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

传统上，制造业的设计是由工程师通过结合计算、实验和经验的结果来完成的。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约束或物理效应的影响。例如，考虑飞机的起落架的最佳形状，其将维持强气流和起飞和着陆的机械冲击，或者植入物，例如人造心脏瓣膜，其必须具有某些特性，但体内实验非常难以进行。在这种情况下，几个物理效应在形成最佳设计中竞争，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，当人们进行测量时，例如使用声波散射，目标是识别几何形状，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，可以以偏微分方程的数学形式应用已知的物理定律，在优化框架中重复求解方程，并找到优化物体性能或最佳拟合测量数据的几何形状。然而，这是一项复杂的工作，程序的每一步都充满困难。为了进行计算机模拟，首先必须将几何体分解成更小的实体，比如立方体或四面体，即所谓的计算网格。在网格上构造物理问题的解并通过优化演化。但是，由于网格是由几何体定义的，因此随着几何体的变化，网格也必须变化。问题是，随着网格的变化，数据结构以及计算方法的属性。由于网格划分是昂贵的，并且传统上已经分别研究了优化的不同构建块，因此到目前为止，很难设计出有效的优化过程，并且可以评估结果的质量。在这个项目中，我们的目标是从以前的EPSRC资助的项目“多物理场接口问题的计算方法”的经验，我们设计的方法中的几何形状是独立的计算网格使用。在这个框架中，仍然有一个计算网格，但它不需要随着几何形状的变化而变化。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。这种方法提出了一个整体的角度来形状优化和逆向识别的几何形状，其中的优化算法的所有不同的步骤可以被证明具有类似的属性方面的准确性和效率，避免了“最薄弱环节”的问题，其中一些表现不佳的方法破坏了整个算法的性能。在该项目中提出的方法是足够的一般适用于一个非常大的范围内的问题和数学的声音，使数学分析可以用来证明，无论是从准确性和效率的角度来看，该方法是最佳的。

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.