Miha Rot

Miha Rot, Aleksandra Rashkovska

Meshless methods are an active and modern branch of numerical analysis with many intriguing benefits. One of the main open research questions related to local meshless methods is how to select the best possible stencil - a collection of neighbouring nodes - to base the calculation on. In this paper, we describe the procedure for generating a labelled stencil dataset and use a variation of pointNet - a deep learning network based on point clouds - to create a classifier for the quality of the stencil. We exploit features of pointNet to implement a model that can be used to classify differently sized stencils and compare it against models dedicated to a single stencil size. The model is particularly good at detecting the best and the worst stencils with a respectable area under the curve (AUC) metric of around 0.90. There is much potential for further improvement and direct application in the meshless domain.

Jon Vehovar, Miha Rot, Matjaž Depolli et al.

Meshless methods are used to solve partial differential equations by approximating differential operators at a node as a weighted sum of values at its neighbours. One of the algorithms for generating nodes suitable for meshless numerical analysis is an n-dimensional Poisson disc sampling based method. It can handle complex geometries and supports variable node density, a crucial feature for adaptive analysis. We modify this method for parallel execution using coupled spatial indexing and work distribution hypertrees. The latter is prebuilt according to the node density function, ensuring that each leaf represents a balanced work unit. Threads advance separate fronts and claim work hypertree leaves as needed while avoiding leaves neighbouring those claimed by other threads. Node placement constraints and the partially prebuilt spatial hypertree are combined to eliminate the need to lock the tree while it is being modified. Thread collision handling is managed by the work hypertree at the leaf level, drastically reducing the number of required mutex acquisitions for point insertion collision checks. We explore the behaviour of the proposed algorithm and compare the performance with existing attempts at parallelisation and consider the requirements for adapting the developed algorithm to distributed systems.

Miha Rot, Žiga Vaupotič, Andrej Kolar-Požun et al.

This paper presents an adaptive hyperviscosity stabilisation procedure for the Radial Basis Function-generated Finite Difference (RBF-FD) method, aimed at solving linear and non-linear advection-dominated transport equations on domains without a boundary. The approach employs a PDE-independent algorithm that adaptively determines the hyperviscosity constant based on the spectral radius of the RBF-FD evolution matrix. The proposed procedure supports general node layouts and is not tailored for specific equations, avoiding the limitations of empirical tuning and von Neumann-based estimates. To reduce computational cost, it is shown that lower monomial augmentation in the approximation of the hyperviscosity operator can still ensure consistent stabilisation, enabling the use of smaller stencils and improving overall efficiency. A hybrid strategy employing different spline orders for the advection and hyperviscosity operators is also implemented to enhance stability. The method is evaluated on pure linear advection and non-linear Burgers' equation, demonstrating stable performance with limited numerical dissipation. The two main contributions are: (1) a general hyperviscosity RBF-FD solution procedure demonstrated on both linear and non-linear advection-dominated problems, and (2) an in-depth analysis of the behaviour of hyperviscosity within the RBF-FD framework, addressing the interplay between key free parameters and their influence on numerical results.

Miha Rot, Gregor Kosec

One of the main challenges in numerically solving partial differential equations is finding a discretisation for the computational domain that balances the accurate representation of the underlying field with computational efficiency. Meshless methods approximate differential operators based on the values of the field in computational nodes, offering a natural approach to adaptivity. The density of computational nodes can either be increased to enhance accuracy or decreased to reduce the number of numerical operations, depending on the properties of the intermediate solution. In this paper, we utilise an adaptive discretisation approach for the numerical simulation of natural convection in non-Newtonian fluid flow. The shear-thinning behaviour is interesting both due to its numerous occurrences in nature, blood being a prime example, and due to its properties, as the decreasing viscosity with increasing shear rate results in sharper flow structures. We focus on the de Vahl Davis test case, a natural convection driven flow in a differentially heated rectangular cavity. The thin boundary layer flow along the vertical boundaries makes this an ideal test case for refinement. We demonstrate that adaptively refining the node density enhances computational efficiency and examine how the parameters for adaptive refinement affect the solution.