Geometrically unfitted finite element methods for inverse identification of geometries and shape optimization

传统上，制造中的设计是由工程师结合计算、实验和经验的结果进行的。然而，在某些情况下，由于问题的复杂性，不可能以这种方式处理所有制约因素或实际影响的影响。例如，考虑飞机起落架的最佳形状，该起落架既能承受强劲的空气流动，又能承受起飞和降落时的机械冲击，或者植入物，例如人造心脏瓣膜，它必须具有某些特性，但在体内进行实验非常困难。在这种情况下，几种物理效应竞争形成最佳设计，经典方法可能过于简单化，并以不必要的昂贵或低效设计的形式导致次优结果。另一种需要重建未知形状或边界的情况是，当一个人进行测量时，例如使用声波散射，目标是识别一个几何图形，这可能是子宫中的婴儿，隐藏在地下或海洋中的东西。在上述形状优化问题和逆重建问题中，可以应用偏微分方程数学形式的已知物理定律，在优化框架中重复求解方程，并找到优化对象性能或与测量数据最佳匹配的几何形状。然而，这是一项复杂的工作，程序的每一步都充满了困难。为了进行计算机模拟，首先要将几何图形分解成更小的实体，比如立方体或四面体，也就是所谓的计算网格。在网格上构造物理问题的解，并通过优化进行进化。但是，由于网格是由几何体定义的，因此随着几何体的更改，网格也必须更改。问题是，随着网格的变化，数据结构以及计算方法的属性也会发生变化。由于网格化是昂贵的，而且优化的不同组成部分传统上是分开研究的，因此到目前为止，很难设计出有效的优化程序，并且有可能评估结果的质量。在这个项目中，我们的目标是借鉴以前EPSRC资助的项目“多物理界面问题的计算方法”的经验，在该项目中，我们设计了几何形状独立于所使用的计算网格的方法。在这个框架中，仍然有一个计算网格，但它不需要随着几何体的变化而改变。相反，所有的几何信息都内置于求解描述物理模型的方程的计算方法中。该方法提出了形状优化和几何的逆识别的整体观点，其中可以证明优化算法的所有不同步骤在精度和效率方面具有相似的特性，从而避免了“最弱环节”问题，在该问题中，一些执行不佳的方法破坏了整个算法的性能。该项目中提出的方法具有足够的通用性，可以应用于非常广泛的问题，并且在数学上是合理的，因此可以用数学分析来证明这些方法从精度和效率的角度都是最优的。

Design in manufacturing has traditionally been made by engineers, by combining results of computation, experiments and experience. In certain situations however the complexity of the problem is such that it is impossible to handle the effect of all the constraints or physical effects this way. Consider for instance the optimal shape of a landing gear of an aircraft that will both sustain strong air flow and the mechanical impacts of take off and landing, or an implant, for instance an artificial heart valve, that must have certain properties, but where experiments in vivo are very difficult to carry out. In such cases where several physical effects compete in shaping the optimal design the classical approach may be too simplistic and lead to suboptimal results in the form of unnecessarily costly or inefficient designs. Another situation where an unknown shape or boundary has to be reconstructed is when one has measurements, for instance using acoustic wave scattering, and the objective is to identify a geometry, this could be a baby in the womb, something hidden under ground or in the sea. Both in the above shape optimization problem and in the inverse reconstruction problem, one may apply known physical laws in the mathematical form of partial differential equations, solve the equations repeatedly in an optimization framework and find the geometry that either optimizes the performance of the object or best fits with the measured data. This however is a complex undertaking, where every step of the procedure is fraught with difficulties. To make the computer simulation, first of all the geometry has to be decomposed into smaller entities, let us say cubes or tetrahedra, the so-called computational mesh. On the mesh the solution of the physical problem is constructed and evolved through the optimization. However since the mesh is defined by the geometry, as the geometry changes, so must the mesh. The problem is that with the mesh changes the data structures as well the properties of the computational methods. Since meshing is costly and the different building blocks of the optimization traditionally have been studied separately it has so far been difficult to design optimization procedure that are efficient and where it is possible to assess the quality of the result. In this project our aim is to draw from the experiences of a previous EPSRC funded project "Computational Methods for Multiphysics Interface Problems" where we designed methods in which the geometries were independent of the computational mesh used. In this framework, there still is a computational mesh, but it does not need to change as the geometry changes. Instead all the geometry information is built in to the computational methods that solves the equations describing the physical model. This approach proposes a holistic perspective to shape optimisation and inverse identification of geometries, where all the different steps of the optimisation algorithm can be shown to have similar properties with respect to accuracy and efficiency, avoiding the "weakest link" problem, where some poorly performing method destroys the performance of the whole algorithm. The methods proposed in the project are sufficiently general to be applied to a very large range of problems and mathematically sound so that mathematical analysis may be used to prove that the methods are optimal both from the point of view of accuracy and efficiency.