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

Lead Research Organisation:
University College London

Department Name: Mathematics

### Abstract

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.

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.

### Planned Impact

Let us first discuss the impact of the proposed research in facilitating the step from experimental data or design to computation and then discuss the special case of optimization and inverse identification and explain why the proposed research could lead to major advancements. Finally we will discuss the impact of the proposed research on the scientific community.

An important bottle neck in the efforts to make cutting edge computational methods to bear on important

problems in applications is the interfacing that has to be done between on the one hand images and experimental data, for instance geometries of arteries or cerebral vascular structure, and on the other hand

the computational codes. An important disadvantage here is the need to go through the double interfacing

process of first creating a computational mesh, and then passing this mesh to the computational code. In particular if the meshing process has to be repeated due to changes in the geometry throught the computation.

Note that the meshing of complex geometries is a complicated process that can fail, or give rise to very poor

quality computational meshes. If the meshing step can be circumvented without compromising accuracy or stability this would be a huge step forward in the efforts to bring computation closer to applications, in particular in the context of optimization or inverse identification of geometries,

This is equally true in other engineering applications. Here computational design is typically performed on pre-CAD models and the geometric description must be flexible enough to allow for modifications through optimization or time evolution. In industry it is important that a records are kept of every computational geometry/mesh that has been used in the design process throughout the life cycle of the workpiece. The use of standard meshes makes such databases unnecessarily unwieldy and large. In the EPSRC workshop ``State of the art in simulation and design workshop'', with representatives from academia and industry on the 16th of July 2015, the efficient management of the geometry lifecycle was identified as one of the main challenges during the discussion with industrial representatives.

Some of the pilot research that led up to the EPSRC project EP/J002313/1 was carried out in the frame of

a masters thesis in collaboration with SKF in the Netherlands. From the collaboration it perspired that, in large scale industrial

computations, the design of simple and robust methods to impose complex geometries on simple grids without sacrificing neither efficiency nor robustness is of great importance and how to compute average quantities such as fluxes accurately. In the collaboration with SKF the final aim was to consider

the air-cooling of a rotating ball-bearing. Flow at approximately Reynolds number 2000 and heat conduction

should be simulated around and inside the rotating geometry. A full meshing of the ball bearing geometry

was considered unfeasible by the engineers.

Shape optimization is currectly becoming an important tool in manufacturing engineering. However the current state of the art relies either on low resolution fictitious domain methods or high complexity meshing/interpolation approaches. The present project proposes a more streamlined methodology, with a rigorous mathematical foundation, ensuring high accuracy and robustness without the need of repeated remeshing during optimization and high efficiency. The methods developed in the present project will allow for faster and more accurate computational design using more complex models than was previously realistic. The next generation software for shape optimization and inverse identification will without doubt be designed around this type of technology if it is available.

An important bottle neck in the efforts to make cutting edge computational methods to bear on important

problems in applications is the interfacing that has to be done between on the one hand images and experimental data, for instance geometries of arteries or cerebral vascular structure, and on the other hand

the computational codes. An important disadvantage here is the need to go through the double interfacing

process of first creating a computational mesh, and then passing this mesh to the computational code. In particular if the meshing process has to be repeated due to changes in the geometry throught the computation.

Note that the meshing of complex geometries is a complicated process that can fail, or give rise to very poor

quality computational meshes. If the meshing step can be circumvented without compromising accuracy or stability this would be a huge step forward in the efforts to bring computation closer to applications, in particular in the context of optimization or inverse identification of geometries,

This is equally true in other engineering applications. Here computational design is typically performed on pre-CAD models and the geometric description must be flexible enough to allow for modifications through optimization or time evolution. In industry it is important that a records are kept of every computational geometry/mesh that has been used in the design process throughout the life cycle of the workpiece. The use of standard meshes makes such databases unnecessarily unwieldy and large. In the EPSRC workshop ``State of the art in simulation and design workshop'', with representatives from academia and industry on the 16th of July 2015, the efficient management of the geometry lifecycle was identified as one of the main challenges during the discussion with industrial representatives.

Some of the pilot research that led up to the EPSRC project EP/J002313/1 was carried out in the frame of

a masters thesis in collaboration with SKF in the Netherlands. From the collaboration it perspired that, in large scale industrial

computations, the design of simple and robust methods to impose complex geometries on simple grids without sacrificing neither efficiency nor robustness is of great importance and how to compute average quantities such as fluxes accurately. In the collaboration with SKF the final aim was to consider

the air-cooling of a rotating ball-bearing. Flow at approximately Reynolds number 2000 and heat conduction

should be simulated around and inside the rotating geometry. A full meshing of the ball bearing geometry

was considered unfeasible by the engineers.

Shape optimization is currectly becoming an important tool in manufacturing engineering. However the current state of the art relies either on low resolution fictitious domain methods or high complexity meshing/interpolation approaches. The present project proposes a more streamlined methodology, with a rigorous mathematical foundation, ensuring high accuracy and robustness without the need of repeated remeshing during optimization and high efficiency. The methods developed in the present project will allow for faster and more accurate computational design using more complex models than was previously realistic. The next generation software for shape optimization and inverse identification will without doubt be designed around this type of technology if it is available.

### Organisations

### Publications

Barrenechea G
(2020)

*Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow*in Mathematical Models and Methods in Applied Sciences
Bertoluzza S
(2022)

*An A Posteriori Error Estimate of the Outer Normal Derivative Using Dual Weights*in SIAM Journal on Numerical Analysis
Betcke T
(2019)

*Boundary Element Methods with Weakly Imposed Boundary Conditions*in SIAM Journal on Scientific Computing
Betcke T
(2022)

*Hybrid coupling of finite element and boundary element methods using Nitsche's method and the Calderon projection*in Numerical Algorithms
Betcke T
(2022)

*Boundary Element Methods for Helmholtz Problems With Weakly Imposed Boundary Conditions*in SIAM Journal on Scientific Computing
Boiveau T
(2017)

*Fictitious domain method with boundary value correction using penalty-free Nitsche method*in Journal of Numerical Mathematics
Boulakia M
(2020)

*Data assimilation finite element method for the linearized Navier-Stokes equations in the low Reynolds regime*in Inverse Problems
Burman E
(2019)

*Numerical Mathematics and Advanced Applications ENUMATH 2017*
Burman E
(2021)

*Convergence Analysis of Hybrid High-Order Methods for the Wave Equation*in Journal of Scientific Computing
Burman E
(2018)

*Fully discrete finite element data assimilation method for the heat equation*in ESAIM: Mathematical Modelling and Numerical AnalysisDescription | In this project we have developed computational methods that can handle problems where different materials, with different material parameters, or even different physical models, are present in different parts of the computational domain. Heterogeneities on lower dimensional subdomains are also allowed, such as flow or transport in fractured porous media. To further resolve solution features due to singularities in the solution adaptive solution methods based on a posteriori error estimates have been proposed. These methods have then been used to develop methods that allows the optimisation of geometries, that is to find the shape of an objects that gives the best properties, or to identify an unknown, hidden object, using some measured data. We have proved the optimality of the proposed computational methods and shown how they can be used for optimisation/identification purposes in new and efficient ways. A software library has been designed that allows for the use of the computational methods designed, in a semi-automatic fashion. |

Exploitation Route | Most of the results in the present projects are taken to a level where it is relatively straightforward to adopt them in engineering practice, whereever relevant for the sectors indicated below. |

Sectors | Aerospace, Defence and Marine,Construction,Digital/Communication/Information Technologies (including Software),Education,Energy,Environment,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology |