REPOSITORY > RESULTS

Abstract

Meshless methods are an increasingly popular alternative to traditional mesh-based numerical
methods, offering the ability to handle complex geometries while avoiding the
resource-intensive and difficult-to-automate process of mesh generation. Further benefits
of meshless methods include dimension independence and natural support for spatially
variable discretisation. The Radial Basis Function-generated Finite Differences (RBF-FD)
method has emerged as a promising strong-form approach, owing to its robustness on scattered
nodes, shape-parameter-free formulation, and explicit control over the convergence
order. Despite these advantages, several open questions limit its practical deployability
for demanding fluid flow problems, where strong nonlinearity, sharp boundary layers,
and instabilities in the convection-dominated regime place demanding requirements on
the solution procedure. This thesis addresses these challenges through the development
and analysis of adaptive discretisation and stabilisation strategies for the RBF-FD method,
demonstrated on natural convection driven flow of a non-Newtonian fluid. We first develop
a dimension-independent solution procedure based on the artificial compressibility method
and validate it against established benchmark results across a wide range of Rayleigh
numbers and power-law indices. We show that variable discretisation density is essential
for computational efficiency in the presence of thin boundary layers, reducing the node
count by an order of magnitude, and that larger stencil sizes permit more aggressive spatial
variation of the discretisation. Building on these findings, we replace the hand-tuned
discretisation density, which requires a priori knowledge of the flow structure, with an automatic
adaptive procedure based on a gradient variability indicator. The adaptive procedure
matches the performance of the hand-refined discretisation without prior knowledge of the
solution and transfers across different geometries without parameter re-tuning. We further
exploit the local nature of the meshless approximation by introducing a hybrid discretisation
that combines computationally efficient collocation on regular nodes with RBF-FD
on scattered nodes, supported by a dimension-agnostic hybrid node placement algorithm,
achieving a significant reduction in computational time with no loss in accuracy. Finally,
we address the stabilisation of the advective term in the inertial regime. We demonstrate
that hyperviscosity stabilisation extends the range of achievable Rayleigh numbers to the
full extent reported in the literature while preserving solution accuracy. To resolve the
persistent difficulty of selecting the hyperviscosity scaling constant, we propose an adaptive
algorithm that determines the stabilisation magnitude from the spectral radius of the
evolution matrix, removing the need for empirical tuning without assumptions about the
governing equations or node layout. We further show that the high-order derivatives in the
hyperviscosity operator can be approximated with a significantly reduced monomial basis,
lowering the computational cost of stabilisation, and that properly parametrised stabilisation
does not degrade the order of accuracy of the solution procedure. Together, these
contributions represent a step towards robust, automatic, and computationally efficient
meshless simulation of fluid flow.