Shape parameterization

The wall shape is controlled by a B-spline FFD block around the wall nodes (WallFFD, created by DG_windtunnel_model.set_up_sim() as model.ffd). Design variables change the block’s control points; the wall moves with them, and the volume mesh follows through IDWarp.

FFD block

ffd_shape sets the number of control points per parametric direction (2 entries in 2D, 3 in 3D) and ffd_degree the B-spline degree. By default the block is the axis-aligned bounding box of the wall. For a tapered or swept wing, pass ffd_block_corner_list to DG_windtunnel_model: two rectangular end surfaces (root and tip), each given by two opposite corners, e.g. [[(-0.1, 0., 0.32), (5.1, 0., -0.3)], [(7.4, 14.1, 0.32), (9.1, 14.1, -0.3)]].

Control-point motions

cp_motions moves control points individually. Its directions and bounds come from one dictionary keyed by spatial direction (0 = x, 1 = y, 2 = z):

dv_spec = {1: 0.01}                     # 2D: vertical motions within +/-0.01
dv_spec = {0: 0.02, 2: (-0.05, 0.10)}   # 3D: x symmetric, z asymmetric
dv_spec = {2: spanwise_linear_bounds(base_bound=0.1, scale_at_root=1.0, scale_at_ref=0.5, y_ref=14.1)}

A bound is a symmetric half-range, a (lower, upper) pair, or a callable of the control-point coordinates such as spanwise_linear_bounds, which varies the bound linearly along the span.

cp_dv_directions(dv_spec) gives the cp_coord_opt_idxs that DG_windtunnel_model needs.
build_cp_motion_dv(model.ffd.baseline_coefficients, dv_spec) builds the variable’s value, bounds and scaling (each direction’s box maps to \(\pm 1\)).
model.ffd.apply_cp_motions(coefficients, cp_motions, cp_coord_opt_idxs) adds the motions to the control points.

Sectional variables

SectionalShape(model.ffd, principal_dim) views the block as a stack of sections: principal_dim=0 in 2D (chordwise stations), 1 for a wing (spanwise stations). Variables are added with add(kind, values):

Kind	Operation	Dimensions
`camber`	translation along the vertical axis	2D
`thickness`	stretch along the vertical axis	2D, 3D
`twist`	rotation about the spanwise axis (radians)	3D
`chord`	stretch along the chordwise axis	3D
`sweep`	translation along x	3D
`dihedral`	translation along z	3D
`span`	translation along y	3D

A variable has one value per section, or fewer values, which are then B-spline coefficients of a profile over the sections. Create its bounds with build_sectional_dv(num_values, bounds).

Apply the sectional layer to the constant baseline first, then the control-point motions:

sectional = SectionalShape(model.ffd, principal_dim=1)
twist = shape_dvs.add('twist', build_sectional_dv(5, np.radians(2.)))
sectional.add('twist', twist)
coefficients = sectional.apply(model.ffd.baseline_variable())
coefficients = model.ffd.apply_cp_motions(coefficients, cp_motions, cp_coord_opt_idxs)

Design-variable set

ShapeDVSet collects all shape variables in declaration order; shape_dvs.flat_variable() is the vector passed to the mesh warper, flow model and postprocessor.

Geometric constraints

model.wall_geometry() returns a WallGeometry for the deformed wall:

Method	Quantity
`enclosed_measure(wall_displacement)`	enclosed area (2D) or volume (3D)
`add_thickness_stations(chord_fractions, span_stations=None)` then `thickness_ratio(coefficients, handle)`	deformed over baseline thickness at the stations
`add_planform_stations(span_stations)` then `planform(coefficients, handle)`	section chords, span and planform area (3D)

geometry = model.wall_geometry()
area = geometry.enclosed_measure(model.ffd.wall_displacement(coefficients))
area.set_as_constraint(lower=geometry.baseline_measure)
stations = geometry.add_thickness_stations(np.linspace(0.1, 0.9, 9))
geometry.thickness_ratio(coefficients, stations).set_as_constraint(lower=0.9)