Discrete Approach (DA)

First a Discrete Approach is adopted, using a subset of the finite element grid point coordinates in the structural model to define the regions for the application of design variables. To allow for a wide range of possible shapes, the cross-sectional areas of twelve sections of the spinal beam are varied by parameterising the thickness distribution along its length (as depicted in Figure 3.3 and note that the width of the spine is held fixed at 8 mm). The resulting design is used to achieve aerodynamic-related shapes, e.g., a four-digit NACA camber definition. An equidistant finite element discretisation scheme is chosen here in such a manner as to provide equidistant mapping of the parameterised design space. This mapping provides a means for controlled displacement field of the spinal structure, which becomes more significant in the vicinity of the maximum camber of the target shape.

Some preliminary results obtained during global optimisation show the variation of pa- rameterised space with structured configuration, as depicted in Figure 3.1 (due to the symmetry, only the upper half of the parameterised spinal structure is shown).

0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 x, mm h, mm 12 section discretisation

Figure 3.1: _{Preliminary semi-thickness distribution, single-shape morphing beam,}

DA. 8.24 8.25 8.26 8.27 8.28 8.29 8.3 8.31 8.32 0 100 200 300 400 500 600 700 800 900 1000 h 9 mm f data points initial solution

Figure 3.2: _{Sensitivity analysis of a 12 section parameterised beam, DA.}

During this initial study, very abrupt transitions between sections were produced that may clearly alter the displacement field required. Therefore, a sensitivity analysis was performed, using Dynamic Hill Climbing. This study outlines the feasibility of the mapping and also the bounds of the parameterised design space. The analysis can be carried out based on the optimal set of design variables and may be used to infer changes in the solution as a result of some parameter variations or constraints, without re-optimising the entire system, Braun et al. (1993). The results (objective function) can be plotted systematically (see Figure 3.2) by varying the parameters through user

selected ranges of values, while other variables are held constant, in order to study the impact on the cost function. The sensitivity analysis is performed here on the 9th design parameter, since it governs the continuum deflection state of the problem and can alter the convergence trend of the functional (see Figure 3.2). Setting the bounds of the local changes of the parameter under investigation, the system not only shows no improvement over the cost function, but exhibits a behavior akin to shear locking (n.b., first-order elements are prone to shear locking for thin elements) and significantly alter the solution vector. Numerical instabilities may also be encountered in systems where:

• adjacent elements have widely varying stiffness and there is a source of insufficient information in the original stiffness coefficients (e.g., a stiff region supported by a much more flexible region);

• large rigid body rotations occur in the system without any significant strain;

• global stiffness becomes singular at a limit point, therefore the transition to post limit is stopped and the Newton-Raphson nonlinear solver under-performs, indi- cating a diminishing of the accuracy of the solution in the vicinity of a critical point.

To overcome such numerical instabilities for this model, a Kirchhoff constraint enforces well-posed numerical solution to anticipate the approximate field. This translates into a choice of C1displacement type finite elements, i.e., Euler-Bernoulli formulation. Such instabilities are also mitigated by augmented eccentricity from 0.1% to 0.3% of chord, without altering the scope of global shape control proposed in this thesis. These insta- bilites can also originate in the numerical procedure employed, i.e., Newton-Raphson, with slow rate convergence in the vicinity of critical points. It is noted that difficul- ties in detecting and traversing critical points have been a challenge for post-buckling and post-collapse analyses since early 1970’s and have led to the development of the arc-length control and alternating load-displacement control methods for handling cases where the response is unstable during part of its loading history (further details can be found in section 2.5).

A second mapping scheme is proposed in Figure 3.3, and the parameterised design space is augmented with two more variables, to yield a more gradual section transformation in regions of high curvature. After 60 generations of the GA, each of 50 members, and alternatively, 300 iterations of SA, Figure 3.4 shows the best designs in terms of deflected shapes. These deflections are governed by the optimised shapes shown in Figure 3.5. The optimised design achieved with the GA presents reduced variations in stiffness of adjacent elements. Consequently, the metric employed here to numerically analyse the fitness of the deflected states, viz. root mean square error (RMSE), in comparison with the target, shows better agreement than with SA (see table 3.1). Note however that

0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 x, mm h, mm old discretization new discretization

Figure 3.3: _{Augmented mapping scheme, semi-thickness distribution, single-shape}

morphing beam, DA.

both the resulting structural geometries lack smoothness and runs the risk of limiting the manufacturability, as pointed out by Braibant and Fleury (1984).

Figure 3.4: _{The deflected states of the optimised single-shape morphing beams, 1%}

chord, stochastic search with GA and SA, DA.