/// <summary>
/// <c>S_Matrix</c> returns stress matrix required to compute nonlinear part of K matrix
/// </summary>
private MatrixST Stress_Matrix(int inc, int GaussPoint)
{
// __ __ __ __ __ __
// | s 0 0 | | s11 s12 s13 | | s0 s3 s5 |
// S = | 0 s 0 | where s = | s21 s22 s23 | => | s3 s1 s4 |
// | 0 0 s | | s31 s32 s33 | | s5 s4 s2 |
// --- --- --- --- --- ---
MatrixST S = new MatrixST(9, 9);
for (int i = 0; i < 3; i++)
{
S.SetFast(3 * i + 0, 3 * i + 0, Stress[inc].GetFast(GaussPoint, 0) + dS[GaussPoint].GetFast(0, 0));
S.SetFast(3 * i + 1, 3 * i + 0, Stress[inc].GetFast(GaussPoint, 3) + dS[GaussPoint].GetFast(3, 0));
S.SetFast(3 * i + 2, 3 * i + 0, Stress[inc].GetFast(GaussPoint, 5) + dS[GaussPoint].GetFast(5, 0));
S.SetFast(3 * i + 0, 3 * i + 1, Stress[inc].GetFast(GaussPoint, 3) + dS[GaussPoint].GetFast(3, 0));
S.SetFast(3 * i + 1, 3 * i + 1, Stress[inc].GetFast(GaussPoint, 1) + dS[GaussPoint].GetFast(1, 0));
S.SetFast(3 * i + 2, 3 * i + 1, Stress[inc].GetFast(GaussPoint, 4) + dS[GaussPoint].GetFast(4, 0));
S.SetFast(3 * i + 0, 3 * i + 2, Stress[inc].GetFast(GaussPoint, 5) + dS[GaussPoint].GetFast(5, 0));
S.SetFast(3 * i + 1, 3 * i + 2, Stress[inc].GetFast(GaussPoint, 4) + dS[GaussPoint].GetFast(4, 0));
S.SetFast(3 * i + 2, 3 * i + 2, Stress[inc].GetFast(GaussPoint, 2) + dS[GaussPoint].GetFast(2, 0));
}
return(S);
}
/// <summary>
/// Inverse of 3x3 matrix. Designed for Jacobian matrix.
/// </summary>
/// <returns>Inverse of this matrix if size is 3x3 and determinant is not 0.
/// <br>Otherwise returns empty 3x3 matrix</br></returns>
public MatrixST Inverse()
{
double det = Det3();
if (det != 0 && Rows == 3 && Cols == 3)
{
MatrixST Inv = new MatrixST(3, 3);
double X = 1.0 / det;
Inv.SetFast(0, 0, X * (GetFast(1, 1) * GetFast(2, 2) - GetFast(1, 2) * GetFast(2, 1)));
Inv.SetFast(0, 1, X * (GetFast(0, 2) * GetFast(2, 1) - GetFast(0, 1) * GetFast(2, 2)));
Inv.SetFast(0, 2, X * (GetFast(0, 1) * GetFast(1, 2) - GetFast(0, 2) * GetFast(1, 1)));
Inv.SetFast(1, 0, X * (GetFast(1, 2) * GetFast(2, 0) - GetFast(1, 0) * GetFast(2, 2)));
Inv.SetFast(1, 1, X * (GetFast(0, 0) * GetFast(2, 2) - GetFast(0, 2) * GetFast(2, 0)));
Inv.SetFast(1, 2, X * (GetFast(0, 2) * GetFast(1, 0) - GetFast(0, 0) * GetFast(1, 2)));
Inv.SetFast(2, 0, X * (GetFast(1, 0) * GetFast(2, 1) - GetFast(1, 1) * GetFast(2, 0)));
Inv.SetFast(2, 1, X * (GetFast(0, 1) * GetFast(2, 0) - GetFast(0, 0) * GetFast(2, 1)));
Inv.SetFast(2, 2, X * (GetFast(0, 0) * GetFast(1, 1) - GetFast(0, 1) * GetFast(1, 0)));
return(Inv);
}
else
{
throw new System.ArgumentException("Inverse matrix error: Matrix size in not 3x3 or det=0", "Inverse");
}
}
/// <summary>
/// <c>BL0_Matrix</c> returns linear strain-displacement transformation matrix without initial displacement effect
/// </summary>
private MatrixST BL0_Matrix(MatrixST dN)
{
// Strain vector: e = { e_xx e_yy e_zz e_xy e_yz e_xz }
//
// BL0 is linear strain-displacement matrix:
// __ __
// | dNi/dx 0 0 |
// | 0 dNi/dy 0 |
// | 0 0 dNi/dz |
// BL0 = L*N = | dNi/dy dNi/dx 0 ... |
// | 0 dNi/dz dNi/dy |
// | dNi/dz 0 dNi/dx |
// --- ---
// Strain-displacement Matrix BL0
MatrixST BL0 = new MatrixST(6, 3 * NList.Count);
for (int i = 0; i < NList.Count; i++)
{
BL0.SetFast(0, 3 * i + 0, dN.GetFast(0, i));
BL0.SetFast(1, 3 * i + 1, dN.GetFast(1, i));
BL0.SetFast(2, 3 * i + 2, dN.GetFast(2, i));
BL0.SetFast(3, 3 * i + 0, dN.GetFast(1, i));
BL0.SetFast(3, 3 * i + 1, dN.GetFast(0, i));
BL0.SetFast(4, 3 * i + 1, dN.GetFast(2, i));
BL0.SetFast(4, 3 * i + 2, dN.GetFast(1, i));
BL0.SetFast(5, 3 * i + 0, dN.GetFast(2, i));
BL0.SetFast(5, 3 * i + 2, dN.GetFast(0, i));
}
return(BL0);
}
/// <summary>
/// <c>BNL_Matrix</c> returns non-linear strain-displacement transformation matrix
/// </summary>
private MatrixST BNL_Matrix(MatrixST dN)
{
// __ __ ___ ___
// | B 0 0 | | dN1/dx 0 0 dN2/dx dNi/dx |
// BNL = | 0 B 0 | where B = | dN1/dy 0 0 dN2/dy ... dNi/dy |
// | 0 0 B | | dN1/dz 0 0 dN2/dz dNi/dz |
// --- --- --- ---
//
// | dN1/dx dN2/dx dNk/dx |
// Input: dN = | dN1/dy dN2/dy ... dNk/dy |
// | dN1/dz dN2/dz dNk/dz |
//
// Strain-displacement Matrix
MatrixST BNL = new MatrixST(9, 3 * NList.Count);
for (int i = 0; i < NList.Count; i++)
{
BNL.SetFast(0, 3 * i + 0, dN.GetFast(0, i));
BNL.SetFast(1, 3 * i + 0, dN.GetFast(1, i));
BNL.SetFast(2, 3 * i + 0, dN.GetFast(2, i));
BNL.SetFast(3, 3 * i + 1, dN.GetFast(0, i));
BNL.SetFast(4, 3 * i + 1, dN.GetFast(1, i));
BNL.SetFast(5, 3 * i + 1, dN.GetFast(2, i));
BNL.SetFast(6, 3 * i + 2, dN.GetFast(0, i));
BNL.SetFast(7, 3 * i + 2, dN.GetFast(1, i));
BNL.SetFast(8, 3 * i + 2, dN.GetFast(2, i));
}
return(BNL);
}
/// <summary>
/// Add nodal value to List
/// <list>
/// <item><c>nid</c></item>
/// <description> - Node ID</description>
/// <item><c>direction</c></item>
/// <description> - X=0; Y=1; Z=2</description>
/// <item><c>value</c></item>
/// <description> - Value of BC (double for Force BC, 0 or 1 for Fix BC)</description>
/// <item><c>NodeLib</c></item>
/// <description> - Database Node library</description>
/// </list>
/// </summary>
public void Add(int nid, double[] value, Dictionary <int, Node> NodeLib)
{
if (NodeLib.ContainsKey(nid))
{
MatrixST nVal = new MatrixST(3, 1);
nVal.SetFast(0, 0, value[0]);
nVal.SetFast(1, 0, value[1]);
nVal.SetFast(2, 0, value[2]);
NodalValues.Add(nid, nVal);
}
}
// ==================== Finite Element Matrix methods ===================================
/// <summary>
/// <c>Jacobian</c> returns Jacobian matrix in Gauss Point.
/// </summary>
public MatrixST Jacobian(Dictionary <int, Node> NodeLib, MatrixST dN_dLocal)
{
// | dN1/d_xi dN2/d_xi ... | |x1 y1 z1|
// J = | dN1/d_eta dN2/d_eta ... | * |x2 y2 z2|
// | dN1/d_zeta dN2/d_zeta ... | |x3 y3 z3|
// | ... |
MatrixST Nodal_Coord = new MatrixST(NList.Count, 3);
for (int i = 0; i < NList.Count; i++)
{
Nodal_Coord.SetFast(i, 0, NodeLib[NList[i]].X);
Nodal_Coord.SetFast(i, 1, NodeLib[NList[i]].Y);
Nodal_Coord.SetFast(i, 2, NodeLib[NList[i]].Z);
}
MatrixST J = dN_dLocal * Nodal_Coord;
return(J);
}
public MatrixST Vector2Tensor(double[] v)
{
MatrixST T = new MatrixST(3, 3);
T.SetFast(0, 0, v[0]);
T.SetFast(0, 1, v[3]);
T.SetFast(0, 2, v[5]);
T.SetFast(1, 0, v[3]);
T.SetFast(1, 1, v[1]);
T.SetFast(1, 2, v[4]);
T.SetFast(2, 0, v[5]);
T.SetFast(2, 1, v[4]);
T.SetFast(2, 2, v[2]);
return(T);
}
/// <summary>
/// <c>BL1_Matrix</c> returns intial displacement part of BL matrix in Total Lagrangian framework
/// </summary>
private MatrixST BL1_Matrix(MatrixST dN, MatrixST dU)
{
// Strain vector: e = { e_xx e_yy e_zz e_xy e_yz e_xz }
// __ __
// | F11 * dNi/dx F21 * dNi/dx F31 * dNi/dx |
// | F12 * dNi/dy F22 * dNi/dy F32 * dNi/dy |
// | F13 * dNi/dy F23 * dNi/dy F33 * dNi/dy |
// BL1 = | F11*dNi/dy + F12*dNi/dx F21*dNi/dy + F22*dNi/dx F11*dNi/dy + F12*dNi/dx ... |
// | F12*dNi/dz + F13*dNi/dy F22*dNi/dz + F23*dNi/dy F12*dNi/dz + F13*dNi/dy |
// | F11*dNi/dz + F13*dNi/dx F21*dNi/dz + F23*dNi/dx F11*dNi/dz + F13*dNi/dx |
// --- ---
//
// | x1 y1 z1 | | dN1/dx dN2/dx dNk/dx |
// F(i,j) = summ ( dN[j, k] * U[k, i] ) where U = | ... | dN = | dN1/dy dN2/dy ... dNk/dy |
// | xk yk zk | | dN1/dz dN2/dz dNk/dz |
//
MatrixST F = dN * dU;
MatrixST BL1 = new MatrixST(6, 3 * NList.Count);
for (int i = 0; i < NList.Count; i++)
{
for (int j = 0; j < 3; j++)
{
BL1.SetFast(0, 3 * i + j, F.GetFast(j, 0) * dN.GetFast(0, i));
BL1.SetFast(1, 3 * i + j, F.GetFast(j, 1) * dN.GetFast(1, i));
BL1.SetFast(2, 3 * i + j, F.GetFast(j, 2) * dN.GetFast(2, i));
BL1.SetFast(3, 3 * i + j, F.GetFast(j, 0) * dN.GetFast(1, i) + F.GetFast(j, 1) * dN.GetFast(0, i));
BL1.SetFast(4, 3 * i + j, F.GetFast(j, 1) * dN.GetFast(2, i) + F.GetFast(j, 2) * dN.GetFast(1, i));
BL1.SetFast(5, 3 * i + j, F.GetFast(j, 0) * dN.GetFast(2, i) + F.GetFast(j, 2) * dN.GetFast(0, i));
}
}
return(BL1);
}
/// <summary>
/// Multiply all element of this matrix by scalar s
/// </summary>
/// <returns>
/// Matrix-Scalar product
/// </returns>
public MatrixST MultiplyScalar(double s)
{
MatrixST C = new MatrixST(Rows, Cols);
for (int i = 0; i < Rows; i++)
{
for (int j = 0; j < Cols; j++)
{
C.SetFast(i, j, GetFast(i, j) * s);
}
}
return(C);
}
/// <returns>
/// Transposition of this matrix
/// </returns>
public MatrixST Transpose()
{
MatrixST C = new MatrixST(Cols, Rows);
for (int r = 0; r < Rows; r++)
{
for (int c = 0; c < Cols; c++)
{
C.SetFast(c, r, GetFast(r, c));
}
}
return(C);
}
/// <summary>
/// Calculate Tangent stiffness matrix
/// </summary>
public MatrixST K_Tangent(int inc, Database DB)
{
// Element Matrix Initialization
MatrixST K = new MatrixST(NList.Count * 3, NList.Count * 3);
// Loop thorugh all Gauss Points
for (int g = 0; g < GaussPointNumb; g++)
{
// Jacobian Matrix at Gauss Point
J[g] = Jacobian(DB.NodeLib, DB.FELib.FE[Type].dN_dLocal[g]);
// Shape function derivatives in Global Coordinates System
MatrixST dN = J[g].Inverse() * DB.FELib.FE[Type].dN_dLocal[g];
// ------ Linear strain-displacement matrix ---------------------------------
// Initial displacement effect - BL1 Matrix
MatrixST U = new MatrixST(NList.Count, 3);
for (int i = 0; i < NList.Count; i++)
{
for (int j = 0; j < 3; j++)
{
U.SetFast(i, j, DB.NodeLib[NList[i]].GetDisp(inc, j));
}
}
BL[g] = BL0_Matrix(dN) + BL1_Matrix(dN, U); // BL = BL0 + BL1
// ------ Constitutive Matrix ---------------------------------
MatrixST D = DB.MatLib[MatID].GetElastic();
// --- LINEAR PART OF STIFFNESS MATRIX -------------------------------------------
// Add K components to Element K matrix (sum from all Gauss points)
K += (BL[g].Transpose() * D * BL[g]).MultiplyScalar(J[g].Det3() * DB.FELib.FE[Type].GaussWeight);
// --- NONLINEAR PART OF STIFFNESS MATRIX ------------------------------------------------------------
// Nonlinear Strain-displacement matrix
MatrixST BNL = BNL_Matrix(dN);
// Initial Stress matrix
MatrixST S = Stress_Matrix(inc, g);
// Nonlinear part of Element Stiffness Matrix
K += (BNL.Transpose() * S * BNL).MultiplyScalar(J[g].Det3() * DB.FELib.FE[Type].GaussWeight);
}
return(K);
}
/// <summary>
/// Get row of this matrix
/// </summary>
/// <returns> Row of this matrix as MatrixST with size 1 x ColumnNumber </returns>
public MatrixST GetRow_MatrixST(int index)
{
if (index >= 0 && index < Rows)
{
MatrixST row = new MatrixST(1, Cols);
for (int i = 0; i < Cols; i++)
{
row.SetFast(0, i, GetFast(index, i));
}
return(row);
}
else
{
throw new System.ArgumentException("GetRow error: Matrix doesn't have specified row", "Index");
}
}
public void SetElastic(double Young, double PoissonRatio)
{
// Set elastic properties
E = Young;
Poisson = PoissonRatio;
// Calculate elastic D matrix
ElasticMatrix = new MatrixST(6, 6);
double lambda = (E * Poisson) / ((1 - 2 * Poisson) * (1 + Poisson));
double G = (0.5 * E) / (1 + Poisson);
ElasticMatrix.SetFast(0, 0, lambda + (2 * G));
ElasticMatrix.SetFast(0, 1, lambda);
ElasticMatrix.SetFast(0, 2, lambda);
ElasticMatrix.SetFast(1, 0, lambda);
ElasticMatrix.SetFast(1, 1, lambda + (2 * G));
ElasticMatrix.SetFast(1, 2, lambda);
ElasticMatrix.SetFast(2, 0, lambda);
ElasticMatrix.SetFast(2, 1, lambda);
ElasticMatrix.SetFast(2, 2, lambda + (2 * G));
ElasticMatrix.SetFast(3, 3, G);
ElasticMatrix.SetFast(4, 4, G);
ElasticMatrix.SetFast(5, 5, G);
}
/// <summary>
/// Finite Element type:
/// <br>Tetrahedral, 4 node, Isoparametric</br>
/// <br><c>Gauss_Point</c> - Integration point natural coordinates {xi, eta, zeta}</br>
/// </summary>
/// <returns>Matrix with partial derivatives (natural) of shape functions in Gauss point
/// <br></br></returns>
private MatrixST TET4_Diff_ShapeFunctions()
{
// Input: Gauss_Point = { xi, eta, zeta}
// | dN1/d_xi dN2/d_xi dN8/d_xi |
// Output: dN_dLocal = | dN1/d_eta dN2/d_eta ... dN8/d_eta |
// | dN1/d_zeta dN2/d_zeta dN8/d_zeta |
// Shape function of TET4 element:
// N1 = 1 - xi - eta - zeta
// N2 = xi
// N3 = eta
// N4 = zeta
// Initialize Matrix with shape function derivatives in local coordinate system
MatrixST dN_dLocal = new MatrixST(3, 4);
// dN/d_xi
dN_dLocal.SetFast(0, 0, -1);
dN_dLocal.SetFast(0, 1, 1);
dN_dLocal.SetFast(0, 2, 0);
dN_dLocal.SetFast(0, 3, 0);
// dN/d_eta
dN_dLocal.SetFast(1, 0, -1);
dN_dLocal.SetFast(1, 1, 0);
dN_dLocal.SetFast(1, 2, 1);
dN_dLocal.SetFast(1, 3, 0);
// dN/d_zeta
dN_dLocal.SetFast(2, 0, -1);
dN_dLocal.SetFast(2, 1, 0);
dN_dLocal.SetFast(2, 2, 0);
dN_dLocal.SetFast(2, 3, 1);
return(dN_dLocal);
}
/// <summary>
/// Finite Element type:
/// <br>Hexahedral, 8 node, Isoparametric</br>
/// <br><c>Gauss_Point</c> - Integration point natural coordinates {xi, eta, zeta}</br>
/// </summary>
/// <returns>Matrix with partial derivatives (natural) of shape functions in Gauss point
/// <br></br></returns>
private MatrixST HEX8_Diff_ShapeFunctions(double[] Gauss_Point)
{
// Input: Gauss_Point = { xi, eta, zeta}
// | dN1/d_xi dN2/d_xi dN8/d_xi |
// Output: dN_dLocal = | dN1/d_eta dN2/d_eta ... dN8/d_eta |
// | dN1/d_zeta dN2/d_zeta dN8/d_zeta |
// Shape function of HEX8 element:
// N1 = 1 / 8 * (1 - xi) * (1 - eta) * (1 - zeta)
// N2 = 1 / 8 * (1 + xi) * (1 - eta) * (1 - zeta)
// N3 = 1 / 8 * (1 + xi) * (1 + eta) * (1 - zeta)
// N4 = 1 / 8 * (1 - xi) * (1 + eta) * (1 - zeta)
// N5 = 1 / 8 * (1 - xi) * (1 - eta) * (1 + zeta)
// N6 = 1 / 8 * (1 + xi) * (1 - eta) * (1 + zeta)
// N7 = 1 / 8 * (1 + xi) * (1 + eta) * (1 + zeta)
// N8 = 1 / 8 * (1 - xi) * (1 + eta) * (1 + zeta)
// Table with signs:
//
// 1 xi eta zeta xi*eta eta*zeta xi*zeta xi*eta*zeta
// N1 + - - - + + + -
// N2 + + - - - + - +
// N3 + + + - + - - -
// N4 + - + - - - + +
// N5 + - - + + - - +
// N6 + + - + - - + -
// N7 + + + + + + + +
// N8 + - + + - + - -
// Gauss point coordinates
double xi = Gauss_Point[0];
double eta = Gauss_Point[1];
double zeta = Gauss_Point[2];
// Initialize Matrix with shape function derivatives in local coordinate system
MatrixST dN_dLocal = new MatrixST(3, 8);
// dN/d_xi
dN_dLocal.SetFast(0, 0, 1.0 / 8.0 * (-1 + eta + zeta - eta * zeta));
dN_dLocal.SetFast(0, 1, 1.0 / 8.0 * (1 - eta - zeta + eta * zeta));
dN_dLocal.SetFast(0, 2, 1.0 / 8.0 * (1 + eta - zeta - eta * zeta));
dN_dLocal.SetFast(0, 3, 1.0 / 8.0 * (-1 - eta + zeta + eta * zeta));
dN_dLocal.SetFast(0, 4, 1.0 / 8.0 * (-1 + eta - zeta + eta * zeta));
dN_dLocal.SetFast(0, 5, 1.0 / 8.0 * (1 - eta + zeta - eta * zeta));
dN_dLocal.SetFast(0, 6, 1.0 / 8.0 * (1 + eta + zeta + eta * zeta));
dN_dLocal.SetFast(0, 7, 1.0 / 8.0 * (-1 - eta - zeta - eta * zeta));
// dN/d_eta
dN_dLocal.SetFast(1, 0, 1.0 / 8.0 * (-1 + xi + zeta - xi * zeta));
dN_dLocal.SetFast(1, 1, 1.0 / 8.0 * (-1 - xi + zeta + xi * zeta));
dN_dLocal.SetFast(1, 2, 1.0 / 8.0 * (1 + xi - zeta - xi * zeta));
dN_dLocal.SetFast(1, 3, 1.0 / 8.0 * (1 - xi - zeta + xi * zeta));
dN_dLocal.SetFast(1, 4, 1.0 / 8.0 * (-1 + xi - zeta + xi * zeta));
dN_dLocal.SetFast(1, 5, 1.0 / 8.0 * (-1 - xi - zeta - xi * zeta));
dN_dLocal.SetFast(1, 6, 1.0 / 8.0 * (1 + xi + zeta + xi * zeta));
dN_dLocal.SetFast(1, 7, 1.0 / 8.0 * (1 - xi + zeta - xi * zeta));
// dN/d_zeta
dN_dLocal.SetFast(2, 0, 1.0 / 8.0 * (-1 + xi + eta - xi * eta));
dN_dLocal.SetFast(2, 1, 1.0 / 8.0 * (-1 - xi + eta + xi * eta));
dN_dLocal.SetFast(2, 2, 1.0 / 8.0 * (-1 - xi - eta - xi * eta));
dN_dLocal.SetFast(2, 3, 1.0 / 8.0 * (-1 + xi - eta + xi * eta));
dN_dLocal.SetFast(2, 4, 1.0 / 8.0 * (1 - xi - eta + xi * eta));
dN_dLocal.SetFast(2, 5, 1.0 / 8.0 * (1 + xi - eta - xi * eta));
dN_dLocal.SetFast(2, 6, 1.0 / 8.0 * (1 + xi + eta + xi * eta));
dN_dLocal.SetFast(2, 7, 1.0 / 8.0 * (1 - xi + eta - xi * eta));
return(dN_dLocal);
}