Possible
PhD projects for 2011
EPSRC funded PhD studentship available for Septermber, 2011.
Project: Finite element-immersed boundary method and its application to mitral valves
Description:
Mathematical modelling can help us to understand mitral valve (MV) diseases and their relationship with left ventricle (LV) functions. This project will study MRI based dynamic MV models, which will include important features such as fluid-structure interaction and nonlinear soft tissue modelling. The work will be carried out through an interdisciplinary collaboration with Prof. Boyce Griffith from NYU, Dr. Berry at the Cardiovascular Research Centre, and Prof. R. W. Ogden. The PhD student will work closely with a post-doc RA working on the the same project, and develop and apply the structure-based constitutive models for human MV and LV. The computations will be carried out using the object oriented C++ finite element version of immersed boundary method with adaptive mesh refinement (IBAMR: http://code.google.com/p/ibamr/). The computational models will be validated using MRI DENSE measurements of 3D temporal displacement and strain vector field of the human MV/LV in vivo. This will tie the computational simulations with clinical applications together and allow us to identify key elements and parameters in our models. The research in this proposal will address important questions about MV mechanics. Deeper understanding of the basic mechanisms of heart valve function could result in improved clinical therapies and therefore has clear social benefit. Ultimately, the project will contribute towards delay or prevent progression of valvular disease, for example, by modulating transvalvular blood flow or engaging pharmacological approaches to modify cardiac output and valve elasticity.
Further information: This is a fully funded, three and half years PhD studentship, including full UK/EU fees plus an annual stipend of 13,590 pounds. Funding for project-related expenses, including attendance at conferences, will also be available. The project is funded by the EPSRC.
Who is eligible:
This studentship is open to applicants from the UK and EU.
Applicants must have a minimum of a 2:1 BSc or BEng degree in Applied Mathematics, Engineering, Physics, or Computer Science.
Knowledge of continuum mechanics, numerical methods, and programming experience would be desirable.
How to Apply: In the first instance, applicants are required to send a CV and covering letter to
Prof. Xiaoyu Luo
E-mail: xiaoyu.luo@glasgow.ac.uk
Project: Instability and self-excited oscillations
in collapsible channels
Description:
Heart disease is the biggest killer in the world. To understand how to best maintain a healthy heart, it is essential for us to understand the mechanical functions of a normal heart. There is growing interest in the research on cardiac tissues in the heart, as reflected in the recent workshop on the ``Cardiac Physiome: Multi-scale and Multi-physics Mathematical Modelling Applied to the Heart'', at the Isaac Newton Institute in July 2009. However, to date, the biggest challenge and the weakest link in our understanding remains the mechanics of large-deformation, nonlinear fluid-structure interactions in the heart. It is for this purpose that we are developing EPSRC and BHF proposals, in conjunction with Prof. G. Smith at the Faculty of Biomedical & Life Sciences, and Dr. Colin Berry at the BHF Glasgow Cardiovascular Research Centre, with the aim of developing a mathematical model of the mechanics of the heart. The models will be built using the most-advanced structure-based nonlinear material model, as well the most-advanced computational immersed boundary methods IBAMR. These novel approaches will enable us to simulate the fluid-structure interactions inside the heart and will be world-leading. One of the principal questions of interest in flow in collapsible tubes is the mechanism of the self-excited oscillations. There are numerous physiological applications which are related to flow in collapsible tubes: arteries compressed by a sphygmomanometer cuff, intra-myocardial coronary blood vessels during systole, pulmonary blood vessels in the lung, the urethra during micturition, and the glottis during phonation. Many experiments with model systems of collapsible tubes in the laboratory have revealed a rich variety of self-excited oscillations. This has stimulated numerous theoretical and numerical studies. Most of the studies, however, have consisted of linear or nonlinear instability theories for flow in a long, parallel-sided channel, so in the basic state the steady flow is unidirectional and the elastic walls are planar.
Since 1994, Dr Luo and Prof. Pedley have embarked on a somewhat less idealized project by using numerical simulations based on a two dimensional model, i.e. flow in an asymmetric collapsible channel that is not parallel-sided in steady flow, where part of the upper wall of the channel is replaced by an elastic segment, and steady flow is assumed upstream. They have successfully shown that self-excited oscillations are initiated by the linear instabilities of the coupled fluid-structure interaction system (Luo & Pedley, 95,96,98,00). However, in their original approach, the wall model has been over simplified as a membrane without bending or stretching stiffness, hence the mechanism of the instability identified may only be of limited value.
In this project, instability of a new
fluid-beam model will be studied and the mechanisms of the instability identified
in the membrane mode will be carefully evaluated.
It is expected that the research will shed new light in the
mechanism of self-excited oscillations of flow in a realistic wall model.
This project is in collaboration with Prof. T.J. Pedley at DAMTP, Cambridge.
The following papers will be useful for the
project:
1. X.Y. Luo, Z.X. Cai, W.G. LI, T.J. Pedley,
The cascade structure of linear stabilities of flow in collapsible channels,
J. Fluid Mechanics, 600,45-76, 2008
3. X.Y.
Luo & T.J. Pedley, Flow limitation and multiple solutions in 2-D
collapsible channel flow. J. of Fluid Mechanics, 420, 301-324, 2000.
4. X.Y.
Luo & T.J. Pedley, The effects of the wall inertia on the 2-D collapsible
channel flow. J. of Fluid Mechanics,
363, 253-280, 1998.
5. T.J.
Pedley & X.Y. Luo, Modelling flow and oscillations in collapsible tubes.
J. of Theoret. Comp. Fluid Dynamics, 10, 277-294, 1998.
6. X.Y.
Luo & T.J. Pedley, A numerical simulation of unsteady flow in a 2-D
collapsible channel. J. of Fluid
Mechanics. 314, 191-225, 1996.
7. X.Y.
Luo & T.J. Pedley, A numerical simulation of steady flow in a 2-D
collapsible channel. J. of Fluids & Structures, 9, 149-174, 1995.
Project: The post-buckling
behaviour of constrained elastic tube under external pressure
Description.
Flow in collapsible
tubes has numerous physiological and clinical applications. When the
transmural (internal minus external) pressure, p, of a Starling Resistor tube
is decreased below a critical value, the structure buckles into a
non-axisymmetric cross-section, commonly a twin-lobed shape (mode 2).
After the collapse, the tube becomes highly compliant until the opposite walls
are in contact. It has been generally thought that the compliance
(dA/A)/dp of a tube under external loading decreases as the thickness of the tube
wall increases. However, in 3D-numerical [1] and experimental [2] studies
of tube collapse, it is observed that after a certain degree of collapse
(around A/A0 ~ 0.8, where A/A0 is the non-dimensional cross-sectional area of
the tube), the thicker-walled tube may become more compliant than the thinner
one.
This work aims to investigate this puzzling phenomenon by studying the asymptotic solution of an infinitely long tube with one end constrained. The impacts of the three-dimensional forces such as bending, shear and longitudinal tension on the post-buckling behaviour of the system will be analyzed. The results will be compared with those for the plane strain problem of tubes with different thicknesses. Key factors responsible for the compliance changes will be identified.
The following research papers will be useful for the project:
[1] Zhu Y F, Luo X Y, R W Ogden,
Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes
under axial loading and external pressure, (in press) Int. J. of Solids and
Structures. 2008.
[2] Marzo, A, Luo, XY, & Bertram,
CD, Three-dimensional collapse and steady flow in thick-walled flexible
tubes. Journal of Fluids and Structures 20, 817-835, 2005.
[3] Bertram, CD,The effects of wall thickness,
axial strain and end proximity on the pressure-area relation of collapsible
tubes. Journal of Biomechanics 20, 863-876, 1987.
[4]
Flaherty, JE, Keller, JB, & Rubinow, SI, Post buckling behavior of elastic
tubes and rings with opposite sides in contact. SIAM Journal of Applied
Mathematics 23, 446-455, 1972.
[5]
Haughton DM & Ogden RW, On the incremental equations in non-linear
elasticity - II. Bifurcation of pressurized spherical shells. Journal of the Mechanics and Physics of Solids 26, 111-138,
1978.
[6] Haughton DM & Ogden RW, Bifurcation of
inflated circular cylinders of elastic material under axial loading - I .
Membrane theory for thin-walled tubes. Journal of the Mechanics and Physics of Solids
27, 179-212, 1979.
[7] Haughton DM
& Ogden RW, Bifurcation of inflated circular cylinders of elastic material
under axial loading - II . Exact theory for thick-walled tubes. Journal of the Mechanics and Physics of Solids 27, 489-512, 1979.
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Project: Large
Mechanical modelling of sleep apnoea
Description:
Obstructive sleep apnoea is becoming a major health care topic. It affects 4 percentage of the adult population, and has many consequences such as excessive daytime sleepiness or hypertension. Obstructive sleep apnoea consists periodic episodes of soft tissue collapse within the upper airway during sleep. From a fluid mechanical point of view, the partial or the total collapse of the upper airway, as observed during obstructive sleep apnoea, can be understood as a spectacular example of fluid-walls interaction. While the most important parameters influencing this effect in-vivo are well known, this phenomenon is still difficult to model and thus to predict. In this PhD project, the aim is to develop and to validate a mechanical model for the flow induced collapse inside an elastic walled conduit.
The project will focus two crucial aspects. One is the description of the flow, and in particular to the movement of the point of flow separation associated with the deformation of the conduit [1]. Several theoretical descriptions (Boundary-layer method, RNSP) have been considered and tested against experimental data obtained on an in vitro replica of the human airways. However, these models suffer from some over-simplifications. Another aspect is related with the description of the deformable tissues. It is quite clear that, for clinical applications, distributed or lumped models are unable to reproduce accurately the behaviour of the human tissues. This is arguably one of the major limitations of all existing models [2-4]. In the current project, a nonlinear elastic solid mechanical model will be developed to explore the key impact of the soft tissue effect. .
This project is in collaboration with
Dr. Z. S. Liu, A*star Institute of High Performance Computing, Singapore,
and Dr. Annemie VAN HIRTUM GIPSA-Lab, Grenoble, France .
Move up to Prof. X. Y. Luo
X.Y. Luo© 2011