The element and its degrees of freedom ====================================== .. currentmodule:: bfsccylinder.bfsccylinder 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 :doc:`kinematics`: - :class:`bfsccylinder.bfsccylinder.BFSCCylinder`, Donnell-type kinematics, given by the ``bfsccylinder`` namespace; - :class:`bfsccylinder.bfsccylinder_sanders.BFSCCylinderSanders`, Sanders-type kinematics, given by the ``bfsccylinder.sanders`` namespace. The interpolation functions, the kinematics and the integration of each matrix are given in the Notes of :mod:`bfsccylinder.bfsccylinder` and :mod:`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: .. list-table:: :header-rows: 1 * - 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 :doc:`assembling`. 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 :class:`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]``. :func:`bfsccylinder.utils.assign_constant_ABD` copies the same values, e.g. of :func:`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 :func:`.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 :doc:`assembling`. ``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 :func:`.update_M` integrate over the element with ``nint`` by ``nint`` Gauss-Legendre points, the products of the one-dimensional points and weights of :func:`bfsccylinder.quadrature.get_points_weights`, given to each function. The number of points that integrates each matrix exactly is given in the Notes of :mod:`bfsccylinder.bfsccylinder`.