Computational methods for multiphysics interface problems

Lead Research Organisation: University College London
Department Name: Mathematics

Abstract

Many problems in science and technology includes a fixed or moving boundary over which
two different physical systems are coupled. This situation is particularly common in systems in
medicine and biology, for instance: in the human arteries the fluid dynamics of the blood couples to the solid dynamics of the arterial wall, in rivers and estuaries the free flow couples to the porous media flow in the infiltrated river bed. Making accurate computational predictions of the evolution of such systems remains an important challenge for engineers and the accurate mathematical analysis of the associated methods is even more daunting. Indeed no known methods allow for rigorous mathematical analysis and many suffer from problems of stability or accuracy depending on the orientation of the interface. Numerical computations are most often performed on a computational mesh, that is a decomposition of the computational domain in a large number of small building blocks, so called elements. An important feature of the methods that we propose is that the interface may cut through the elements of the computational mesh, or in other words, the computational mesh does not need to fit the interface.

In multiphysics problems the situation is often complicated by the fact that the computational mesh may not be adapted to fit the interface, but the coupling of the two systems must take place independent of the mesh. This is the type of situation that we aim to study in the present project. New approaches will be designed for multiphysics couplings over moving interfaces. The mathematical methods will be designed so as to be robust and accurate and we will also explore the possibility to decouple the two systems for efficient time advancement. This may lead to very important savings in computational time, in particular for nonlinear problems.

Three important model cases will be considered: the coupling of two fluids of which one or both may be viscoelastic, the coupling of free flow and porous media flow and finally the coupling of a fluid and an elastic structure. All of these applications have important applications in the modeling of the human cardiovascular system, but also in a wide variety of other applications such as ink-jet printers, environmental science, chemical industry and so on.

Planned Impact

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. 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 the present framework where the mesh does not need to be fitted to interfaces and boundaries this meshing problem is circumvented, since it is always possible to work on a simple structured mesh.

The same double interfacing problem exists in engineering applications where the geometry is given by a CAD drawing. CAD-drawings are in general too imprecise to be used directly in computation or production. Moreover they do not use a representation of surfaces that is suitable for mesh design. Both these problems are solved in our framework, since we only need information on where the surfaces are in order to integrate over them. The representation is irrelevant, as is the precision, provided the mismatch between surfaces remain smaller than the mesh size.

There are countless applications where the present technology is of importance, but let us discuss two. Firstly, in the computation of cardiovascular flows, geometries are patient specific and obtained from experimental data (scans). Here meshing is very awkward, in particular since the experimental 3D image does not necessarily represent an outer domain, but an internal boundary separating for instance the fluid domain from the solid domain of the arterial wall. During the simulation the interaction between the fluid and the solid is extremely important and results in a displacement of the wall, requiring a displacement of the computational mesh if it is to remained fitted to the interface. The flow most also be simulated both in the artery and in the arterial wall resulting in the coupling of Navier-Stokes' equations and Darcy's equation across the interface. In our framework no mesh movement is necessary since the underlying approximation space and variational formulation adapts to fit the moving interface.

Another example in the same framework is the accurate simulation of heartvalves. The design and optimization of artificial heartvalves require accurate tools for simulation, here the valves will move with the fluid, but also open and close, leading to complex contact problems. The approach of re-meshing in order to follow the valves movements is practically impossible, due to the closing of the valve. In general remeshing will fail for problems experiencing some topological change or contact in the interface configuration. Computer based simulations of blood flows, in patient-specific geometries, can provide valuable information to physicians in order to enhance therapy planning. Such
simulations can also be a major ingredient in the design/optimization of medical devices. Moreover, the emerging interest in improving clinical diagnosis through model
personalization (i.e., solving inverse problems coupling clinical data and FSI models)
clearly demands further developments of efficient numerical methods.

Publications

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Burman E (2017) Galerkin least squares finite element method for the obstacle problem in Computer Methods in Applied Mechanics and Engineering

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Burman E (2017) A cut finite element method for the Bernoulli free boundary value problem in Computer Methods in Applied Mechanics and Engineering

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Burman E (2017) A cut discontinuous Galerkin method for the Laplace-Beltrami operator in IMA Journal of Numerical Analysis

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Burman E (2017) A cut finite element method with boundary value correction in Mathematics of Computation

Related Projects

Project Reference Relationship Related To Start End Award Value
EP/J002313/1 31/07/2012 31/12/2012 £420,615
EP/J002313/2 Transfer EP/J002313/1 01/01/2013 31/01/2016 £373,568
 
Description Computational method for the approximation of solutions to problems in mechanics typically use a computational mesh that organises the unknowns and defines the computational domain. Sometimes the geometry can be very difficult to mesh, this is for instance often the case for geometries obtained using MRI scans in biomedical imaging. Situations can also arise where the domain changes under the computation due to time evolution or due to some optimisation that depends on the domain shape. In such cases repeated re-meshing can be very costly and sometimes impossible due to topological changes, i.e. for instance formation of drops in a two phase flow. For cases where meshing is too expensive a well-known tool are so called fictitious domain methods. They allow computations on a fully structured mesh and conditions on boundaries or interfaces are imposed implicitly through the computational method. Traditionally these methods have suffered from the drawback of either strongly reduced accuracy, or problems with the stability. In the present project we have developed and analysed new techniques that make it possible to use fictitious domain methods without loss neither of accuracy nor of stability. The framework allows for general couplings between systems with different physical character as well as the solution of problems defined on surfaces and boundaries and the coupling between problems in the bulk and on the surface. The developed methods have been implemented in a computational code that is destined for the public domain.
Exploitation Route These results are being adopted by the engineering community for use in advanced computational mechanics applications such as fluid structure interaction, multi phase flows or shape optimisation.

A strong indication of this is that the emblematic publication of this project:

CutFEM: discretizing geometry and partial differential equations
E Burman, S Claus, P Hansbo, MG Larson, A Massing
International Journal for Numerical Methods in Engineering 104 (7), 472-501

has been cited 217 times on scholar google since its publication. It is now drawing close to 100 citations per year, which
is a very good score for a paper in Computational Mathematics. It is also listed as the most downloaded article from IJNME.
Sectors Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Education,Energy,Healthcare,Manufacturing, including Industrial Biotechology

 
Description Collaboration on fluid structure interaction methods 
Organisation The National Institute for Research in Computer Science and Control (INRIA)
Country France 
Sector Public 
PI Contribution In a collaboration with the research group of Miguel Fernandez at INRIA Rocquencourt we have developed fast solvers for fluid structure interaction and fluid structure interaction algorithms that are robust and accurate on unfitted meshes.
Collaborator Contribution In the recent joint publication, Burman, Fernandez, An unfitted Nitsche method for incompressible fluid-structure interaction using overlapping meshes, Computer Methods in Mechanics and Engineering, DOI: 10.1016/j.cma.2014.07.007, 2014 we proposed and analysed a framework for fluid-structure interaction on unfitted meshes. These methods are robust and accurate in spite of the non-conforming coupling. Both the case of thin and thick structures were included. The INRIA group continued the work on thin structures, whereas my group at UCL continued the work on the general framework and the non-symmetric penalty free Nitsche method for thick structures.
Impact Burman, Fernandez, An unfitted Nitsche method for incompressible fluid-structure interaction using overlapping meshes, Computer Methods in Mechanics and Engineering, DOI: 10.1016/j.cma.2014.07.007, 2014 Burman, Fernandez, Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling, International Journal of Numerical Methods in Engineering, DOI: 10.1002/nme.4607, 2013
 
Description Fenics/CutFEM 
Organisation Jönköping University
Department School of Engineering
Country Sweden 
Sector Academic/University 
PI Contribution We have participated in the development of a module to the software package FENICS that allows for automatic finite element computations using cut meshes. Susanne Claus has in collaboration with Andre Massing developed the cutFEM module that will soon be included in the Fenics package. On the theoretical side we have contributed with the development of a number of theoretical results that underpins this technology and the development of the theory necessary for the discretisation of partial differential equations on surfaces in collaboration with the partners. The first milestone that was finished this spring considered fictitious domain methods on cut meshes, discretisation of partial differential equations on surfaces on cut meshes and bulk-surface coupling on cut meshes. These results were reported in the review article (IJNME): CutFEM: Discretizing geometry and partial differential equations. and in the technical reports (arxiv) A Stabilized Cut Finite Element Method for the Three Field Stokes Problem A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator Cut Finite Element Methods for Coupled Bulk-Surface Problems Ongoing work considers the extension of these results to multi physics problems and discretisation of more complicated partial differential equations on surfaces discretised independently of the computational mesh. The former includes visco-elastic flow problems with internal boundaries and fluid structure interaction problems the latter includes the surface Helmholtz equation (as a model problem for vibrating structures) and the surface transport equation.
Collaborator Contribution Andre Massing had developed a beta version for this type of methods for his thesis using overlapping meshes and this work formed the basis of the development reported above. He made important contributions to the development of the cutFEM module and has taken the lead in the extension to multi physics coupling problems. Mats Larson has been research leader for the development of discretisation methods for partial differential equations on surfaces. Peter Hansbo has developed the initial research codes that were used for the discretisation on odes on surfaces and wrote the introductory chapters of the review paper. He also contributed to the theoretical development in all the papers that he co-authored.
Impact cutFEM/Fenics software module Burman, E. Claus, S. Hansbo, H., Larson, M., Massing, A. CutFEM: Discretizing geometry and partial differential equations. International Journal on Numerical Methods in Engineering. in press. Burman, E. Claus, S., Massing, A. A Stabilized Cut Finite Element Method for the Three Field Stokes Problem. arXiv:1408.5165 [math.NA] Burman, E., Hansbo, H., Larson, M., A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator, arXiv:1312.1097 [math.NA] Burman, E., Hansbo, H., Larson, M., Zahedi, S., Cut Finite Element Methods for Coupled Bulk-Surface Problems, arXiv:1403.6580 [math.NA]
Start Year 2012
 
Description Fenics/CutFEM 
Organisation Simula Research Laboratory
Country Norway 
Sector Academic/University 
PI Contribution We have participated in the development of a module to the software package FENICS that allows for automatic finite element computations using cut meshes. Susanne Claus has in collaboration with Andre Massing developed the cutFEM module that will soon be included in the Fenics package. On the theoretical side we have contributed with the development of a number of theoretical results that underpins this technology and the development of the theory necessary for the discretisation of partial differential equations on surfaces in collaboration with the partners. The first milestone that was finished this spring considered fictitious domain methods on cut meshes, discretisation of partial differential equations on surfaces on cut meshes and bulk-surface coupling on cut meshes. These results were reported in the review article (IJNME): CutFEM: Discretizing geometry and partial differential equations. and in the technical reports (arxiv) A Stabilized Cut Finite Element Method for the Three Field Stokes Problem A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator Cut Finite Element Methods for Coupled Bulk-Surface Problems Ongoing work considers the extension of these results to multi physics problems and discretisation of more complicated partial differential equations on surfaces discretised independently of the computational mesh. The former includes visco-elastic flow problems with internal boundaries and fluid structure interaction problems the latter includes the surface Helmholtz equation (as a model problem for vibrating structures) and the surface transport equation.
Collaborator Contribution Andre Massing had developed a beta version for this type of methods for his thesis using overlapping meshes and this work formed the basis of the development reported above. He made important contributions to the development of the cutFEM module and has taken the lead in the extension to multi physics coupling problems. Mats Larson has been research leader for the development of discretisation methods for partial differential equations on surfaces. Peter Hansbo has developed the initial research codes that were used for the discretisation on odes on surfaces and wrote the introductory chapters of the review paper. He also contributed to the theoretical development in all the papers that he co-authored.
Impact cutFEM/Fenics software module Burman, E. Claus, S. Hansbo, H., Larson, M., Massing, A. CutFEM: Discretizing geometry and partial differential equations. International Journal on Numerical Methods in Engineering. in press. Burman, E. Claus, S., Massing, A. A Stabilized Cut Finite Element Method for the Three Field Stokes Problem. arXiv:1408.5165 [math.NA] Burman, E., Hansbo, H., Larson, M., A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator, arXiv:1312.1097 [math.NA] Burman, E., Hansbo, H., Larson, M., Zahedi, S., Cut Finite Element Methods for Coupled Bulk-Surface Problems, arXiv:1403.6580 [math.NA]
Start Year 2012
 
Description Fenics/CutFEM 
Organisation Umea University
Country Sweden 
Sector Academic/University 
PI Contribution We have participated in the development of a module to the software package FENICS that allows for automatic finite element computations using cut meshes. Susanne Claus has in collaboration with Andre Massing developed the cutFEM module that will soon be included in the Fenics package. On the theoretical side we have contributed with the development of a number of theoretical results that underpins this technology and the development of the theory necessary for the discretisation of partial differential equations on surfaces in collaboration with the partners. The first milestone that was finished this spring considered fictitious domain methods on cut meshes, discretisation of partial differential equations on surfaces on cut meshes and bulk-surface coupling on cut meshes. These results were reported in the review article (IJNME): CutFEM: Discretizing geometry and partial differential equations. and in the technical reports (arxiv) A Stabilized Cut Finite Element Method for the Three Field Stokes Problem A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator Cut Finite Element Methods for Coupled Bulk-Surface Problems Ongoing work considers the extension of these results to multi physics problems and discretisation of more complicated partial differential equations on surfaces discretised independently of the computational mesh. The former includes visco-elastic flow problems with internal boundaries and fluid structure interaction problems the latter includes the surface Helmholtz equation (as a model problem for vibrating structures) and the surface transport equation.
Collaborator Contribution Andre Massing had developed a beta version for this type of methods for his thesis using overlapping meshes and this work formed the basis of the development reported above. He made important contributions to the development of the cutFEM module and has taken the lead in the extension to multi physics coupling problems. Mats Larson has been research leader for the development of discretisation methods for partial differential equations on surfaces. Peter Hansbo has developed the initial research codes that were used for the discretisation on odes on surfaces and wrote the introductory chapters of the review paper. He also contributed to the theoretical development in all the papers that he co-authored.
Impact cutFEM/Fenics software module Burman, E. Claus, S. Hansbo, H., Larson, M., Massing, A. CutFEM: Discretizing geometry and partial differential equations. International Journal on Numerical Methods in Engineering. in press. Burman, E. Claus, S., Massing, A. A Stabilized Cut Finite Element Method for the Three Field Stokes Problem. arXiv:1408.5165 [math.NA] Burman, E., Hansbo, H., Larson, M., A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator, arXiv:1312.1097 [math.NA] Burman, E., Hansbo, H., Larson, M., Zahedi, S., Cut Finite Element Methods for Coupled Bulk-Surface Problems, arXiv:1403.6580 [math.NA]
Start Year 2012
 
Title the cutFEM module for fictitious domain computations in Fenics 
Description The cutFEM library consists of a number of modules that allows the user to perform computations using unfitted finite element methods in two or three dimensions. The current version supports interface and boundary geometric data that is defined implicitly by a level set function. Both overlapping meshes and interfaces defined on a background mesh may be used. The software has been developed within the Fenics project in a fairly general setting making multi-physics models easy to include. 
Type Of Technology Software 
Year Produced 2014 
Open Source License? Yes  
Impact The development of the cutFEM software has allowed us to test and develop several new methods for multi physics coupling and we expect it to be adopted by the wider research community for the solution of optimisation problems, inverse problems or free surface problems. 
URL https://bitbucket.org/sclaus/dolfin-cutfem-1.3.0