Numerical investigation of lymph flow through pumping lymphatics

This thesis aims at numerically modelling the lymphatic network at different scales. A failure of this network results in an accumulation of fluid in the area concerned, which can lead to lymphoedema.With the objective to understand this network and how it develops its ca-pacity to move lymph, we are interested in a comparison between different description and models on the lymphatic network from the literature. We first study the discrete approach in zero dimension and in one dimension. In addi-tion, the different constitutive equations observed in the literature are detailed and analysed. In order to understand the interactions between the basic motile elements of the lymphatic network, different methods for coupling calculations in fluid-structure simulation are presented. Then, with the help of different recent articles, we compare different approaches and geometries for the study of a lymphangion.In this zero-dimensional approach, a new numerical formulation is used for the calculation of lymph flow in the collecting network. Equations of this model are detailed in this document, and the number of parameters generally used in the constitutive equation is reduced. Moreover, these equations allow for a variable contraction frequency depending on the load imposed by the boundary condition applied. Different specific cases such as divergent and convergent bifurcations, elementary units of a network are first studied. Furthermore, lymphangions at the end of a channel appear to deliver more pumping energy than the initial ones. The results from the simulations are compared with experimental data. Finally, a specific and realistic network geometry extracted from an anatomical drawing of a leg is used to simulate the model and show complex synchronization behaviours between lymphangions. Three different regimes of synchronization between lymphangion in a channel are identified.For the last chapter, a two-dimensional model of a lymphangion is proposed, which will then be used to study the behaviour of valves, lymphocytes and walls. The operation of the multi-physics fluid-structure code is explained, it is based on a method called: Immersed Structural Potential Method (ISPM). Initially, the equations of fluid and solid mechanics are introduced, how they are coupled, as well as the details of their implementation. Then, we study the behaviour of a group of lymphocytes in the lymphatic channel using a fluid-structure interaction code. Using the geometry of a lymphangion, we compare the displacement of the lymphocytes in different cases, first with or without valve, pulsation of the fluid and then the moving walls. We observe that a poiseuille flow is maintained across the range of lymphocite density considered here. This type of flow is similar to that of red blood cells in a blood stream

Asynchronous Stabilisation and Assembly Techniques for Additive Multigrid

Multigrid solvers are among the best solvers in the world, but once applied in the real world there are issues they must overcome. Many multigrid phases exhibit low concurrency. Mesh and matrix assembly are challenging to parallelise and introduce algorithmic latency. Dynamically adaptive codes exacerbate these issues. Multigrid codes require the computation of a cascade of matrices and dynamic adaptivity means these matrices are recomputed throughout the solve. Existing methods to compute the matrices are expensive and delay the solve. Non- trivial material parameters further increase the cost of accurate equation integration. We propose to assemble all matrix equations as stencils in a delayed element-wise fashion. Early multigrid iterations use cheap geometric approximations and more accurate updated stencil integrations are computed in parallel with the multigrid cycles. New stencil integrations are evaluated lazily and asynchronously fed to the solver once they become available. They do not delay multigrid iterations. We deploy stencil integrations as parallel tasks that are picked up by cores that would otherwise be idle. Coarse grid solves in multiplicative multigrid also exhibit limited concurrency. Small coarse mesh sizes correspond to small computational workload and require costly synchronisation steps. This acts as a bottleneck and delays solver iterations. Additive multigrid avoids this restriction, but becomes unstable for non-trivial material parameters as additive coarse grid levels tend to overcorrect. This leads to oscillations. We propose a new additive variant, adAFAC-x, with a stabilisation parameter that damps coarse grid corrections to remove oscillations. Per-level we solve an additional equation that produces an auxiliary correction. The auxiliary correction can be computed additively to the rest of the solve and uses ideas similar to smoothed aggregation multigrid to anticipate overcorrections. Pipelining techniques allow adAFAC-x to be written using single-touch semantics on a dynamically adaptive mesh