 1
var ls = GetLocalPoints();
var xs = new[] { ls[0].X, ls[1].X, ls[2].X };
var ys = new[] { ls[0].Y, ls[1].Y, ls[2].Y };
//step 2
var kl = CstElement.GetStiffnessMatrix(xs, ys, this._thickness, this.ElasticModulus, this.PoissonRatio,
this._formulationType);
//step 3
var currentOrder = new[]
{
new ElementPermuteHelper.ElementLocalDof(0, DoF.Dx),
new ElementPermuteHelper.ElementLocalDof(0, DoF.Dy),
new ElementPermuteHelper.ElementLocalDof(1, DoF.Dx),
new ElementPermuteHelper.ElementLocalDof(1, DoF.Dy),
new ElementPermuteHelper.ElementLocalDof(2, DoF.Dx),
new ElementPermuteHelper.ElementLocalDof(2, DoF.Dy),
};
var kle = ElementPermuteHelper.FullyExpand(kl, currentOrder, 3);
return(kle);
}
private MembraneStressTensor GetMembraneInternalForce(LoadCombination combination)
{
//Note: membrane internal force is constant
//step 1 : get transformation matrix
//step 2 : convert globals points to locals
//step 3 : convert global displacements to locals
//step 4 : calculate B matrix and D matrix
//step 5 : M=D*B*U
//Note : Steps changed...
var trans = this.GetTransformationMatrix();
var lp = GetLocalPoints();
var g2l = new Func <Vector, Vector>(glob => (trans.Transpose() * glob.ToMatrix()).ToVector());
//var l2g = new Func<Vector, Vector>(local => (trans*local.ToMatrix()).ToPoint());
var d1g = this.nodes[0].GetNodalDisplacement(combination);
var d2g = this.nodes[1].GetNodalDisplacement(combination);
var d3g = this.nodes[2].GetNodalDisplacement(combination);
//step 3
var d1l = new Displacement(g2l(d1g.Displacements), g2l(d1g.Rotations));
var d2l = new Displacement(g2l(d2g.Displacements), g2l(d2g.Rotations));
var d3l = new Displacement(g2l(d3g.Displacements), g2l(d3g.Rotations));
var uCst = Matrix.OfVector(new[]
{ d1l.DX, d1l.DY, d2l.DX, d2l.DY, /**/ d3l.DX, d3l.DY });
var dbCst = CstElement.GetDMatrix(_elasticModulus, _poissonRatio, _formulationType);
var bCst = CstElement.GetBMatrix(lp.Select(i => i.X).ToArray(),
lp.Select(i => i.Y).ToArray());
var sCst = dbCst * bCst * uCst;
var buf = new MembraneStressTensor();
buf.Sx = sCst[0, 0];
buf.Sy = sCst[1, 0];
buf.Txy = sCst[2, 0];
return(buf);
}
private PlateBendingStressTensor GetBendingInternalForce(double localX, double localY, LoadCombination cmb)
{
//step 1 : get transformation matrix
//step 2 : convert globals points to locals
//step 3 : convert global displacements to locals
//step 4 : calculate B matrix and D matrix
//step 5 : M=D*B*U
//Note : Steps changed...
var trans = this.GetTransformationMatrix();
var lp = GetLocalPoints();
var g2l = new Func <Vector, Vector>(glob => (trans.Transpose() * glob.ToMatrix()).ToVector());
//var l2g = new Func<Vector, Vector>(local => (trans*local.ToMatrix()).ToPoint());
var d1g = this.nodes[0].GetNodalDisplacement(cmb);
var d2g = this.nodes[1].GetNodalDisplacement(cmb);
var d3g = this.nodes[2].GetNodalDisplacement(cmb);
//step 3
var d1l = new Displacement(g2l(d1g.Displacements), g2l(d1g.Rotations));
var d2l = new Displacement(g2l(d2g.Displacements), g2l(d2g.Rotations));
var d3l = new Displacement(g2l(d3g.Displacements), g2l(d3g.Rotations));
var uDkt = Matrix.OfVector(new[]
{ d1l.DZ, d1l.RX, d1l.RY, /**/ d2l.DZ, d2l.RX, d2l.RY, /**/ d3l.DZ, d3l.RX, d3l.RY });
var dbDkt = DktElement.GetDMatrix(this._thickness, this._elasticModulus, this._poissonRatio);
var b = DktElement.GetBMatrix(localX, localY,
lp.Select(i => i.X).ToArray(),
lp.Select(i => i.Y).ToArray());
var mDkt = dbDkt * b * uDkt; //eq. 32, batoz article
var buf = new PlateBendingStressTensor();
buf.Mx = mDkt[0, 0];
buf.My = mDkt[1, 0];
buf.Mxy = mDkt[2, 0];
return(buf);
}
#endregion
#region stresses
/// <summary>
/// Gets the internal stress at defined location.
/// tensor is in local coordinate system.
/// </summary>
/// <param name="localX">The X in local coordinate system (see remarks).</param>
/// <param name="localY">The Y in local coordinate system (see remarks).</param>
/// <param name="combination">The load combination.</param>
/// <param name="probeLocation">The probe location for the stress.</param>
/// <param name="thickness">The location for the bending stress. Maximum at the shell thickness</param>
/// <returns>Stress tensor of flat shell, in local coordination system</returns>
/// <remarks>
/// for more info about local coordinate of flat shell see page [72 of 166] (page 81 of pdf) of "Development of Membrane, Plate and Flat Shell Elements in Java" thesis by Kaushalkumar Kansara freely available on the web
/// </remarks>
public FlatShellStressTensor GetInternalStress(double localX, double localY, LoadCombination combination, double thickness, SectionPoints probeLocation)
{
var buf = new FlatShellStressTensor();
if ((this._behaviour & PlaneElementBehaviour.ThinPlate) != 0)
{
buf.MembraneTensor = GetMembraneInternalForce(combination);
}
if ((this._behaviour & PlaneElementBehaviour.Membrane) != 0)
{
buf.BendingTensor = GetBendingInternalForce(localX, localY, combination);
}
buf.UpdateTotalStress(thickness, probeLocation);
return(buf);
}
#endregion
#region strains
public StrainTensor GetMembraneInternalStrain(LoadCombination combination)
{
//Note: membrane internal force is constant
//step 1 : get transformation matrix
//step 2 : convert globals points to locals
//step 3 : convert global displacements to locals
//step 4 : calculate B matrix
//step 5 : e=B*U
//Note : Steps changed...
var trans = this.GetTransformationMatrix();
var lp = GetLocalPoints();
var g2l = new Func <Vector, Vector>(glob => (trans.Transpose() * glob.ToMatrix()).ToVector());
//var l2g = new Func<Vector, Vector>(local => (trans*local.ToMatrix()).ToPoint());
var d1g = this.nodes[0].GetNodalDisplacement(combination);
var d2g = this.nodes[1].GetNodalDisplacement(combination);
var d3g = this.nodes[2].GetNodalDisplacement(combination);
//step 3
var d1l = new Displacement(g2l(d1g.Displacements), g2l(d1g.Rotations));
var d2l = new Displacement(g2l(d2g.Displacements), g2l(d2g.Rotations));
var d3l = new Displacement(g2l(d3g.Displacements), g2l(d3g.Rotations));
var uCst = Matrix.OfVector(new[]
{ d1l.DX, d1l.DY, d2l.DX, d2l.DY, /**/ d3l.DX, d3l.DY });
var bCst = CstElement.GetBMatrix(lp.Select(i => i.X).ToArray(),
lp.Select(i => i.Y).ToArray());
var ECst = bCst * uCst;
var buf = new StrainTensor();
buf.S11 = ECst[0, 0];
buf.S22 = ECst[1, 0];
buf.S12 = ECst[2, 0];
return(buf);
}
