public Matrix GetLocalStifnessMatrix()
{
var helpers = GetHelpers();
var buf = MatrixPool.Allocate(18, 18);
for (var i = 0; i < helpers.Length; i++)
{
var helper = helpers[i];
var ki = helper.CalcLocalStiffnessMatrix(this);// ComputeK(helper, transMatrix);
var dofs = helper.GetDofOrder(this);
for (var ii = 0; ii < dofs.Length; ii++)
{
var bi = dofs[ii].NodeIndex * 6 + (int)dofs[ii].Dof;
for (var jj = 0; jj < dofs.Length; jj++)
{
var bj = dofs[jj].NodeIndex * 6 + (int)dofs[jj].Dof;
buf[bi, bj] += ki[ii, jj];
}
}
}
return(buf);
}
/// <inheritdoc/>
public Matrix GetDMatrixAt(Element targetElement, params double[] isoCoords)
{
var tri = targetElement as TriangleElement;
if (tri == null)
{
throw new Exception();
}
var mat = tri._material.GetMaterialPropertiesAt(isoCoords);
var t = tri.Section.GetThicknessAt(isoCoords);
var d = MatrixPool.Allocate(3, 3);
{
var cf = t * t * t / 12;
d[0, 0] = mat.Ex / (1 - mat.NuXy * mat.NuYx);
d[1, 1] = mat.Ey / (1 - mat.NuXy * mat.NuYx);
d[0, 1] = d[1, 0] =
mat.Ex * mat.NuYx / (1 - mat.NuXy * mat.NuYx);
d[2, 2] = mat.Ex / (2 * (1 + mat.NuXy));
//p55 http://www.code-aster.org/doc/v11/en/man_r/r3/r3.07.03.pdf
d.MultiplyByConstant(cf);
}
return(d);
}
public Matrix GetLocalMassMatrix()
{
var helpers = new List <IElementHelper>();
if ((this._behavior & PlateElementBehaviour.ThinPlate) != 0)
{
helpers.Add(new DktHelper());
}
var buf = MatrixPool.Allocate(18, 18);
for (var i = 0; i < helpers.Count; i++)
{
var helper = helpers[i];
var ki = helper.CalcLocalMassMatrix(this);// ComputeK(helper, transMatrix);
var dofs = helper.GetDofOrder(this);
for (var ii = 0; ii < dofs.Length; ii++)
{
var bi = dofs[ii].NodeIndex * 6 + (int)dofs[ii].Dof;
for (var jj = 0; jj < dofs.Length; jj++)
{
var bj = dofs[jj].NodeIndex * 6 + (int)dofs[jj].Dof;
buf[bi, bj] += ki[ii, jj];
}
}
}
return(buf);
}
/// <summary>
/// Gets the stifness matrix in local coordination system.
/// </summary>
/// <returns>stiffness matrix</returns>
public virtual Matrix GetLocalStifnessMatrix()
{
var helpers = GetElementHelpers();
var buf =
MatrixPool.Allocate(6 * nodes.Length, 6 * nodes.Length);
//var transMatrix = GetTransformationMatrix();
for (var i = 0; i < helpers.Count; i++)
{
var helper = helpers[i];
var ki = helper.CalcLocalStiffnessMatrix(this);// ComputeK(helper, transMatrix);
var dofs = helper.GetDofOrder(this);
for (var ii = 0; ii < dofs.Length; ii++)
{
var bi = dofs[ii].NodeIndex * 6 + (int)dofs[ii].Dof;
for (var jj = 0; jj < dofs.Length; jj++)
{
var bj = dofs[jj].NodeIndex * 6 + (int)dofs[jj].Dof;
buf[bi, bj] += ki[ii, jj];
}
}
ki.ReturnToPool();
}
return(buf);
}
public override Matrix GetLambdaMatrix()
{
//local coor system is same as global one
var buf = MatrixPool.Allocate(3, 3);
buf[0, 0] = buf[1, 1] = buf[2, 2] = 1;
return(buf);
}
/// <inheritdoc/>
public Matrix GetJMatrixAt(Element targetElement, params double[] isoCoords)
{
var tmgr = targetElement.GetTransformationManager();
var tri = targetElement as TriangleElement;
if (tri == null)
{
throw new Exception();
}
var xi = isoCoords[0];
var eta = isoCoords[1];
var mgr = tmgr;
var p1l = mgr.TransformGlobalToLocal(tri.Nodes[0].Location);
var p2l = mgr.TransformGlobalToLocal(tri.Nodes[1].Location);
var p3l = mgr.TransformGlobalToLocal(tri.Nodes[2].Location);
var p23 = p2l - p3l;
var p31 = p3l - p1l;
var p12 = p1l - p2l;
var x23 = p23.X;
var x31 = p31.X;
var x12 = p12.X;
var y23 = p23.Y;
var y31 = p31.Y;
var y12 = p12.Y;
var buf = MatrixPool.Allocate(2, 2);
buf[0, 0] = x31;
buf[1, 1] = y12;
buf[0, 1] = x12;
buf[1, 0] = y31;
return(buf);
}
public Matrix GetTransformationMatrix()
{
var cxx = 0.0;
var cxy = 0.0;
var cxz = 0.0;
var cyx = 0.0;
var cyy = 0.0;
var cyz = 0.0;
var czx = 0.0;
var czy = 0.0;
var czz = 0.0;
var teta = _webRotation;
var s = Math.Sin(teta * Math.PI / 180.0);
var c = Math.Cos(teta * Math.PI / 180.0);
var v = this.EndNode.Location - this.StartNode.Location;
if (MathUtil.Equals(0, v.X) && MathUtil.Equals(0, v.Y))
{
if (v.Z > 0)
{
czx = 1;
cyy = 1;
cxz = -1;
}
else
{
czx = -1;
cyy = 1;
cxz = 1;
}
}
else
{
var l = v.Length;
cxx = v.X / l;
cyx = v.Y / l;
czx = v.Z / l;
var d = Math.Sqrt(cxx * cxx + cyx * cyx);
cxy = -cyx / d;
cyy = cxx / d;
cxz = -cxx * czx / d;
cyz = -cyx * czx / d;
czz = d;
}
var pars = new double[9];
pars[0] = cxx;
pars[1] = cxy * c + cxz * s;
pars[2] = -cxy * s + cxz * c;
pars[3] = cyx;
pars[4] = cyy * c + cyz * s;
pars[5] = -cyy * s + cyz * c;
pars[6] = czx;
pars[7] = czy * c + czz * s;
pars[8] = -czy * s + czz * c;
var buf =
//new Matrix(3, 3);
MatrixPool.Allocate(3, 3);
buf.FillColumn(0, pars[0], pars[1], pars[2]);
buf.FillColumn(1, pars[3], pars[4], pars[5]);
buf.FillColumn(2, pars[6], pars[7], pars[8]);
return(buf);
}
/// <summary>
/// get the polynomial that takes iso coord as input and return local coord as output
/// </summary>
/// <returns>X(ξ) (ξ input, X output)</returns>
public virtual Mathh.Polynomial GetIsoToLocalConverter()
{
var cachekey = "{54CEC6B2-F882-4505-9FC0-E7844C99F249}";
Mathh.Polynomial chd;
if (this.TryGetCache(cachekey, out chd)) //prevent double calculation
{
if (IsValidIsoToLocalConverter(chd)) //Validation of chached polynomial to see if is outdated due to change in node locations
{
return(chd);
}
}
chd = null;
var targetElement = this;
var bar = this;
Mathh.Polynomial x_xi = null;//x(ξ)
var n = targetElement.Nodes.Length;
var xs = new double[n];
var xi_s = new double[n];
{
//var conds = new List<Tuple<double, double>>();//x[i] , ξ[i]
for (var i = 0; i < n; i++)
{
var deltaXi = 2.0 / (n - 1);
var xi = (bar.Nodes[i].Location - bar.Nodes[0].Location).Length;
var xi_i = -1 + deltaXi * i;
xs[i] = xi;
xi_s[i] = xi_i;
//conds.Add(Tuple.Create(xi, xi_i));
}
//polinomial degree of shape function is n-1
var mtx =
//new Matrix(n, n);
MatrixPool.Allocate(n, n);
var right =
//new Matrix(n, 1);
MatrixPool.Allocate(n, 1);
//var o = n - 1;
for (var i = 0; i < n; i++)
{
//fill row i'th of mtx
//x[i] = { ξ[i]^o, ξ[i]^o-1 ... ξ[i]^1 ξ[i]^0} * {a[o] a[o-1] ... a[1] a[0]}'
var kesi_i = xi_s[i];
for (var j = 0; j < n; j++)
{
mtx[i, j] = Math.Pow(kesi_i, n - j - 1);
right[i, 0] = xs[i];
}
}
//var as_ = mtx.Inverse() * right;
var as_ = mtx.Solve(right.CoreArray);
x_xi = new Mathh.Polynomial(as_);
right.ReturnToPool();
mtx.ReturnToPool();
}
SetCache(cachekey, x_xi);
return(x_xi);
}
/// <summary>
/// Computes the I.
/// </summary>
/// <returns>The I</returns>
public Matrix Integrate()
{
if (MatrixPool == null)
{
MatrixPool = new MatrixPool();
}
if (XiPointCount < 1 || EtaPointCount < 1 || GammaPointCount < 1)
{
throw new NotSupportedException();
}
double a1 = A1, a2 = A2;
var f1 = F1;
var f2 = F2;
var g1 = G1;
var g2 = G2;
var vk = GaussPoints.GetGaussianValues(GammaPointCount);
var wk = GaussPoints.GetGaussianWeights(GammaPointCount);
var vj = GaussPoints.GetGaussianValues(EtaPointCount);
var wj = GaussPoints.GetGaussianWeights(EtaPointCount);
var vi = GaussPoints.GetGaussianValues(XiPointCount);
var wi = GaussPoints.GetGaussianWeights(XiPointCount);
Matrix I = null;//we do not know dimensions yet!
for (var k = 0; k < GammaPointCount; k++)
{
var gammaK = (a2 - a1) / 2 * vk[k] + (a2 + a1) / 2;
Matrix phi = null;// = new Matrix(H, W);
for (var j = 0; j < EtaPointCount; j++)
{
var noj = (f2(gammaK) - f1(gammaK)) / 2 * vj[j] + (f2(gammaK) + f1(gammaK)) / 2;
Matrix beta = null;//= new Matrix(H, W);
for (var i = 0; i < XiPointCount; i++)
{
var xii = (g2(noj, gammaK) - g1(noj, gammaK)) / 2 * vi[i] + (g2(noj, gammaK) + g1(noj, gammaK)) / 2;
var val = H.GetMatrix(xii, noj, gammaK);
//initiate I, phi and beta
if (beta == null)
{
beta =
//new Matrix(val.RowCount, val.ColumnCount);
MatrixPool.Allocate(val.RowCount, val.ColumnCount);
}
if (phi == null)
{
phi =
//new Matrix(val.RowCount, val.ColumnCount);
MatrixPool.Allocate(val.RowCount, val.ColumnCount);
}
if (I == null)
{
I =
//new Matrix(val.RowCount, val.ColumnCount);
MatrixPool.Allocate(val.RowCount, val.ColumnCount);
}
var valSc = (g2(noj, gammaK) - g1(noj, gammaK)) / 2 * wi[i];
//beta += val;
beta.AddToThis(val, valSc);
val.ReturnToPool();
}
var phiSc = (f2(gammaK) - f1(gammaK)) / 2 * wj[j];
phi.AddToThis(beta, phiSc);
beta.ReturnToPool();
}
var iSc = (a2 - a1) / 2 * wk[k];
I.AddToThis(phi, iSc);
phi.ReturnToPool();
//I += (a2 - a1) / 2 * wk[k] * phi;
}
return(I);
}
public static Matrix CalcLocalKMatrix_Triangle(IElementHelper helper, Element targetElement)
{
var qq = targetElement as TriangleElement;
if (qq == null)
{
throw new Exception();
}
var trans = qq.GetLambdaMatrix().Transpose();
var nb = helper.GetBMaxOrder(targetElement);
var nd = qq.Material.GetMaxFunctionOrder();
var nt = qq.Section.GetMaxFunctionOrder();
var nj = helper.GetDetJOrder(targetElement);
var sum = new int[3];
foreach (var i in new int[][] { nb, nd, nb, nt, nj })
{
for (int j = 0; j < 3; j++)
{
sum[j] += i[j];
}
}
var nXi = sum[0] / 2 + 1;
var nEta = sum[1] / 2 + 1;
var nGama = sum[2] / 2 + 1;
var intg = new GaussianIntegrator();
intg.GammaPointCount = nGama;
intg.XiPointCount = nXi;
intg.EtaPointCount = nEta;
intg.A2 = 1;
intg.A1 = 0;
intg.F2 = (gama => 1);
intg.F1 = (gama => 0);
intg.G2 = ((eta, gama) => 1 - eta);
intg.G1 = ((eta, gama) => 0);
intg.H = new FunctionMatrixFunction((xi, eta, gama) =>
{
var b = helper.GetBMatrixAt(qq, xi, eta);
var d = helper.GetDMatrixAt(qq, xi, eta);
var j = helper.GetJMatrixAt(qq, xi, eta);
var buf = MatrixPool.Allocate(b.ColumnCount, b.ColumnCount);
CalcUtil.Bt_D_B(b, d, buf);
buf.MultiplyByConstant((j.Determinant()));
return(buf);
});
var res = intg.Integrate();
return(res);
}
/// <inheritdoc/>
public Matrix GetBMatrixAt(Element targetElement, params double[] isoCoords)
{
var tri = targetElement as TriangleElement;
if (tri == null)
{
throw new Exception();
}
var xi = isoCoords[0];
var eta = isoCoords[1];
#region inits
var mgr = targetElement.GetTransformationManager();//TransformManagerL2G.MakeFromTransformationMatrix(transformMatrix);
var p1l = mgr.TransformGlobalToLocal(tri.Nodes[0].Location);
var p2l = mgr.TransformGlobalToLocal(tri.Nodes[1].Location);
var p3l = mgr.TransformGlobalToLocal(tri.Nodes[2].Location);
var p23 = p2l - p3l;
var p31 = p3l - p1l;
var p12 = p1l - p2l;
var x23 = p23.X;
var x31 = p31.X;
var x12 = p12.X;
var y23 = p23.Y;
var y31 = p31.Y;
var y12 = p12.Y;
var a = 0.5 * Math.Abs(x31 * y12 - x12 * y31);
var l23_2 = y23 * y23 + x23 * x23;
var l31_2 = y31 * y31 + x31 * x31;
var l12_2 = y12 * y12 + x12 * x12;
var P4 = -6 * x23 / l23_2;
var P5 = -6 * x31 / l31_2;
var P6 = -6 * x12 / l12_2;
var q4 = 3 * x23 * y23 / l23_2;
var q5 = 3 * x31 * y31 / l31_2;
var q6 = 3 * x12 * y12 / l12_2;
var r4 = 3 * y23 * y23 / l23_2;
var r5 = 3 * y31 * y31 / l31_2;
var r6 = 3 * y12 * y12 / l12_2;
var t4 = -6 * y23 / l23_2;
var t5 = -6 * y31 / l31_2;
var t6 = -6 * y12 / l12_2;
#endregion
#region h{x,y}{kesi,no}
var hx_xi = new double[]//eq. 4.27 ref [1], also noted in several other references
{
P6 *(1 - 2 * xi) + (P5 - P6) * eta,
q6 *(1 - 2 * xi) - (q5 + q6) * eta,
-4 + 6 * (xi + eta) + r6 * (1 - 2 * xi) - eta * (r5 + r6),
-P6 * (1 - 2 * xi) + eta * (P4 + P6),
q6 *(1 - 2 * xi) - eta * (q6 - q4),
-2 + 6 * xi + r6 * (1 - 2 * xi) + eta * (r4 - r6),
-eta * (P5 + P4),
eta *(q4 - q5),
-eta * (r5 - r4)
};
var hy_xi = new double[] //eq. 4.28 ref [1], also noted in several other references
{
t6 *(1 - 2 * xi) + eta * (t5 - t6),
1 + r6 * (1 - 2 * xi) - eta * (r5 + r6),
-q6 * (1 - 2 * xi) + eta * (q5 + q6),
-t6 * (1 - 2 * xi) + eta * (t4 + t6),
-1 + r6 * (1 - 2 * xi) + eta * (r4 - r6),
-q6 * (1 - 2 * xi) - eta * (q4 - q6),
-eta * (t4 + t5),
eta *(r4 - r5),
-eta * (q4 - q5)
};
var hx_eta = new double[] //eq. 4.29 ref [1], also noted in several other references
{
-P5 * (1 - 2 * eta) - xi * (P6 - P5),
q5 *(1 - 2 * eta) - xi * (q5 + q6),
-4 + 6 * (xi + eta) + r5 * (1 - 2 * eta) - xi * (r5 + r6),
xi *(P4 + P6),
xi *(q4 - q6),
-xi * (r6 - r4),
P5 *(1 - 2 * eta) - xi * (P4 + P5),
q5 *(1 - 2 * eta) + xi * (q4 - q5),
-2 + 6 * eta + r5 * (1 - 2 * eta) + xi * (r4 - r5)
};
var hy_eta = new double[] //eq. 4.30 ref [1], also noted in several other references
{
-t5 * (1 - 2 * eta) - xi * (t6 - t5),
1 + r5 * (1 - 2 * eta) - xi * (r5 + r6),
-q5 * (1 - 2 * eta) + xi * (q5 + q6),
xi *(t4 + t6),
xi *(r4 - r6),
-xi * (q4 - q6),
t5 *(1 - 2 * eta) - xi * (t4 + t5),
-1 + r5 * (1 - 2 * eta) + xi * (r4 - r5),
-q5 * (1 - 2 * eta) - xi * (q4 - q5)
};
#endregion
var buf = MatrixPool.Allocate(3, 9);
for (var i = 0; i < 9; i++)
{
buf[0, i] = y31 * hx_xi[i] + y12 * hx_eta[i];
buf[1, i] = -x31 * hy_xi[i] - x12 * hy_eta[i];
buf[2, i] = -x31 * hx_xi[i] - x12 * hx_eta[i] + y31 * hy_xi[i] + y12 * hy_eta[i];
}//eq. 4.26 page 46 ref [1]
buf.MultiplyByConstant(1 / (2 * a));
return(buf);
}