The element and its degrees of freedom#

The single-curvature BFSC element is a four-node rectangle of the developed mid-surface of a cylinder of radius R, with \(x\) the axial coordinate and \(y = R \theta\) the arc length along the circumference, of sides lex along \(x\) and ley along \(y\). The displacements \(u\), \(v\) and \(w\) are interpolated with products of cubic Hermite polynomials, whose nodal values are the degrees of freedom below. Two classes share the same nodes, degrees of freedom and attributes, and differ in their kinematics, see Donnell-type and Sanders-type kinematics:

The interpolation functions, the kinematics and the integration of each matrix are given in the Notes of bfsccylinder.bfsccylinder and bfsccylinder.bfsccylinder_sanders.

Node ordering#

The nodes are numbered counterclockwise from the corner with the smallest \(x\) and \(y\), as the CQUAD4 element of Nastran:

Node

Natural coordinates

Position

1

\(\xi = -1, \eta = -1\)

\((x_1, y_1)\)

2

\(\xi = +1, \eta = -1\)

\((x_1 + lex, y_1)\)

3

\(\xi = +1, \eta = +1\)

\((x_1 + lex, y_1 + ley)\)

4

\(\xi = -1, \eta = +1\)

\((x_1, y_1 + ley)\)

with \(x = x_1 + lex (\xi + 1)/2\) and \(y = y_1 + ley (\eta + 1)/2\). The element knows neither \(x_1\) nor \(y_1\): its matrices depend on lex, ley, R, its properties and its degrees of freedom, not on its position.

Degrees of freedom of a node#

Each node has DOF = 10 degrees of freedom, at the positions c to c + 9 of the global vectors and matrices, with c the attribute c1, c2, c3 or c4 of the element for the nodes 1 to 4:

Position

Degree of freedom

Dimension

c + 0

\(u\)

length

c + 1

\(u_{,x}\)

dimensionless

c + 2

\(u_{,y}\)

dimensionless

c + 3

\(v\)

length

c + 4

\(v_{,x}\)

dimensionless

c + 5

\(v_{,y}\)

dimensionless

c + 6

\(w\)

length

c + 7

\(w_{,x}\)

dimensionless

c + 8

\(w_{,y}\)

dimensionless

c + 9

\(w_{,xy}\)

one over length

\(u\) is the axial displacement, \(v\) the circumferential displacement and \(w\) the normal displacement, positive outwards, away from the axis of the cylinder. The cross derivative \(w_{,xy}\) is a degree of freedom of \(w\) only; \(u\) and \(v\) have no cross derivative. With c = DOF*i for the node of index i, the usual choice of the tests, the slice u[6::DOF] of a global vector gives \(w\) at all the nodes, u[0::DOF] gives \(u\), and so on, which is how the boundary conditions and the loads are applied, see Assembling a model.

The 40 degrees of freedom of the element, e.g. its attribute u, are those of the nodes 1, 2, 3 and 4 in sequence, 10 per node in the order of the table.

Continuity between elements#

Along an edge, \(u\), \(v\), \(w\) and their derivatives normal to the edge depend only on the degrees of freedom of the two nodes of that edge, such that \(u\), \(v\), \(w\) and their first derivatives are continuous between elements that share an edge. Along an edge \(x\) = constant, for instance, \(w\) is interpolated from \(w\) and \(w_{,y}\) of the two nodes, and the normal slope \(w_{,x}\) from \(w_{,x}\) and \(w_{,xy}\).

Properties of the element#

The constructor takes nint, the number of integration points in each direction, the size of the points and weights later given to the functions, e.g. BFSCCylinder(nint). Before the functions are called, the attributes below are set for each element, see bfsccylinder.bfsccylinder.BFSCCylinder for their definitions:

  • c1, c2, c3, c4: positions of the nodes in the global vectors;

  • lex, ley and R: lengths of the sides and radius, R nonzero;

  • A11, …, D66: the laminate stiffnesses at each of the nint by nint integration points, arrays of shape (nint, nint), the entry [i, j] at xi = points[i] and eta = points[j]. bfsccylinder.utils.assign_constant_ABD() copies the same values, e.g. of composites.laminated_plate(), to all the points; they can also be set point by point, e.g. for fiber angles that vary along \(x\), as in tests/test_linear_buckling_FW_VAT_P_VT.py;

  • h and rho: thickness and density, used only by the mass matrix of update_M(), which with the Sanders-type kinematics also uses R;

  • init_k_KC0, init_k_KCNL, init_k_KG, init_k_M, init_k_KCfollower: positions of the element in the sparse arrays of each matrix, see Assembling a model.

n1, n2, n3 and n4 are identification numbers for the user, not read by the functions. The methods update_Su, …, update_Sw_y, update_Bm and update_Bb of the element evaluate the interpolation functions and the strain operators at a point \((\xi, \eta)\), e.g. to integrate a load or to recover the strains; the update_* functions of the modules do not use them.

Integration points#

All the functions except update_M() integrate over the element with nint by nint Gauss-Legendre points, the products of the one-dimensional points and weights of bfsccylinder.quadrature.get_points_weights(), given to each function. The number of points that integrates each matrix exactly is given in the Notes of bfsccylinder.bfsccylinder.