Performance of Parallel Approximate Ideal Restriction Multigrid for Transport Applications

Algebraic multigrid (AMG) methods have been widely used to solve systems arising from the discretization of elliptic partial differential equations. In serial, AMG algorithms scale linearly with problem size. In parallel, communication costs scale logarithmically with the number of processors. Recently, a classical AMG method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices. AIR has already been shown to be effective for solving the linear systems arising from upwind discontinuous Galerkin (DG) finite element discretization of advection-diffusion problems, including the hyperbolic limit of pure advection. A new parallel version of AIR, pAIR, has been implemented in the hypre library. In this thesis, pAIR is tested for use solving the source iteration equations of the SN approximations to the transport equation. The performance is investigated with various meshes in two and three dimensions. Detailed profiling of parallel performance is also conducted to identify the most important areas for algorithm improvements. An improvement to the Local Ideal Approximate Restriction algorithm is introduced and discussed. Weak scaling results to 4,096 processors are presented. These results show total solve growing logarithmically with the number of processors used. Importantly, this result is shown on both uniform grids and unstructured grids in three dimensions. The unstructured mesh did not include reentrant cells

Optimization-based algebraic multigrid coarsening using reinforcement learning

Large sparse linear systems of equations are ubiquitous in science and engineering, such as those arising from discretizations of partial differential equations. Algebraic multigrid (AMG) methods are one of the most common methods of solving such linear systems, with an extensive body of underlying mathematical theory. A system of linear equations defines a graph on the set of unknowns and each level of a multigrid solver requires the selection of an appropriate coarse graph along with restriction and interpolation operators that map to and from the coarse representation. The efficiency of the multigrid solver depends critically on this selection and many selection methods have been developed over the years. Recently, it has been demonstrated that it is possible to directly learn the AMG interpolation and restriction operators, given a coarse graph selection. In this dissertation, we consider the complementary problem of learning to coarsen graphs for a multigrid solver. We study a method using a reinforcement learning (RL) agent based on two different neural network type; convolutional neural network (CNN) and graph neural networks (GNNs). The agent with CNN architecture is trained and tested on structured grids; however, the agent with GNN architecture can learn to perform graph coarsening on small training graphs and then be applied to unstructured large graphs. We demonstrate that this method can produce better coarse graphs than existing algorithms, even as the graph size increases and other properties of the graph are varied. We also propose an efficient inference procedure for performing graph coarsening that results in linear time complexity in graph size

Towards a robust Terra

In this work mantle convection simulation with Terra is investigated from a numerical point of view, theoretical analysis as well as practical tests are performed. The stability criteria for the numerical formulation of the physical model will be made clear. For the incompressible case and the Terra specific treatment of the anelastic approximation, two inf-sup stable grid modifications are presented, which are both compatible with hanging nodes. For the Q1hQ12h element pair a simple numeric test is introduced to prove the stability for any given grid. For the Q1h Pdisc 12h element pair and 1-regular refinements with hangig nodes an existing general proof can be adopted. The influence of the slip boundary condition is found to be destabilizing. For the incompressible case a cure can be adopted from the literature. The necessary conditions for the expansion of the stability results to the anelastic approximation will be pointed out. A numerical framework is developed in order to measure the effect of different numerical approaches to improve the handling of strongly varying viscosity. The framework is applied to investigate how block smoothers with different block sizes, combination of different block smoothers, different prolongation schemes and semi coarsening influence the multigrid performance. A regression-test framework for Terra will be briefly introduced

Coarse-grid selection for parallel algebraic multigrid

The need to solve linear systems arising from problems posed on extremely large, unstructured grids has sparked great interest in parallelizing algebraic multigrid (AMG) To date, however, no parallel AMG algorithms exist We introduce a parallel algorithm for the selection of coarse-grid points, a crucial component of AMG, based on modifications of certain paallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids A prototype serial version of the algorithm is implemented, and tests are conducted to determine its effect on multigrid convergence, and AMG complexit

Experimental and computational analysis of bubble generation combining oscillating fields and microfluidics

Microbubbles generated by microfluidic techniques have gained substantial interest in various fields such as food engineering, biosensors and the biomedical field. Recently, T-Junction geometries have been utilised for this purpose due to the exquisite control they offer over the processing parameters. However, this only relies on pressure driven flows; therefore bubble size reduction is limited, especially for very viscous solutions. The idea of combining microfluidics with electrohydrodynamics has recently been investigated using DC fields, however corona discharge was recorded at very high voltages with detrimental effects on the bubble size and stability. In order to overcome the aforementioned limitation, a novel set-up to superimpose an AC oscillation on a DC field is presented in this work with the aim of introducing additional parameters such as frequency, AC voltage and waveform type to further control bubble size, capitalising on well documented bubble resonance phenomena and properties. Firstly, the effect of applied AC voltage magnitude and the applied frequency were investigated. This was followed by investigating the effect of the mixing region and electric field strength on the microbubble diameter. A capillary embedded T-junction microfluidic device fitted with a stainless steel capillary was utilised for microbubble formation. A numerical model of the T-Junction was developed using a computational fluid dynamics-based multiphysics technique, combining the solution of transport equations for mass and momentum (Navier-Stokes Equations), a Volume of Fluid algorithm for tracking the gas-liquid interfaces, and a Maxwell Equations solver, all in a coupled manner. Simulation results were attained for the formation of the microbubbles with particular focus on the flow fields along the detachment of the emerging bubble. Experimental results indicated that frequencies between 2-10 kHz have a pronounced effect on the bubble size, whereas elevated AC voltages of 3-4 〖kV〗_(P-P) promoted bubble elongation and growth. It was observed that reducing the mixing region gap to 100 μm facilitated the formation of smaller bubbles due to the reduction of surface area, which increases the shear stresses experienced at the junction. Reducing the tip-to-collector distance causes a further reduction in the bubble size due to an increase in the electric field strength. Computational simulations suggest that there is a uniform velocity field distribution along the bubble upon application of a superimposed field. Microbubble detachment is facilitated by the recirculation of the dispersed phase. A decrease in velocity was observed upstream as the gas column occupies the junction suggesting the build-up in pressure, which corresponds to the widely reported ‘squeezing regime’ before the emerging bubble breaks off from the main stream. The novel set-up described in this work provides a viable processing methodology for preparing microbubbles that offers superior control and precision. In conjunction with optimised processing parameters, microbubbles of specific sizes can be generated to suit specific industrial applications