Deformed Shape Optimisation for Elastic Bodies

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Elastic objects move and change shape in response to applied forces. Certain phenomena have the same effect, including heat, which causes objects to expand. A well-known example of this phenomenon is the bimetallic strip, a flat plate formed by attaching two metal strips together, one more expansible than the other. On the application of heat, the strip bends in the direction of the less-expansible strip, due to the mismatch in expansion. This propensity to bend is an example of a control field that may be programmed into the material. One could, for instance, consider making one or both layers variable in thickness, meaning that the degree of bending experienced at each location in the strip would be different. The natural question to ask is how the control field should be chosen in order to produce the desired shape in the strip. The present proposal introduces a general-purpose algorithm for such problems in arbitrary geometries. The method is not limited to thermal expansion mismatch as above; here a control field refers to any non-elastic effect that couples with elasticity to create a shape change, including piezoelectric coupling and shape memory effects, among others.The mathematical problem to be solved is a PDE-constrained optimisation, in which a certain function is to be minimised by varying the displacement field and control field in the elastic body, subject to the constraint that the displacement field solves the elastic equations describing the object's deformation. Developing this approach into a functional algorithm requires the resolution of several issues, which will necessitate the synthesis of techniques from elasticity theory, optimisation theory, and image processing. Perhaps most important is the choice of objective function, i.e. what function is to be minimised? One would like the shape of the deformed object to be as close as possible to a target shape, so the proposed choice is to use the target shape to set up a distance function in the computational domain, which can be used to determine the distance of each point on the elastic object to the nearest point on the target. This particular formulation has advantageous properties for the numerical calculation of the optimisation problem. The proposal also requires the determination of the most appropriate numerical method for solving the problem, which is inherently nonlinear, thus making it a non-trivial one to solve.Experimental restrictions lead to a further objective. Typically, optimisation problems need to be regularised in order to make them mathematically well-behaved. The usual choice is Tikhonov regularisation, which results in control fields that vary as smoothly as possible over the domain. Smooth control fields, however, are naturally difficult to program into an elastic object; it's much more straightforward to manufacture a control field which is piecewise constant over the domain, by gluing components with different (constant) properties together. Mathematically, this is achieved by appealing to total variational regularisation.The final goal of the proposal is to seek configurations which are multiply-stable, rather than simply one shape. For instance, a metal ruler fixed at both ends and heated will expand and buckle either up or down. More complicated multistable configurations can be found by more complicated control fields or object geometries, and the proposal seeks to find the practical issues involved in their calculation.Finally the aspects discussed above will be brought together with the practical example of a composite plate whose fibre-reinforced layers can be oriented to produce a control field sufficient to cause multistable configurations. The algorithm developed in this proposal will find the fibre configurations sufficient to cause a predetermined multistable configuration, which will be experimentally investigated to validate the approach of this proposal.