Deformed Shape Optimisation for Elastic Bodies

弹性物体在外力作用下移动和改变形状。某些现象也有同样的效果，包括热，它会使物体膨胀。这种现象的一个著名的例子是双金属条，由两条金属条连接在一起形成的平板，其中一条比另一条更容易膨胀。在加热时，由于膨胀不匹配，带材向不容易膨胀的带材方向弯曲。这种倾向的弯曲是一个例子，一个控制领域，可以编程到材料。例如，人们可以考虑使一层或两层厚度可变，这意味着在带材的每个位置经历的弯曲程度将是不同的。自然要问的问题是，为了在带材中产生所需的形状，应该如何选择控制场。本文提出了一种通用算法来解决任意几何中的这类问题。该方法不限于上述热膨胀失配；这里的控制场是指与弹性耦合产生形状变化的任何非弹性效应，包括压电耦合和形状记忆效应等。所要解决的数学问题是一个pde约束优化问题，即通过改变弹性体中的位移场和控制场来实现某一函数的最小化，但必须满足位移场求解描述物体变形的弹性方程的约束。将这种方法发展为函数算法需要解决几个问题，这将需要综合弹性理论，优化理论和图像处理技术。也许最重要的是目标函数的选择，即哪个函数要最小化？人们希望变形物体的形状尽可能接近目标形状，因此建议的选择是在计算域中使用目标形状建立距离函数，该函数可用于确定弹性物体上的每个点到目标上最近点的距离。这种特殊的公式对于优化问题的数值计算具有有利的性质。该方案还要求确定最合适的数值方法来解决固有的非线性问题，从而使其成为一个非平凡的问题。实验限制导致进一步的目标。通常，优化问题需要被规范化，以使它们在数学上表现良好。通常的选择是Tikhonov正则化，它导致控制域在域上尽可能平滑地变化。然而，平滑控制域自然很难编程成弹性对象；通过将具有不同（常数）属性的组件粘合在一起，可以更直接地制造一个控制场，该控制场在域上分段恒定。在数学上，这是通过诉诸总变分正则化来实现的。该方案的最终目标是寻求多重稳定的结构，而不是简单的一种形状。例如，两端固定的金属尺子加热后会膨胀并扣上或扣下。更复杂的多稳定结构可以通过更复杂的控制域或物体几何来找到，并且该提案寻求找到涉及其计算的实际问题。最后，上面讨论的方面将与复合板的实际例子结合在一起，该复合板的纤维增强层可以定向以产生足以引起多稳定配置的控制场。本提案中开发的算法将发现光纤配置足以产生预定的多稳态配置，这将通过实验研究来验证本提案的方法。

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.