# Computational methods for multiphysics interface problems

Lead Research Organisation:
University of Sussex

Department Name: Sch of Mathematical & Physical Sciences

### 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.

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.

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

Barrenechea GR
(2017)

*Edge-based nonlinear diffusion for finite element approximations of convection-diffusion equations and its relation to algebraic flux-correction schemes.*in Numerische mathematik
Boiveau T
(2017)

*Fictitious domain method with boundary value correction using penalty-free Nitsche method*in Journal of Numerical Mathematics
Boiveau T
(2016)

*A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity*in IMA Journal of Numerical Analysis
Burman E
(2014)

*Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part II: Hyperbolic Equations*in SIAM Journal on Scientific Computing
Burman E
(2013)

*Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling*in International Journal for Numerical Methods in 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
(2016)

*Local CIP Stabilization for Composite Finite Elements*in SIAM Journal on Numerical Analysis
Burman E
(2013)

*Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces and boundaries*in Numerical Methods for Partial Differential Equations
Burman E
(2017)

*Error estimates for transport problems with high Péclet number using a continuous dependence assumption*in Journal of Computational and Applied Mathematics
Burman E
(2015)

*Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation*in Mathematical Models and Methods in Applied Sciences
Burman E
(2015)

*Robust error estimates for stabilized finite element approximations of the two dimensional Navier-Stokes' equations at high Reynolds number*in Computer Methods in Applied Mechanics and Engineering
Burman E
(2014)

*CutFEM: Discretizing geometry and partial differential equations*in International Journal for Numerical Methods in Engineering
Burman E
(2013)

*Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part I: Elliptic Equations*in SIAM Journal on Scientific Computing
Burman E
(2015)

*A monotonicity preserving, nonlinear, finite element upwind method for the transport equation*in Applied Mathematics Letters
Burman E
(2014)

*An unfitted Nitsche method for incompressible fluid-structure interaction using overlapping meshes*in Computer Methods in Applied Mechanics and Engineering
Burman E
(2016)

*Full gradient stabilized cut finite element methods for surface partial differential equations*in Computer Methods in Applied Mechanics and Engineering
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
(2014)

*Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem*in ESAIM: Mathematical Modelling and Numerical Analysis
Burman E
(2015)

*Cut finite element methods for coupled bulk-surface problems*in Numerische Mathematik
Burman Erik
(2014)

*Projection Stabilization of Lagrange Multipliers for the Imposition of Constraints on Interfaces and Boundaries*in NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
Cantin P
(2016)

*A vertex-based scheme on polyhedral meshes for advection-reaction equations with sub-mesh stabilization*in Computers & Mathematics with Applications
Pueyo JI
(2016)

*New Peptides Under the s(ORF)ace of the Genome.*in Trends in biochemical sciencesDescription | 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 | The methods are implemented in a software library that will soon be put in the public domain and can then be used by engineers and scientists. Other engineering groups are also implementing our theoretical results into their computational codes. |

Sectors | Aerospace, Defence and Marine,Electronics,Energy,Environment,Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology |

Description | The findings have been used in methods implemented in several computational codes. One of them is the developed cutFEM module for FENICS, that will be put in the public domain soon. |

First Year Of Impact | 2014 |

Sector | Digital/Communication/Information Technologies (including Software),Education |

Impact Types | Economic |

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 |