Ignition of a reactive solid by an inert hot spot

A theoretical analysis is presented for the description of the ignition of a reactive media by inert hot bodies of finite size, when the activation energy of the reaction is large. The analysis leads to closed-form relations for the minimum "critical" size of the hot spot resulting in ignition and for the ignition time by hot spots of supercritical size. The analysis is carried out, first, for inert spots with heat conductivities and diff usivities of the order of those of the reactive media, and, second, for inert bodies of large conductivity

Gasification effects in the heterogeneous ignition of a condensed fuel by a hot gas

An analysis is presented to describe the heterogeneous ignition of a condensed fuel suddenly exposed to a hot oxidizing atmosphere. The exothermic heterogeneous reaction, generating gaseous products, is considered to be of the Arrhenius type with a large activation energy compared with the initial thermal energy of the fuel. Instantaneously after contact with the gases, the surface temperature rises to a jump value which is calculated allowing for variable transport properties of fuel and gas. The effect of the chemical heat release and the cooling effect due to the gasification flow are taken into account in obtaining an integral equation which is solved to describe the evolution of surface temperature with time

Propagation of premixed laminar flames in 3D narrow open ducts using RBF-generated finite differences

Laminar flame propagation is an important problem in combustion modelling for which great advances have been achieved both in its theoretical understanding and in the numerical solution of the governing equations in 2D and 3D. Most of these numerical simulations use finite difference techniques on simple geometries (channels, ducts, ...) with equispaced nodes. The objective of this work is to explore the applicability of the radial basis function generated finite difference (RBF-FD) method to laminar flame propagation modelling. This method is specially well suited for the solution of problems with complex geometries and irregular boundaries. Another important advantage is that the method is independent of the dimension of the problem and, therefore, it is very easy to apply in 3D problems with complex geometries. In this work we use the RBF-FD method to compute 2D and 3D numerical results that simulate premixed laminar flames with different Lewis numbers propagating in open ducts.This work has been supported by Spanish MICINN grants FIS2010-18473, FIS2011-28838 and CSD2010-00011

Optimal shape parameter for the solution of elastostatic problems with the RBF method

Radial basis functions (RBFs) have become a popular method for the solution of partial differential equations. In this paper we analyze the applicability of both the global and the local versions of the method for elastostatic problems. We use multiquadrics as RBFs and describe how to select an optimal value of the shape parameter to minimize approximation errors. The selection of the optimal shape parameter is based on analytical approximations to the local error using either the same shape parameter at all nodes or a node-dependent shape parameter. We show through several examples using both equispaced and nonequispaced nodes that significant gains in accuracy result from a proper selection of the shape parameter.This work was supported by Spanish MICINN Grants FIS2011-28838 and CSD2010-00011 and by Madrid Autonomous Region Grant S2009-1597. M.K. acknowledges Fundación Caja Madrid for its financial support

Radial basis function interpolation in the limit of increasingly flat basis functions

We propose a new approach to study Radial Basis Function (RBF) interpolation in the limit of increasingly flat functions. The new approach is based on the semi-analytical computation of the Laurent series of the inverse of the RBF interpolation matrix described in a previous paper [3]. Once the Laurent series is obtained, it can be used to compute the limiting polynomial interpolant, the optimal shape parameter of the RBFs used for interpolation, and the weights of RBF finite difference formulas, among other things.This work has been supported by Spanish MICINN Grants FIS2010-18473, FIS2013-41802-R, CSD2010-00011

RBF-FD Formulas and Convergence Properties

The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.This work has been supported by Spanish MECD Grants FIS2007-62673, FIS2008-04921 and by Madrid Autonomous Region Grant S2009-1597

Optimal constant shape parameter for multiquadric based RBF-FD method

Radial basis functions (RBFs) have become a popular method for interpolation and solution of partial differential equations (PDEs). Many types of RBFs used in these problems contain a shape parameter, and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter. In this paper, we focus on PDE problems solved with a multiquadric based RBF finite difference (RBF-FD) method. We propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error. The algorithm is based on analytical approximations to the local RBF-FD error derived in [1]. We show through several examples in 1D and 2D, both with structured and unstructured nodes, that very accurate solutions (compared to finite differences) can be achieved using the optimal value of the constant shape parameter.This work has been supported by Spanish MICINN grants FIS2010-18473, CSD2010-00011 and by Madrid Autonomous Region grant S2009-1597

Gaussian RBF-FD weights and its corresponding local truncation errors

In this work we derive analytical expressions for the weights of Gaussian RBF-FD and Gaussian RBF-HFD formulas for some differential operators. These weights are used to derive analytical expressions for the leading order approximations to the local truncation error in powers of the inter-node distance h and the shape parameter є. We show that for each differential operator, there is a range of values of the shape parameter for which RBF-FD formulas and RBF-HFD formulas are significantly more accurate than the corresponding standard FD formulas. In fact, very often there is an optimal value of the shape parameter є+ for which the local error is zero to leading order. This value can be easily computed from the analytical expressions for the leading order approximations to the local error. Contrary to what is generally believed, this value is, to leading order, independent of the internodal distance and only dependent on the value of the function and its derivatives at the node.This work has been supported by Spanish MICINN Grants FIS2010-18473, CSD2010-00011 and by Madrid Autonomous Region Grant S2009-1597. M.K. acknowledges Fundacion Caja Madrid for its financial support