Preconditioning for radial basis function partition of unity methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF--PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner

Maximizing Minimum Pressure in Fluid Dynamic Bearings of Hard Disk Drives

We focus on the central spindle which supports the rotating magnetic platters which hold all of the data. The spindle must operate with great precision and stability at high rotational speeds. Design practice has converged on oil-lubricated hydrodynamic journal bearings as the most common choice for spindles. That is, a layer of viscous oil separates a rotating shaft (the bearing) from the fixed outer sleeve (the journal). In hard drives, it is very important for the shaft to be centered within the sleeve. Plain journal bearings (i.e. both surfaces are circular cylinders) are unstable to perturbations that push the shaft off-center. It was found that this stability problem can be overcome by cutting diagonal grooves into the journal in a pattern called a herring-bone. Another consequence of this design is that very high pressures are generated by the grooves as they drive the oil to the middle of the bearing, away from the top/bottom ends of the spindle. This pumping action generally works to oppose leakage out of the bearing. We examine how choices for the groove pattern can influence the key properties of the bearing. The focus is to understand the effect of the groove geometry on the pumping action. In particular the undesirable behavior caused by the low pressures created near the top/bottom ends of the bearing which, under many conditions, may result in the pressure becoming negative, relative to atmospheric pressure. Negative pressure can result in cavitation or, when it occurs near an air-oil interface, can cause air to be ingested and hence create bubbles. Any bubbles in the oil can corrupt the lubricating layer in the bearing and, as they are created and collapse, can cause significant undesirable vibrations. The negative pressures have therefore been identified as one of the key problems in design of hard disk drive bearings. We will use numerical computations and some analysis to show that by modifying the groove geometry we can reduce the negative pressure while retaining good stability characteristics

Adaptive residual subsampling methods for radial basis function interpolation and collocation problems

We construct a new adaptive algorithm for radial basis func-tions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications

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