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
Burman E
(2020)
Cut Bogner-Fox-Schmit elements for plates
in Advanced Modeling and Simulation in Engineering Sciences
Burman E
(2023)
The Augmented Lagrangian Method as a Framework for Stabilised Methods in Computational Mechanics
in Archives of Computational Methods in Engineering
Burman E
(2019)
Dirichlet boundary value correction using Lagrange multipliers
in BIT Numerical Mathematics
Burman E
(2021)
Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain
in Communications on Applied Mathematics and Computation
Burman E
(2018)
A simple finite element method for elliptic bulk problems with embedded surfaces
in Computational Geosciences
Burman E
(2022)
Unfitted hybrid high-order methods for the wave equation
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2020)
A stable cut finite element method for partial differential equations on surfaces: The Helmholtz-Beltrami operator
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2017)
A cut finite element method for the Bernoulli free boundary value problem
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2022)
A mechanically consistent model for fluid-structure interactions with contact including seepage
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2023)
Extension operators for trimmed spline spaces
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2019)
Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions
in Computer Methods in Applied Mechanics and Engineering
Moura R
(2022)
Gradient jump penalty stabilisation of spectral/ h p element discretisation for under-resolved turbulence simulations
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2020)
A stabilized cut streamline diffusion finite element method for convection-diffusion problems on surfaces
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2018)
Shape optimization using the cut finite element method
in Computer Methods in Applied Mechanics and Engineering
Burman E
(2019)
Cut finite elements for convection in fractured domains
in Computers & Fluids
Burman E
(2023)
Spacetime finite element methods for control problems subject to the wave equation
in ESAIM: Control, Optimisation and Calculus of Variations
Burman E
(2020)
A Nitsche-based formulation for fluid-structure interactions with contact
in ESAIM: Mathematical Modelling and Numerical Analysis
Burman E
(2019)
Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions
in ESAIM: Mathematical Modelling and Numerical Analysis
Burman E
(2021)
Space time stabilized finite element methods for a unique continuation problem subject to the wave equation
in ESAIM: Mathematical Modelling and Numerical Analysis
Capatina D
(2021)
Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation
in ESAIM: Mathematical Modelling and Numerical Analysis
Burman E
(2018)
Fully discrete finite element data assimilation method for the heat equation
in ESAIM: Mathematical Modelling and Numerical Analysis
Burman E
(2018)
A simple approach for finite element simulation of reinforced plates
in Finite Elements in Analysis and Design
Burman E
(2019)
A cut finite element method for elliptic bulk problems with embedded surfaces
in GEM - International Journal on Geomathematics
Burman E
(2021)
An unfitted hybrid high-order method for the Stokes interface problem
in IMA Journal of Numerical Analysis
Burman E
(2017)
A cut discontinuous Galerkin method for the Laplace-Beltrami operator
in IMA Journal of Numerical Analysis
Burman E
(2022)
A posteriori error estimates with boundary correction for a cut finite element method
in IMA Journal of Numerical Analysis
Burman E
(2018)
Augmented Lagrangian and Galerkin least-squares methods for membrane contact
in International Journal for Numerical Methods in Engineering
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)
Unique continuation for the Helmholtz equation using stabilized finite element methods
in Journal de Mathématiques Pures et Appliquées
Burman E
(2023)
Stability estimate for scalar image velocimetry
in Journal of Inverse and Ill-posed Problems
Burman E
(2021)
Error Estimates for the Smagorinsky Turbulence Model: Enhanced Stability Through Scale Separation and Numerical Stabilization
in Journal of Mathematical Fluid Mechanics
Burman E
(2022)
Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow
in Journal of Numerical Mathematics
Boiveau T
(2017)
Fictitious domain method with boundary value correction using penalty-free Nitsche method
in Journal of Numerical Mathematics
Burman E
(2021)
Comparison of Shape Derivatives Using CutFEM for Ill-posed Bernoulli Free Boundary Problem
in Journal of Scientific Computing
Burman E
(2021)
Convergence Analysis of Hybrid High-Order Methods for the Wave Equation
in Journal of Scientific Computing
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
Burman E
(2020)
A finite element data assimilation method for the wave equation
in Mathematics of Computation
Burman E
(2023)
Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
in Mathematics of Computation
Burman E
(2017)
A cut finite element method with boundary value correction
in Mathematics of Computation
Betcke T
(2022)
Hybrid coupling of finite element and boundary element methods using Nitsche's method and the Calderon projection
in Numerical Algorithms
Burman E
(2020)
A cut finite element method for a model of pressure in fractured media
in Numerische Mathematik
Burman E
(2022)
CutFEM based on extended finite element spaces
in Numerische Mathematik
Burman E
(2019)
Finite element approximation of the Laplace-Beltrami operator on a surface with boundary.
in Numerische mathematik
Burman E
(2022)
Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains
in Numerische Mathematik
Burman E
(2019)
A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime
in Numerische Mathematik
Burman E
(2018)
Stabilized CutFEM for the convection problem on surfaces
in Numerische Mathematik
Burman E
(2018)
Data assimilation for the heat equation using stabilized finite element methods
in Numerische Mathematik
Burman E
(2020)
A Fully Discrete Numerical Control Method for the Wave Equation
in SIAM Journal on Control and Optimization
Burman E
(2018)
Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
in SIAM Journal on Numerical Analysis
Burman E
(2019)
Primal Dual Mixed Finite Element Methods for Indefinite Advection-Diffusion Equations
in SIAM Journal on Numerical Analysis
Description | 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 |