Reconstruction of density functions by sk-splines

Reconstruction of density functions and their characteristic functions by radial basis functions with scattered data points is a popular topic in the theory of pricing of basket options. Such functions are usually entire or admit an analytic extension into an appropriate tube and "bell-shaped" with rapidly decaying tails. Unfortunately, the domain of such functions is not compact which creates various technical difficulties. We solve interpolation problem on an infinite rectangular grid for a wide range of kernel functions and calculate explicitly their Fourier transform to obtain representations for the respective density functions

Stable multispeed lattice Boltzmann methods

We demonstrate how to produce a stable multispeed lattice Boltzmann method (LBM) for a wide range of velocity sets, many of which were previously thought to be intrinsically unstable. We use non-Gauss--Hermitian cubatures. The method operates stably for almost zero viscosity, has second-order accuracy, suppresses typical spurious oscillation (only a modest Gibbs effect is present) and introduces no artificial viscosity. There is almost no computational cost for this innovation. DISCLAIMER: Additional tests and wide discussion of this preprint show that the claimed property of coupled steps: no artificial dissipation and the second-order accuracy of the method are valid only on sufficiently fine grids. For coarse grids the higher-order terms destroy coupling of steps and additional dissipation appears. The equations are true.Comment: Disclaimer about the area of applicability is added to abstrac

Enhancing SPH using moving least-squares and radial basis functions

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.Comment: 10 pages, 3 figures, In Proc. A4A5, Chester UK, Jul. 18-22 200

Stabilisation of the lattice-Boltzmann method using the Ehrenfests' coarse-graining

The lattice-Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modelling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple and novel technique to stabilise the lattice-Boltzmann method by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests' steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests' idea of coarse-graining. The one-dimensional shock tube for a compressible isothermal fluid is a standard benchmark test for hydrodynamic codes. We observe that, of all the LBMs considered in the numerical experiment with the one-dimensional shock tube, only the method which includes Ehrenfests' steps is capable of suppressing spurious post-shock oscillations.Comment: 4 pages, 9 figure

Development of a mechatronic sorting system for removing contaminants from wool

Automated visual inspection (AVI) systems have been extended to many fields, such as agriculture and the food, plastic and textile industries. Generally, most visual systems only inspect product defects, and then analyze and grade them due to the lack of any sorting function. This main reason rests with the difficulty of using the image data in real time. However, it is increasingly important to either sort good products from bad or grade products into separate groups usingAVI systems. This article describes the development of a mechatronic sorting system and its integration with a vision system for automatically removing contaminants from wool in real time. The integration is implemented by a personal computer, which continuously processes live images under the Windows 2000 operating system. The developed real-time sorting approach is also applicable to many other AVI systems