Thanh Tran-Cong

Approximation of function and its derivatives using radial basis function networks

Abstract This paper presents a numerical approach, based on radial basis function networks (RBFNs), for the approximation of a function and its derivatives (scattered data interpolation). The approach proposed here is called the indirect radial basis function network (IRBFN) approximation which is compared with the usual direct approach. In the direct method (DRBFN) the closed form RBFN approximating function is first obtained from a set of training points and the derivative functions are then calculated directly by differentiating such closed form RBFN. In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs. The derivative expression is then integrated to yield an expression for the original function, which is then solved via the general linear least squares principle, given an appropriate set of discrete data points. The IRBFN method allows the filtering of noise arisen from the interpolation of the original function from a discrete set of data points and produces a greatly improved approximation of its derivatives. In both cases the input data consists of a set of unstructured discrete data points (function values), which eliminates the need for a discretisation of the domain into a number of finite elements. The results obtained are compared with those obtained by the feed forward neural network approach where appropriate and the “finite element” methods. In all examples considered, the IRBFN approach yields a superior accuracy. For example, all partial derivatives up to second order of the function of three variables y = x 1 2 + x 1 x 2 −2 x 2 2 − x 2 x 3 + x 3 2 are approximated with at least an order of magnitude better in the L 2 -norm in comparison with the usual DRBFN approach.

Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations

This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poissons equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN–DD approach facilitates coarse-grained parallelisation of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poissons equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution.

Manufacturing polymer/carbon nanotube composite using a novel direct process

A direct process for manufacturing polymer carbon nanotube (CNT)-based composite yarns is reported. The new approach is based on a modified dry spinning method of CNT yarn and gives a high alignment of the CNT bundle structure in yarns. The aligned CNT structure was combined with a polymer resin and, after being stressed through the spinning process, the resin was cured and polymerized, with the CNT structure acting as reinforcement in the composite. Thus the present method obviates the need for special and complex treatments to align and disperse CNTs in a polymer matrix. The new process allows us to produce a polymer/CNT composite with properties that may satisfy various engineering specifications. The structure of the yarn was investigated using scanning electron microscopy coupled with a focused-ion-beam system. The tensile behavior was characterized using a dynamic mechanical analyzer. Fourier transform infrared spectrometry was also used to chemically analyze the presence of polymer on the composites. The process allows development of polymer/CNT-based composites with different mechanical properties suitable for a range of applications by using various resins.

Compact local integrated-RBF approximations for second-order elliptic differential problems

This paper presents a new compact approximation method for the discretisation of second-order elliptic equations in one and two dimensions. The problem domain, which can be rectangular or non-rectangular, is represented by a Cartesian grid. On stencils, which are three nodal points for one-dimensional problems and nine nodal points for two-dimensional problems, the approximations for the field variable and its derivatives are constructed using integrated radial basis functions (IRBFs). Several pieces of information about the governing differential equation on the stencil are incorporated into the IRBF approximations by means of the constants of integration. Numerical examples indicate that the proposed technique yields a very high rate of convergence with grid refinement.

A compact five-point stencil based on integrated RBFs for 2D second-order differential problems

In this paper, a compact 5-point stencil for the discretisation of second-order partial differential equations (PDEs) in two space dimensions is proposed. We employ integrated radial basis functions in one dimension (1D-IRBFs) to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil. Certain nodal values of the second-order derivatives are incorporated into the approximations with the help of the integration constants. In the case of elliptic PDEs, one algebraic equation is formed at each interior node, and the obtained final system, of which each row has 5 non-zero entries, is solved iteratively using a Picard scheme. In the case of parabolic PDEs discretised with a Crank-Nicolson procedure, a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach. Linear and non-linear test problems, including lid-driven cavity flow and natural convection between the outer square and the inner cylinder, are considered to verify the proposed stencil.

An Effective Integrated-RBFN Cartesian-Grid Discretization for the Stream Function–Vorticity–Temperature Formulation in Nonrectangular Domains

This article presents a new numerical collocation procedure, based on Cartesian grids and one-dimensional integrated radial-basis-function networks (1D-IRBFNs), for the simulation of natural convection defined in two-dimensional, multiply connected domains and governed by the stream function–vorticity–temperature formulation. Special emphasis is placed on the handling of vorticity values at boundary points that do not coincide with grid nodes. A suitable formula for computing vorticity boundary conditions, which is based on the approximations with respect to one coordinate direction only, is proposed. Normal derivative boundary conditions for the stream function are forced to be satisfied identically. Several test problems, including natural convection in the annulus between square and circular cylinders, are considered to investigate the accuracy of the proposed technique.

Nonlinear Optimisation Using Production Functions to Estimate Economic Benefit of Conjunctive Water Use for Multicrop Production

Uncertainty and shortages of surface water supplies, as a result of global climate change, necessitate development of groundwater in many canal commands. Groundwater can be expensive to pump, but provides a reliable supply if managed sustainably. Groundwater can be used optimally in conjunction with surface water supplies. The use of such conjunctive systems can significantly decrease the risk associated with a stochastic availability of surface water supply. However, increasing pumping cost due to groundwater drawdown and energy prices are key concerns. We propose an innovative nonlinear programing model for the optimisation of profitability and productivity in an irrigation command area, with conjunctive water use options. The model, rather than using exogenous yields and gross margins, uses crop water production and profit functions to endogenously determine yields and water uses, and associated gross margins, respectively, for various conjunctive water use options. The model allows the estimation of the potential economic benefits of conjunctive water use and derives an optimal use of regional level land and water resources by maximising the net benefits and water productivity under various physical and economic constraints, including escalating energy prices. The proposed model is applied to the Coleambally Irrigation Area (CIA) in southeastern Australia to explore potential of conjunctive water use and evaluate economic implication of increasing energy prices. The results show that optimal conjunctive water use can offer significant economic benefit especially at low levels of surface water allocation and pumping cost. The results show that conjunctive water use potentially generates additional AUD 57.3 million if groundwater price is the same as surface water price. The benefit decreases significantly with increasing pumping cost.

BEM-NN computation of generalised Newtonian flows

Abstract This paper presents a boundary-element-only method (BEM) for the calculation of generalised Newtonian fluid (GNF) flows. The volume integral arising from non-linear effects is approximated via a particular solution technique. Multilayer perceptron networks (MLPN) and radial basis function networks (RBFN) are used for global approximation of field variables and hence volume discretisation is not required. The iterative numerical formulation is achieved by viewing the material as being composed of a Newtonian base (artificially assigned with a constant, but maybe different from subdomain to subdomain, viscosity) and the remaining component, which is accordingly defined from the original constitutive equation. This decoupling of the non-linear effects allows a Picard-type iterative procedure to be employed by treating the non-linear term as a known forcing function. However, convergence is sensitive to the estimate of this forcing function and an adaptive subregioning of the domain is adopted to control the accuracy of the estimate of this non-linear term. The criterion for subregioning is that the velocity gradient should not vary significantly in each subdomain. This strategy enables convergence of the present method (BEM-NN) at power-law index as low as 0.2 for the difficult power law fluid. The use of MLPNs (instead of single layer perceptrons) and RBFNs is another contributing factor to the improved convergence performance. The overall scheme is very suitable for coarse-grain parallelisation as each subdomain can be independently analysed within an iteration. Furthermore, within each subdomain process, there are other parallelisable computations. The present method is verified with circular Couette and planar Poiseuille flows of the power-law, Carreau–Yasuda and Cross fluids.

Finite strip elements for laminated composite plates with transverse shear strain discontinuities

Abstract A finite strip solution for laminated plates made of composite materials is presented. Simple two-node and three-node strips are developed for the general analysis of laminated plates using a layerwise model. The plate is locally in the xy -plane and the displacement field is denoted by ( u , v , w ) where u , v are the in-plane displacements and w is the transverse displacement. The displacement field w is assumed to be constant through the thickness while the displacements u and v are assumed to vary linearly through each lamina thickness and therefore have C 0 continuity through the composite thickness. This is achieved using the kinematic hypothesis of Mindlin plate theory. The resulting displacements allow realistically for the warping of the composite cross-section and the shear strain field is discontinuous at the lamina interfaces. Some examples are considered and the results show very good agreement with the corresponding exact solutions.

A Galerkin approach incorporating integrated radial basis function networks for the solution of 2D biharmonic equations

This paper is concerned with the use of integrated radial basis function networks (IRBFNs) for the discretisation of Galerkin approximations for Dirichlet biharmonic problems in two dimensions. The field variable is approximated by global high-order IRBFNs on uniform grids without suffering from Runges phenomenon. Double boundary conditions, which can be of complicated shapes, are both satisfied identically. The proposed technique is verified through the solution of linear and nonlinear problems, including a benchmark buoyancy-driven flow in a square slot. Good accuracy and fast convergence are obtained.