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The numerical simulation of deformable solids has many applications in fluid-solid interaction such as in aerospace structures and civil and marine engineering. However, the area where it can have its greatest application is in medical engineering in which it will be possible to simulate the movement of tube like structures such as the oesophagus, intestines, bile duct, fallopian tube, uterus, urethra and blood vessels etc. It will also be possible to simulate the movement and interaction of blood cells and lead to the design of better stents. With the support of this project, we established computational model for general fluid-solid interactions by combining a finite volume method based fluid solver, a finite-discrete element based solver and an immersed boundary technique to serve as fluid-solid interfaces. This model was used for the numerical simulation of the movement of individual blood cells in flows, as well as the deformation and interaction of red blood cells.

The work performed is as the follows.

(a) We have modified and developed our in-house computational fluid dynamics code--CgLes so that solids within the fluid can have deformable boundaries. To make the code suitable for simulation of blood flow with multiple red blood cells, which is usually laminar flow with very low Reynolds numbers, a second order accuracy implicit time stepping scheme was introduced and programmed. Verifications against theoretical solutions were carried out for laminar flows to validate the simulation results. Immersed boundary method (IBM) was used to represent the moving and deforming boundaries inside the fluid.

(b) We have coupled the CgLes code for fluid and the DEM code for transport and deformation of solids. A complete fluid-structural interaction (FSI) code package was developed to model flow with deformable discrete coarse particles and their movement and interaction with themselves and the fluid. Besides the combination with CgLes, other new contributions to the DEM code include: we have developed an adhesion model to account for the aggregation of red blood cells and incorporated it into the DEM code; because all red blood cells share the same topography (node connectivity specified for finite elements), a particle based data structure was developed, which saves computer memory and improves efficiency; a volume conservative method was developed in DEM to keep the volume of individual red blood cells conservative by applying pressure inside the red blood cell membrane.

(c) We have carried out necessary modifications, optimisation and parallelisation of the coupled code in order to achieve maximum efficiency. The original DEM code was serial, which can only run on one core of a computer. We have successfully made it fully parallelized based on spatial decomposition with only block hosted cells simulated by local processor. The immersed boundary method was also fully parallelized. Now it can run on the UK's high-end computing resource—HECToR with thousands of red blood cells involved.

(d) We have carried out a series of verifications step by step for both fluid and solids including: deformation of red blood cell was verified against stretch test of human red blood cell in laboratory; adhesion of red blood cells was verified against an experiment on separation of two red blood cells in shear flow; fluid-solid interaction was verified against an experiment of flow induced vibration of an elastic sheet.

(e) We have investigated the influence of different flow parameters on the movement, interaction and deposition of the particles.

(f) We have carried out large scale simulations of red blood cells and its behaviour including deformation, aggregation, cell-free-layer formation were investigated.

(g) We have carried out simulation of the red blood cell motion within peristaltic pumping by wavelike contractions.

The main results achieved so far are as the follows.

(a) A computational package for general purpose fluid-solid interactions was well established. Adhesion force was introduced into the model to simulate the adhesion of individual red blood cells when contact occurs. The established model was widely verified with a series of experimental data.

(b) The computational code developed was fully parallelized based on spatial decomposition, which enable the program to run on the largest super computers of the UK. An advanced block adaptive load balancing scheme was developed to achieve the highest performance on supercomputers.

(c) The numerical simulation shows that in flow with high shear rate (100/s) showed no major aggregation and the cells tended to move independently under entrainment by the flow and were uniformly distributed in space. However, when the shear rate decreased from 100/s to 10/s, significant aggregation occurs. The RBCs aggregated face-to-face to form coin-stack-like structures (rouleaux).

(d) The results from the simulation show that the shear rates have significant influence on the velocity profiles of the flow. Owing to the existence of RBCs, all the simulation results show velocity profiles deviating from the linear profile corresponding to a single phase Newtonian flow.

(e) The appearance of large scale aggregation clusters contributed greatly to the bi-phasic velocity profile by forming a plug-flow like status.

(f) Statistical analysis based on simulation results show that at high shear rates (γ=100/s, γ=60/s), the angle distribution has an obvious peak at around 15-20º and the majority of the cells adopt an angle less than 30ºs, indicating that most of the RBCs are moving with a more-or-less fixed angle to the top plate.

(g) RBC aggregation and the formation of the cell-free layer (CFL), may significantly affect the rheological characteristics of blood in the circulatory systems as the flow and the viscosity of the fluid within the CFL region is lower than this in the core of the flow.

The simulation of deformable solids in laminar and turbulence flow will greatly help us understand the nature of many blood flow problems and one of the potential outcomes of the project which will be of great benefit to both the EU and China will be better stent design. It has been reported by the Public Health Portal of the EU that cardiovascular diseases (CVD) are the largest cause of death in the EU and account for approximately 40% of deaths or 2 million deaths per year. Our long-term objectives are to develop upon the numerical simulation models developed into a world-class model for the diagnosis and treatment of cardiovascular diseases as well as analyses of biofluids which will have implications for instance in prenatal care and genito-urinary medicine.