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

用于几何反演和形状优化的几何不拟合有限元方法

基本信息

批准号：
EP/P01576X/1
负责人：
Erik Burman
金额：
$ 60.09万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP01576X%2F1
关键词：
Geometrically unfitted finite element methods

项目摘要

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.

传统上，制造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，问题的复杂性使得不可能以这种方式处理所有约束或物理效应的影响。例如，考虑飞机起落架的最佳形状，它既能承受强大的气流，又能承受起飞和着陆的机械冲击，或者植入物，例如人造心脏瓣膜，它必须具有某些特性，但在实验中体内很难进行。在这种情况下，几种物理效应在塑造最佳设计方面相互竞争，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。必须重建未知形状或边界的另一种情况是当人们进行测量时，例如使用声波散射，并且目标是识别几何形状，这可能是子宫中的婴儿，隐藏在地下或地下的东西。海。在上述形状优化问题和逆重构问题中，人们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对象性能或与测量数据最吻合。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，首先必须将几何形状分解为更小的实体，例如立方体或四面体，即所谓的计算网格。在网格上，物理问题的解决方案是通过优化构建和演化的。然而，由于网格是由几何体定义的，随着几何体的变化，网格也必须发生变化。问题在于网格改变了数据结构以及计算方法的属性。由于网格划分成本高昂，而且传统上优化的不同构建模块是单独研究的，因此迄今为止很难设计出高效且可以评估结果质量的优化程序。在这个项目中，我们的目标是借鉴之前 EPSRC 资助的项目“多物理场接口问题的计算方法”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，仍然存在计算网格，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。这种方法提出了形状优化和几何逆向识别的整体视角，其中优化算法的所有不同步骤都可以显示在准确性和效率方面具有相似的属性，避免了“最薄弱环节”问题，其中一些表现不佳方法破坏了整个算法的性能。该项目中提出的方法足够通用，可以应用于非常大范围的问题，并且在数学上是合理的，因此可以使用数学分析来证明这些方法从准确性和效率的角度来看都是最优的。