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

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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.